Add Journal to My Library
IMA Journal of Mathematical Control and Information
, Volume Advance Article – Feb 8, 2018

15 pages

/lp/ou_press/the-approximate-solution-of-non-linear-time-delay-fractional-optimal-blkvF6q2D3

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnx063
- Publisher site
- See Article on Publisher Site

Abstract In this paper, a class of time-delay fractional optimal control problems (TDFOCPs) is studied. Delays may appear in state or control (or both) functions. By an embedding process and using conformable fractional derivative as a new definition of fractional derivative and integral, the class of admissible pair (state, control) is replaced by a class of positive Radon measures. The optimization problem found in measure space is then approximated by a linear programming problem (LPP). The optimal measure which is representing optimal pair is approximated by the solution of a LPP. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to non-linear TDFOCPs. The usefulness of the used idea in this paper is that the method is not iterative, quite straightforward and can be applied to non-linear dynamical systems. 1. Introduction Fractional optimal control problems (FOCPs) have been investigated by many researchers due to their applications over the simulation and modeled in mathematics and natural physical processes (see Agrawal, 2008; Tricaud & Chen, 2010). One can see these applications in signal processing, complex dynamic, mechanics, viscoelasticity and so on (see Bagley & Torvic, 1984; Magin, 2010; Merala et al., 2010; Sabatier et al., 2011). As a special case of FOCP when time delay in control and (or) state has occurred, we can find widespread fields to observe and study, such as life sciences, populations biology, physiology, production of red blood cells by the stem cells in the bone marrow, thermal systems and heat conduction. Also in (Belair et al., 1995; Crauste, 2009) many applications of time delay fractional systems that describe natural phenomena are introduced. So, an effective technique for the solution of time-delay fractional optimal control problem (TDFOCP) is mostly required (see Agrawal, 2008). During recent years, optimal control problems (OCPs) have been considered in many researches and also different approaches have been developed for time-delay optimal control systems. In (Bellen & Zennaro, 2003; Dadkhah et al., 2015; Dadkhah et al., 2016; Ghomanjani et al., 2013; Dehghan & Keyanpour, 2015), the reader can find some methods which are introduced. In this article, we use an embedding process to solve TDFOCPs. In this process, measure theory approach is used and we show that this straightforward and systematic method can be applied for various classes of TDFOCPs. Measure theory approach is used successfully by many researchers, here we mention only some of them (Farahi et al., 1996; Nazemi et al., 2007; Koshkouei et al., 2012). Koshkouei et al., 2012 applied measure theory for solving a class of time-delay optimal control systems governed by integer-order dynamical systems. In the study by Ziaei et al. (2017), we extend this embedding technique to solve FOCPs, where we used conformable fractional calculus, introduced by (Khalil et al., 2014). Conformable fractional calculus versus the most famous fractional derivatives such as Riemann–Liouville, Caputo and Grunwald–Letinkov has some advantages like conforming the chain rule, product and Leibniz rule, where these advantages will be of great benefits in using measure theory approach to solve TDFOCPs. In this paper, we propose a new powerful method in generalization of measure theory approach to solve the systems of non-linear fractional order with time delay in state and control. To achieve this goal, first we define a fractional positive measure, then by using functional analysis and linear programming, the TDFOCP is replaced with a minimizing linear form subject to linear equalities where the optimal solution of this linear programming problem (LPP) gives the approximate optimal pair of state and control. The rest of the paper is organized as follows: in Section 2, some definitions on conformable fractional calculus and related properties are presented. Sections 3 and 4 addresses the system description and functional space and approximation process by embedding method. Two examples are given in Section 5 and finally Section 6 consists of the conclusion. 