Optimal control of longitudinal deformations of a thermoelastic rod with unilateral contact condition of the Signorini type

Optimal control of longitudinal deformations of a thermoelastic rod with unilateral contact... Abstract In this article, we study the optimal control for longitudinal deformations of a thermoelastic rod. The controlled system is equipped with unilateral contact condition of the Signorini type. By the Dubovitskii and Milyutin functional analytical approach, we derive the necessary optimality conditions for optimal control problems with equality and inequality constraints in both fixed and free final horizon cases. Finally, the applicability of the derived Pontryagin maximum principles is simply addressed in the conclusion. 1. Introduction Thermoelasticity is concerned with elastic solids under conditions that are neither isothermal nor adiabatic. As one of four basic models that represent how a solid responds to an applied stress, thermoelasticity describes the coupling between the elastic field in the structure caused by deformation and the temperature field. Other three models are, respectively, the elasticity, the viscoelasticity, and the plasticity (Ignaczak & Ostoja-Starzewski, 2010). In diverse areas of engineering, there are considerable and successful applications involving thermoelastic contact. People can find them in the study of automotive disk brakes, the process of metal forming, and the analysis of heat exchangers, among others (Pelesko, 1999). Due to its importance, the thermoelastic contact problem has drawn lots of attention of not only engineers but also mathematicians in recent years. Moreover, research fellows centre on it and have established many original results. In view of our research focus, here we only present some interesting works obtained from the mathematical perspective, to name just a few. In Andrews et al. (1993), a one-dimensional thermoelastic system that may come into contact with a rigid obstacle is considered and the existence of both strong solution and weak solution is obtained. Copetti & French (2005) consider a numerical method for a partial differential equation problem involving thermoelastic contact and derive a posteriori error estimate by exploiting the variational framework. Based on the Crank–Nicolson discretization, Shi et al. (1991) use a numerical method to one-dimensional thermoelastic contact problems and prove the method converges. Similar works can be obtained from the extensive literatures. Nevertheless, it is not so optimistic when we encounter the optimal control problems of the thermoelastic contact. There are few works available especially in which the contact problem is equipped with the inequality constraints. Here, we give two nice investigations on the optimal control of a thermoelastic plate model. Bucci & Lasiecka (2004) consider a thermoelastic plate model with rotational inertia subject to thermal boundary control, and present well-posedness of the corresponding Riccati equations of the corresponding optimal control problems in both finite and infinite time horizon cases. Furthermore, as its continuation, Acquistapace et al. (2005) deal with a model for a Kirchoff thermoelastic plate with clamped boundary conditions and establish a regularity result for the outer normal derivative of the thermal velocity on the boundary, which is crucial to guarantee well-posedness for the associated differential Riccati equations in the study of optimal control problems. In this article, we investigate the optimal control for longitudinal deformations of a thermoelastic rod. Of special interest is that the controlled system is equipped with unilateral contact condition of the Signorini type. By the Dubovitskii and Milyutin functional analytical approach, we derive the necessary optimality conditions for optimal control problems with equality and inequality state constraints in both fixed and free final horizon cases. The obtained Pontryagin maximum principles in two cases constitute the main results of this investigation. As far as we know, this is the first attempt in optimal control investigation of a thermoelastic contact problem with multiple inequality constraints. The Pontryagin maximum principle, as a cornerstone of optimal control theory, tells us that the control Hamiltonian must take an extreme value over controls in the set of all permissible controls. It presents one first-order necessary optimality condition in optimal control problems. For optimal control problems with ordinary differential equations, the question of obtaining this condition is now completely solved due to the works of Dubovitskii and Milyutin (Dmitruk, 2009). However, for the infinite-dimensional optimal control problems governed by partial differential equations, the maximum principle does not generally hold as a necessary condition for optimal controls (Fattorini, 2005). In this field, such kinds of investigations are clearly of great significance and worthy of more attention. The remaining part of the article is organized as follows. In Section 2, we present some preliminaries which will be used throughout the article and list the first main result of this article. In fixed final horizon case, an optimal control problem for longitudinal deformations of a thermoelastic rod is formulated with unilateral contact condition of the Signorini type. The corresponding necessary optimality condition, namely the Pontryagin maximum principle, is established. Section 3, which is composed of three subsections, is devoted to the proof of the first main result. In Section 4, we investigate the optimal control of controlled thermoelastic rod in free final horizon case. The second main result is established in Subsection 4.1 and the corresponding proof is presented in Subsection 4.2. Section 5 concludes the paper with simple discussion on the applicability of the obtained results. 2. Preliminaries and the first main result We consider the longitudinal deformations of a one-dimensional homogeneous elastic rod that can be expressed as the following partial differential equations   \begin{align}\label{thermoelastic} \left\{ \begin{array}{l} \theta_{t}(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + f(x, t), \\ \displaystyle {{\sigma_x(x, t) = }} \; u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle u(1, t) \leq g, \; u_x(1, t) - a \theta(1, t) \leq 0, \; [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{align} (2.1) in which $$\theta(x, t)$$, $$u(x, t)$$, $$\sigma(x, t) = u_x(x, t) - a \theta(x, t)$$, are, respectively, the temperature, displacement and stress of the rod at location $$x \in {\it{\Omega}} = (0, 1)$$ and time $$t \in I_T = [0, T]$$ for $$T > 0$$ (Copetti & French, 2005). The subscripted variable indicates partial differentiation $$\theta_t \equiv \partial \theta / \partial t,\; u_{xt} \equiv \partial^2 u / \partial x \partial t$$, etc. At the left end, the rod is fixed to a wall while the right end is free to expand or contract but may come into contact with a rigid wall at temperature $$\theta = 0$$. We assume that the deformations are due to thermal effects and act under the regime of linear thermoelasticity. And the thermal exchange coefficient $$\beta$$ is assumed to be a positive constant. Moreover, $$f(x, t)$$ is a heat source and $$0 < a < 1$$ is a small constant. In equations (2.1), we see that there is a unilateral contact condition of the Signorini type at the right end of the bar. And the constant $$0 < g < 1$$ represents the width of the gap between the right end of the bar and the wall. The initial condition $$\varphi$$ is a smooth function. In fact, the system (2.1) models a long, thin, and homogeneous elastic rod which is situated between two walls that are kept at different temperatures (Andrews et al., 1993). One edge of the rod is permanently attached to one of the walls, while the other edge is free to expand or contract as a result of the evolution of the temperature and the stresses. However, the expansion is limited by the existence of the other wall, which acts as an obstacle and blocks any further expansion, once the rod comes into contact with it. We depict this physical setting in Fig. 1. Fig. 1. View largeDownload slide A thermoelastic rod, with the Signorini unilateral contact condition at $$x = 1,$$ between two walls. Fig. 1. View largeDownload slide A thermoelastic rod, with the Signorini unilateral contact condition at $$x = 1,$$ between two walls. Furthermore, we can decouple the system (2.1) into to a simpler problem involving only one equation and one unknown function, the temperature $$\theta(x, t)$$. People can read Andrews et al. (1993), Copetti (1999) and Shi et al. (1991) for the information. Here, we directly present the decoupled system defined on $${\it{\Omega}}_T = {\it{\Omega}} \times I_T$$ as follows.   \begin{align}\label{decoupled} \left\{ \begin{array}{l} \displaystyle (1 + a^2) \theta_{t}(x, t) - \theta_{xx}(x, t) = a^2 \frac{d}{dt} F(\gamma) + f(x, t), \\ \displaystyle \theta(0, t) = 1, \; - \theta_x(1, t) = \beta \theta(1, t), \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{align} (2.2) in which   \[ \displaystyle F(s) = \max\{s, 0\}, \textrm{ and } \gamma(t) = \int_{{\it{\Omega}}} \theta(\xi, t) \; d \xi - \frac{g}{a}, \] and $$d F / dt$$ denotes the first-order derivative of $$F(\cdot)$$ with respect to $$t$$. Take the Hilbert space $$W^{2, 1}_2 ({\it{\Omega}}_T)$$, which consists of all $$L^2({\it{\Omega}}_T)$$-summable functions that possess generalized $$L^2({\it{\Omega}}_T)$$-summable second-order space and first-order time derivatives, equipped with the norm   \[ \|\theta\|^2_{W^{2, 1}_2 ({\it{\Omega}}_T)} = \int_{{\it{\Omega}}_T} [\theta_t^2 + \theta_x^2 + \theta_{xx}^2 + \theta^2] \; dx\;dt. \] Assume that $$\varphi(x) \in H^1({\it{\Omega}})$$ with $$\varphi(0) = 1$$, $$f(x, t) \in L^2({\it{\Omega}}_T)$$. It has been proven that there exists a unique $$\theta(x, t) \in W^{2, 1}_2 ({\it{\Omega}}_T)$$ to the system (2.2) (Andrews et al., 1993; Copetti & French, 2005). Moreover, the displacement $$u(x, t)$$ and stress $$\sigma(x, t)$$ in (2.1) can be computed in a direct manner from $$\theta(x, t)$$ (Shi et al., 1991). Specially, we can present the displacement at the right end of the rod as below.   $$ \displaystyle u(1, t) = \min \left\{a \int_{{\it{\Omega}}} \theta(\xi, t)\; d\xi, \; g\right\}\!. $$ In this way, we suppose, unless otherwise stated, in what follows when we speak of a solution to (2.1), we shall always mean it solves (2.2) in this sense. Now we proceed the optimal control formulation of the investigated system (2.1). Firstly focus on the fixed final horizon case. Let $$f(x, t) = \rho(x, t) + \omega(t)$$, in which $$\rho(x, t) \in L^2({\it{\Omega}}_T)$$ and $$\omega(t) \in L^2(I_T)$$ is the control. Consider an optimal control problem for the system (2.1) with the general cost functional   \begin{equation}\label{op} \min_{\omega(\cdot)\in U_{ad}} J(\theta, u, \omega) = \min_{\omega(\cdot)\in U_{ad}} \int^T_0\int^{1}_0 L(\theta(x, t), u(x, t), \omega(t), x, t)\, dx\,dt, \end{equation} (2.3) in which the control constraint $$U_{ad}$$ is a non-empty closed convex set of $$L^2(I_T)$$. Take $$(\theta, u) \in W^{2, 1}_2 ({\it{\Omega}}_T) \times W^{1, 1}_2 ({\it{\Omega}}_T)$$ as the state of the system. The control space is $$L^2(I_T)$$ and the control function $$\omega(t)$$ satisfies a convex constraint $$\omega(\cdot) \in U_{ad}$$. Here we assume that the set of $$U_{ad}$$ of admissible controls has the non-empty interior with respect to $$L^2(I_T)$$ topology, i.e., $$\textrm{int}_{L^2(I_T)} U_{ad} \not = \emptyset$$. Of course, this is the normal assumption on the admissible control set, which is often used in literatures (Tröltzsch, 2010). Additionally, we make the following two assumptions for the cost functional. (a) $$L$$ is a functional defined on $$(L^2({\it{\Omega}}))^2 \times U_{ad} \times [0, 1] \times I_T$$ and   $$ \frac{\partial L(\theta, u, \omega, x, t)}{\partial \theta},\; \frac{\partial L(\theta, u, \omega, x, t)}{\partial u},\; \frac{\partial L(\theta, u, \omega, x, t)}{\partial \omega} $$ exist for every $$(\theta, u, \omega)\in (L^2({\it{\Omega}}))^2 \times U_{ad}$$ and $$L$$ is continuous in its variables. (b)   $$ \displaystyle \int^{1}_0\left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial \theta}\right|\,dx, \; \int^{1}_0\left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial u}\right|\,dx, \; \int^{1}_0 \left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial \omega}\right|\,dx $$ are bounded for $$t\in I_T$$. Subsequently, define $$X_T = W^{2, 1}_2({\it{\Omega}}_T) \times W^{1, 1}_2({\it{\Omega}}_T) \times L^2(I_T)$$. Let $$(\theta^*, u^*, \omega^*)$$ be the optimal solution to the optimal control problem (2.3) subject to the equations (2.1). Besides, we set   $$ \begin{array}{ll} {\it{\Xi}}_1 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! \omega(t) \in U_{ad}, \; t\in I_T \textrm{ a.e.}\}, \\ \displaystyle {\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! u(1, t) \leq g \}, \\ \displaystyle {\it{\Xi}}_3 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! u_x(1, t) - a \theta(1, t) \leq 0 \}, \\ \displaystyle {\it{\Xi}}_4 = \{(\theta, u, \omega) \in X_T\;|& \!\!\! \theta_t(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + \rho(x, t) + \omega(t), \\ \displaystyle & \!\!\! \displaystyle u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle & \!\!\! [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle & \!\!\! - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle & \!\!\! \theta(x, 0) = \varphi(x), \; \theta(x, T) = \theta^*(x, T), \; u(x, T) = u^*(x, T)\}. \end{array} $$ And the problem (2.3) is equivalent to questing for $$(\theta^*, u^*, \omega^*) \in {\it{\Xi}} = \bigcap\limits_{i = 1}^4 {\it{\Xi}}_i$$ such that   $$ J(\theta^*, u^*, \omega^*) = \min_{(\theta, u, \omega) \in {{\it{\Xi}}}} J(\theta, u, \omega). $$ As a result, we see that the problem (2.3) is an extremum problem on one constraint of inclusion type $${\it{\Xi}}_1$$, two inequality constraints $${\it{\Xi}}_2$$ and $${\it{\Xi}}_3$$, as well as one equality constraint $${\it{\Xi}}_4$$. Moreover, the Dubovitskii and Milyutin functional analytical approach has been turned out to be very powerful to solve such kind of extremum problems (Girsanov, 1972; Chan & Guo, 1990; Sun & Guo, 2005; Dmitruk, 2009). Now we present the general Dubovitskii and Milyutin theorem for the problem (2.3) in the form of Theorem 1. Theorem 1 (Dubovitskii–Milyutin) Suppose the functional $$J(\theta, u, \omega)$$ assumes a minimum at point $$(\theta^*, u^*, \omega^*)$$ in $${\it{\Xi}}$$. Assume that $$J(\theta, u, \omega)$$ is regularly decreasing at $$(\theta^*, u^*, \omega^*)$$ with the cone of directions of decrease $$K_0;$$ one inclusion constraint $${\it{\Xi}}_1$$ is regular at $$(\theta^*, u^*, \omega^*)$$ with the cone of feasible directions $$K_1;$$ two inequality constraints $${\it{\Xi}}_2$$, $${\it{\Xi}}_3$$, are regular at $$(\theta^*, u^*, \omega^*)$$ with two cones of feasible directions $$K_2, \; K_3;$$ and that the equality constraint $${\it{\Xi}}_4$$ is also regular at $$(\theta^*, u^*, \omega^*)$$ with the cone of tangent directions $$K_4$$. Then there exist continuous linear functionals $$f_i,$$ not all identically zero, such that $$f_i \in K^*_i,$$ the dual cone of $$K_i,$$$$i = 0, 1, \cdots, 4,$$ which satisfy the condition   \begin{equation}\label{dmth} \sum^4_{i = 0} f_i = 0. \end{equation} (2.4) By Theorem 1, we can establish the following Theorem 2, which is the Pontryagin maximum principle of the optimal control problem (2.3) for the longitudinal deformations of a thermoelastic rod with unilateral contact condition of the Signorini type. Theorem 2 Suppose $$(\theta^*, u^*, \omega^*)$$ is a solution to the optimal control problem (2.3). Then there exist a $$\kappa_0 \geq 0$$ and $$(\phi(x, t), v(x, t))$$, not identically zero, such that the following maximum principle holds true:   \begin{align*} \begin{array}{l} \displaystyle \left\{\int^{1}_0 \left[ \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right]dx \right\}\cdot \left[\omega(t) - \omega^*(t)\right] \geq 0, \quad \\ \displaystyle \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall\; \omega(t) \in U_{ad},\; t \in I_T \textrm{ a.e.}, \end{array} \end{align*} where the function $$(\phi(x, t), v(x, t))$$ satisfies the following adjoint equation   \begin{equation}\label{3.24} \left\{ \begin{array}{l} \displaystyle \phi_{t}(x, t) + \phi_{xx}(x, t) - a v_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \phi_x(1, t) - \theta_x(1, t) \phi(1, t) = [a \theta(1, t) - u_x(1, t)] [v(1, t) - \frac{d n(t)}{dt}], \\ \displaystyle a \phi_t(1, t) + v_x(1, t) = \frac{d m(t)}{dt}, \; v(0,t) = 0, \; \phi(0, t) = 0, \\ \displaystyle \phi(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \; \phi(x, 0) = 0. \end{array}\right. \end{equation} (2.5) Here, $$dm(t)$$ is a non-negative measure with support on $${\it{\Lambda}}_1$$ in Lemma 1 of Section 3.1, and $$dn(t)$$ a non-negative measure with support on $${\it{\Lambda}}_2$$ in Lemma 2 of Section 3.2. And $$\mu(x) \in H^{-2}({\it{\Omega}})$$ and $$\nu(x) \in H^{-1}({\it{\Omega}})$$ are given through (3.13). Theorem 2 is the first main result of this article and its proof is in Section 3 below. 3. Proof of Theorem 2 It is noted that the investigated optimal control problem (2.3) in this article is with multiple inequality constraints, which makes this constrained optimization problem more difficult to handle (Sun & Guo, 2015; Boccia et al., 2016). That means that, in the framework of the Dubovitskii–Milyutin approach, we need to consider the extra two cones of feasible directions $$K_2$$, $$K_3$$. By Theorem 1, we proceed as follows: to determine all cones $$K_i$$ and their dual cones $$K_i^*,\; i=0,1, \cdots, 4,$$ one by one; under the guidance of equation (2.4), to derive the final result step by step. First of all, let us find the cone of directions of decrease $$K_0$$. By assumption, $$J(\theta, u, \omega)$$ is differentiable at any point $$(\theta^0, u^0, \omega^0)$$ in any direction $$(\theta, u, \omega)$$ and its directional derivative is   \begin{align*} &\displaystyle J^\prime(\theta^0, u^0, \omega^0; \theta, u, \omega) \\ \displaystyle &\quad= \lim\limits_{\varepsilon\rightarrow 0+}\frac{1}{\varepsilon}\left[J(\theta^0 + \varepsilon \theta, u^0 + \varepsilon u, \omega^0 + \varepsilon \omega) - J(\theta^0, u^0, \omega^0)\right]\\ \displaystyle &\quad= \lim\limits_{\varepsilon\rightarrow 0+}\frac{1}{\varepsilon}\left\{\int^T_0\int^{1}_{0} \left[L(\theta^0 + \varepsilon \theta, u^0 + \varepsilon u, \omega^0 + \varepsilon \omega, x, t) - L(\theta^0, u^0, \omega^0, x, t)\right]\, dx\,dt\right\}\\ \displaystyle &\quad= \int^T_0\int^{1}_{0} \left[\frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial u} \; u + \frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial \omega} \; \omega \right]\,dx\, dt. \end{align*} The cone of directions of decrease of the functional $$J(\theta, u, \omega)$$ at point $$(\theta^*, u^*, \omega^*)$$ is determined by   \begin{align*} K_0 \displaystyle &= \left\{(\theta, u, \omega)\in X_T \,\Big|\, J^\prime(\theta^*, u^*, \omega^*; \theta, u, \omega) < 0 \right\}\\ \displaystyle &= \bigg\{(\theta, u, \omega)\in X_T \,\Big|\, \int^T_0 \int^{1}_{0}\bigg[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u \\ \displaystyle &\qquad\qquad\qquad\qquad\qquad\qquad\quad + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega \bigg]\,dx\,dt < 0 \bigg\}. \end{align*} If $$K_0\neq \emptyset$$, then for any $$f_0\in K^*_0$$, there exists a $$\kappa_0\geq 0 $$ such that   $$ \displaystyle f_0(\theta, u, \omega)= -\kappa_0\int^T_0 \int^{1}_{0}\left[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega \right]\,dx\,dt. $$ Second, the cone of feasible directions $$K_1$$ of $${\it{\Xi}}_1$$ is determined by   \begin{align*} K_1 & =\left\{\kappa\left(\stackrel{o}{{\it{\Omega}}}_1 - \, (\theta^*, u^*, \omega^*)\right) \;\big|\;\kappa>0 \right\}\\ \displaystyle & = \left\{h \;|\; h = \kappa(\theta - \theta^*, u - u^*, \omega - \omega^*), \; (\theta, u, \omega)\in\;\stackrel{o}{{\it{\Omega}}}_1, \;\kappa > 0\right\}\!, \end{align*} in which $$\stackrel{o}{{\it{\Omega}}}_1$$ is the interior of $${\it{\Omega}}_1$$. What is more, for an arbitrary $$f_1\in{K^*_1}$$, if there is an $$\bar{a}(t)\in L^2(I_T),$$ such that the linear functional defined by   \begin{equation}\label{f1} f_1(\theta, u, \omega) = \int_0^T \bar{a}(t) \omega(t)\, dt \end{equation} (3.6) is a support to $$U_{ad}$$ at point $$\omega^*$$, then   \begin{equation}\label{3.9} \bar{a}(t)\left[\omega(t) - \omega^*(t)\right]\geq 0,\quad \forall \; \omega(t) \in U_{ad}, \; t \in I_T \textrm{ a.e.}. \end{equation} (3.7) 3.1. The cone of feasible directions $$K_2$$ Now, we come to treat the inequality constraint $${\it{\Xi}}_2 = \left\{(\theta, u, \omega) \in X_T \; | \; u(1, t) \leq g \right\}$$. Let   \begin{equation}\label{3.10} F_1(u) = \max\limits_{0 \leq t \leq T} \{u(1, t) - g\}. \end{equation} (3.8) Then   $$ {\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \:|\: F_1(u) \leq 0\}. $$ Here we only consider $$F_1(u^*) = \max\limits_{0 \leq t \leq T} \{u^*(1, t) - g\} = 0$$. Since otherwise $$F_1(u^*) < 0$$, and $$(\theta^*, u^*, \omega^*)$$ is an inner point of $${\it{\Xi}}_2$$. In this case, any direction is feasible and hence the cone of feasible directions $$K_2$$ of $${\it{\Xi}}_2$$ at $$(\theta^*, u^*, \omega^*)$$ is the whole space, i.e., $$K_2 = X_T$$. So $${\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \:|\: F_1(u) \leq F_1(u^*) \}.$$ Then, we have the following Lemma 1. Lemma 1 Let $$F_1(u)$$ be defined by (3.8). Then $$F_1(u)$$ is differentiable at any point $$\hat{u}$$ in any direction $$u$$ and its directional derivative $$F^\prime_1(\hat{u}; u) = \max\limits_{t \in {\it{\Lambda}}_1}\{u(1, t)\}$$, here $${\it{\Lambda}}_1 = \{t \in I_T \:|\: \hat{u}(1, t) - g = F_1(\hat{u})\},$$ and $$F_1(u)$$ satisfies the Lipschitz condition in any ball. Here we assume that $$F^\prime_1(u; h)\neq 0$$ in the direction $$h$$ provided that $$F_1(u) = 0.$$ Note that for $$F_1(u)$$ given by (3.8), the directional derivative of $$F_1$$ at $$u^*$$ in the direction $$u + 1$$ is   $$ F^\prime_1(u^*; u + 1) = \max\limits_{t \in {\it{\Lambda}}_1}\{u(1, t) + 1\} < 0. $$ Hence   $$ K_2 = \{(\theta, u, \omega) \in X_T \:|\: F^\prime_1(u^*; u) < 0\}. $$ Define the linear operator $$A: X_T \rightarrow H^{1}(I_T)$$ by   $$ A u = - u(1, t) $$ and   $$Y_1 = \{\xi\in H^{1}(I_T) \:|\: \xi(t) \geq 0, \; \forall \; t \in {\it{\Lambda}}_1\}. $$ Then   $$ K_2 = \{(\theta, u, \omega)\in X_T \:|\: A u(x, t) \in Y_1 \}. $$ In view of $$A(u + 1) = -u(1, t) - 1 \in \stackrel{o}{Y_1}$$, the interior of $$Y_1$$, one has   $$ K^*_2=A^* Y^*_1, $$ i.e., for any $$f_2\in K^*_2,$$ there exists a non-negative measure $$dm(t)$$ with support on $${\it{\Lambda}}_1$$ such that   $$ \begin{array}{ll} f_2(\theta, u, \omega) \!\!\!\!\!\; & \displaystyle = \int^T_0 A u(x, t)\, dm(t) = \int_{{\it{\Lambda}}_1} A u(x, t) \, dm(t) \\ & \displaystyle = - \int_{{\it{\Lambda}}_1} u(1, t) \, dm(t) = - \int^T_0 u(1, t) \, dm(t). \end{array} $$ 3.2. The cone of feasible directions $$K_3$$ Then, we proceed to determine the cone of feasible directions $$K_3$$. In fact, the arguments for the determination of the cone of feasible directions of $${\it{\Xi}}_3$$ is similar to that of $${\it{\Xi}}_2$$. Let   \begin{equation}\label{3.13} F_2(\theta, u) = \max\limits_{0 \leq t \leq T} \{u_x(1, t) - a \theta(1, t) \}. \end{equation} (3.9) Then   $$ \begin{array}{ll} {\it{\Xi}}_3\;\!\!\!\!\! & = \left\{(\theta, u, \omega)\in X_T \: |\: u_x(1, t) - a \theta(1, t) \leq 0 \right\}\\ & = \{(\theta, u, \omega)\in X_T\: |\: F_2(\theta, u) \leq 0\}. \end{array} $$ Again we consider only $$F_2(\theta^*, u^*) = \max\limits_{0 \leq t \leq T} \{u^*_x(1, t) - a \theta^*(1, t)\} = 0$$. Since otherwise $$F_2(\theta^*, u^*) < 0$$ and $$(\theta^*, u^*, \omega^*)$$ is an interior point of $${\it{\Xi}}_3$$. Hence any direction is feasible and the cone of feasible directions $$K_3$$ of $${\it{\Xi}}_3$$ at $$(\theta^*, u^*, \omega^*)$$ is the whole space, i.e. $$K_3 = X_T$$. So $${\it{\Xi}}_3 = \{(\theta, u, \omega)\in X_T \: |\: F_2(\theta, u) \leq F_2(\theta^*, u^*) \}$$. Similarly, we have the following Lemma 2. Lemma 2 Let $$F_2(\theta, u)$$ be given by (3.9). Then $$F_2$$ is differentiable at any point $$(\hat{\theta}, \hat{u})$$ in any direction $$(\theta, u)$$ and its directional derivative is given by   $$ F^\prime_2(\hat{\theta}, \hat{u}; \theta, u) = \max\limits_{t \in {\it{\Lambda}}_2}\{u_x(1, t) - a \theta(1, t)\}, $$ where $${\it{\Lambda}}_2 = \{t \in I_T \, | \, \hat{u}_x(1, t) - a \hat{\theta}(1, t)) = F_2(\hat{\theta}, \hat{u})\}$$ and $$F_2(\theta, u)$$ satisfies the Lipschitz condition in any ball, here again we assume that $$F^\prime_2(\theta, u; h, p)\neq 0$$ in the direction $$(h, p)$$ provided that $$F_2(\theta, u) = 0$$. Next since   $$ F^\prime_2(\theta^*, u^*; \theta + 1, u + 1)<0, $$ we have   $$ K_3 = \left\{(\theta, u, \omega) \in X_T \:|\: F^\prime_2(\theta^*, u^*; \theta, u) < 0 \right\}\!. $$ Define the linear operator $$B: X_T \rightarrow H^{1}(I_T)$$ by   $$ B(\theta, u) = a \theta(1, t) - u_x(1, t) $$ and   $$ Y_2 = \{\xi \in H^1(I_T) \:|\: \xi(t) \geq 0, \forall \; t \in {\it{\Lambda}}_2\}. $$ Then   $$ K_3 = \left\{(\theta, u, \omega) \in X_T \:|\: B(\theta(x,t), u(x, t)) \in Y_2 \right\}\!. $$ By virtue of the fact that $$B(\theta + 1, u + 1) = a \theta(1, t) - u_x(1, t) + 1 \in \stackrel{o}{Y_2}$$, the interior of $$Y_2$$, we have $$K^*_3=B^* Y^*_2.$$ Namely, for any $$f_3\in K^*_3 $$, there exists a non-negative measure $$dn(t)$$ with support on $${\it{\Lambda}}_2$$ such that   $$ \begin{array}{ll} f_3(\theta, u, \omega)\!\!\!\!\!\! &\displaystyle =\int^T_0 B(\theta(x, t), u(x, t)) \, dn(t) = \int_{{\it{\Lambda}}_2} B(\theta(x, t), u(x, t)) \, {dn(t)} \\ &\displaystyle = \int_{{\it{\Lambda}}_2} [a \theta(1, t) - u_x(1, t)] \, dn(t) = - \int^T_0 [a \theta(1, t) - u_x(1, t)] \, dn(t). \end{array} $$ So far, we have obtained several cones including the cone of directions of decrease $$K_0$$ and the cones of feasible directions $$K_1, K_2, K_3$$. Furthermore, in their dual cones $$K^*_i,$$ the continuous linear functionals $$f_i \in K^*_i, \; i = 0, 1, 2, 3,$$ are respectively constructed. 3.3. The cone of tangent directions $$K_4$$ Next, we proceed to derive the cone of tangent directions $$K_4$$. Define the operator $$G: X_T \rightarrow (L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2$$ by   $$ \begin{array}{ll} G(\theta, u, \omega) = & \Big(\theta_t(x, t) - \theta_{xx}(x, t) + a u_{xt}(x, t) - \rho(x, t) - \omega(t), \; u_{xx}(x, t) - a \theta_x(x, t), \\ \displaystyle & \quad [u_x(1, t) - a \theta(1, t)] [u(1, t) - g], \; - \theta_x(1, t) - \beta \theta(1, t), \; u(0, t), \; \theta(0, t) - 1, \\ \displaystyle & \quad \theta(x, 0) - \varphi(x), \; \theta(x, T) - \theta^*(x, T), \; u(x, T) - u^*(x, T) \Big). \end{array} $$ Then   $$ {\it{\Xi}}_4 = \left\{(\theta, u, \omega) \in X_T \; | \; G(\theta(x, t), u(x, t), \omega(t)) = 0 \right\}\!. $$ The Fréchet derivative of the operator $$G(\theta, u, \omega)$$ is   \begin{align*} &G^\prime(\theta, u, \omega)(\hat{\theta}, \hat{u}, \hat{\omega}) \\ \displaystyle &\quad= \Big(\hat{\theta}_{t}(x, t) - \hat{\theta}_{xx}(x, t) + a \hat{u}_{xt}(x, t) - \hat{\omega}(t), \; \hat{u}_{xx}(x, t) - a \hat{\theta}_x(x, t),\\ \displaystyle &\qquad\quad [\hat{u}_x(1, t) - a \hat{\theta}(1, t)] [u(1, t) - g] + [u_x(1, t) - a \theta(1, t)] \hat{u}(1, t), - \hat{\theta}_x(1, t) - \beta \hat{\theta}(1, t), \\ \displaystyle &\qquad\quad \hat{u}(0, t), \; \hat{\theta}(0, t), \; \hat{\theta}(x, 0), \; \hat{\theta}(x, T), \; \hat{u}(x, T) \Big). \end{align*} Since $$(\theta^*, u^*, \omega^*)$$ is the solution to problem (2.3), it has $$G^\prime(\theta^*, u^*, \omega^*) = 0$$. Choosing arbitrary   $$ \big(p, q, q_0, q_1, q_2, q_3, q_4, q_5, q_6\big) \in (L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2 $$ and solving the equation   $$ G^\prime(\theta^*, u^*, \omega^*)(\hat{\theta}, \hat{u}, \hat{\omega}) = \big(p(x, t), q(x, t), q_1(t), q_2(t), q_3(t), q_4(t), q_5(x), q_6(x), q_7(x)\big), $$ we obtain   \begin{equation}\label{3.10&#x2013;&#x2013;} \left\{ \begin{array}{l} \displaystyle \hat{\theta}_{t}(x, t) - \hat{\theta}_{xx}(x, t) + a \hat{u}_{xt}(x, t) - \hat{\omega}(t) = p(x, t), \\ \displaystyle \hat{u}_{xx}(x, t) - a \hat{\theta}_x(x, t) = q(x, t), \\ \displaystyle [\hat{u}_x(1, t) - a \hat{\theta}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] \hat{u}(1, t) = q_1(t), \\ \displaystyle - \hat{\theta}_x(1, t) - \beta \hat{\theta}(1, t) = q_2(t), \; \hat{u}(0, t) = q_3(t), \; \hat{\theta}(0, t) = q_4(t), \\ \displaystyle \hat{\theta}(x, 0) = q_5(x), \; \hat{\theta}(x, T) = q_6(x), \; \hat{u}(x, T) = q_7(x). \end{array} \right. \end{equation} (3.10) Next, assume that the linearized system   \begin{equation}\label{3.20} \left\{ \begin{array}{l} \displaystyle {\theta}_{t}(x, t) - {\theta}_{xx}(x, t) + a {u}_{xt}(x, t) = {\omega}(t), \\ \displaystyle {u}_{xx}(x, t) - a {\theta}_x(x, t) = 0, \\ \displaystyle [{u}_x(1, t) - a {\theta}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] {u}(1, t) = 0, \\ \displaystyle - {\theta}_x(1, t) - \beta {\theta}(1, t) = 0, \; {u}(0, t) = 0, \; {\theta}(0, t) = 0, \\ \displaystyle {\theta}(x, 0) = 0, \end{array} \right. \end{equation} (3.11) is controllable. (Please refer to Chan & Guo (1989, 1990) for the information of the linearization.) Then choose $$\omega(t) = \hat{\omega}(t) \in L^2(I_T)$$ such that $${\theta}(x, T) = q_6(x) - \xi(x, T)$$, $${u}(x, T) = q_7(x) - \eta(x, T)$$ and let $$(\theta, u)$$ be the solution to the linearized system (3.11). Choose $$\hat{\theta}(x, t) = \theta(x, t) + \xi(x, t)$$, $$\hat{u}(x, t) = u(x, t) + \eta(x, t)$$, where $$(\xi, \eta)$$ satisfies the following equation   $$ \left\{ \begin{array}{l} \displaystyle {\xi}_{t}(x, t) - {\xi}_{xx}(x, t) + a {\eta}_{xt}(x, t) = p(x, t), \\ \displaystyle {\eta}_{xx}(x, t) - a {\xi}_x(x, t) = q(x, t), \\ \displaystyle [{\eta}_x(1, t) - a {\xi}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] {\eta}(1, t) = q_1(t), \\ \displaystyle - {\xi}_x(1, t) - \beta {\xi}(1, t) = q_2(t), \; {\eta}(0, t) = q_3(t), \; {\xi}(0, t) = q_4(t), \\ \displaystyle {\xi}(x, 0) = q_5(x). \end{array} \right. $$ In this way, it suffices for $$(\hat{\theta}, \hat{u}, \hat{\omega})$$ satisfying (3.10). Therefore $$G^\prime(\theta^*, u^*, \omega^*)$$ maps $$X_T$$ onto $$(L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2$$. Moreover, the cone of the tangent directions $$K_4$$ to the constraint $${\it{\Xi}}_4$$ at point $$(\theta^*, u^*, \omega^*)$$ consists of the solution to (3.11) in $$X_T$$ and   \begin{equation}\label{3.22} \theta(x, T) = 0, \; u(x, T) = 0. \end{equation} (3.12) Let   \begin{gather*} K_{41}=\{(\theta, u, \omega) \in X_T \; |\; (\theta(x, t), u(x, t), \omega(t)) \textrm{ satisfies } (3.11)\},\\ K_{42}=\{(\theta, u, \omega) \in X_T \; |\; (\theta(x, t), u(x, t), \omega(t)) \textrm{ satisfies } (3.12)\}. \end{gather*} Then the cone of tangent directions $$K_4 = K_{41} \bigcap K_{42}$$. (Under the assumption that (3.11) is controllable, $$K_4 \not = \emptyset.