2. Some preliminaries of conformable fractional calculus This section consists of some basic concepts in fractional calculus. We assume w = f(t) (t ⩾ 0) be a real valued and continuous function, and α ⩾ 0 is a given real number. Definition 2.1 Let $$f:\left [ 0, \infty \right ] \longrightarrow R$$, then the conformable fractional derivative of f(t) is defined as follows (Benkhettou et al., 2016; Khalil et al., 2014): \begin{align} T_{\alpha}\,f(t)=\lim_{\varepsilon\to 0} \frac{f(t+\varepsilon t^{1-\alpha})-f(t)}{\varepsilon}\,\,\,, \quad 0\leqslant\alpha <1,\, t\geqslant 0. \end{align} (2.1) We write sometimes f(α)(t) for Tαf(t) to denote conformable fractional derivative of order α, also if Tαf(t) exists, then we say f is α-differentiable. The classic fractional derivatives, Riemann–Liouville and Caputo sense lost some of the basic properties such as product rule and chain rule, while conformable fractional derivative obeys these conventional properties, so this new concept is author’s motivation to use conformable fractional derivative in solving TDFOCPs. Let α ∈ (0, 1] and f, g be α-differentiable for t > 0, then the following properties can be resulted by the Definition 2.4 in the study by Khalil et al., 2014: \begin{align} T_{\alpha}(af+bg)=aT_{\alpha}(\,f)+bT_{\alpha}(g),\,\,\, a, b \in R, \end{align} (2.2) \begin{align} T_{\alpha}(t^{p})=pt^{p-\alpha}, \quad p\in R, \end{align} (2.3) \begin{align} T_{\alpha}(\lambda) =0, \quad \lambda\ \textrm{is a constant number,} \end{align} (2.4) \begin{align} T_{\alpha}(\,fg)=f\,T_{\alpha}(g)+gT_{\alpha}(\,f), \end{align} (2.5) \begin{align} T_{\alpha}\left(\dfrac{f}{g}\right)=\dfrac{gT_{\alpha}(\,f)-f\,T_{\alpha}(g)}{g^{2}}. \end{align} (2.6) Moreover, if f be a differentiable function, then one can prove that \begin{align} T_{\alpha}\,f(t)=t^{1-\alpha}\dfrac{df(t)}{dt}. \end{align} (2.7) Definition 2.2 Let $$f:\left ( 0, t\right ) \longrightarrow R$$ (t ⩾ 0), be a continuous function and $$\alpha \in \left ( 0, 1 \right )$$, the conformable α-fractional integral of a function f is defined as: \begin{align} I_{\alpha}\,f(t)={\int_{0}^{t}}\tau^{\alpha-1}f(\tau) \, \textrm{d}\tau. \end{align} (2.8) Theorem 2.3 Let $$f:[0, \infty ]\longrightarrow R$$ be a function such that f is differentiable and also α-differentiable. Let g be a function defined in the range of f and also differentiable, then we have the following rule: \begin{align} T_{\alpha}(\,fog)(t)=(T_{\alpha}f(g(t))\cdot(T_{\alpha}g(t))\cdot g(t)^{\alpha-1}. \end{align} (2.9) Proof. (see Benkhettou et al., 2016) Theorem 2.4 Let f be a differentiable function for t > 0, and 0 < α ⩽ 1, then \begin{align} I_{\alpha}T_{\alpha}\,f(t)=f(t)-f(0), \end{align} (2.10) Proof. Since f is differentiable by (2.7) and (2.8) we have $$ I_{\alpha}T_{\alpha}\,f(t)={\int_{0}^{t}} (\tau)^{\alpha-1}T_{\alpha}\,f(\tau)\, \textrm{d}\tau={\int_{0}^{t}} (\tau )^{\alpha -1}\left((\tau )^{1-\alpha} \dfrac{\textrm{d}f(\tau)}{\textrm{d}\tau} \right) \textrm{d}\tau= f(t)-f(0). $$ 3 System description and variational formulation Consider the following time-delay fractional system: \begin{align} \,\,\,\,\,\,\,\,\,\,\,\,Minimize\qquad J(x(.), u(.))=\int_{I} f_{0}(t,x(t),u(t),x(t-\tau_{1}), u(t-\tau_{2}))\, \textrm{d}t, \end{align} (3.1) S. to: \begin{align} x^{(\alpha)}(t)=g(t, x(t), u(t),x(t-\tau_{1}),u(t-\tau_{2})), \quad t \in I=[t_{0}, t_{f}], \end{align} (3.2) \begin{align} x(t_{0})&=x_{0},\, x(t_{f})=x_{f},\nonumber\\ x( t)&=\phi(t),\qquad t\in[-\tau_{1}, t_{0}], \nonumber\\ u( t)&=\theta(t),\qquad t\in[-\tau_{2}, t_{0}]. \end{align} (3.3) We mention that x(.) : I→A ⊆ Rn is the system function that is absolutely continuous on I, the set A is closed and bounded in Rn, u(.) : I→U ⊆ Rm is a piecewise continuous measurable control function on I, also the set U is a closed and bounded subset in Rm. Also, ϕ(t) = [ϕ1(t)⋯ϕn(t)]T, θ(t) = [θ1(t)⋯θm(t)]T are given vector functions and τ1, τ2 are known positive constant real number. Define τ = max{τ1, τ2}, I1 = [t0 − τ, tf], and assume tf − t0 ⩾ τ. Also, we assume $$ f_{0}\in \mathbb{R}$$ and $$ g \in \mathbb{R}^{n}$$ are respectively smooth function and vector field. Suppose \begin{array}{c} \alpha_{1j}\leqslant \phi_{j}(t)\leqslant \alpha_{2j},\\ \\ \beta_{1i} \leqslant \theta_{i}(t) \leqslant \beta_{2i},\\ \\ \delta_{1j} \leqslant x_{j}(t) \leqslant \delta_{2j},\\ \\ \lambda_{1i} \leqslant u_{i}(t) \leqslant \lambda_{2i},\end{array} for j = 1, 2, ⋯ , n i = 1, 2, ⋯ , m and t ∈ I1. Define: \begin{align}\mathit{\Omega}_{1}&=[t_{0}-\tau, t_{0}]\times\prod_{j=1}^{j=n}[\alpha_{1j}, \alpha_{2j}]\times\prod_{i=1}^{i=m}[\beta_{1i}, \beta_{2i}],\nonumber\\ \Omega_{2}&=[t_{0}, t_{f}]\times\prod_{j=1}^{j=n}[\delta_{1_{j}}, \delta_{2_{j}}]\times\prod_{i=1}^{i=m}[\lambda_{1_{i}}, \lambda_{2_{i}}],\nonumber\\ \Omega & =\Omega_{1}\cup\Omega_{2} . \end{align} (3.4) We need to recall that t0 is the initial time and x0 = x(t0) is initial condition. Since in fractional optimal control delay systems (3.1)–(3.3), we have delay in state x(t) and control u(t), so we define: \begin{align*} x( t)&=\phi(t),\qquad t\in[-\tau_{1}, t_{0}],\\ u( t)&=\theta(t),\qquad t\in[-\tau_{2}, t_{0}], \end{align*} where vector functions