$$ We can deal with other case in the latter proof on non-degeneracy (3.21) of the system.) Hence   $$ K^*_4=K^*_{41}+K^*_{42}. $$ For any $$f_4\in K^*_4$$, decompose $$f_4 = f_{41} + f_{42},\; f_{4i} \in K^*_{4i}$$, the dual cone of $$K_{4i}, \; i = 1, 2$$. Then $$f_{41}(\theta, u, \omega) = 0$$ and for all $$(\theta, u) \in W^{2, 1}_2({\it{\Omega}}_T) \times W^{1, 1}_2({\it{\Omega}}_T)$$ satisfying $$\theta(x, T) = 0$$, $$u(x, T) = 0$$, there exists a $$\mu(x) \in H^{-2}({\it{\Omega}})$$ and a $$\nu(x) \in H^{-1}({\it{\Omega}})$$ such that   \begin{equation}\label{f42} f_{42}(\theta, u, \omega) = \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x) \right] \; dx. \end{equation} (3.13) By the Dubovitskii–Milyutin theorem, there exist continuous linear functionals, not all identically zero, such that   $$ \sum^3_{i = 0} f_i + f_{41} + f_{42} = 0. $$ Therefore, when selecting $$(\theta, u, \omega)$$ satisfies (3.11), $$f_{41}(\theta, u, \omega) = 0$$. Furthermore,   \begin{align} f_1(\theta, u, \omega) & \displaystyle = - f_0(\theta, u, \omega) - f_2(\theta, u, \omega) - f_3(\theta, u, \omega) - f_{42}(\theta, u, \omega) \notag \\ \displaystyle &= \kappa_0\int^T_0\int^{1}_{0}\bigg[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t) \notag\\ \displaystyle &\quad + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega(t) \bigg] \, dx \, dt \label{f-1}\\ \displaystyle &\quad + \int^T_0 u(1, t) \; dm(t) + \int^T_0 [a \theta(1, t) - u_x(1, t)] \; dn(t) \notag\\ \displaystyle &\quad - \int^{1}_0 [\theta(x, T) \mu(x) + u(x, T) \nu(x)] \, dx.\notag \end{align} (3.14) Finally, define the adjoint system of (3.11) as   \begin{equation}\label{adjoint-e} \left\{ \begin{array}{l} \displaystyle \phi_{t}(x, t) + \phi_{xx}(x, t) - a v_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \phi_x(1, t) - \theta_x(1, t) \phi(1, t) = [a \theta(1, t) - u_x(1, t)] [v(1, t) - \frac{d n(t)}{dt}], \\ \displaystyle a \phi_t(1, t) + v_x(1, t) = \frac{d m(t)}{dt}, \; v(0,t) = 0, \; \phi(0, t) = 0, \\ \displaystyle \phi(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \end{array}\right. \end{equation} (3.15) with   \begin{equation}\label{adjoint-e2} \phi(x, 0) = 0. \end{equation} (3.16) As with (2.1), the existence of solution to (3.15) can be obtained similarly. Subsequently, multiply the first two equations in (3.15), (3.16) by $$\theta(x, t)$$, $$u(x, t)$$, respectively, and integrate the product by parts over $${\it{\Omega}}_T$$ with respect to $$x$$ and $$t$$. For the derived integral, we transfer the derivatives from $$\phi(x, t)$$ to $$\theta(x, t)$$ and $$v(x, t)$$ to $$u(x, t)$$. Theorem 3 below then follows the sum of these two integrals. Theorem 3 The solution of system (3.11) and that of its adjoint system (3.15), (3.16) have the following relationship   \begin{align*} &\displaystyle \kappa_0 \int^T_0 \int^{1}_0 \left[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t)\right] \, dx\,dt \\ \displaystyle &\qquad + \int^T_0 u(1, t) \; d m(t) + \int^T_0 \left[a \theta(1, t) - u_x(1, t)\right]\; d n(t) - \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x)\right] \,dx \\ \displaystyle &\quad= - \int^T_0 \int^{1}_0 \omega(t) \phi(x, t) \, dx \, dt. \end{align*} Now we come to prove the first main result of this article. Proof of Theorem 2. By Theorem 3, we can rewrite $$f_1(\theta, u, \omega)$$ in (3.14) as   $$ \begin{array}{l} f_1(\theta, u, \omega) = \displaystyle \int^T_0 \left\{\int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t)\right]dx \right\} \omega(t) \, dt. \end{array} $$ In view of (3.6), we have   $$\begin{array}{c} \displaystyle \bar{a}(t) = \int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \,dx \end{array} $$ and (3.7) then reads   \begin{equation}\label{3.28} \begin{array}{l} \displaystyle \left\{\int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right]dx \right\}\cdot \left[\omega(t) - \omega^*(t)\right] \geq 0, \\ \displaystyle \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall\; \omega(t)\in U_{ad},\; t \in I_T \textrm{ a.e.}, \end{array} \end{equation} (3.17) where $$\kappa_0$$ and $$\phi(x, t)$$ are not identical to zero simultaneously. Since otherwise, there are definitely $$f_i = 0, \; i = 0, 1, 2, 3,$$ and $$f_{42} = 0$$, which contradict with the fact in Theorem 1 that these continuous linear functionals are not all identically zero. In addition, when $$K_0$$ is a null set, there is   $$ \begin{array}{r} \displaystyle \int^T_0\int^{1}_0 \bigg[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t) \\ \displaystyle + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega(t) \bigg] \, dx \, dt = 0, \quad \forall \; (\theta, u, \omega) \in X_T. \end{array} $$ In this case, if we choose $$\kappa_0 = 1,$$$$\mu(x) = \nu(x) = 0$$ and $$d m(t)$$, $$d n(t)$$ satisfying   \begin{equation}\label{mtnt} \int^T_0 u(1, t) \; d m(t) = \int^T_0 [u_x(1, t) - a \theta(1, t)] \; d n(t), \end{equation} (3.18) it then follows from Theorem 3 that   \begin{align*} &\displaystyle \int^T_0 \int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}\; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}\; u(x, t) \right] dx \,dt \\ \displaystyle &\quad= - \int^T_0 \int^{1}_0 \omega(t) \phi(x, t) \,dx\,dt. \end{align*} Thus   $$ \displaystyle \int^T_0 \left\{ \int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \, dx \right\} \omega(t) \, dt = 0, \quad \forall \; \omega(t) \in U_{ad}, $$ from which we obtain   $$ \displaystyle\int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e}. $$ Hence (3.17) still holds. Eventually, if there is a non-zero solution $$(\hat{\phi}(x, t), \hat{v}(x, t))$$ to the adjoint system   \begin{equation}\label{3.29} \left\{ \begin{array}{l} \displaystyle \hat{\phi}_{t}(x, t) + \hat{\phi}_{xx}(x, t) - a \hat{v}_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle \hat{v}_{xx}(x, t) + a \hat{\phi}_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \hat{\phi}_x(1, t) - \theta_x(1, t) \hat{\phi}(1, t) = [a \theta(1, t) - u_x(1, t)] \left[\hat{v}(1, t) - \frac{d n(t)}{dt}\right], \\ \displaystyle a \hat{\phi}_t(1, t) + \hat{v}_x(1, t) = \frac{d m(t)}{dt}, \; \hat{v}(0,t) = 0, \; \hat{\phi}(0, t) = 0, \\ \displaystyle \hat{\phi}(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \end{array}\right. \end{equation} (3.19) with   \begin{equation}\label{3.29-1} \hat{\phi}(x, 0) = 0, \end{equation} (3.20) such that the following equality holds true   $$ \displaystyle \int^{1}_0 \omega(t) \hat{\phi}(x, t) \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e.}, $$ then when we choose $$\kappa_0=0$$, $$\mu(x) = \hat{\phi}(x, T)$$, and $$\nu(x)$$ satisfying $$a u_x(x, T) \mu(x) = u(x, T) \nu(x)$$, (3.17) is still valid. Since otherwise, if for any non-zero solution $$(\hat{\phi}, \hat{v})$$ to (3.19), (3.20), it has   \begin{equation}\label{nondegeneracy} \int^{1}_0 \omega(t) \hat{\phi}(x, t) \, dx \not \equiv 0, \end{equation} (3.21) in this case we say the situation is non-degenerate. Then the linearized system (3.11) is controllable. In fact, if (3.11) is not controllable, then there exist a $$\mu(x) \in H^{- 2}({\it{\Omega}})$$ and a $$\nu(x) \in H^{- 1}({\it{\Omega}})$$ such that   $$ \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x) \right] \, dx = 0, \quad (\mu(x), \nu(x))\not\equiv 0. $$ Choose $$\kappa_0 = 0$$, $$(\hat{\phi}, \hat{v})$$ to be the solution of (3.19), (3.20) and $$d m(t)$$, $$d n(t)$$ satisfying (3.18). Then it follows from Theorem 3 that   $$ \int^T_0 \left[\int^{1}_0 \hat{\phi}(x, t)\, dx \right] \omega(t) \,dt = 0, \quad \forall \; \omega(t) \in U_{ad}. $$ Hence   $$ \int^1_0 \hat{\phi}(x, t) \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e.} $$ This is a contradiction. Therefore, under the case of (3.19), (3.20), the system (3.11) is controllable. Combining the results above, we have obtained the Pontryagin maximum principle for the problem (2.3) subject to the system (2.1). This completes the proof of the first main result. □ 4. Optimal control in free final horizon case In the preceding section, we give the Pontryagin maximum principle for optimal control problem (2.3) of the system (2.1) in fixed final horizon case. The result is derived under two additional conditions. The first one is that the admissible control set $$U_{ad}$$ must be convex and contains interior points, i.e., $$\textrm{int}_{L^2(0, T)} U_{ad} \not = \emptyset$$, and the second requires the cost functional $$L$$ to be differentiable with respect to the control variable $$\omega$$. In this section, we consider the system with free final time without these assumptions. Moreover, we still adopt the same symbols to denote these functionals in the case of no confusions caused. 4.1. The second main result Let $$t_1 > 0$$ and define $$I_{t_1} = [0, t_1]$$. Consider the following control system defined in the fixed domain $${\it{\Omega}}_{t_1} = [0, 1] \times I_{t_1}$$,   \begin{equation}\label{fh1} \left\{ \begin{array}{l} \theta_{t}(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + \rho(x, t) + \omega(t), \\ \displaystyle u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle u(1, t) \leq g, \; u_x(1, t) - a \theta(1, t) \leq 0, \; [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle \theta(x, 0) = \varphi(x), \; \theta(x, t_1) = \theta_1(x), \; u(x, t_1) = u_1(x), \\ \displaystyle (x, t)\in {\it{\Omega}}_{t_1} = {\it{\Omega}} \times (0, t_1), \; \omega \in M \subset \mathbb{R}, \end{array} \right. \end{equation} (4.22) and formulate the optimal control problem (4.23) below. Surely it is worth emphasizing the cancellation of assumptions imposed on the preceding fixed final horizon problem. That is to say, in this section the admissible control set $$M$$ is not necessarily convex as well as the cost functional $$L(\theta, u, \omega)$$ need not be differentiable with respect to the control variable $$\omega$$. The optimal control problem with free final horizon $$t_1$$ is presented as follows.   \begin{equation}\label{fop} \textrm{Problem I: Minimize }J(\theta, u, \omega) = \int^{t_1}_0\int^{1}_{0} L(\theta(x, t), u(x, t), \omega(t)) \;dx\,dt \end{equation} (4.23) for $$(\theta, u) \in X_{t_1} = W^{2, 1}_2({\it{\Omega}}_{t_1}) \times W^{1, 1}_2({\it{\Omega}}_{t_1})$$, $$\omega(t) \in L^2(I_{t_1})$$, under the constraints (4.22), where the functional $$L$$ defined on $$L^2({\it{\Omega}}) \times \mathbb{R}$$ satisfies (c) $$L(\theta, u, \omega)$$ is continuous in $$\omega$$. (d) $$ \displaystyle \left|\frac{\partial L(\theta, u, \omega)}{\partial \theta}\right|$$, $$ \displaystyle \left|\frac{\partial L(\theta, u, \omega)}{\partial u}\right|$$ are bounded for every bounded subset of $$L^2({\it{\Omega}}) \times \mathbb{R}$$. And thereafter we will establish the necessary optimality condition of optimal control problem (4.23) for the coupled system (4.22). In the same way, we list main result in advance, which is formulated as Theorem 4 below. Theorem 4 Suppose $$(\theta^*, u^*, \omega^*, t_1)$$ is a solution to Problem I (4.23), then there exist a $$\kappa_0 \geq 0$$ and a pair $$(\phi(x, t)$$, $$v(x, t))$$, not identically zero, such that:   \begin{align*} \displaystyle &\kappa_0 \int^1_{0} L(\theta^*(x, t), u^*(x, t), \omega^*(t)) \; d x + [u^*(1, t) - g] \frac{d \bar{m}(t)}{d t} + [u^*_x(1, t) - a \theta^*(1, t)] \frac{d \bar{n}(t)}{d t} \\ \displaystyle &\quad+ \int^1_0 [- \theta^*_{xx}(x, t) - \rho(x, t) - \omega^*(t)] \phi(x, t) \; dx = 0, \\ \displaystyle &\qquad \forall\; t\in [0,t_1] \textrm{ a.e.},\\ \displaystyle &\kappa_0 \int^1_{0} L(\theta^*(x, t), u^*(x, t), \omega(t)) \; d x + [u^*(1, t) - g] \frac{d \bar{m}(t)}{d t} + [u^*_x(1, t) - a \theta^*(1, t)] \frac{d \bar{n}(t)}{d t} \\ \displaystyle &\quad+ \int^1_0 [- \theta^*_{xx}(x, t) - \rho(x, t) - \omega(t)] \phi(x, t) \; dx = 0, \\ \displaystyle &\qquad\forall\; \omega \in M, \; t \in[0, t_1] \textrm{ a.e.}, \end{align*} where the function pair $$(\phi(x, t), \; v(x, t))$$ satisfies   $$ \left\{ \begin{array}{l} \displaystyle \phi_t(x, t) - \phi_{xx}(x, t) - a v(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta}, \\ \displaystyle \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u}, \\ \displaystyle - \theta(1, t) \phi_x(1, t) - \beta \theta(1, t) = \left[- u_x(1, t) + a \theta(1, t)\right] \left[v(1, t) + \frac{d \bar{n}(t)}{d t}\right]\!, \\ \displaystyle a \phi_t(1, t) = \frac{d \bar{m}(t)}{d t} - v_x(1, t), \\ \displaystyle \theta_x(0, t) \phi(0, t) = [\theta(0, t) + u(0, t)] \phi_x(0, t), \; v(0, t) = 0, \\ \displaystyle \phi(x, t_1) = \mu(x), \; a u_x(x, t_1) \mu(x) = u(x, t_1) \nu(x), \; \phi(x, 0) = 0. \end{array} \right. $$ Here, two non-negative measures $$d \bar{m}(t)$$, $$d \bar{n}(t)$$, with the functions $$\mu(x) \in H^{- 2}({\it{\Omega}})$$ and $$\nu(x) \in H^{- 1}({\it{\Omega}})$$, will be presented in (4.28) in the next subsection. The proof of Theorem 4 is given in next subsection. 4.2. Maximum principle of problem (4.23) Now we introduce a time transformation $$t \rightarrow s$$, mapping $$I_{t_1}$$ onto $$[0, 1]$$, defined by a certain function $$\varpi(\cdot) \geq 0$$,   $$ t(s) = \int^{s}_0 \varpi(\varsigma) \, d \varsigma, \quad t(1) = t_1, $$ and let $$\theta(x, s) = \theta(x, t(s))$$, $$u(x, s) = u(x, t(s))$$,   \begin{equation}\label{f2.5} \omega(s) = \left\{\displaystyle \begin{array}{cc} \omega(t(s)), & s \in {\it{\Upsilon}}_1 = \{s\;|\; s \in[0,1],\; \varpi(s) > 0\}, \\ \displaystyle \textrm{arbitrary}, & s \in {\it{\Upsilon}}_2 = \{s\;|\; s\in[0,1],\; \varpi(s) = 0\}, \end{array} \right. \end{equation} (4.24) Then $$(\theta(x, s), u(x, s), \omega(s))$$ satisfies the following equations   \begin{equation}\label{f2.6} \left\{ \begin{array}{l} \theta_{s}(x, s) - \theta_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi(s) - a \theta_x(x, s) \varpi(s) = 0, \\ \displaystyle u(1, s) \varpi(s) \leq g \varpi(s), \; u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s) \leq 0, \\ \displaystyle [u_x(1, s) - a \theta(1, s)] [u(1, s) - g] \varpi(s) = 0, \\ \displaystyle - \theta_x(1, s) \varpi(s) = \beta \theta(1, s) \varpi(s), \; u(0, s) \varpi(s) = 0, \; \theta(0, s) \varpi(s) = \varpi(s), \\ \displaystyle \theta(x, 0) = \varphi(x), \; \theta(x, 1) = \theta_1(x), \; u(x, 1) = u_1(x), \end{array} \right. \end{equation} (4.25) in which $$\rho(x, s) = \rho(x, t(s))$$. To make the definition of $$s(t)$$ one-to-one, we shall assume that   $$ s(t) = \inf\{s \;|\; t(s)=t\}. $$ Define $${\it{\Omega}}_1 = {\it{\Omega}} \times (0, 1)$$. And then we can formulate a new problem:   $$ \textrm{Problem II: Minimize }J(\theta, u, \omega, \varpi) = \int^1_0 \int^{1}_0 \varpi(s) L(\theta(x, s), u(x, s), \omega(s)) \; dx\,ds $$ for $$(\theta, u) \in W^{2, 1}_2({\it{\Omega}}_1) \times W^{1, 1}_2({\it{\Omega}}_1)$$, $$\omega(s) \in L^2(0, 1)$$, $${\varpi(s)} \in L^2(0, 1)$$, under the constraints (4.25) with $$\varpi(s)\geq 0$$, $$\omega(s) \in M$$, for almost all $$0 \leq s \leq 1$$. If $$(\theta^*, u^*, \omega^*)$$ is an optimal solution to the control problem (4.23) subject to the equations (4.22), then for any $$\varpi^*(s) \geq 0$$ satisfying $$\int^1_0 \varpi^*(\varsigma)\, d\varsigma = t_1$$, $$\omega^*(s)$$ defined similar to (4.24), $$(\theta^*, u^*, \omega^*, \varpi^*)$$ solves Problem II (Girsanov, 1972). Fixing $$\omega = \omega^*$$, another optimal control problem can be formulated as:   $$ \textrm{Problem III: Minimize }J(\theta, u, \omega^*, \varpi) = \int^1_0\int^{1}_0 \varpi(s) L(\theta(x, s), u(x, s), \omega^*(s)) \; dx\,ds $$ for $$(\theta(x, s), u(x, s), \varpi(s)) \in X_1 = W^{2, 1}_2({\it{\Omega}}_1) \times W^{1, 1}_2({\it{\Omega}}_1) \times L^{2}(0, 1)$$ subject to   \begin{equation}\label{p3} \left\{ \begin{array}{l} \theta_{s}(x, s) - \theta_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega^*(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi(s) - a \theta_x(x, s) \varpi(s) = 0, \\ \displaystyle u(1, s) \varpi(s) \leq g \varpi(s), \; u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s) \leq 0, \\ \displaystyle [u_x(1, s) - a \theta(1, s)] [u(1, s) - g] \varpi(s) = 0, \\ \displaystyle - \theta_x(1, s) \varpi(s) = \beta \theta(1, s) \varpi(s), \; u(0, s) \varpi(s) = 0, \; \theta(0, s) \varpi(s) = \varpi(s), \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{equation} (4.26) and   $$ \theta(x, 1) = \theta_1(x), \; u(x, 1) = u_1(x), $$ in which $$\varpi(s)$$ plays the role of control. Again, we assume that the set of new admissible controls has the non-empty interior with respect to $$L^2(0, 1)$$ topology (Tröltzsch, 2010). In what follows, we will prove the second main result. Proof of Theorem 4. We observe that Problem III is an optimal control problem with fixed final horizon, which can be tackled by the same method adopted in the investigation of the preceding optimal control problem (2.3) under the direction of the similar theorem to Theorem 1 (in this case, by assumption (c), (d), the new cost functional in Problem III naturally satisfies the similar conditions to assumption (a), (b)). For the sake of brevity, here we only list the key results and omit the detailed procedures. The linearized system of system (4.26) in this case reads as   \begin{equation}\label{ls2} \left\{ \begin{array}{l} \displaystyle \theta_s(x, s) - \theta_{xx}(x, s) \varpi^*(s) - \theta^*_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega^*(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi^*(s) - a \theta_x(x, s) \varpi^*(s) + u^*_{xx}(x, s) \varpi(s) - a \theta^*_x(x, s) \varpi(s) = 0, \\ \displaystyle [u^*_x(1, s) - a \theta^*(1, s)] [u^*(1, s) - g] \varpi(s) + [u^*_x(1, s) - a \theta^*(1, s)] u(1, s) \varpi^*(s) \\ \displaystyle \quad + [u_x(1, s) - a \theta(1, s)] [u^*(1, s) - g] \varpi^*(s) = 0, \\ \displaystyle [\theta_x(1, s) + \beta \theta(1, s)] \varpi^*(s) + [\theta^*_x(1, s) + \beta \theta^*(1, s)] \varpi(s) = 0, \\ \displaystyle u^*(0, s) \varpi(s) + u(0, s) \varpi^*(s) = 0, \; \theta^*(0, s) \varpi(s) + \theta(0, s) \varpi^*(s) = 0, \\ \displaystyle \theta(x, 0) = 0. \end{array} \right. \end{equation} (4.27) There exist $$\kappa_0 \geq 0$$, $$\bar{a}(s) \in L^2(0, 1)$$, $$\phi(x) \in H^2_0(0, 1)$$ and $$\psi(x) \in H^1_0(0, 1)$$ such that those continuous linear functionals in the general Dubovitskii and Milyutin theorem can be respectively determined as   \begin{equation}\label{fs} \left\{ \begin{array}{l} \displaystyle f_0(\theta, u, \omega^*, \varpi) = -\kappa_0 \int^1_0 \int^1_{0} \bigg[ \varpi^*(s)\frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta} \; \theta(x, s) + \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u} \; u(x, s) \\ \displaystyle \hspace{2.11in} + L(\theta^*, u^*, \omega^*) \varpi(s) \bigg] \; dx\,ds, \\ \displaystyle f_1(\theta, u, \omega^*, \varpi) = \int^1_0 \bar{a}(s) \varpi(s) \; ds, \\ \displaystyle f_2(\theta, u, \omega^*, \varpi) = \int^1_0 \left[- u^*(1, s) \varpi(s) - u(1, s) \varpi^*(s) + g \varpi(s)\right]\; d \bar{m}(s), \\ \displaystyle f_3(\theta, u, \omega^*, \varpi) = \int^1_0 \bigg[- u^*_x(1, s) \varpi(s) - u_x(1, s) \varpi^*(s) \\ \displaystyle \hspace{1.65in} + a \theta^*(1, s) \varpi(s) + a \theta(1, s) \varpi^*(s)\bigg]\; d \bar{n}(s), \\ \displaystyle f_{41}(\theta, u, \omega^*, \varpi) = 0, \\ \displaystyle f_{42}(\theta, u, \omega^*, \varpi) = \int^1_{0} \left[\theta(x, 1) \mu(x) + u(x, 1) \nu(x)\right] \; dx. \end{array}\right. \end{equation} (4.28) Here, there exists a non-negative measure $$d \bar{m}(s)$$ with support on   $$ {\it{\Lambda}}_3 = \left\{s \in [0, 1] \; | \; \hat{u}(1, s) \hat{\varpi}(s) - g \hat{\varpi}(s) = F_3(\hat{u}, \hat{\varpi})\right\} $$ with   $$ F_3(u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{u(1, s) \varpi(s) - g \varpi(s)\right\} $$ and its directional derivative   $$ F^\prime_3(\hat{u}, \hat{\varpi}; u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{\hat{u}(1, s) \varpi(s) + u(1, s) \hat{\varpi}(s) - g \varpi(s)\right\} $$ at any point $$(\hat{u}, \hat{\varpi})$$ in any direction $$(u, \varpi)$$. Similarly, there exists a non-negative measure $$d \bar{n}(s)$$ with support on   $$ {\it{\Lambda}}_4 = \left\{s \in [0, 1] \; | \; \hat{u}_x(1, s) \hat{\varpi}(s) - a \hat{\theta}(1, s) \hat{\varpi}(s) = F_4(\hat{\theta}, \hat{u}, \hat{\varpi})\right\} $$ with   $$ F_4(\theta, u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s)\right\} $$ and its directional derivative   $$ F^\prime_4(\hat{\theta}, \hat{u}, \hat{\varpi}; \theta, u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{\hat{u}_x(1, s) \varpi(s) + u_x(1, s) \hat{\varpi}(s) - a \hat{\theta}(1, s) \varpi(s) - a \theta(1, s) \hat{\varpi}(s)\right\} $$ at any point $$(\hat{\theta}, \hat{u}, \hat{\varpi})$$ in any direction $$(\theta, u, \varpi)$$. Correspondingly, the adjoint system of the linearized system (4.27) is   \begin{equation}\label{as2} \left\{ \begin{array}{l} \displaystyle \phi_s(x, s) - \varpi^*(s) \phi_{xx}(x, s) - a \varpi^*(s) v(x, s) = \kappa_0 \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta}, \\ \displaystyle \displaystyle \varpi^*(s) v_{xx}(x, s) + a \phi_{xs}(x, s) = - \kappa_0 \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u}, \\ \displaystyle - \theta(1, s) \phi_x(1, s) - \beta \theta(1, s) = \left[- u_x(1, s) + a \theta(1, s)\right] \left[v(1, s) + \frac{d \bar{n}(s)}{d s}\right]\!, \\ \displaystyle a \phi_s(1, s) = \varpi^*(s) \left[\frac{d \bar{m}(s)}{d s} - v_x(1, s)\right]\!, \\ \displaystyle \theta_x(0, s) \varpi^*(s) \phi(0, s) = [\theta(0, s) \varpi^*(s) + u(0, s)] \phi_x(0, s), \; v(0, s) = 0, \\ \displaystyle \phi(x, 1) = \mu(x), \; a u_x(x, 1) \mu(x) = u(x, 1) \nu(x), \end{array} \right. \end{equation} (4.29) with   \begin{equation}\label{as2-1} \phi(x, 0) = 0. \end{equation} (4.30) Furthermore, the relationship between the solution of the linearized system (4.27) and that of its adjoint system (4.29) with (4.30), is   \begin{align*} \displaystyle &\kappa_0 \int^1_0 \int^1_{0} \varpi^*(s) \left[\frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta} \; \theta(x, s) + \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u} \; u(x, s)\right]\; dx\,ds \\ \displaystyle &\quad + \int^1_{0} u(1, s) \varpi^*(s) \; d \bar{m}(s) - \int^1_{0} \left[- u_x(1, s) + a \theta(1, s) \right] \varpi^*(s) \; d \bar{n}(s) \\ \displaystyle &\quad - \int^1_{0} \left[\theta(x, 1) \mu(x) + u(x, 1) \nu(x)\right]\, dx\\ \displaystyle &= \int^1_0 \int^1_{0} \Big\{\left[- \theta^*_{xx}(x, s) - \rho(x, s) - \omega^*(s)\right] \phi(x, s) + \left[u^*_{xx}(x, s) - a \theta^*_x(x, s)\right] v(x, s) \Big\} \varpi(s) \; dx \, ds \\ \displaystyle &\quad - \int^1_0 \Big\{ [\theta^*_{x}(1, s) + \beta \theta^*(1, s)] \phi(1, s) + u^*(0, s) v_{x}(0, s)\Big\} \varpi(s) \; ds. \end{align*} By this relationship expression, we can get the Pontryagin maximum principle of Problem III. After that, the maximum principle of Problem I with free final horizon can be easily obtained (Girsanov, 1972; Dmitruk, 2009). And the obtained result is stated as Theorem 4, which is none other than the Pontryagin maximum principle of Problem I with free final horizon. This completes the proof of the second main result. □ 5. Conclusions In control theory, the maximum principle does not always hold as a necessary condition for optimal control of the infinite-dimensional system. Thus, the infinite-dimensional generalization of the maximum principle is an important and interesting topic in this field. For this purpose, this article considers the optimal control for longitudinal deformations of a thermoelastic rod. Specially, the investigational controlled system is equipped with unilateral contact condition of the Signorini type. And in both the fixed and free final horizon cases, we establishes the necessary optimality conditions given in the form of the Pontryagin maximum principle. Frankly speaking, the immediately following step is to numerically solve this optimal control problem. However, note that the investigational object in this article is an optimal control problem of the distributed parameter system governed by partial differential equations equipped with the Signorini unilateral boundary conditions. Even in fixed final horizon case, in terms of optimal control problem (2.3), the solution process includes not only the numerical approximation to the state equation but also solving the corresponding adjoint equation (2.5). We still need to determine the non-negative measures $$dm(t)$$ and $$dn(t)$$, which further aggravate this procedure. It is definitely not an easy job to get the numerical solutions for the optimal control-trajectory pair. In Sun & Wu (2013), the efficient numerical approaches are discussed and then a min-H iterative method with the higher convergence rate are developed to present the numerical solution of optimal control. It does show the applicability of obtained Pontryagin maximum principle although it does not give the detailed numerical simulation. People can refer to it for the details. On other numerical investigations under the Dubovitskii–Milyutin formalism, people can read Gayte et al. (2010) and Kotarski (1997) for the information. In conclusion, we investigate the optimal control for longitudinal deformations of a thermoelastic rod and prove the Pontryagin maximum principles in two cases, which constitute main results of this paper. The investigational model is deterministic without any uncertainties and unknown parameters. However, in real-world applications, there are a lots of distributed parameter systems where the exact model of the system is unknown and contains uncertainties. The articles such as Kulkarni et al. (2006) and Rebiai & Zinober (1993) can be read for the easy reference. Funding National Natural Science Foundation of China (Grant No. 11471036) (in part). Acknowledgements The authors would like to thank the editor and the anonymous referees for their very careful reading and constructive suggestions that improve substantially the article. References Acquistapace P., Bucci F. & Lasiecka I. ( 2005) A trace regularity result for thermoelastic equations with application to optimal boundary control. J. Math. Anal. Appl.,  310, 262– 277. Google Scholar CrossRef Search ADS   Andrews K. T., Shi P., Shillor M. & Wright S. ( 1993) Thermoelastic contact with Barber’s heat exchange condition. Appl. Math. Optim. , 28, 11– 48. Google Scholar CrossRef Search ADS   Boccia A., de Pinho M. D. R. & Vinter R. B. ( 2016) Optimal control problems with mixed and pure state constraints. SIAM J. Control Optim. , 54, 3061– 3083. Google Scholar CrossRef Search ADS   Bucci F. & Lasiecka I. ( 2004) Singular estimates and Riccati theory for thermoelastic plate models with boundary thermal control. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. , 11, 545– 568. Chan W. L. & Guo B. Z. ( 1989) Optimal birth control of population dynamics. J. Math. Anal. Appl. , 144, 532– 552. Google Scholar CrossRef Search ADS PubMed  Chan W. L. & Guo B. Z. ( 1990) Optimal birth control of population dynamics. II. Problems with free final time, phase constraints, and mini-max costs. J. Math. Anal. Appl. , 146, 523– 539. Google Scholar CrossRef Search ADS PubMed  Copetti M. I. M. ( 1999) Finite element approximation to a contact problem in linear thermoelasticity. Math. Comput. , 68, 1013– 1024. Google Scholar CrossRef Search ADS   Copetti M. I. M. & French D. A. ( 2005) Numerical approximation and error control for a thermoelastic contact problem. Appl. Numer. Math. , 55, 439– 457. Google Scholar CrossRef Search ADS   Dmitruk A. V. ( 2009) On the development of Pontryagin’s maximum principle in the works of A. Ya. Dubovitskii and A. A. Milyutin. Control and Cybernet. , 38, 923– 957. Fattorini H. O. ( 2005) Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems . North-Holland Mathematics Studies, Vol. 201. Amsterdam: Elsevier Science B.V. Gayte I., Guillén-González F. & Rojas-Medar M. ( 2010) Dubovitskii-Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data. IMA J. Math. Control Inform. , 27, 57– 76. Google Scholar CrossRef Search ADS   Girsanov I. V. ( 1972) Lectures on Mathematical Theory of Extremum Problems . Lecture Notes in Economics and Mathematical Systems, Vol. 67. Berlin: Springer. Google Scholar CrossRef Search ADS   Ignaczak J. & Ostoja-Starzewski M. ( 2010) Thermoelasticity with Finite Wave Speeds . Oxford Mathematical Monographs. Oxford: Oxford University Press. Kotarski W. ( 1997) Some Problem of Optimal and Pareto Optimal Control for Distributed Parameter Systems . Katowice: Wydawinictwo Uniwersytetu Śla̧skiego. Kulkarni K., Zhang L. & Linninger A. A. ( 2006) Model and parameter uncertainty in distributed systems. Ind. Eng. Chem. Res. , 45, 7832– 7840. Google Scholar CrossRef Search ADS   Pelesko J. A. ( 1999) Nonlinear stability considerations in thermoelastic contact. Trans. ASME J. Appl. Mech. , 66, 109– 116. Google Scholar CrossRef Search ADS   Rebiai S. E. & Zinober A. S. I. ( 1993) Stabilization of uncertain distributed parameter systems. Int. J. Control , 57, 1167– 1175. Google Scholar CrossRef Search ADS   Shi P., Shillor M. & Zou X. L. ( 1991) Numerical solutions to one-dimensional problems in thermoelastic contact. Comput. Math. Appl. , 22, 65– 78. Google Scholar CrossRef Search ADS   Sun B. & Guo B. Z. ( 2005) Maximum principle for the optimal control of an ablation-transpiration cooling system. J. Syst. Sci. Complex.  18, 285– 301. Sun B. & Guo B. Z. ( 2015) Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control. IEEE Trans. Automat. Control , 60, 3012– 3017. Google Scholar CrossRef Search ADS   Sun B. & Wu M. X. ( 2013) Optimal control of age-structured population dynamics for spread of universally fatal diseases. Appl. Anal. , 92, 901– 921. Google Scholar CrossRef Search ADS   Tröltzsch F. ( 2010) Optimal Control of Partial Differential Equations: Theory, Methods and Applications . Graduate Studies in Mathematics, vol. 112. Providence, RI: American Mathematical Society. Translated from the 2005 German original by J. Sprekels. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Optimal control of longitudinal deformations of a thermoelastic rod with unilateral contact condition of the Signorini type

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Abstract