ϕ(t) and θ(t) are known vector functions. Thus, while the time variable t starts from t0, t − τ1 for x(t − τ1) starts from t0 − τ1 and u(t − τ2) time t starts from t0 − τ2. So one needs to define τ = max{τ1, τ2} and define the new time interval as I1 = [t0 − τ, tf]. The TDFOCP (3.1)–(3.3) is the fractional form of the delay OCP discussed in (Koshkoue et al., 2012). The pair p = (x(.), u(.)) is said to be admissible if satisfies the fractional system (3.2) and conditions (3.3). Let W be the set of admissible pairs and assume to be non-empty. To change the classical TDFOCP (3.1)–(3.3) and using the embedding process, the system (3.2) is first converted to a scalar integral equation problem by using an auxiliary non-zero continuously α-differentiable n-vector function. Select the auxiliary function \begin{align} \eta(t)=[\eta_{1}(t)\,\, \eta_{2}(t) \cdots\,\, \eta_{n}(t)]^{T}, \end{align} (3.5) where $$ \eta_{j}(t) \in C\,^{\alpha}({I_{1}}) ,\,\,\,j=1, 2, \cdots,n, $$ are arbitrary non-zero continuously α-differentiable functions on I1. Multiplying of the left- and right-hand sides of the differential equation (3.2) in η(t) and adding η(α)(t) · x(t) to both sides of (3.2) and getting fractional integration of the both sides over the time interval I = [t0, tf] yields \begin{align} \eta(t)\cdot x^{(\alpha)}(t)+\eta^{(\alpha)}(t)\cdot x(t)=\eta(t)\cdot g+\eta^{(\alpha)}(t)\cdot x(t) \end{align} (3.6) \begin{align} I_{\alpha}(\eta(t)\cdot x^{(\alpha)}(t)+\eta^{(\alpha)}(t)\cdot x(t))= I_{\alpha}(\eta(t)\cdot g+\eta^{(\alpha)}(t)\cdot x(t)). \end{align} (3.7) Now by fractional integration as stated in (2.8) and using relations (2.5) and (2.7) respectively, the left- hand side of (3.7) is as follows: \begin{align} \int_{t_{0}}^{t_{f}}\!\!\!\tau^{\alpha-1}\!\left(\eta(\tau)\!\cdot \!x^{(\alpha)}(\tau)\!+\!\eta^{(\alpha)}(\tau)\!\cdot \!x(\tau)\right) {d\tau} \!&=\!\!\int_{t_{0}}^{t_{f}}\!\!\!\tau^{\alpha-1}T_{\alpha}(\eta(\tau)\!\cdot \!x(\tau))\,d\tau\! =\!\! \int_{t_{0}}^{t_{f}}{\!\!\!\tau^{\alpha-1\!}}\!\left({\!\tau^{1-\alpha}\dfrac{d(\eta(\tau)\!\cdot \!x(\tau)\!)}{d\tau} }\right)\,d\tau \nonumber\\ &=\int_{t_{0}}^{t_{f}}d(\eta(\tau)\cdot x(\tau))\,d{\tau} =\varDelta\eta, \end{align} (3.8) where \begin{align} \varDelta\eta=\eta(t_{f}).x(t_{f})-\eta(t_{0}).x(t_{0}). \end{align} (3.9) and the right-hand side of (3.7) is as follows: \begin{align} \int_{t_{0}}^{t_{f}}{\tau^{\alpha-1}}\left(\eta(\tau)\cdot g+\eta^{(\alpha)}(\tau)\cdot x(\tau)\right)\textrm{d}\tau , \end{align} (3.10) Note that both η(t) and x(t) are n-vector functions and the multiplications in the above relations are in fact inner product operators. Thus, (3.2) now is changed to the following form: \begin{align} \int_{t_{0}}^{t_{f}}{\tau^{\alpha-1}}\left(\eta(\tau)\cdot g+\eta^{(\alpha)}(\tau)\cdot x(\tau)\right)\textrm{d}\tau =\varDelta\eta . \end{align} (3.11) Let I0 = (t0, tf) and D(I0) be the space of all infinitely differentiable functions as ψ(t), with compact support in I0 and zero value at the initial point t0 and the terminal point tf, i. e. ψ(t0) = ψ(tf) = 0. Thus, (3.11) for function ψ’s changes to \begin{align} \int_{t_{0}}^{t_{f}}{\tau^{\alpha-1}}\left(\psi(\tau)\cdot g+\psi^{(\alpha)}(\tau)\cdot x(\tau)\right)\textrm{d}\tau =0. \end{align} (3.12) For each pair, p = (x(.), u(.)) ∈ W, we define the mapping \begin{align} \Lambda_{p}^{\alpha}:f\in C\,^{\alpha}(\Omega)\longrightarrow\int_{t_{0}}^{t_{f}}\tau^{\alpha-1}f(t,x(t), u(t), x(t-\tau_{1}), u(t-\tau_{2}))\,\textrm{d}\tau = I_{\alpha}(f), \end{align} (3.13) where C α(Ω) indicates the space of all real-valued continuously α-differentiable functions on Ω. The functional $$\Lambda _{p}^{\alpha }$$ is well-defined, linear, non-negative and continuous (Rubio, 1986). Consider now the transformation $$p\longrightarrow \Lambda _{p}^{\alpha }$$, of an admissible pair into a continuous, positive, linear functional. We can easily show the following proposition: Proposition 3.1 The transformation $$p\longrightarrow \Lambda _{p}^{\alpha }$$, of the admissible pair in W into the linear mapping defined in (3.13) is an injection. Proof. (see Rubio,1986). So in embedding process, we identify each pair p with the linear functional $$\Lambda _{p}^{\alpha }$$ (sometimes is called as Radon measure). We need to emphasize that the left-hand sides of equality (3.11) is integral of exactly the same type as that appearing in the definition of $$\Lambda _{p}^{\alpha }$$ in (3.13), so these equalities can then be written using the definition of $$\Lambda _{p}^{\alpha }$$ as follows: \begin{align} \qquad \Lambda_{p}^{\alpha}\left(f_{\eta}+x_{\eta}^{(\alpha)}\right)=\varDelta&\,\eta,\,\,\,\,\,\,\,\,\eta\in C^{\alpha}(\Omega),\nonumber\\ \Lambda_{p}^{\alpha}\left(f_{\psi}+x_{\psi}^{(\alpha)}\right)=&\,0,\,\,\,\,\,\,\,\,\,\,\psi \in