Abstract In this article, we study the optimal control for longitudinal deformations of a thermoelastic rod. The controlled system is equipped with unilateral contact condition of the Signorini type. By the Dubovitskii and Milyutin functional analytical approach, we derive the necessary optimality conditions for optimal control problems with equality and inequality constraints in both fixed and free final horizon cases. Finally, the applicability of the derived Pontryagin maximum principles is simply addressed in the conclusion. 1. Introduction Thermoelasticity is concerned with elastic solids under conditions that are neither isothermal nor adiabatic. As one of four basic models that represent how a solid responds to an applied stress, thermoelasticity describes the coupling between the elastic field in the structure caused by deformation and the temperature field. Other three models are, respectively, the elasticity, the viscoelasticity, and the plasticity (Ignaczak & Ostoja-Starzewski, 2010). In diverse areas of engineering, there are considerable and successful applications involving thermoelastic contact. People can find them in the study of automotive disk brakes, the process of metal forming, and the analysis of heat exchangers, among others (Pelesko, 1999). Due to its importance, the thermoelastic contact problem has drawn lots of attention of not only engineers but also mathematicians in recent years. Moreover, research fellows centre on it and have established many original results. In view of our research focus, here we only present some interesting works obtained from the mathematical perspective, to name just a few. In Andrews et al. (1993), a one-dimensional thermoelastic system that may come into contact with a rigid obstacle is considered and the existence of both strong solution and weak solution is obtained. Copetti & French (2005) consider a numerical method for a partial differential equation problem involving thermoelastic contact and derive a posteriori error estimate by exploiting the variational framework. Based on the Crank–Nicolson discretization, Shi et al. (1991) use a numerical method to one-dimensional thermoelastic contact problems and prove the method converges. Similar works can be obtained from the extensive literatures. Nevertheless, it is not so optimistic when we encounter the optimal control problems of the thermoelastic contact. There are few works available especially in which the contact problem is equipped with the inequality constraints. Here, we give two nice investigations on the optimal control of a thermoelastic plate model. Bucci & Lasiecka (2004) consider a thermoelastic plate model with rotational inertia subject to thermal boundary control, and present well-posedness of the corresponding Riccati equations of the corresponding optimal control problems in both finite and infinite time horizon cases. Furthermore, as its continuation, Acquistapace et al. (2005) deal with a model for a Kirchoff thermoelastic plate with clamped boundary conditions and establish a regularity result for the outer normal derivative of the thermal velocity on the boundary, which is crucial to guarantee well-posedness for the associated differential Riccati equations in the study of optimal control problems. In this article, we investigate the optimal control for longitudinal deformations of a thermoelastic rod. Of special interest is that the controlled system is equipped with unilateral contact condition of the Signorini type. By the Dubovitskii and Milyutin functional analytical approach, we derive the necessary optimality conditions for optimal control problems with equality and inequality state constraints in both fixed and free final horizon cases. The obtained Pontryagin maximum principles in two cases constitute the main results of this investigation. As far as we know, this is the first attempt in optimal control investigation of a thermoelastic contact problem with multiple inequality constraints. The Pontryagin maximum principle, as a cornerstone of optimal control theory, tells us that the control Hamiltonian must take an extreme value over controls in the set of all permissible controls. It presents one first-order necessary optimality condition in optimal control problems. For optimal control problems with ordinary differential equations, the question of obtaining this condition is now completely solved due to the works of Dubovitskii and Milyutin (Dmitruk, 2009). However, for the infinite-dimensional optimal control problems governed by partial differential equations, the maximum principle does not generally hold as a necessary condition for optimal controls (Fattorini, 2005). In this field, such kinds of investigations are clearly of great significance and worthy of more attention. The remaining part of the article is organized as follows. In Section 2, we present some preliminaries which will be used throughout the article and list the first main result of this article. In fixed final horizon case, an optimal control problem for longitudinal deformations of a thermoelastic rod is formulated with unilateral contact condition of the Signorini type. The corresponding necessary optimality condition, namely the Pontryagin maximum principle, is established. Section 3, which is composed of three subsections, is devoted to the proof of the first main result. In Section 4, we investigate the optimal control of controlled thermoelastic rod in free final horizon case. The second main result is established in Subsection 4.1 and the corresponding proof is presented in Subsection 4.2. Section 5 concludes the paper with simple discussion on the applicability of the obtained results. 2. Preliminaries and the first main result We consider the longitudinal deformations of a one-dimensional homogeneous elastic rod that can be expressed as the following partial differential equations   \begin{align}\label{thermoelastic} \left\{ \begin{array}{l} \theta_{t}(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + f(x, t), \\ \displaystyle {{\sigma_x(x, t) = }} \; u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle u(1, t) \leq g, \; u_x(1, t) - a \theta(1, t) \leq 0, \; [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{align} (2.1) in which $$\theta(x, t)$$, $$u(x, t)$$, $$\sigma(x, t) = u_x(x, t) - a \theta(x, t)$$, are, respectively, the temperature, displacement and stress of the rod at location $$x \in {\it{\Omega}} = (0, 1)$$ and time $$t \in I_T = [0, T]$$ for $$T > 0$$ (Copetti & French, 2005). The subscripted variable indicates partial differentiation $$\theta_t \equiv \partial \theta / \partial t,\; u_{xt} \equiv \partial^2 u / \partial x \partial t$$, etc. At the left end, the rod is fixed to a wall while the right end is free to expand or contract but may come into contact with a rigid wall at temperature $$\theta = 0$$. We assume that the deformations are due to thermal effects and act under the regime of linear thermoelasticity. And the thermal exchange coefficient $$\beta$$ is assumed to be a positive constant. Moreover, $$f(x, t)$$ is a heat source and $$0 < a < 1$$ is a small constant. In equations (2.1), we see that there is a unilateral contact condition of the Signorini type at the right end of the bar. And the constant $$0 < g < 1$$ represents the width of the gap between the right end of the bar and the wall. The initial condition $$\varphi$$ is a smooth function. In fact, the system (2.1) models a long, thin, and homogeneous elastic rod which is situated between two walls that are kept at different temperatures (Andrews et al., 1993). One edge of the rod is permanently attached to one of the walls, while the other edge is free to expand or contract as a result of the evolution of the temperature and the stresses. However, the expansion is limited by the existence of the other wall, which acts as an obstacle and blocks any further expansion, once the rod comes into contact with it. We depict this physical setting in Fig. 1. Fig. 1. View largeDownload slide A thermoelastic rod, with the Signorini unilateral contact condition at $$x = 1,$$ between two walls. Fig. 1. View largeDownload slide A thermoelastic rod, with the Signorini unilateral contact condition at $$x = 1,$$ between two walls. Furthermore, we can decouple the system (2.1) into to a simpler problem involving only one equation and one unknown function, the temperature $$\theta(x, t)$$. People can read Andrews et al. (1993), Copetti (1999) and Shi et al. (1991) for the information. Here, we directly present the decoupled system defined on $${\it{\Omega}}_T = {\it{\Omega}} \times I_T$$ as follows.   \begin{align}\label{decoupled} \left\{ \begin{array}{l} \displaystyle (1 + a^2) \theta_{t}(x, t) - \theta_{xx}(x, t) = a^2 \frac{d}{dt} F(\gamma) + f(x, t), \\ \displaystyle \theta(0, t) = 1, \; - \theta_x(1, t) = \beta \theta(1, t), \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{align} (2.2) in which   \[ \displaystyle F(s) = \max\{s, 0\}, \textrm{ and } \gamma(t) = \int_{{\it{\Omega}}} \theta(\xi, t) \; d \xi - \frac{g}{a}, \] and $$d F / dt$$ denotes the first-order derivative of $$F(\cdot)$$ with respect to $$t$$. Take the Hilbert space $$W^{2, 1}_2 ({\it{\Omega}}_T)$$, which consists of all $$L^2({\it{\Omega}}_T)$$-summable functions that possess generalized $$L^2({\it{\Omega}}_T)$$-summable second-order space and first-order time derivatives, equipped with the norm   \[ \|\theta\|^2_{W^{2, 1}_2 ({\it{\Omega}}_T)} = \int_{{\it{\Omega}}_T} [\theta_t^2 + \theta_x^2 + \theta_{xx}^2 + \theta^2] \; dx\;dt. \] Assume that $$\varphi(x) \in H^1({\it{\Omega}})$$ with $$\varphi(0) = 1$$, $$f(x, t) \in L^2({\it{\Omega}}_T)$$. It has been proven that there exists a unique $$\theta(x, t) \in W^{2, 1}_2 ({\it{\Omega}}_T)$$ to the system (2.2) (Andrews et al., 1993; Copetti & French, 2005). Moreover, the displacement $$u(x, t)$$ and stress $$\sigma(x, t)$$ in (2.1) can be computed in a direct manner from $$\theta(x, t)$$ (Shi et al., 1991). Specially, we can present the displacement at the right end of the rod as below.   $$ \displaystyle u(1, t) = \min \left\{a \int_{{\it{\Omega}}} \theta(\xi, t)\; d\xi, \; g\right\}\!. $$ In this way, we suppose, unless otherwise stated, in what follows when we speak of a solution to (2.1), we shall always mean it solves (2.2) in this sense. Now we proceed the optimal control formulation of the investigated system (2.1). Firstly focus on the fixed final horizon case. Let $$f(x, t) = \rho(x, t) + \omega(t)$$, in which $$\rho(x, t) \in L^2({\it{\Omega}}_T)$$ and $$\omega(t) \in L^2(I_T)$$ is the control. Consider an optimal control problem for the system (2.1) with the general cost functional   \begin{equation}\label{op} \min_{\omega(\cdot)\in U_{ad}} J(\theta, u, \omega) = \min_{\omega(\cdot)\in U_{ad}} \int^T_0\int^{1}_0 L(\theta(x, t), u(x, t), \omega(t), x, t)\, dx\,dt, \end{equation} (2.3) in which the control constraint $$U_{ad}$$ is a non-empty closed convex set of $$L^2(I_T)$$. Take $$(\theta, u) \in W^{2, 1}_2 ({\it{\Omega}}_T) \times W^{1, 1}_2 ({\it{\Omega}}_T)$$ as the state of the system. The control space is $$L^2(I_T)$$ and the control function $$\omega(t)$$ satisfies a convex constraint $$\omega(\cdot) \in U_{ad}$$. Here we assume that the set of $$U_{ad}$$ of admissible controls has the non-empty interior with respect to $$L^2(I_T)$$ topology, i.e., $$\textrm{int}_{L^2(I_T)} U_{ad} \not = \emptyset$$. Of course, this is the normal assumption on the admissible control set, which is often used in literatures (Tröltzsch, 2010). Additionally, we make the following two assumptions for the cost functional. (a) $$L$$ is a functional defined on $$(L^2({\it{\Omega}}))^2 \times U_{ad} \times [0, 1] \times I_T$$ and   $$ \frac{\partial L(\theta, u, \omega, x, t)}{\partial \theta},\; \frac{\partial L(\theta, u, \omega, x, t)}{\partial u},\; \frac{\partial L(\theta, u, \omega, x, t)}{\partial \omega} $$ exist for every $$(\theta, u, \omega)\in (L^2({\it{\Omega}}))^2 \times U_{ad}$$ and $$L$$ is continuous in its variables. (b)   $$ \displaystyle \int^{1}_0\left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial \theta}\right|\,dx, \; \int^{1}_0\left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial u}\right|\,dx, \; \int^{1}_0 \left|\frac{\partial L(\theta, u, \omega, x, t)}{\partial \omega}\right|\,dx $$ are bounded for $$t\in I_T$$. Subsequently, define $$X_T = W^{2, 1}_2({\it{\Omega}}_T) \times W^{1, 1}_2({\it{\Omega}}_T) \times L^2(I_T)$$. Let $$(\theta^*, u^*, \omega^*)$$ be the optimal solution to the optimal control problem (2.3) subject to the equations (2.1). Besides, we set   $$ \begin{array}{ll} {\it{\Xi}}_1 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! \omega(t) \in U_{ad}, \; t\in I_T \textrm{ a.e.}\}, \\ \displaystyle {\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! u(1, t) \leq g \}, \\ \displaystyle {\it{\Xi}}_3 = \{(\theta, u, \omega) \in X_T \; |& \!\!\! u_x(1, t) - a \theta(1, t) \leq 0 \}, \\ \displaystyle {\it{\Xi}}_4 = \{(\theta, u, \omega) \in X_T\;|& \!\!\! \theta_t(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + \rho(x, t) + \omega(t), \\ \displaystyle & \!\!\! \displaystyle u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle & \!\!\! [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle & \!\!\! - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle & \!\!\! \theta(x, 0) = \varphi(x), \; \theta(x, T) = \theta^*(x, T), \; u(x, T) = u^*(x, T)\}. \end{array} $$ And the problem (2.3) is equivalent to questing for $$(\theta^*, u^*, \omega^*) \in {\it{\Xi}} = \bigcap\limits_{i = 1}^4 {\it{\Xi}}_i$$ such that   $$ J(\theta^*, u^*, \omega^*) = \min_{(\theta, u, \omega) \in {{\it{\Xi}}}} J(\theta, u, \omega). $$ As a result, we see that the problem (2.3) is an extremum problem on one constraint of inclusion type $${\it{\Xi}}_1$$, two inequality constraints $${\it{\Xi}}_2$$ and $${\it{\Xi}}_3$$, as well as one equality constraint $${\it{\Xi}}_4$$. Moreover, the Dubovitskii and Milyutin functional analytical approach has been turned out to be very powerful to solve such kind of extremum problems (Girsanov, 1972; Chan & Guo, 1990; Sun & Guo, 2005; Dmitruk, 2009). Now we present the general Dubovitskii and Milyutin theorem for the problem (2.3) in the form of Theorem 1. Theorem 1 (Dubovitskii–Milyutin) Suppose the functional $$J(\theta, u, \omega)$$ assumes a minimum at point $$(\theta^*, u^*, \omega^*)$$ in $${\it{\Xi}}$$. Assume that $$J(\theta, u, \omega)$$ is regularly decreasing at $$(\theta^*, u^*, \omega^*)$$ with the cone of directions of decrease $$K_0;$$ one inclusion constraint $${\it{\Xi}}_1$$ is regular at $$(\theta^*, u^*, \omega^*)$$ with the cone of feasible directions $$K_1;$$ two inequality constraints $${\it{\Xi}}_2$$, $${\it{\Xi}}_3$$, are regular at $$(\theta^*, u^*, \omega^*)$$ with two cones of feasible directions $$K_2, \; K_3;$$ and that the equality constraint $${\it{\Xi}}_4$$ is also regular at $$(\theta^*, u^*, \omega^*)$$ with the cone of tangent directions $$K_4$$. Then there exist continuous linear functionals $$f_i,$$ not all identically zero, such that $$f_i \in K^*_i,$$ the dual cone of $$K_i,$$$$i = 0, 1, \cdots, 4,$$ which satisfy the condition   \begin{equation}\label{dmth} \sum^4_{i = 0} f_i = 0. \end{equation} (2.4) By Theorem 1, we can establish the following Theorem 2, which is the Pontryagin maximum principle of the optimal control problem (2.3) for the longitudinal deformations of a thermoelastic rod with unilateral contact condition of the Signorini type. Theorem 2 Suppose $$(\theta^*, u^*, \omega^*)$$ is a solution to the optimal control problem (2.3). Then there exist a $$\kappa_0 \geq 0$$ and $$(\phi(x, t), v(x, t))$$, not identically zero, such that the following maximum principle holds true:   \begin{align*} \begin{array}{l} \displaystyle \left\{\int^{1}_0 \left[ \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right]dx \right\}\cdot \left[\omega(t) - \omega^*(t)\right] \geq 0, \quad \\ \displaystyle \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall\; \omega(t) \in U_{ad},\; t \in I_T \textrm{ a.e.