D(I^{0}), \end{align} (3.14) where we recall that \begin{align} f_{\eta}=\eta(t)\cdot g(t, x(t), u(t), x(t-\tau_{1}), u(t-\tau_{2})), \end{align} (3.15) and \begin{align} x_{\eta}^{(\alpha)}=\eta^{(\alpha)}(t)\cdot x(t), \end{align} (3.16) in (3.14) fψ and $$x_{\psi }^{(\alpha )}$$ are defined similarly. By the Riesz representation theorem (see Royden, 1970), every Radon measure $$\Lambda _{p}^{\alpha }$$ can be corresponding to regular, finite and unique Borel measure. So there exists a Borel measure $$\mu _{\alpha }^{p}$$ on M+(Ω), the space of all positive Radon measures on Ω, such that \begin{align} \Lambda_{p}^{\alpha}(f)=\int_{\Omega}\tau^{\alpha-1} f(t, x(t), u(t), x(t-\tau_{1}), u(t-\tau_{2}))\,\textrm{d}\tau =\mu_{\alpha}^{p}(\,f),\,\,\, f\in C^{\alpha}(\Omega), \end{align} (3.17) (see Royden, 1970; Rubio, 1986). Using this concept, we can put the TDFOCP (3.1)–(3.3) in its definite form. Consider \begin{align} I :\mu_{\alpha}^{p}\longrightarrow \mu_{\alpha}^{p}\left(\,f_{0}^{*}\right), \end{align} (3.18) where from (3.1) and considering (2.8) $$ f_{0}^{*}=t^{1-\alpha }\,\,f_{0}$$. Thus, the TDFOCP is now equivalent to the following problem: \begin{align} \qquad Minimize\,\,\,J=\mu^{p}_{\alpha}\left(\,f_{0}^{*}\right), \end{align} (3.19) S. to \begin{align} \mu^{p}_{\alpha}\big(\,f_{\eta}&+x_{\eta}^{(\alpha)}\big)=\varDelta\eta,\quad\eta\in C\,^{\alpha}(\Omega),\nonumber\\ \mu^{p}_{\alpha}\big(\,f_{\psi}&+x_{\psi}^{(\alpha)}\big)=0,\quad\,\,\,\,\psi\in D(I^{0}). \end{align} (3.20) Assume that Q ⊂ M+(Ω) is the set of all positive Borel measures which satisfy infinite constraints (3.20). One needs to mention that since η and ψ belong respectively to infinite dimensional spaces Cα(Ω) and D(I0), the numbers of constraints in (3.20) are not finite. If we equip the space M+(Ω) by the weak*topology, then it can be proved that Q is compact and the functional $$J:\,\,Q\longrightarrow \mathbb{R}$$ is continuous. Also, the space M+(Ω) with weak*topology is a Hausdorff space. These conditions guarantee that there exists an optimal measure $$\mu ^{p*}_{\alpha }$$ in the set Q in which $$\mu ^{p*}_{\alpha }(\,f_{0}^{*}) \leqslant \mu ^{p}_{\alpha }(\,f_{0}^{*})$$, for any $$\mu ^{p}_{\alpha } \in Q$$ (for more details and proofs see Koshkouei et al., 2012). 4. Approximation The LPP consisting of minimizing the functional (3.19) on the subset Q of M+(Ω) described by the constraints (3.20), is an infinite-dimensional LPP. In fact, the underlying space M+(Ω) is not finite dimensional, and the number of constraints in (3.20) is not finite. (see Fattorini, 2005). We show that one can approximate the infinite dimensional LPP (3.19), (3.20), by a finite dimensional LPP, while the solution of this new problem, can give accurate pair of trajectory-control for the main problem (3.1)–(3.3). Assume that the set consisting of the vector functions ηi in the first constraints (3.20) is total in Cα[t0 − τ, tf], that is, the linear combinations of the vector functions ηi’s are uniformly dense in Cα[t0 − τ, tf]. Select M1 of these functions. For the second of the constraints in (3.20), we choose the function {ψj : j = 1, 2, ⋯ , M2} that are total in D(I0). Under these conditions following theorem holds: Theorem 4.1 Consider the LPP of minimizing $$\mu ^{p}_{\alpha }(\,f_{0}^{*})$$ over the set Q(M1, M2) of measures in M+(Ω) satisfying \begin{align} \mu^{p}_{\alpha}\big(\,f_{\eta_{i}}&+x_{\eta_{i}}^{(\alpha)}\big)=\varDelta\,\eta_{i},\quad\eta_{i}\in C^{\alpha}(\Omega),\quad i=1, 2, \cdots, M_{1}\nonumber\\ \mu^{p}_{\alpha}\big(\,f_{\psi_{j}}&+x_{\psi_{j}}^{(\alpha)}\big)=0,\quad psi_{j}\in D(I^{0}),\quad\,\,\, j=1, 2, \cdots, M_{2} \end{align} (4.1) if $$M_{1} \rightarrow \infty $$, $$M_{2} \rightarrow \infty $$ then $$ {\mu^{p}_{{\alpha}_{Q(M_{1},M_{2})}}}\big(\,f_{0}^{*}\big) \longrightarrow{\mu^{p}_{{\alpha}_{Q}}}\left(\,f_{0}^{*}\right).$$ Proof. (see Effati et al., 2014). So, we have limited the number of constraints in the infinite-dimensional LPP (3.19), (3.20), while the underlying space is not finite dimensional. Now, we are going to develop a finite dimensional LPP whose solution could give a good approximation of the original LPP (3.19), (3.20). First, we characterize a measure in the set Q(M1, M2), where $$\mu ^{p}_{\alpha }\longrightarrow \mu ^{p}_{\alpha }(\,f_{0}^{*})$$ attaints its minimum, it follows from Rosenbloom (see Rosenbloom, 1952) that we bring it here as the following proposition: Proposition 4.2 The measure $$\mu ^{p*}_{\alpha }$$ in the set Q(M1, M2) at which (3.19) attains its minimum has the form \begin{align} \qquad \mu^{p*}_{\alpha}=\sum_{k=1}^{M_{1}+M_{2}}\,\,\, \alpha_{k}^{\ast}\delta(z_{k}^{\ast}), \end{align} (4.2) where $$z_{k}^{\ast } \in \Omega $$, and the coefficients $$\alpha _{k}^{\ast }\geqslant 0$$, for k = 1, 2, ⋯, M1 + M2, Proof. (see Farahi et al., 2006). We need to mention that δ(.) in (4.2) is unitary atomic measure with support the singleton set {z*}, where z* ∈ Ω and characterized by \begin{align} \qquad \delta(z^{\ast})F=F(z^{\ast}),\,\,\, F\in C^{\alpha}(\Omega),\ z^{\ast}\in \Omega. \end{align} (4.3) Thus, the corresponding optimization problem (3.19), (3.20) now can be rewritten as the following form: \begin{align} \qquad Minimize \,\,\,\sum_{k=1}^{M_{1}+M_{2}}\,\alpha^{\ast}_{k}f^{*}_{0}\left(z^{\ast}_{k}\right), \end{align} (4.4) S. to \begin{align} \qquad \sum\nolimits_{k=1}^{M_{1}+M_{2}} \alpha^{\ast}_{k} \left(\,f_{\eta_{i}} +x_{\eta_{i}}^{(\alpha)}\right)\big(z_{k}^{*}\big)&=\varDelta\eta_{i},\quad i=1, 2, \cdots, M_{1}, \nonumber\\ \qquad \sum\nolimits_{k=1}^{M_{1}+M_{2}} \alpha^{\ast}_{k} \left(\,f_{\psi_{j}}+x_{\psi_{j}}^{(\alpha)}\right)\big(z_{k}^{*}\big)&=0,\quad j=1, 2, \cdots, M_{2}, \end{align} (4.5) where \begin{align} f^{*}_{0}\left(z_{k}^{*}\right)&=t^{{*}^{1-\alpha}}_{k}f_{0}\left(t^{*}_{k}, x^{*}_{k}, u^{*}_{k}\right) \nonumber\\ f_{\eta_{i}}\left(z_{k}^{*}\right)&=\eta_{i}\left(t^{*}_{k}\right)\cdot g\left(t^{*}_{k}, x^{*}_{k},x\left(t^{*}_{k}-\tau_{1}\right), u^{*}_{k}, u\left(t^{*}_{k}-\tau_{2}\right)\right) \nonumber\\ x_{\eta_{i}}^{(\alpha)}\left(z^{*}_{k}\right) &= \eta_{i}^{(\alpha)}\left(t^{*}_{k}\right)\cdot x\left(t^{*}_{k}\right) \\ f_{\psi_{j}}\left(z_{k}^{*}\right)&=\psi_{j}\left(t^{*}_{k}\right)\cdot g\left(t^{*}_{k}, x^{*}_{k},x\left(t^{*}_{k}-\tau_{1}\right), u^{*}_{k}, u\left(t^{*}_{k}-\tau_{2}\right)\right) \nonumber\\ x_{\psi_{j}}^{(\alpha)}\left(z^{*}_{k}\right) &= \psi_{j}^{(\alpha)}\left(t^{*}_{k}\right)\cdot x\left(t^{*}_{k}\right)\nonumber \end{align} (4.6) Let ω = {z1, ⋯ , zN} be a countable approximately dense subset of Ω, that is every element of Ω can be showed approximately by the linear combination of the elements of ω. A measure $$\nu _{\alpha }^{*} \in Q(M_{1}, M_{2})$$ as a good approximation for $$\mu _{\alpha }^{*}$$ can be found such that \begin{align} \nu_{\alpha}^{\ast}=\sum_{k=1}^{N} \alpha_{k}^{*}\delta (z_{k}), \end{align} (4.7) where the coefficients $$\alpha _{k}^{*}$$ are the same as in the optimal measure $$\mu _{\alpha }^{*}$$ in (4.2) (see Barati, 2012), and zk ∈ ω, k = 1, 2, …, N. By selecting zi, i = 1, ..., N for sufficiently large N in ω, and considering (4.7), then the non-linear optimization problem (4.4), (4.5) can be approximated by the following LPP: \begin{align} \qquad Minimize\,\,\,\,\,\sum\nolimits_{k=1}^{N}{\alpha^{*}}_{k}{ \,f^{*}}_{0}(z_{k}), \end{align} (4.8) S. to: \begin{align} \sum\nolimits_{k=1}^{N}{\alpha^{*}}_{k} \left(\,f_{\eta_{i}} +x_{\eta_{i}}^{(\alpha)}\right)(z_{k})&=\varDelta\,\eta_{i}, \quad i=1, 2, \cdots, M_{1},\nonumber\\ \sum\nolimits_{k=1}^{N}{\alpha^{*}}_{k} \left(\,f_{\psi_{j}}+x_{\psi_{j}}^{(\alpha)}\right)(\,z_{k})&=0,\quad j=1, 2, \cdots, M_{2}, \end{align} (4.9) where \begin{align} \qquad{ f^{*}}_{0}(z_{k})&={ t_{k}}^{1-\alpha}f_{0}(t_{k}, x_{k}, u_{k}), \nonumber\\ \qquad f_{\eta_{i}}(z_{k})&=\eta_{i}(t_{k})\cdot g(t_{k}, x_{k},x(t_{k}-{\tau_{1}}), u_{k}, u(t_{k}-{\tau_{2}})), \nonumber\\ \qquad x_{\eta_{i}}^{(\alpha)}(z_{k}) &= \eta_{i}^{(\alpha)}(t_{k})x_{k}, \\ \qquad f_{\psi_{j}}(z_{k}) &=\psi_{j}(t_{k})\cdot g(t_{k}, x_{k},x(t_{k}-{\tau_{1}}), u_{k}, u(t_{k}-{\tau_{2}})), \nonumber\\ \qquad x_{\psi_{j}}^{(\alpha)}(z_{k}) &= \psi_{j}^{(\alpha)}(t_{k})x_{k}.\nonumber \end{align} (4.10) Now ω = {zk, k = 1, 2⋯ , N} is constructed by dividing the time interval I1 = [t0 − τ, tf], sets A and U into a number of equal subintervals, defining in this way a grid of points. We need to mention the following Remark: Remark 4.3 Let $$L=D_{I_{1}}$$ and n = DA × DU, where $$D_{I_{1}}$$, DA and DU are the number of divisions of I1, A and U, respectively. For p = 1, 2, ⋯ , L, one may select t(p−1)n+1 = t(p−1)n+2 = ⋯ = tpn, where L is selected such that $$ \dfrac{L\tau _{1}}{\varDelta t}$$ and $$\dfrac{L\tau _{2}}{\varDelta t}$$ be natural numbers, then the problem (4.8), (4.9) is converted to the following linear programming problem. Note τ1, τ2 are the constant delays in state and control respectively and $$\varDelta t=\frac{(t_{f}-t_{0})}{L}$$ (see Barati, 2012). So the linear programming problem (4.8), (4.9) now is as follows: \begin{align} \qquad Minimize \,\,\sum_{i=0}^{n-1}\sum_{p=1}^{L}\,{\alpha^{*}}_{np-i}{\,\,f^{*}}_{0}(z_{np-i}), \end{align} (4.11) S. to \begin{align} &\qquad \,\,\,\sum\nolimits_{i=0}^{n-1}\sum\nolimits_{p=1}^{L}\,{\alpha^{*}}_{np-i}\left(\,f_{\eta_{i}+x_{\eta_{i}}^{(\alpha)}}\right)(z_{np-i}) =\varDelta\,\eta_{i},\quad i=1, 2, \cdots, M_{1} \nonumber\\ &\qquad \,\,\,\sum\nolimits_{i=0}^{n-1}\sum\nolimits_{p=1}^{L}\,{\alpha^{*}}_{np-i}\left(\,f_{\psi_{j}+x_{\psi_{j}}^{(\alpha)}}\right)(z_{np-i}) =0,\quad j=1, 