}, \end{array} \end{align*} where the function $$(\phi(x, t), v(x, t))$$ satisfies the following adjoint equation   \begin{equation}\label{3.24} \left\{ \begin{array}{l} \displaystyle \phi_{t}(x, t) + \phi_{xx}(x, t) - a v_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \phi_x(1, t) - \theta_x(1, t) \phi(1, t) = [a \theta(1, t) - u_x(1, t)] [v(1, t) - \frac{d n(t)}{dt}], \\ \displaystyle a \phi_t(1, t) + v_x(1, t) = \frac{d m(t)}{dt}, \; v(0,t) = 0, \; \phi(0, t) = 0, \\ \displaystyle \phi(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \; \phi(x, 0) = 0. \end{array}\right. \end{equation} (2.5) Here, $$dm(t)$$ is a non-negative measure with support on $${\it{\Lambda}}_1$$ in Lemma 1 of Section 3.1, and $$dn(t)$$ a non-negative measure with support on $${\it{\Lambda}}_2$$ in Lemma 2 of Section 3.2. And $$\mu(x) \in H^{-2}({\it{\Omega}})$$ and $$\nu(x) \in H^{-1}({\it{\Omega}})$$ are given through (3.13). Theorem 2 is the first main result of this article and its proof is in Section 3 below. 3. Proof of Theorem 2 It is noted that the investigated optimal control problem (2.3) in this article is with multiple inequality constraints, which makes this constrained optimization problem more difficult to handle (Sun & Guo, 2015; Boccia et al., 2016). That means that, in the framework of the Dubovitskii–Milyutin approach, we need to consider the extra two cones of feasible directions $$K_2$$, $$K_3$$. By Theorem 1, we proceed as follows: to determine all cones $$K_i$$ and their dual cones $$K_i^*,\; i=0,1, \cdots, 4,$$ one by one; under the guidance of equation (2.4), to derive the final result step by step. First of all, let us find the cone of directions of decrease $$K_0$$. By assumption, $$J(\theta, u, \omega)$$ is differentiable at any point $$(\theta^0, u^0, \omega^0)$$ in any direction $$(\theta, u, \omega)$$ and its directional derivative is   \begin{align*} &\displaystyle J^\prime(\theta^0, u^0, \omega^0; \theta, u, \omega) \\ \displaystyle &\quad= \lim\limits_{\varepsilon\rightarrow 0+}\frac{1}{\varepsilon}\left[J(\theta^0 + \varepsilon \theta, u^0 + \varepsilon u, \omega^0 + \varepsilon \omega) - J(\theta^0, u^0, \omega^0)\right]\\ \displaystyle &\quad= \lim\limits_{\varepsilon\rightarrow 0+}\frac{1}{\varepsilon}\left\{\int^T_0\int^{1}_{0} \left[L(\theta^0 + \varepsilon \theta, u^0 + \varepsilon u, \omega^0 + \varepsilon \omega, x, t) - L(\theta^0, u^0, \omega^0, x, t)\right]\, dx\,dt\right\}\\ \displaystyle &\quad= \int^T_0\int^{1}_{0} \left[\frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial u} \; u + \frac{\partial L(\theta^0, u^0, \omega^0, x, t)}{\partial \omega} \; \omega \right]\,dx\, dt. \end{align*} The cone of directions of decrease of the functional $$J(\theta, u, \omega)$$ at point $$(\theta^*, u^*, \omega^*)$$ is determined by   \begin{align*} K_0 \displaystyle &= \left\{(\theta, u, \omega)\in X_T \,\Big|\, J^\prime(\theta^*, u^*, \omega^*; \theta, u, \omega) < 0 \right\}\\ \displaystyle &= \bigg\{(\theta, u, \omega)\in X_T \,\Big|\, \int^T_0 \int^{1}_{0}\bigg[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u \\ \displaystyle &\qquad\qquad\qquad\qquad\qquad\qquad\quad + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega \bigg]\,dx\,dt < 0 \bigg\}. \end{align*} If $$K_0\neq \emptyset$$, then for any $$f_0\in K^*_0$$, there exists a $$\kappa_0\geq 0 $$ such that   $$ \displaystyle f_0(\theta, u, \omega)= -\kappa_0\int^T_0 \int^{1}_{0}\left[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega \right]\,dx\,dt. $$ Second, the cone of feasible directions $$K_1$$ of $${\it{\Xi}}_1$$ is determined by   \begin{align*} K_1 & =\left\{\kappa\left(\stackrel{o}{{\it{\Omega}}}_1 - \, (\theta^*, u^*, \omega^*)\right) \;\big|\;\kappa>0 \right\}\\ \displaystyle & = \left\{h \;|\; h = \kappa(\theta - \theta^*, u - u^*, \omega - \omega^*), \; (\theta, u, \omega)\in\;\stackrel{o}{{\it{\Omega}}}_1, \;\kappa > 0\right\}\!, \end{align*} in which $$\stackrel{o}{{\it{\Omega}}}_1$$ is the interior of $${\it{\Omega}}_1$$. What is more, for an arbitrary $$f_1\in{K^*_1}$$, if there is an $$\bar{a}(t)\in L^2(I_T),$$ such that the linear functional defined by   \begin{equation}\label{f1} f_1(\theta, u, \omega) = \int_0^T \bar{a}(t) \omega(t)\, dt \end{equation} (3.6) is a support to $$U_{ad}$$ at point $$\omega^*$$, then   \begin{equation}\label{3.9} \bar{a}(t)\left[\omega(t) - \omega^*(t)\right]\geq 0,\quad \forall \; \omega(t) \in U_{ad}, \; t \in I_T \textrm{ a.e.}. \end{equation} (3.7) 3.1. The cone of feasible directions $$K_2$$ Now, we come to treat the inequality constraint $${\it{\Xi}}_2 = \left\{(\theta, u, \omega) \in X_T \; | \; u(1, t) \leq g \right\}$$. Let   \begin{equation}\label{3.10} F_1(u) = \max\limits_{0 \leq t \leq T} \{u(1, t) - g\}. \end{equation} (3.8) Then   $$ {\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \:|\: F_1(u) \leq 0\}. $$ Here we only consider $$F_1(u^*) = \max\limits_{0 \leq t \leq T} \{u^*(1, t) - g\} = 0$$. Since otherwise $$F_1(u^*) < 0$$, and $$(\theta^*, u^*, \omega^*)$$ is an inner point of $${\it{\Xi}}_2$$. In this case, any direction is feasible and hence the cone of feasible directions $$K_2$$ of $${\it{\Xi}}_2$$ at $$(\theta^*, u^*, \omega^*)$$ is the whole space, i.e., $$K_2 = X_T$$. So $${\it{\Xi}}_2 = \{(\theta, u, \omega) \in X_T \:|\: F_1(u) \leq F_1(u^*) \}.$$ Then, we have the following Lemma 1. Lemma 1 Let $$F_1(u)$$ be defined by (3.8). Then $$F_1(u)$$ is differentiable at any point $$\hat{u}$$ in any direction $$u$$ and its directional derivative $$F^\prime_1(\hat{u}; u) = \max\limits_{t \in {\it{\Lambda}}_1}\{u(1, t)\}$$, here $${\it{\Lambda}}_1 = \{t \in I_T \:|\: \hat{u}(1, t) - g = F_1(\hat{u})\},$$ and $$F_1(u)$$ satisfies the Lipschitz condition in any ball. Here we assume that $$F^\prime_1(u; h)\neq 0$$ in the direction $$h$$ provided that $$F_1(u) = 0.$$ Note that for $$F_1(u)$$ given by (3.8), the directional derivative of $$F_1$$ at $$u^*$$ in the direction $$u + 1$$ is   $$ F^\prime_1(u^*; u + 1) = \max\limits_{t \in {\it{\Lambda}}_1}\{u(1, t) + 1\} < 0. $$ Hence   $$ K_2 = \{(\theta, u, \omega) \in X_T \:|\: F^\prime_1(u^*; u) < 0\}. $$ Define the linear operator $$A: X_T \rightarrow H^{1}(I_T)$$ by   $$ A u = - u(1, t) $$ and   $$Y_1 = \{\xi\in H^{1}(I_T) \:|\: \xi(t) \geq 0, \; \forall \; t \in {\it{\Lambda}}_1\}. $$ Then   $$ K_2 = \{(\theta, u, \omega)\in X_T \:|\: A u(x, t) \in Y_1 \}. $$ In view of $$A(u + 1) = -u(1, t) - 1 \in \stackrel{o}{Y_1}$$, the interior of $$Y_1$$, one has   $$ K^*_2=A^* Y^*_1, $$ i.e., for any $$f_2\in K^*_2,$$ there exists a non-negative measure $$dm(t)$$ with support on $${\it{\Lambda}}_1$$ such that   $$ \begin{array}{ll} f_2(\theta, u, \omega) \!\!\!\!\!\; & \displaystyle = \int^T_0 A u(x, t)\, dm(t) = \int_{{\it{\Lambda}}_1} A u(x, t) \, dm(t) \\ & \displaystyle = - \int_{{\it{\Lambda}}_1} u(1, t) \, dm(t) = - \int^T_0 u(1, t) \, dm(t). \end{array} $$ 3.2. The cone of feasible directions $$K_3$$ Then, we proceed to determine the cone of feasible directions $$K_3$$. In fact, the arguments for the determination of the cone of feasible directions of $${\it{\Xi}}_3$$ is similar to that of $${\it{\Xi}}_2$$. Let   \begin{equation}\label{3.13} F_2(\theta, u) = \max\limits_{0 \leq t \leq T} \{u_x(1, t) - a \theta(1, t) \}. \end{equation} (3.9) Then   $$ \begin{array}{ll} {\it{\Xi}}_3\;\!\!\!\!\! & = \left\{(\theta, u, \omega)\in X_T \: |\: u_x(1, t) - a \theta(1, t) \leq 0 \right\}\\ & = \{(\theta, u, \omega)\in X_T\: |\: F_2(\theta, u) \leq 0\}. \end{array} $$ Again we consider only $$F_2(\theta^*, u^*) = \max\limits_{0 \leq t \leq T} \{u^*_x(1, t) - a \theta^*(1, t)\} = 0$$. Since otherwise $$F_2(\theta^*, u^*) < 0$$ and $$(\theta^*, u^*, \omega^*)$$ is an interior point of $${\it{\Xi}}_3$$. Hence any direction is feasible and the cone of feasible directions $$K_3$$ of $${\it{\Xi}}_3$$ at $$(\theta^*, u^*, \omega^*)$$ is the whole space, i.e. $$K_3 = X_T$$. So $${\it{\Xi}}_3 = \{(\theta, u, \omega)\in X_T \: |\: F_2(\theta, u) \leq F_2(\theta^*, u^*) \}$$. Similarly, we have the following Lemma 2. Lemma 2 Let $$F_2(\theta, u)$$ be given by (3.9). Then $$F_2$$ is differentiable at any point $$(\hat{\theta}, \hat{u})$$ in any direction $$(\theta, u)$$ and its directional derivative is given by   $$ F^\prime_2(\hat{\theta}, \hat{u}; \theta, u) = \max\limits_{t \in {\it{\Lambda}}_2}\{u_x(1, t) - a \theta(1, t)\}, $$ where $${\it{\Lambda}}_2 = \{t \in I_T \, | \, \hat{u}_x(1, t) - a \hat{\theta}(1, t)) = F_2(\hat{\theta}, \hat{u})\}$$ and $$F_2(\theta, u)$$ satisfies the Lipschitz condition in any ball, here again we assume that $$F^\prime_2(\theta, u; h, p)\neq 0$$ in the direction $$(h, p)$$ provided that $$F_2(\theta, u) = 0$$. Next since   $$ F^\prime_2(\theta^*, u^*; \theta + 1, u + 1)<0, $$ we have   $$ K_3 = \left\{(\theta, u, \omega) \in X_T \:|\: F^\prime_2(\theta^*, u^*; \theta, u) < 0 \right\}\!. $$ Define the linear operator $$B: X_T \rightarrow H^{1}(I_T)$$ by   $$ B(\theta, u) = a \theta(1, t) - u_x(1, t) $$ and   $$ Y_2 = \{\xi \in H^1(I_T) \:|\: \xi(t) \geq 0, \forall \; t \in {\it{\Lambda}}_2\}. $$ Then   $$ K_3 = \left\{(\theta, u, \omega) \in X_T \:|\: B(\theta(x,t), u(x, t)) \in Y_2 \right\}\!. $$ By virtue of the fact that $$B(\theta + 1, u + 1) = a \theta(1, t) - u_x(1, t) + 1 \in \stackrel{o}{Y_2}$$, the interior of $$Y_2$$, we have $$K^*_3=B^* Y^*_2.$$ Namely, for any $$f_3\in K^*_3 $$, there exists a non-negative measure $$dn(t)$$ with support on $${\it{\Lambda}}_2$$ such that   $$ \begin{array}{ll} f_3(\theta, u, \omega)\!\!\!\!\!\! &\displaystyle =\int^T_0 B(\theta(x, t), u(x, t)) \, dn(t) = \int_{{\it{\Lambda}}_2} B(\theta(x, t), u(x, t)) \, {dn(t)} \\ &\displaystyle = \int_{{\it{\Lambda}}_2} [a \theta(1, t) - u_x(1, t)] \, dn(t) = - \int^T_0 [a \theta(1, t) - u_x(1, t)] \, dn(t). \end{array} $$ So far, we have obtained several cones including the cone of directions of decrease $$K_0$$ and the cones of feasible directions $$K_1, K_2, K_3$$. Furthermore, in their dual cones $$K^*_i,$$ the continuous linear functionals $$f_i \in K^*_i, \; i = 0, 1, 2, 3,$$ are respectively constructed. 3.3. The cone of tangent directions $$K_4$$ Next, we proceed to derive the cone of tangent directions $$K_4$$. Define the operator $$G: X_T \rightarrow (L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2$$ by   $$ \begin{array}{ll} G(\theta, u, \omega) = & \Big(\theta_t(x, t) - \theta_{xx}(x, t) + a u_{xt}(x, t) - \rho(x, t) - \omega(t), \; u_{xx}(x, t) - a \theta_x(x, t), \\ \displaystyle & \quad [u_x(1, t) - a \theta(1, t)] [u(1, t) - g], \; - \theta_x(1, t) - \beta \theta(1, t), \; u(0, t), \; \theta(0, t) - 1, \\ \displaystyle & \quad \theta(x, 0) - \varphi(x), \; \theta(x, T) - \theta^*(x, T), \; u(x, T) - u^*(x, T) \Big). \end{array} $$ Then   $$ {\it{\Xi}}_4 = \left\{(\theta, u, \omega) \in X_T \; | \; G(\theta(x, t), u(x, t), \omega(t)) = 0 \right\}\!. $$ The Fréchet derivative of the operator $$G(\theta, u, \omega)$$ is   \begin{align*} &G^\prime(\theta, u, \omega)(\hat{\theta}, \hat{u}, \hat{\omega}) \\ \displaystyle &\quad= \Big(\hat{\theta}_{t}(x, t) - \hat{\theta}_{xx}(x, t) + a \hat{u}_{xt}(x, t) - \hat{\omega}(t), \; \hat{u}_{xx}(x, t) - a \hat{\theta}_x(x, t),\\ \displaystyle &\qquad\quad [\hat{u}_x(1, t) - a \hat{\theta}(1, t)] [u(1, t) - g] + [u_x(1, t) - a \theta(1, t)] \hat{u}(1, t), - \hat{\theta}_x(1, t) - \beta \hat{\theta}(1, t), \\ \displaystyle &\qquad\quad \hat{u}(0, t), \; \hat{\theta}(0, t), \; \hat{\theta}(x, 0), \; \hat{\theta}(x, T), \; \hat{u}(x, T) \Big). \end{align*} Since $$(\theta^*, u^*, \omega^*)$$ is the solution to problem (2.3), it has $$G^\prime(\theta^*, u^*, \omega^*) = 0$$. Choosing arbitrary   $$ \big(p, q, q_0, q_1, q_2, q_3, q_4, q_5, q_6\big) \in (L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2 $$ and solving the equation   $$ G^\prime(\theta^*, u^*, \omega^*)(\hat{\theta}, \hat{u}, \hat{\omega}) = \big(p(x, t), q(x, t), q_1(t), q_2(t), q_3(t), q_4(t), q_5(x), q_6(x), q_7(x)\big), $$ we obtain   \begin{equation}\label{3.10&#x2013;&#x2013;} \left\{ \begin{array}{l} \displaystyle \hat{\theta}_{t}(x, t) - \hat{\theta}_{xx}(x, t) + a \hat{u}_{xt}(x, t) - \hat{\omega}(t) = p(x, t), \\ \displaystyle \hat{u}_{xx}(x, t) - a \hat{\theta}_x(x, t) = q(x, t), \\ \displaystyle [\hat{u}_x(1, t) - a \hat{\theta}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] \hat{u}(1, t) = q_1(t), \\ \displaystyle - \hat{\theta}_x(1, t) - \beta \hat{\theta}(1, t) = q_2(t), \; \hat{u}(0, t) = q_3(t), \; \hat{\theta}(0, t) = q_4(t), \\ \displaystyle \hat{\theta}(x, 0) = q_5(x), \; \hat{\theta}(x, T) = q_6(x), \; \hat{u}(x, T) = q_7(x). \end{array} \right. \end{equation} (3.10) Next, assume that the linearized system   \begin{equation}\label{3.20} \left\{ \begin{array}{l} \displaystyle {\theta}_{t}(x, t) - {\theta}_{xx}(x, t) + a {u}_{xt}(x, t) = {\omega}(t), \\ \displaystyle {u}_{xx}(x, t) - a {\theta}_x(x, t) = 0, \\ \displaystyle [{u}_x(1, t) - a {\theta}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] {u}(1, t) = 0, \\ \displaystyle - {\theta}_x(1, t) - \beta {\theta}(1, t) = 0, \; {u}(0, t) = 0, \; {\theta}(0, t) = 0, \\ \displaystyle {\theta}(x, 0) = 0, \end{array} \right. \end{equation} (3.11) is controllable. (Please refer to Chan & Guo (1989, 1990) for the information of the linearization.) Then choose $$\omega(t) = \hat{\omega}(t) \in L^2(I_T)$$ such that $${\theta}(x, T) = q_6(x) - \xi(x, T)$$, $${u}(x, T) = q_7(x) - \eta(x, T)$$ and let $$(\theta, u)$$ be the solution to the linearized system (3.11). Choose $$\hat{\theta}(x, t) = \theta(x, t) + \xi(x, t)$$, $$\hat{u}(x, t) = u(x, t) + \eta(x, t)$$, where $$(\xi, \eta)$$ satisfies the following equation   $$ \left\{ \begin{array}{l} \displaystyle {\xi}_{t}(x, t) - {\xi}_{xx}(x, t) + a {\eta}_{xt}(x, t) = p(x, t), \\ \displaystyle {\eta}_{xx}(x, t) - a {\xi}_x(x, t) = q(x, t), \\ \displaystyle [{\eta}_x(1, t) - a {\xi}(1, t)] [u^*(1, t) - g] + [u^*_x(1, t) - a \theta^*(1, t)] {\eta}(1, t) = q_1(t), \\ \displaystyle - {\xi}_x(1, t) - \beta {\xi}(1, t) = q_2(t), \; {\eta}(0, t) = q_3(t), \; {\xi}(0, t) = q_4(t), \\ \displaystyle {\xi}(x, 0) = q_5(x). \end{array} \right. $$ In this way, it suffices for $$(\hat{\theta}, \hat{u}, \hat{\omega})$$ satisfying (3.10). Therefore $$G^\prime(\theta^*, u^*, \omega^*)$$ maps $$X_T$$ onto $$(L^2({\it{\Omega}}_T))^2 \times (L^2(I_T))^4 \times H^1({\it{\Omega}}) \times (L^2({\it{\Omega}}))^2$$. Moreover, the cone of the tangent directions $$K_4$$ to the constraint $${\it{\Xi}}_4$$ at point $$(\theta^*, u^*, \omega^*)$$ consists of the solution to (3.11) in $$X_T$$ and   \begin{equation}\label{3.22} \theta(x, T) = 0, \; u(x, T) = 0. \end{equation} (3.12) Let   \begin{gather*} K_{41}=\{(\theta, u, \omega) \in X_T \; |\; (\theta(x, t), u(x, t), \omega(t)) \textrm{ satisfies } (3.11)\},\\ K_{42}=\{(\theta, u, \omega) \in X_T \; |\; (\theta(x, t), u(x, t), \omega(t)) \textrm{ satisfies } (3.12)\}. \end{gather*} Then the cone of tangent directions $$K_4 = K_{41} \bigcap K_{42}$$. (Under the assumption that (3.11) is controllable, $$K_4 \not = \emptyset.