2, \cdots, M_{2} \\ &\qquad\,\,\,\,\alpha^{*}_{np-i} \geqslant \,0,\quad p=1, 2, \cdots, L , i=0, 1, \cdots, n-1,\nonumber \end{align} (4.12) where \begin{align} \qquad{f^{*}}_{0}(z_{np-i})&=t^{1-\alpha}_{np-i}\,\,f_{0}(t_{np-i}, x_{np-i}, u_{np-i} ),\nonumber\\ \qquad f_{\eta_{i}}(z_{np-i})&=\eta_{i}(t_{np-i})\cdot g\left(t_{np-i}, x_{np-i}, x_{np-i-\frac{nL\tau_{1}}{\varDelta t}}, u_{np-i}, u_{np-i-\frac{nL\tau_{2}}{\varDelta t}}\right),\\ \qquad x_{\eta_{i}^{(\alpha)}}(z_{np-i})&=\eta_{i}^{(\alpha)}(t_{np-i})x_{np-i},\nonumber \end{align} (4.13) and \begin{align} \qquad x_{np-i-\frac{nL\tau_{1}}{\varDelta t}}&=\phi(t_{np-i-\tau_{1}}),\,\,\, p=1, 2, \cdots, \frac{L\tau_{1}}{\varDelta t},\quad\,\,\, i=0, 1, \cdots, n-1,\nonumber\\ \qquad u_{np-i-\frac{nL\tau_{2}}{\varDelta t}}&=\theta(t_{np-i-\tau_{2}}),\,\,\, p=1, 2, \cdots, \frac{L\tau_{2}}{\varDelta t}\,\, \quad\,\,\,\,i=0, 1, \cdots, n-1. \end{align} (4.14) The procedure to construct a piecewise constant control function from the solution of $$\alpha ^{*}_{k}$$, k = 1, 2, ⋯ , N, where N = nL, of the linear programming problem (4.11), (4.12) which approximates the action of the optimal measure $$\mu ^{\ast }_{\alpha }$$, is based on the analysis in (Koshkouei et al., 2012). The trajectory is then simply found by solving the differential equation (3.2). (see Bazara et al., 2009; Shampine & Thompson, 2000; Podlubny, 1999). Table 1. The objective values of Example 5.1 for different values of N and α N α = 1 α = 0.9 α = 0.8 1920 3.0117 2.6888 2.5487 2400 2.8959 2.6125 2.4842 N α = 1 α = 0.9 α = 0.8 1920 3.0117 2.6888 2.5487 2400 2.8959 2.6125 2.4842 Table 1. The objective values of Example 5.1 for different values of N and α N α = 1 α = 0.9 α = 0.8 1920 3.0117 2.6888 2.5487 2400 2.8959 2.6125 2.4842 N α = 1 α = 0.9 α = 0.8 1920 3.0117 2.6888 2.5487 2400 2.8959 2.6125 2.4842 Table 2. The objective values of Example 5.2 for different values of N and α N α = 1 α = 0.9 α = 0.8 4000 59.9831 63.0838 66.4016 12000 57.5870 65.8138 70.7380 N α = 1 α = 0.9 α = 0.8 4000 59.9831 63.0838 66.4016 12000 57.5870 65.8138 70.7380 Table 2. The objective values of Example 5.2 for different values of N and α N α = 1 α = 0.9 α = 0.8 4000 59.9831 63.0838 66.4016 12000 57.5870 65.8138 70.7380 N α = 1 α = 0.9 α = 0.8 4000 59.9831 63.0838 66.4016 12000 57.5870 65.8138 70.7380 5. Numerical Example Here, we use our approach to obtain approximate optimal solutions of the following two non-linear time-delayed OCPs by solving its LPP via simplex method. These examples in classic form (non-fractional) were mentioned in (Barati, 2012). All the problems are programmed in MATLAB 9.0. Example 5.1 \begin{align} \qquad Minimize\,\,\,{\int_{0}^{6}} \left(x^{2}(t)+u^{2}(t)\right)\, \textrm{d}t, \end{align} (5.1) S. to \begin{align} \qquad x^{(\alpha)}(t) &=x(t-1)u(t-2)\quad t \in[0, 6],\nonumber\\ \qquad x(t)&=1,\quad t\in[-1, 0],\quad\,\, x(6)=0.21,\\ \qquad u(t)&=0,\quad t\in[-2, 0],\nonumber \qquad \\\nonumber 0\le \alpha&\le 1. \end{align} (5.2) Let I1 = [−2, 6], A = [0.2, 1] and U = [−0.8, 0.2]. The set Ω = I1 × A × U is covered with a grid, where the grid is defined by taking points zk = (tk, xk, uk) in Ω. The points in the grid are sequentially numbered from 1 to N. In this example, the sets I1, A and U are respectively divided into 24, 8 and 10 subintervals, so 24 × 8 × 10 = 1920 nodes are created in the grid. In solving LPP (4.11), (4.12), we have chosen M1 = 2, M2 = 22. The functions ηi’s are selected as monomials and ψj’s are normally chosen as follows: \begin{align} \qquad \psi_{j} (t)=\textrm{sin}\left(\frac{2\pi j(t-t_{0})}{{t_{f}}{-t_{0}}}\right), & \qquad j=1,2,...\dfrac{M_{2}}{2}, \nonumber\\ \qquad \psi_{j} (t)=1-\textrm{cos}\left(\frac{2\pi j(t-t_{0})}{{t_{f}}{-t_{0}}}\right), & \qquad j=\dfrac{M_{2}}{2}+1,...M_{2}. \end{align} (5.3) The minimum value of the cost functional by using the proposed method is J*= 3.0117 for α=1 (see Table 1). The piecewise continuous control functions and the state functions are shown in Figs 1–4 for α = 0.8, 0.9 and 1. Fig. 1. View largeDownload slide Approximate solution u(t) for α = 0.8 in Example 5.1. Fig. 1. View largeDownload slide Approximate solution u(t) for α = 0.8 in Example 5.1. Fig. 2. View largeDownload slide Approximate solution u(t) for α = 0.9 in Example 5.1. Fig. 2. View largeDownload slide Approximate solution u(t) for α = 0.9 in Example 5.1. Fig. 3. View largeDownload slide Approximate solution u(t) for α = 1 in Example 5.1. Fig. 3. View largeDownload slide Approximate solution u(t) for α = 1 in Example 5.1. Fig. 4. View largeDownload slide Approximate solution x(t) for α = 0.8, 0.9 and 1 in Example 5.1. Fig. 4. View largeDownload slide Approximate solution x(t) for α = 0.8, 0.9 and 1 in Example 5.1. Fig. 5. View largeDownload slide Approximate solution