$$ We can deal with other case in the latter proof on non-degeneracy (3.21) of the system.) Hence   $$ K^*_4=K^*_{41}+K^*_{42}. $$ For any $$f_4\in K^*_4$$, decompose $$f_4 = f_{41} + f_{42},\; f_{4i} \in K^*_{4i}$$, the dual cone of $$K_{4i}, \; i = 1, 2$$. Then $$f_{41}(\theta, u, \omega) = 0$$ and for all $$(\theta, u) \in W^{2, 1}_2({\it{\Omega}}_T) \times W^{1, 1}_2({\it{\Omega}}_T)$$ satisfying $$\theta(x, T) = 0$$, $$u(x, T) = 0$$, there exists a $$\mu(x) \in H^{-2}({\it{\Omega}})$$ and a $$\nu(x) \in H^{-1}({\it{\Omega}})$$ such that   \begin{equation}\label{f42} f_{42}(\theta, u, \omega) = \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x) \right] \; dx. \end{equation} (3.13) By the Dubovitskii–Milyutin theorem, there exist continuous linear functionals, not all identically zero, such that   $$ \sum^3_{i = 0} f_i + f_{41} + f_{42} = 0. $$ Therefore, when selecting $$(\theta, u, \omega)$$ satisfies (3.11), $$f_{41}(\theta, u, \omega) = 0$$. Furthermore,   \begin{align} f_1(\theta, u, \omega) & \displaystyle = - f_0(\theta, u, \omega) - f_2(\theta, u, \omega) - f_3(\theta, u, \omega) - f_{42}(\theta, u, \omega) \notag \\ \displaystyle &= \kappa_0\int^T_0\int^{1}_{0}\bigg[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t) \notag\\ \displaystyle &\quad + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega(t) \bigg] \, dx \, dt \label{f-1}\\ \displaystyle &\quad + \int^T_0 u(1, t) \; dm(t) + \int^T_0 [a \theta(1, t) - u_x(1, t)] \; dn(t) \notag\\ \displaystyle &\quad - \int^{1}_0 [\theta(x, T) \mu(x) + u(x, T) \nu(x)] \, dx.\notag \end{align} (3.14) Finally, define the adjoint system of (3.11) as   \begin{equation}\label{adjoint-e} \left\{ \begin{array}{l} \displaystyle \phi_{t}(x, t) + \phi_{xx}(x, t) - a v_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \phi_x(1, t) - \theta_x(1, t) \phi(1, t) = [a \theta(1, t) - u_x(1, t)] [v(1, t) - \frac{d n(t)}{dt}], \\ \displaystyle a \phi_t(1, t) + v_x(1, t) = \frac{d m(t)}{dt}, \; v(0,t) = 0, \; \phi(0, t) = 0, \\ \displaystyle \phi(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \end{array}\right. \end{equation} (3.15) with   \begin{equation}\label{adjoint-e2} \phi(x, 0) = 0. \end{equation} (3.16) As with (2.1), the existence of solution to (3.15) can be obtained similarly. Subsequently, multiply the first two equations in (3.15), (3.16) by $$\theta(x, t)$$, $$u(x, t)$$, respectively, and integrate the product by parts over $${\it{\Omega}}_T$$ with respect to $$x$$ and $$t$$. For the derived integral, we transfer the derivatives from $$\phi(x, t)$$ to $$\theta(x, t)$$ and $$v(x, t)$$ to $$u(x, t)$$. Theorem 3 below then follows the sum of these two integrals. Theorem 3 The solution of system (3.11) and that of its adjoint system (3.15), (3.16) have the following relationship   \begin{align*} &\displaystyle \kappa_0 \int^T_0 \int^{1}_0 \left[\frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t)\right] \, dx\,dt \\ \displaystyle &\qquad + \int^T_0 u(1, t) \; d m(t) + \int^T_0 \left[a \theta(1, t) - u_x(1, t)\right]\; d n(t) - \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x)\right] \,dx \\ \displaystyle &\quad= - \int^T_0 \int^{1}_0 \omega(t) \phi(x, t) \, dx \, dt. \end{align*} Now we come to prove the first main result of this article. Proof of Theorem 2. By Theorem 3, we can rewrite $$f_1(\theta, u, \omega)$$ in (3.14) as   $$ \begin{array}{l} f_1(\theta, u, \omega) = \displaystyle \int^T_0 \left\{\int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t)\right]dx \right\} \omega(t) \, dt. \end{array} $$ In view of (3.6), we have   $$\begin{array}{c} \displaystyle \bar{a}(t) = \int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \,dx \end{array} $$ and (3.7) then reads   \begin{equation}\label{3.28} \begin{array}{l} \displaystyle \left\{\int^{1}_0 \left[\kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right]dx \right\}\cdot \left[\omega(t) - \omega^*(t)\right] \geq 0, \\ \displaystyle \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall\; \omega(t)\in U_{ad},\; t \in I_T \textrm{ a.e.}, \end{array} \end{equation} (3.17) where $$\kappa_0$$ and $$\phi(x, t)$$ are not identical to zero simultaneously. Since otherwise, there are definitely $$f_i = 0, \; i = 0, 1, 2, 3,$$ and $$f_{42} = 0$$, which contradict with the fact in Theorem 1 that these continuous linear functionals are not all identically zero. In addition, when $$K_0$$ is a null set, there is   $$ \begin{array}{r} \displaystyle \int^T_0\int^{1}_0 \bigg[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta} \; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u} \; u(x, t) \\ \displaystyle + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} \; \omega(t) \bigg] \, dx \, dt = 0, \quad \forall \; (\theta, u, \omega) \in X_T. \end{array} $$ In this case, if we choose $$\kappa_0 = 1,$$$$\mu(x) = \nu(x) = 0$$ and $$d m(t)$$, $$d n(t)$$ satisfying   \begin{equation}\label{mtnt} \int^T_0 u(1, t) \; d m(t) = \int^T_0 [u_x(1, t) - a \theta(1, t)] \; d n(t), \end{equation} (3.18) it then follows from Theorem 3 that   \begin{align*} &\displaystyle \int^T_0 \int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}\; \theta(x, t) + \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}\; u(x, t) \right] dx \,dt \\ \displaystyle &\quad= - \int^T_0 \int^{1}_0 \omega(t) \phi(x, t) \,dx\,dt. \end{align*} Thus   $$ \displaystyle \int^T_0 \left\{ \int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \, dx \right\} \omega(t) \, dt = 0, \quad \forall \; \omega(t) \in U_{ad}, $$ from which we obtain   $$ \displaystyle\int^{1}_0 \left[ \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \omega} - \phi(x, t) \right] \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e}. $$ Hence (3.17) still holds. Eventually, if there is a non-zero solution $$(\hat{\phi}(x, t), \hat{v}(x, t))$$ to the adjoint system   \begin{equation}\label{3.29} \left\{ \begin{array}{l} \displaystyle \hat{\phi}_{t}(x, t) + \hat{\phi}_{xx}(x, t) - a \hat{v}_{x}(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial \theta}, \\ \displaystyle \hat{v}_{xx}(x, t) + a \hat{\phi}_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*, x, t)}{\partial u}, \\ \displaystyle \theta(1, t) \hat{\phi}_x(1, t) - \theta_x(1, t) \hat{\phi}(1, t) = [a \theta(1, t) - u_x(1, t)] \left[\hat{v}(1, t) - \frac{d n(t)}{dt}\right], \\ \displaystyle a \hat{\phi}_t(1, t) + \hat{v}_x(1, t) = \frac{d m(t)}{dt}, \; \hat{v}(0,t) = 0, \; \hat{\phi}(0, t) = 0, \\ \displaystyle \hat{\phi}(x, T) = \mu(x), \; a u_x(x, T) \mu(x) = u(x, T) \nu(x), \end{array}\right. \end{equation} (3.19) with   \begin{equation}\label{3.29-1} \hat{\phi}(x, 0) = 0, \end{equation} (3.20) such that the following equality holds true   $$ \displaystyle \int^{1}_0 \omega(t) \hat{\phi}(x, t) \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e.}, $$ then when we choose $$\kappa_0=0$$, $$\mu(x) = \hat{\phi}(x, T)$$, and $$\nu(x)$$ satisfying $$a u_x(x, T) \mu(x) = u(x, T) \nu(x)$$, (3.17) is still valid. Since otherwise, if for any non-zero solution $$(\hat{\phi}, \hat{v})$$ to (3.19), (3.20), it has   \begin{equation}\label{nondegeneracy} \int^{1}_0 \omega(t) \hat{\phi}(x, t) \, dx \not \equiv 0, \end{equation} (3.21) in this case we say the situation is non-degenerate. Then the linearized system (3.11) is controllable. In fact, if (3.11) is not controllable, then there exist a $$\mu(x) \in H^{- 2}({\it{\Omega}})$$ and a $$\nu(x) \in H^{- 1}({\it{\Omega}})$$ such that   $$ \int^{1}_0 \left[\theta(x, T) \mu(x) + u(x, T) \nu(x) \right] \, dx = 0, \quad (\mu(x), \nu(x))\not\equiv 0. $$ Choose $$\kappa_0 = 0$$, $$(\hat{\phi}, \hat{v})$$ to be the solution of (3.19), (3.20) and $$d m(t)$$, $$d n(t)$$ satisfying (3.18). Then it follows from Theorem 3 that   $$ \int^T_0 \left[\int^{1}_0 \hat{\phi}(x, t)\, dx \right] \omega(t) \,dt = 0, \quad \forall \; \omega(t) \in U_{ad}. $$ Hence   $$ \int^1_0 \hat{\phi}(x, t) \, dx = 0, \quad \forall \; t \in I_T \textrm{ a.e.} $$ This is a contradiction. Therefore, under the case of (3.19), (3.20), the system (3.11) is controllable. Combining the results above, we have obtained the Pontryagin maximum principle for the problem (2.3) subject to the system (2.1). This completes the proof of the first main result. □ 4. Optimal control in free final horizon case In the preceding section, we give the Pontryagin maximum principle for optimal control problem (2.3) of the system (2.1) in fixed final horizon case. The result is derived under two additional conditions. The first one is that the admissible control set $$U_{ad}$$ must be convex and contains interior points, i.e., $$\textrm{int}_{L^2(0, T)} U_{ad} \not = \emptyset$$, and the second requires the cost functional $$L$$ to be differentiable with respect to the control variable $$\omega$$. In this section, we consider the system with free final time without these assumptions. Moreover, we still adopt the same symbols to denote these functionals in the case of no confusions caused. 4.1. The second main result Let $$t_1 > 0$$ and define $$I_{t_1} = [0, t_1]$$. Consider the following control system defined in the fixed domain $${\it{\Omega}}_{t_1} = [0, 1] \times I_{t_1}$$,   \begin{equation}\label{fh1} \left\{ \begin{array}{l} \theta_{t}(x, t) - \theta_{xx}(x, t) = - a u_{xt}(x, t) + \rho(x, t) + \omega(t), \\ \displaystyle u_{xx}(x, t) - a \theta_x(x, t) = 0, \\ \displaystyle u(1, t) \leq g, \; u_x(1, t) - a \theta(1, t) \leq 0, \; [u_x(1, t) - a \theta(1, t)] [u(1, t) - g] = 0, \\ \displaystyle - \theta_x(1, t) = \beta \theta(1, t), \; u(0, t) = 0, \; \theta(0, t) = 1, \\ \displaystyle \theta(x, 0) = \varphi(x), \; \theta(x, t_1) = \theta_1(x), \; u(x, t_1) = u_1(x), \\ \displaystyle (x, t)\in {\it{\Omega}}_{t_1} = {\it{\Omega}} \times (0, t_1), \; \omega \in M \subset \mathbb{R}, \end{array} \right. \end{equation} (4.22) and formulate the optimal control problem (4.23) below. Surely it is worth emphasizing the cancellation of assumptions imposed on the preceding fixed final horizon problem. That is to say, in this section the admissible control set $$M$$ is not necessarily convex as well as the cost functional $$L(\theta, u, \omega)$$ need not be differentiable with respect to the control variable $$\omega$$. The optimal control problem with free final horizon $$t_1$$ is presented as follows.   \begin{equation}\label{fop} \textrm{Problem I: Minimize }J(\theta, u, \omega) = \int^{t_1}_0\int^{1}_{0} L(\theta(x, t), u(x, t), \omega(t)) \;dx\,dt \end{equation} (4.23) for $$(\theta, u) \in X_{t_1} = W^{2, 1}_2({\it{\Omega}}_{t_1}) \times W^{1, 1}_2({\it{\Omega}}_{t_1})$$, $$\omega(t) \in L^2(I_{t_1})$$, under the constraints (4.22), where the functional $$L$$ defined on $$L^2({\it{\Omega}}) \times \mathbb{R}$$ satisfies (c) $$L(\theta, u, \omega)$$ is continuous in $$\omega$$. (d) $$ \displaystyle \left|\frac{\partial L(\theta, u, \omega)}{\partial \theta}\right|$$, $$ \displaystyle \left|\frac{\partial L(\theta, u, \omega)}{\partial u}\right|$$ are bounded for every bounded subset of $$L^2({\it{\Omega}}) \times \mathbb{R}$$. And thereafter we will establish the necessary optimality condition of optimal control problem (4.23) for the coupled system (4.22). In the same way, we list main result in advance, which is formulated as Theorem 4 below. Theorem 4 Suppose $$(\theta^*, u^*, \omega^*, t_1)$$ is a solution to Problem I (4.23), then there exist a $$\kappa_0 \geq 0$$ and a pair $$(\phi(x, t)$$, $$v(x, t))$$, not identically zero, such that:   \begin{align*} \displaystyle &\kappa_0 \int^1_{0} L(\theta^*(x, t), u^*(x, t), \omega^*(t)) \; d x + [u^*(1, t) - g] \frac{d \bar{m}(t)}{d t} + [u^*_x(1, t) - a \theta^*(1, t)] \frac{d \bar{n}(t)}{d t} \\ \displaystyle &\quad+ \int^1_0 [- \theta^*_{xx}(x, t) - \rho(x, t) - \omega^*(t)] \phi(x, t) \; dx = 0, \\ \displaystyle &\qquad \forall\; t\in [0,t_1] \textrm{ a.e.},\\ \displaystyle &\kappa_0 \int^1_{0} L(\theta^*(x, t), u^*(x, t), \omega(t)) \; d x + [u^*(1, t) - g] \frac{d \bar{m}(t)}{d t} + [u^*_x(1, t) - a \theta^*(1, t)] \frac{d \bar{n}(t)}{d t} \\ \displaystyle &\quad+ \int^1_0 [- \theta^*_{xx}(x, t) - \rho(x, t) - \omega(t)] \phi(x, t) \; dx = 0, \\ \displaystyle &\qquad\forall\; \omega \in M, \; t \in[0, t_1] \textrm{ a.e.}, \end{align*} where the function pair $$(\phi(x, t), \; v(x, t))$$ satisfies   $$ \left\{ \begin{array}{l} \displaystyle \phi_t(x, t) - \phi_{xx}(x, t) - a v(x, t) = \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta}, \\ \displaystyle \displaystyle v_{xx}(x, t) + a \phi_{xt}(x, t) = - \kappa_0 \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u}, \\ \displaystyle - \theta(1, t) \phi_x(1, t) - \beta \theta(1, t) = \left[- u_x(1, t) + a \theta(1, t)\right] \left[v(1, t) + \frac{d \bar{n}(t)}{d t}\right]\!, \\ \displaystyle a \phi_t(1, t) = \frac{d \bar{m}(t)}{d t} - v_x(1, t), \\ \displaystyle \theta_x(0, t) \phi(0, t) = [\theta(0, t) + u(0, t)] \phi_x(0, t), \; v(0, t) = 0, \\ \displaystyle \phi(x, t_1) = \mu(x), \; a u_x(x, t_1) \mu(x) = u(x, t_1) \nu(x), \; \phi(x, 0) = 0. \end{array} \right. $$ Here, two non-negative measures $$d \bar{m}(t)$$, $$d \bar{n}(t)$$, with the functions $$\mu(x) \in H^{- 2}({\it{\Omega}})$$ and $$\nu(x) \in H^{- 1}({\it{\Omega}})$$, will be presented in (4.28) in the next subsection. The proof of Theorem 4 is given in next subsection. 4.2. Maximum principle of problem (4.23) Now we introduce a time transformation $$t \rightarrow s$$, mapping $$I_{t_1}$$ onto $$[0, 1]$$, defined by a certain function $$\varpi(\cdot) \geq 0$$,   $$ t(s) = \int^{s}_0 \varpi(\varsigma) \, d \varsigma, \quad t(1) = t_1, $$ and let $$\theta(x, s) = \theta(x, t(s))$$, $$u(x, s) = u(x, t(s))$$,   \begin{equation}\label{f2.5} \omega(s) = \left\{\displaystyle \begin{array}{cc} \omega(t(s)), & s \in {\it{\Upsilon}}_1 = \{s\;|\; s \in[0,1],\; \varpi(s) > 0\}, \\ \displaystyle \textrm{arbitrary}, & s \in {\it{\Upsilon}}_2 = \{s\;|\; s\in[0,1],\; \varpi(s) = 0\}, \end{array} \right. \end{equation} (4.24) Then $$(\theta(x, s), u(x, s), \omega(s))$$ satisfies the following equations   \begin{equation}\label{f2.6} \left\{ \begin{array}{l} \theta_{s}(x, s) - \theta_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi(s) - a \theta_x(x, s) \varpi(s) = 0, \\ \displaystyle u(1, s) \varpi(s) \leq g \varpi(s), \; u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s) \leq 0, \\ \displaystyle [u_x(1, s) - a \theta(1, s)] [u(1, s) - g] \varpi(s) = 0, \\ \displaystyle - \theta_x(1, s) \varpi(s) = \beta \theta(1, s) \varpi(s), \; u(0, s) \varpi(s) = 0, \; \theta(0, s) \varpi(s) = \varpi(s), \\ \displaystyle \theta(x, 0) = \varphi(x), \; \theta(x, 1) = \theta_1(x), \; u(x, 1) = u_1(x), \end{array} \right. \end{equation} (4.25) in which $$\rho(x, s) = \rho(x, t(s))$$. To make the definition of $$s(t)$$ one-to-one, we shall assume that   $$ s(t) = \inf\{s \;|\; t(s)=t\}. $$ Define $${\it{\Omega}}_1 = {\it{\Omega}} \times (0, 1)$$. And then we can formulate a new problem:   $$ \textrm{Problem II: Minimize }J(\theta, u, \omega, \varpi) = \int^1_0 \int^{1}_0 \varpi(s) L(\theta(x, s), u(x, s), \omega(s)) \; dx\,ds $$ for $$(\theta, u) \in W^{2, 1}_2({\it{\Omega}}_1) \times W^{1, 1}_2({\it{\Omega}}_1)$$, $$\omega(s) \in L^2(0, 1)$$, $${\varpi(s)} \in L^2(0, 1)$$, under the constraints (4.25) with $$\varpi(s)\geq 0$$, $$\omega(s) \in M$$, for almost all $$0 \leq s \leq 1$$. If $$(\theta^*, u^*, \omega^*)$$ is an optimal solution to the control problem (4.23) subject to the equations (4.22), then for any $$\varpi^*(s) \geq 0$$ satisfying $$\int^1_0 \varpi^*(\varsigma)\, d\varsigma = t_1$$, $$\omega^*(s)$$ defined similar to (4.24), $$(\theta^*, u^*, \omega^*, \varpi^*)$$ solves Problem II (Girsanov, 1972). Fixing $$\omega = \omega^*$$, another optimal control problem can be formulated as:   $$ \textrm{Problem III: Minimize }J(\theta, u, \omega^*, \varpi) = \int^1_0\int^{1}_0 \varpi(s) L(\theta(x, s), u(x, s), \omega^*(s)) \; dx\,ds $$ for $$(\theta(x, s), u(x, s), \varpi(s)) \in X_1 = W^{2, 1}_2({\it{\Omega}}_1) \times W^{1, 1}_2({\it{\Omega}}_1) \times L^{2}(0, 1)$$ subject to   \begin{equation}\label{p3} \left\{ \begin{array}{l} \theta_{s}(x, s) - \theta_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega^*(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi(s) - a \theta_x(x, s) \varpi(s) = 0, \\ \displaystyle u(1, s) \varpi(s) \leq g \varpi(s), \; u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s) \leq 0, \\ \displaystyle [u_x(1, s) - a \theta(1, s)] [u(1, s) - g] \varpi(s) = 0, \\ \displaystyle - \theta_x(1, s) \varpi(s) = \beta \theta(1, s) \varpi(s), \; u(0, s) \varpi(s) = 0, \; \theta(0, s) \varpi(s) = \varpi(s), \\ \displaystyle \theta(x, 0) = \varphi(x), \end{array} \right. \end{equation} (4.26) and   $$ \theta(x, 1) = \theta_1(x), \; u(x, 1) = u_1(x), $$ in which $$\varpi(s)$$ plays the role of control. Again, we assume that the set of new admissible controls has the non-empty interior with respect to $$L^2(0, 1)$$ topology (Tröltzsch, 2010). In what follows, we will prove the second main result. Proof of Theorem 4. We observe that Problem III is an optimal control problem with fixed final horizon, which can be tackled by the same method adopted in the investigation of the preceding optimal control problem (2.3) under the direction of the similar theorem to Theorem 1 (in this case, by assumption (c), (d), the new cost functional in Problem III naturally satisfies the similar conditions to assumption (a), (b)). For the sake of brevity, here we only list the key results and omit the detailed procedures. The linearized system of system (4.26) in this case reads as   \begin{equation}\label{ls2} \left\{ \begin{array}{l} \displaystyle \theta_s(x, s) - \theta_{xx}(x, s) \varpi^*(s) - \theta^*_{xx}(x, s) \varpi(s) = - a u_{xs}(x, s) + \rho(x, s) \varpi(s) + \omega^*(s) \varpi(s), \\ \displaystyle u_{xx}(x, s) \varpi^*(s) - a \theta_x(x, s) \varpi^*(s) + u^*_{xx}(x, s) \varpi(s) - a \theta^*_x(x, s) \varpi(s) = 0, \\ \displaystyle [u^*_x(1, s) - a \theta^*(1, s)] [u^*(1, s) - g] \varpi(s) + [u^*_x(1, s) - a \theta^*(1, s)] u(1, s) \varpi^*(s) \\ \displaystyle \quad + [u_x(1, s) - a \theta(1, s)] [u^*(1, s) - g] \varpi^*(s) = 0, \\ \displaystyle [\theta_x(1, s) + \beta \theta(1, s)] \varpi^*(s) + [\theta^*_x(1, s) + \beta \theta^*(1, s)] \varpi(s) = 0, \\ \displaystyle u^*(0, s) \varpi(s) + u(0, s) \varpi^*(s) = 0, \; \theta^*(0, s) \varpi(s) + \theta(0, s) \varpi^*(s) = 0, \\ \displaystyle \theta(x, 0) = 0. \end{array} \right. \end{equation} (4.27) There exist $$\kappa_0 \geq 0$$, $$\bar{a}(s) \in L^2(0, 1)$$, $$\phi(x) \in H^2_0(0, 1)$$ and $$\psi(x) \in H^1_0(0, 1)$$ such that those continuous linear functionals in the general Dubovitskii and Milyutin theorem can be respectively determined as   \begin{equation}\label{fs} \left\{ \begin{array}{l} \displaystyle f_0(\theta, u, \omega^*, \varpi) = -\kappa_0 \int^1_0 \int^1_{0} \bigg[ \varpi^*(s)\frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta} \; \theta(x, s) + \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u} \; u(x, s) \\ \displaystyle \hspace{2.11in} + L(\theta^*, u^*, \omega^*) \varpi(s) \bigg] \; dx\,ds, \\ \displaystyle f_1(\theta, u, \omega^*, \varpi) = \int^1_0 \bar{a}(s) \varpi(s) \; ds, \\ \displaystyle f_2(\theta, u, \omega^*, \varpi) = \int^1_0 \left[- u^*(1, s) \varpi(s) - u(1, s) \varpi^*(s) + g \varpi(s)\right]\; d \bar{m}(s), \\ \displaystyle f_3(\theta, u, \omega^*, \varpi) = \int^1_0 \bigg[- u^*_x(1, s) \varpi(s) - u_x(1, s) \varpi^*(s) \\ \displaystyle \hspace{1.65in} + a \theta^*(1, s) \varpi(s) + a \theta(1, s) \varpi^*(s)\bigg]\; d \bar{n}(s), \\ \displaystyle f_{41}(\theta, u, \omega^*, \varpi) = 0, \\ \displaystyle f_{42}(\theta, u, \omega^*, \varpi) = \int^1_{0} \left[\theta(x, 1) \mu(x) + u(x, 1) \nu(x)\right] \; dx. \end{array}\right. \end{equation} (4.28) Here, there exists a non-negative measure $$d \bar{m}(s)$$ with support on   $$ {\it{\Lambda}}_3 = \left\{s \in [0, 1] \; | \; \hat{u}(1, s) \hat{\varpi}(s) - g \hat{\varpi}(s) = F_3(\hat{u}, \hat{\varpi})\right\} $$ with   $$ F_3(u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{u(1, s) \varpi(s) - g \varpi(s)\right\} $$ and its directional derivative   $$ F^\prime_3(\hat{u}, \hat{\varpi}; u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{\hat{u}(1, s) \varpi(s) + u(1, s) \hat{\varpi}(s) - g \varpi(s)\right\} $$ at any point $$(\hat{u}, \hat{\varpi})$$ in any direction $$(u, \varpi)$$. Similarly, there exists a non-negative measure $$d \bar{n}(s)$$ with support on   $$ {\it{\Lambda}}_4 = \left\{s \in [0, 1] \; | \; \hat{u}_x(1, s) \hat{\varpi}(s) - a \hat{\theta}(1, s) \hat{\varpi}(s) = F_4(\hat{\theta}, \hat{u}, \hat{\varpi})\right\} $$ with   $$ F_4(\theta, u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{u_x(1, s) \varpi(s) - a \theta(1, s) \varpi(s)\right\} $$ and its directional derivative   $$ F^\prime_4(\hat{\theta}, \hat{u}, \hat{\varpi}; \theta, u, \varpi) = \max\limits_{0 \leq s \leq 1} \left\{\hat{u}_x(1, s) \varpi(s) + u_x(1, s) \hat{\varpi}(s) - a \hat{\theta}(1, s) \varpi(s) - a \theta(1, s) \hat{\varpi}(s)\right\} $$ at any point $$(\hat{\theta}, \hat{u}, \hat{\varpi})$$ in any direction $$(\theta, u, \varpi)$$. Correspondingly, the adjoint system of the linearized system (4.27) is   \begin{equation}\label{as2} \left\{ \begin{array}{l} \displaystyle \phi_s(x, s) - \varpi^*(s) \phi_{xx}(x, s) - a \varpi^*(s) v(x, s) = \kappa_0 \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta}, \\ \displaystyle \displaystyle \varpi^*(s) v_{xx}(x, s) + a \phi_{xs}(x, s) = - \kappa_0 \varpi^*(s) \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u}, \\ \displaystyle - \theta(1, s) \phi_x(1, s) - \beta \theta(1, s) = \left[- u_x(1, s) + a \theta(1, s)\right] \left[v(1, s) + \frac{d \bar{n}(s)}{d s}\right]\!, \\ \displaystyle a \phi_s(1, s) = \varpi^*(s) \left[\frac{d \bar{m}(s)}{d s} - v_x(1, s)\right]\!, \\ \displaystyle \theta_x(0, s) \varpi^*(s) \phi(0, s) = [\theta(0, s) \varpi^*(s) + u(0, s)] \phi_x(0, s), \; v(0, s) = 0, \\ \displaystyle \phi(x, 1) = \mu(x), \; a u_x(x, 1) \mu(x) = u(x, 1) \nu(x), \end{array} \right. \end{equation} (4.29) with   \begin{equation}\label{as2-1} \phi(x, 0) = 0. \end{equation} (4.30) Furthermore, the relationship between the solution of the linearized system (4.27) and that of its adjoint system (4.29) with (4.30), is   \begin{align*} \displaystyle &\kappa_0 \int^1_0 \int^1_{0} \varpi^*(s) \left[\frac{\partial L(\theta^*, u^*, \omega^*)}{\partial \theta} \; \theta(x, s) + \frac{\partial L(\theta^*, u^*, \omega^*)}{\partial u} \; u(x, s)\right]\; dx\,ds \\ \displaystyle &\quad + \int^1_{0} u(1, s) \varpi^*(s) \; d \bar{m}(s) - \int^1_{0} \left[- u_x(1, s) + a \theta(1, s) \right] \varpi^*(s) \; d \bar{n}(s) \\ \displaystyle &\quad - \int^1_{0} \left[\theta(x, 1) \mu(x) + u(x, 1) \nu(x)\right]\, dx\\ \displaystyle &= \int^1_0 \int^1_{0} \Big\{\left[- \theta^*_{xx}(x, s) - \rho(x, s) - \omega^*(s)\right] \phi(x, s) + \left[u^*_{xx}(x, s) - a \theta^*_x(x, s)\right] v(x, s) \Big\} \varpi(s) \; dx \, ds \\ \displaystyle &\quad - \int^1_0 \Big\{ [\theta^*_{x}(1, s) + \beta \theta^*(1, s)] \phi(1, s) + u^*(0, s) v_{x}(0, s)\Big\} \varpi(s) \; ds. \end{align*} By this relationship expression, we can get the Pontryagin maximum principle of Problem III. After that, the maximum principle of Problem I with free final horizon can be easily obtained (Girsanov, 1972; Dmitruk, 2009). And the obtained result is stated as Theorem 4, which is none other than the Pontryagin maximum principle of Problem I with free final horizon. This completes the proof of the second main result. □ 5. Conclusions In control theory, the maximum principle does not always hold as a necessary condition for optimal control of the infinite-dimensional system. Thus, the infinite-dimensional generalization of the maximum principle is an important and interesting topic in this field. For this purpose, this article considers the optimal control for longitudinal deformations of a thermoelastic rod. Specially, the investigational controlled system is equipped with unilateral contact condition of the Signorini type. And in both the fixed and free final horizon cases, we establishes the necessary optimality conditions given in the form of the Pontryagin maximum principle. Frankly speaking, the immediately following step is to numerically solve this optimal control problem. However, note that the investigational object in this article is an optimal control problem of the distributed parameter system governed by partial differential equations equipped with the Signorini unilateral boundary conditions. Even in fixed final horizon case, in terms of optimal control problem (2.3), the solution process includes not only the numerical approximation to the state equation but also solving the corresponding adjoint equation (2.5). We still need to determine the non-negative measures $$dm(t)$$ and $$dn(t)$$, which further aggravate this procedure. It is definitely not an easy job to get the numerical solutions for the optimal control-trajectory pair. In Sun & Wu (2013), the efficient numerical approaches are discussed and then a min-H iterative method with the higher convergence rate are developed to present the numerical solution of optimal control. It does show the applicability of obtained Pontryagin maximum principle although it does not give the detailed numerical simulation. People can refer to it for the details. On other numerical investigations under the Dubovitskii–Milyutin formalism, people can read Gayte et al. (2010) and Kotarski (1997) for the information. In conclusion, we investigate the optimal control for longitudinal deformations of a thermoelastic rod and prove the Pontryagin maximum principles in two cases, which constitute main results of this paper. The investigational model is deterministic without any uncertainties and unknown parameters. However, in real-world applications, there are a lots of distributed parameter systems where the exact model of the system is unknown and contains uncertainties. The articles such as Kulkarni et al. (2006) and Rebiai & Zinober (1993) can be read for the easy reference. Funding National Natural Science Foundation of China (Grant No. 11471036) (in part). Acknowledgements The authors would like to thank the editor and the anonymous referees for their very careful reading and constructive suggestions that improve substantially the article. References Acquistapace P., Bucci F. & Lasiecka I. ( 2005) A trace regularity result for thermoelastic equations with application to optimal boundary control. J. Math. Anal. Appl.,  310, 262– 277. Google Scholar CrossRef Search ADS   Andrews K. T., Shi P., Shillor M. & Wright S. ( 1993) Thermoelastic contact with Barber’s heat exchange condition. Appl. Math. Optim. , 28, 11– 48. Google Scholar CrossRef Search ADS   Boccia A., de Pinho M. D. R. & Vinter R. B. ( 2016) Optimal control problems with mixed and pure state constraints. SIAM J. Control Optim. , 54, 3061– 3083. Google Scholar CrossRef Search ADS   Bucci F. & Lasiecka I. ( 2004) Singular estimates and Riccati theory for thermoelastic plate models with boundary thermal control. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. , 11, 545– 568. Chan W. L. & Guo B. Z. ( 1989) Optimal birth control of population dynamics. J. Math. Anal. Appl. , 144, 532– 552. Google Scholar CrossRef Search ADS PubMed  Chan W. L. & Guo B. Z. ( 1990) Optimal birth control of population dynamics. II. Problems with free final time, phase constraints, and mini-max costs. J. Math. Anal. Appl. , 146, 523– 539. Google Scholar CrossRef Search ADS PubMed  Copetti M. I. M. ( 1999) Finite element approximation to a contact problem in linear thermoelasticity. Math. Comput. , 68, 1013– 1024. Google Scholar CrossRef Search ADS   Copetti M. I. M. & French D. A. ( 2005) Numerical approximation and error control for a thermoelastic contact problem. Appl. Numer. Math. , 55, 439– 457. Google Scholar CrossRef Search ADS   Dmitruk A. V. ( 2009) On the development of Pontryagin’s maximum principle in the works of A. Ya. Dubovitskii and A. A. Milyutin. Control and Cybernet. , 38, 923– 957. Fattorini H. O. ( 2005) Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems . North-Holland Mathematics Studies, Vol. 201. Amsterdam: Elsevier Science B.V. Gayte I., Guillén-González F. & Rojas-Medar M. ( 2010) Dubovitskii-Milyutin formalism applied to optimal control problems with constraints given by the heat equation with final data. IMA J. Math. Control Inform. , 27, 57– 76. Google Scholar CrossRef Search ADS   Girsanov I. V. ( 1972) Lectures on Mathematical Theory of Extremum Problems . Lecture Notes in Economics and Mathematical Systems, Vol. 67. Berlin: Springer. Google Scholar CrossRef Search ADS   Ignaczak J. & Ostoja-Starzewski M. ( 2010) Thermoelasticity with Finite Wave Speeds . Oxford Mathematical Monographs. Oxford: Oxford University Press. Kotarski W. ( 1997) Some Problem of Optimal and Pareto Optimal Control for Distributed Parameter Systems . Katowice: Wydawinictwo Uniwersytetu Śla̧skiego. Kulkarni K., Zhang L. & Linninger A. A. ( 2006) Model and parameter uncertainty in distributed systems. Ind. Eng. Chem. Res. , 45, 7832– 7840. Google Scholar CrossRef Search ADS   Pelesko J. A. ( 1999) Nonlinear stability considerations in thermoelastic contact. Trans. ASME J. Appl. Mech. , 66, 109– 116. Google Scholar CrossRef Search ADS   Rebiai S. E. & Zinober A. S. I. ( 1993) Stabilization of uncertain distributed parameter systems. Int. J. Control , 57, 1167– 1175. Google Scholar CrossRef Search ADS   Shi P., Shillor M. & Zou X. L. ( 1991) Numerical solutions to one-dimensional problems in thermoelastic contact. Comput. Math. Appl. , 22, 65– 78. Google Scholar CrossRef Search ADS   Sun B. & Guo B. Z. ( 2005) Maximum principle for the optimal control of an ablation-transpiration cooling system. J. Syst. Sci. Complex.  18, 285– 301. Sun B. & Guo B. Z. ( 2015) Convergence of an upwind finite-difference scheme for Hamilton-Jacobi-Bellman equation in optimal control. IEEE Trans. Automat. Control , 60, 3012– 3017. Google Scholar CrossRef Search ADS   Sun B. & Wu M. X. ( 2013) Optimal control of age-structured population dynamics for spread of universally fatal diseases. Appl. Anal. , 92, 901– 921. Google Scholar CrossRef Search ADS   Tröltzsch F. ( 2010) Optimal Control of Partial Differential Equations: Theory, Methods and Applications . Graduate Studies in Mathematics, vol. 112. Providence, RI: American Mathematical Society. Translated from the 2005 German original by J. Sprekels. © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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