u(t) for α = 0.8 in Example 5.2. Fig. 5. View largeDownload slide Approximate solution u(t) for α = 0.8 in Example 5.2. Fig. 6. View largeDownload slide Approximate solution u(t) for α = 0.9 in Example 5.2. Fig. 6. View largeDownload slide Approximate solution u(t) for α = 0.9 in Example 5.2. Fig. 7. View largeDownload slide Approximate solution u(t) for α = 1 in Example 5.2. Fig. 7. View largeDownload slide Approximate solution u(t) for α = 1 in Example 5.2. Fig. 8. View largeDownload slide Approximate solution x(t) for α = 0.8, 0.9 and 1 in Example 5.2. Fig. 8. View largeDownload slide Approximate solution x(t) for α = 0.8, 0.9 and 1 in Example 5.2. Example 5.2 Consider the following non-linear TDFOCP. In this example, only the state variable x(t) has a delay τ ⩾ 0 (see Barati, 2012). \begin{align} \qquad Maximize \,\,\,\int_{0}^{20} e^{-0.05t}(2u(t)-0.2x(t)^{-1}u(t)^{3})\, \textrm{d}t, \end{align} (5.4) S. to \begin{align} \qquad x^{(\alpha)}(t) &= 3x(t)\bigg(1-\frac{x(t-\tau)}{5}\bigg)-u(t),\nonumber\\ \qquad x(t) &= 2,\quad t\in[-0.5, 0],\nonumber\\ \qquad x(t)&\geqslant 2, \quad t \in[0,20],\\ \qquad u(t)&\geqslant 0, \quad t\in [0, 20],\nonumber\\ \qquad 0\leqslant \alpha&\leqslant 1.\nonumber \end{align} (5.5) We assume that I1 = [−0.5, 20], A = [2, 4] and U = [0, 4], the sets I1, A and U are respectively divided into 40, 10 and 10 sections, so we have 40 × 10 × 10 = 4000 subintervals. We choose M1 = 2, M2 = 5, then by solving a linear programming problem (4.11), (4.12), the optimal control and the state function are obtained. The minimum value of the cost functional by using the proposed method is J* = 59.9831 for α=1 (see Table 2). The piecewise continuous control functions and the state functions are shown in Figs 5–8 for α = 0.8, 0.9 and 1. 6. Conclusion In this paper, the problem of designing optimal pair of control and state for a class of TDFOCP is investigated. The embedding methodology used is based on variational approach and involves an approximation of the non-linear problem by a linear optimization problem that leads to a piecewise constant control signals. By using the conformable derivative and integral, we get more compatibility to extend this idea in comparison with the traditional definitions of fractional calculus. We need to emphasize that the benefits of using conformable as a modelling tool are that we now can use product and chain rules in conformable derivative and we also can benefit conformable integral that enable us on reformulation of fractional dynamical system as state space representation. This representation is necessary to use embedding technique, that is the main goal in this paper. We have proposed a new method and as an interesting consequence of this development one may apply this method to many problems such as delay optimal control or delay control systems governed by partial differential equations. Acknowledgements The authors are grateful to reviewers for their constructive and helpful comments, which helped to improve the paper. References Agrawal , O. P. ( 2008 ) A formulation and a numerical scheme for fractional optimal control problems . J. Vibration Control , 14 , 1291 -- 1299 . Google Scholar CrossRef Search ADS Bagley , R. L. & Torvik , P. J. ( 1984 ) On the appearance of the fractional derivative in the behavior of real materials . J. Appl. Mech. , 51 , 294 -- 298 . Google Scholar CrossRef Search ADS Barati , S. ( 2012 ) Optimal control of constrained time delay systems. J. AMO, 14 , 103 -- 115. Bazara , M. S. , Jarvis , J. J. & Sherali , H. D. ( 2009 ) Linear Programming and Network Flows , 4th ed. New York : Wiley . Google Scholar CrossRef Search ADS Belair , J. M. , Mackey C. & Maffay J. ( 1995 ) Age-structured and two delay models for erythropoiesis . Math. Biosci., 128 , 317 -- 346 . Google Scholar CrossRef Search ADS Bellen , A. & Zennaro , M. ( 2003 ) Numerical Methods for Delay Differential Equations . Oxford : Clarendon Press . Google Scholar CrossRef Search ADS Benkhettou , N. , Hassani , S. & Torres Delfim , F. M. ( 2016 ) A conformable fractional calculus on arbitrary time scales . J. King Saud University-Science , 28 , 93 -- 98 . Google Scholar CrossRef Search ADS Crauste , F . ( 2009 ) Delay model of hematopoietic stem cell dynamics: asymptotic stability and stability switch. Math. Model. Nat. Phenomena, 4 , 28 -- 47 . Google Scholar CrossRef Search ADS Dadkhah , M. , Farahi , M. H. & Heydari , A. ( 2015 ) Optimal control of a class of non-linear time-delay systems via hybrid functions. IMA J. Math. Control Inf. , 34 , 255 -- 270 . Dadkhah , M. , Farahi , M. H. & Heydari , A. ( 2016 ) Numerical solution of time delay optimal control problems by hybrid of block-pulse functions and Bernstein polynomials . IMA J. Math. Control Inf. , doi:10.1093/imamci/dnw057 . Dehghan , R. & Keyanpour , M. ( 2015 ) A numerical approximation for delay fractional optimal control problems based on the method of moments. IMA J. Math. Control Inf. , 34 , 77 -- 92 . Effati , S. , Nazemi , A. , & Shabani , H. ( 2014 ) Time optimal control problem of the heat equation with thermal source . IMA J. Math. Control Inf. , 31 , 385 -- 402 . Google Scholar CrossRef Search ADS Farahi , M. H. , Rubio , J. E. & Wilson , D. A. ( 1996 ) The global control of a nonlinear wave equation . Int. J. Cont. , 65 , 1 - 15 . Google Scholar CrossRef Search ADS Farahi , M. H. , Mehne , H. H. & Borzabadi , A. H. ( 2006 ) Wing drag minimization by using measure theory . J. Optim. Meth. Soft. , 21 , 169 -- 177 . Google Scholar CrossRef Search ADS Fattorini , H. O. ( 2005 ) Infinite Dimensional Linear System: The Time Optimal and Norm Optimal. Amsterdam : Elsevier Sciences . Ghomanjani , F. , Farahi , M. H. & Vahidian Kamyad , A. ( 2013 ) Numerical solution of some linear optimal control systems with pantograph delays. IMA J. Math. Control Inf. , 32 , 225 -- 243 . Google Scholar CrossRef Search ADS Khalil, R. , Al Horani, M. , Yousef, A. & Sababheh, M. ( 2014 ) A new definition of fractional derivative . J. Comput. Appl. Math. , 264 , 65 -- 70 . Google Scholar CrossRef Search ADS Koshkouei , A. J. , Farahi , M. H. & Burnham , K. J. ( 2012 ) An almost optimal control design method for nonlinear time-delay systems. Int. J. Cont. , 85 , 147 -- 158 . Google Scholar CrossRef Search ADS Magin , R . ( 2010 ) Fractional calculus models of complex dynamics in biological tissues . Comput. Math. Appl. , 59 , 1586 -- 1593 . Google Scholar CrossRef Search ADS Merala , F. C. , Roystona T. J. & Magin , R. ( 2010 ) Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. , 15 , 939 -- 945 . Google Scholar CrossRef Search ADS Nazemi , A. R. , Farahi , M. H. & Kamyad A. V. ( 2007 ) A new technique for approximate solution of the nonlinear volterra integral equation at the second kind. J. Scientia Iranica , 14 , 579 -- 585 . Podlubny , I. ( 1999 ) Fractional Differential Equations. An Introduction to Fractional Derivatives . San Diego : Academic Press . Rosenbloom , P. C. ( 1952 ) Quelques classes de problemes extremaux. Bulletin de la Societe Mathematique de France , 80 , 183 -- 216 . Google Scholar CrossRef Search ADS Royden , H. L. ( 1970 ) Real Analysis. London : The Macmillan Company . Rubio , J. E. ( 1986 ) Control and Optimization: The Linear Treatment of Nonlinear Problem. Manchester and John Wiely, New York and Landon : Manchester University press . Sabatier , J. , Nguyen, HHC. , Farges, C. , Deletage, J-Y. , Moreau, X. , Guillemard, F. & Bavoux, B. ( 2011 ) Fractional models for thermal modeling and temperature estimation of a transistor junction. Adv. Difference Equations, doi:10.1155/2011/687363 . Shampine , L. F. & Thompson , S. ( 2000 ) Solving Delay Differential Equations with DDE , 23. Technical Report . Tricaud , C. & Chen , Y. Q. ( 2010 ) An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. , 59 , 1644 -- 1655 . Google Scholar CrossRef Search ADS Ziaei , E. , Farahi , M. H. , & Safaie , E. ( 2017 ) The approximate solution of nonlinear fractional optimal control problems by measure theory approach . Prog. Fractional Differ. Appl. , doi:10.18576/pfda/paper . © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

IMA Journal of Mathematical Control and Information – Oxford University Press

**Published: ** Feb 8, 2018

Loading...

personal research library

It’s your single place to instantly

**discover** and **read** the research

that matters to you.

Enjoy **affordable access** to

over 18 million articles from more than

**15,000 peer-reviewed journals**.

All for just $49/month

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Read from thousands of the leading scholarly journals from *SpringerNature*, *Elsevier*, *Wiley-Blackwell*, *Oxford University Press* and more.

All the latest content is available, no embargo periods.

## “Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”

Daniel C.

## “Whoa! It’s like Spotify but for academic articles.”

@Phil_Robichaud

## “I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”

@deepthiw

## “My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”

@JoseServera

DeepDyve ## Freelancer | DeepDyve ## Pro | |
---|---|---|

Price | FREE | $49/month |

Save searches from | ||

Create lists to | ||

Export lists, citations | ||

Read DeepDyve articles | Abstract access only | Unlimited access to over |

20 pages / month | ||

PDF Discount | 20% off | |

Read and print from thousands of top scholarly journals.

System error. Please try again!

or

By signing up, you agree to DeepDyve’s Terms of Service and Privacy Policy.

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.

ok to continue