Regional optimal control problem with minimum energy for a class of bilinear distributed systems

Regional optimal control problem with minimum energy for a class of bilinear distributed systems Abstract The aim of this paper is to propose a method for solving a regional control problem with minimum energy for a system governed by a wave equation via bi-linear bounded control. Using optimal control techniques, we show how to bring the solution for a two-dimensional wave equation with control act as multiplier of the state, close to a desired profile only on a subregion of the system domain. 1. Introduction Let the system described by the wave bi-linear equation {∂2y∂t2+Δ2y=u(t)yQy(x,0)=y0(x),∂y∂t(x,0)=y1(x)Ωy=0Σ  (1) where $${\it{\Omega}}$$ is an open bounded domain of $$I\!\!R^2$$ with $$C^2$$ boundary, $$ Q = {\it{\Omega}} \times (0,T)$$ for $$T> 0$$ and $$ {\it{\Sigma}}= \partial{\it{\Omega}} \times (0,T)$$. The control $$ u\in C_b= \{ u\in L^{\infty} (0,T); -b\leq u(t) \leq b\}, $$ for a constant positive $$b$$. We rewrite the system (1) in the parabolic form. {∂w(x,t)∂t=Aw(x,t)+Bw(x,t)Qw(x,0)=w0(x)Ω  (2) where w=(yyt),A=(0I−Δ20),Bw=(0u(t)y)andw0=(y0y1). $${\mathcal A}$$ generates a $${\mathcal C}^{0}$$ semigroup $$S(t)_{t\geq 0 } $$ of isometries on $$H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$. Consequently, there exists a $$K>0$$ such that $$\big\|S(t)\big\| \leq K$$, $$\forall t\in [0,T]$$ see Lasiecka & Triggiani (2008) and Pazy (1983). For $$w_{0}\in H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$, the mild solution of (2) is written as wu(x,t)=S(t)w0(x)+∫0tS(t−s)Bwu(x,s)ds. (3) The existence of a unique solution $$w_u(x,t)$$ in $${\mathcal C}([0,T]; H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}}))$$ satisfying (3) follows from standard results in Brezis (1983) and Lions (1988). Let $$w_{u}(T)=(y_{u}(T) ,\displaystyle{\frac{\partial y_{u}}{\partial t}}(T) )$$ the solution of (3) at time $$T$$, for $$\omega \subset{\it{\Omega}}$$, $$w^d=(y_1^d,y_2^d)$$ is the desired observations on $$L^2(\omega)\times L^2(\omega)$$ and $$\chi_{_\omega}: L^2({\it{\Omega}}) \longrightarrow L^2(\omega)$$ is the restriction operator to $$\omega $$. Given $$\tilde{w}=(\tilde{w}_1^d$$, $$\tilde{w}_2^d)$$, we define the scale product on the space product $$L^2(\omega)\times L^2(\omega)$$ by ⟨wd,w~⟩(L2(ω))2=(w1d,w~1d)L2(ω)+(w2d,w~2d)L2(ω), and denoted by $$\big\|\cdot \big\|_{(L^2(\omega))^{2}}$$ its corresponding norm. We have ‖pωw(T)−wd‖(L2(ω))22 = ⟨pωw(T)−wd,pωw(T)−wd⟩(L2(ω))2 = ⟨χωyu(T)−y1d,χωyu(T)−y1d⟩L2(ω)  +⟨χω(∂yu(T)∂t)−y2d,χω(∂yu(T)∂t)−y2d⟩L2(ω) = ‖χωyu(T)−y1d‖L2(ω)2+‖χω(∂yu(T)∂t)−y2d‖L2(ω)2, where $$p_{\omega}$$ is the restriction map pω:L2(Ω)×L2(Ω)⟶L2(ω)×L2(ω)(y1,y2)⟶(χωy1,χωy2). Our main objective is to solve the regional with minimum energy control problem governed by the bi-linear distributed wave equation (1) {Minimize ‖u‖L2(0,T)2subject to u∈C(ω).  (4) where C(ω)={u∈Cb:‖pωwu(T)−wd‖L2(ω)×L2(ω)2 is minimum }. $$C(\omega)$$ is the set of bounded control with corresponding state $$w_u(T)$$ in a neighborhood of the desired state $$w^d$$. In terms of applications, a problem like this arises, e.g., in the context of “smart materials”, whose properties can be altered by applying various external factors such as temperature, electrical current or magnetic field (Banks & Wang, 1994; Khapalov, 2010). For the background on controllability of bi-linear distributed systems, Ball et al. (1982) have proved for the first time the global approximate controllability in $$H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$ of bi-linear beam and rod equations in the mono-dimensional case. Positive results on exact and approximative controllability of wave equations are given by Zuazua (1990), Lenhart et al. (1999, 2000), Khapalov (2010) and Ouzahra (2014) and negative results by Ball et al. (1982) and Beauchard (2011) due to the choice of functional spaces. Optimal control problem of wave equation type (1), has been treated in Liang (1999) where a bi-linear control has been used to bring the state solutions close to a desired profile on the whole domain. Then the author established the existence and uniqueness of the solution of the optimality system and thus, determined the unique optimal control in terms of the solution of the optimality system. Bradly & Lenhart (1997) and Bradly et al. (1999) treated $${\it{\Delta}}^{2}$$ type of bi-linear control for the Kirchhoff plate equation. In this paper our goal is to solve the regional control problem (4), which consists in finding a bounded control $$u\in C_{b}$$ with minimum energy such its corresponding solution $$w_{u}(T)$$ at time $$T$$ very close to the observations $$w^d$$ only on a subregion on $$\omega \subset{\it{\Omega}}$$. This can be regarded as an extension of the research work in El Jai et al. (1995) and Zerrik & Ould Sidi (2010a,b, 2011, 2013). Besides, focusing on regional controllability would allow for a reduction in the number of physical actuators, offer the potential to reduce computational requirements in some cases, and also offer the possibility to discuss those systems which are not controllable on the whole domain, etc. The paper is organized as follows: we begin by proving the existence of a solution for the associate regional penalized problem in Section 2. In Section 3, we give necessary conditions for the problem (5). In Section 4, we connect the regional penalized problem (5) to the regional control problem (4) as $$\varepsilon \rightarrow 0$$. Specifically, we show that when $$\varepsilon \rightarrow 0$$, the optimal control solution of (5) converges to a feasible solution of (4). The paper ends with open problem. 2. Regional penalized problem To solve the optimal control problem (4), we propose an approach based on quadratic cost control problem, which involves the minimization of the control norm, the final state and speed errors. For $$\varepsilon>0$$, let associate to (4) the problem: minu ∈CbQε(u). (5) where the non-penalized cost functional $$Q_{\varepsilon}$$ is defined by Qε(u) =12‖pωw(T)−wd‖(L2(ω))22+ε‖u(t)‖L2([0,T])2 =12‖χ ωyu(T)−y1d‖L2(ω)2+12‖χ ω(∂yu(T)∂t0)−y2d‖L2(ω)2+ε‖u(t)‖L2([0,T])2. (6) The next theorem justifies that the minimum of the problem (5) exists, Theorem 2.1 There exists an optimal control $$u_{\varepsilon} \in C_b$$, which minimizes the non-penalized cost functional $$Q_{\varepsilon}(u)$$ over $$u$$ in $$C_b$$. Proof. For given $$ w(0) \in H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$ and $$u\in C_{b}$$ the weak solution of (1) satisfies ‖w‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT‖w(0)‖H01(Ω)×L2(Ω). (7) Using (3), and the bound $$K$$ of the strongly continuous semigroup $$(S(t))_{t\geq 0}$$ in all finite interval of $$[0,T]$$ see Pazy (1983), we have ‖wu(t)‖C([0,T];H01(Ω)×L2(Ω))≤K‖w(0)‖H01(Ω)×L2(Ω)+Kb∫0t‖wu(s)‖H01(Ω)×L2(Ω)ds. Using Gronwall inequality see Ball et al. (1982), we obtain (7). Let $$ ( u )_{ n } \in C_{b} $$ be a minimizing sequence such that limn→∞Qε(un)=infu∈CbQε(u)=Q∗, Equation (7) gives a priori estimate of $$ w_{n}=w(u_{n})$$ with bounds independent of $$n$$. On a subsequence, by weak compactness, there exists $$ w_{n}=w(u_{ n })=\big(y_n, {\frac{\partial y_{n}}{\partial t}} \big)$$ such that un⇀u∗weakly in L2(0,T).yn⇀yweakly in L2([0,T];H01(Ω)).∂yn∂t⇀∂y∂tweakly in L2([0,T];L2(Ω)).∂2yn∂t2⇀∂2y∂t2weakly in L2([0,T];H−1(Ω)). (8) By classical argument see Lions & Magenes (1968) and using the fact that $$w_{ n }(0)= w_{ 0 }$$. Taking the limit in the system (1), associated with $$(u_n, w(u_{ n }))$$ as $$n \longrightarrow \infty$$ from this, we obtain $$w^{*}=w(u^{*})$$ which is the weak solution of (1) with control $$u^*$$. Since the objective functional is lower semi-continuous with respect to weak convergence, we obtain Qε(u∗)≤limn→+∞infQε(un)=Q∗≤Qε(u∗). We conclude that $$u^{*}$$ is an optimal control. □ Remark 2.2 If we consider the state equation with a source term $$f\in L^{\infty}(0,T;L^2({\it{\Omega}}))$$, ∂2y∂t2+Δ2y=u(t)y+f(t,x)on Q. The same well-posedness and regularity results as (1) hold, and we have ‖w‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT(‖w(0)‖H01(Ω)×L2(Ω)+‖f‖L∞(0,T;L2(Ω))). (9) where $$w=\big(y, {\frac{\partial y}{\partial t}}\big)$$ and $$w(0)=(w_0, w_1)$$. 3. Necessary conditions We derive necessary conditions that every optimal control must satisfy by differentiate our functional criterion $$ Q_{\varepsilon}(u) $$ with respect to $$u$$. Firstly, we examine the differentiability of $$u\longrightarrow w(u)$$ with respect to u and the differentiation of $$ Q_{\varepsilon} $$ gives a characterization of optimal controls in terms of the optimality system. Lemma 3.1 The mapping u∈Cb→w(u)∈C([0,T];H01(Ω)×L2(Ω)), is differentiable in the following sense: w(u+εh)−w(u)ε⇀ψ~weakly in L∞([0,T];H01(Ω)×L2(Ω)), as $$ \varepsilon \rightarrow 0 , $$ for any $$ u , u + \varepsilon h \in C_{b}. $$ Moreover, the limit $$ \tilde{\psi} = ( \psi , \psi t ) $$ is a weak state of the wave bi-linear equation {∂2ψ∂t2+Δ2ψ=u(t)ψ+h(t)yQψ(x,0)=0,∂ψ∂t(x,0)=0Ωψ=0Σ  (10) Proof. Let $$ w^{ \varepsilon } = w ( u + \varepsilon h ) $$ and $$ w = w( u )$$. From (1), $$(w ^{ \varepsilon } - w ) / \varepsilon $$ is a weak solution of (yε−yε)tt+Δ2(yε−yε)=u(t)(yε−yε)+hyε in Q. (11) Using Remark 2.2 with source term $$ hy ^{\varepsilon }, $$ we obtain ‖(wε−wε)‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT‖hyε‖L∞(0,T;L2(Ω))≤C3. (12) where $$C_{3}$$ is independent of $$\epsilon$$ since the bound on $$ \big\| y^{ \varepsilon }\big\| _{L^{\infty}(0,T;L^2({\it{\Omega}}))}$$ is independent of $$\epsilon$$, and the weak convergence to $$\psi$$ is obtained. □ Next, consider the optimality system {∂2p∂t2+Δ2p=u(t)pQp(x,T)=−(∂y∂t(T)−χω∗y2d)Ω∂p∂t(x,T)=(y(T)−χω∗y1d)Ωp=0Σ  (13) where $$\chi _{\omega }^{*}$$ is the adjoint of $$\chi_\omega$$ defined from $$ L^2 (\omega ) \longrightarrow L^2 ({\it{\Omega}} )$$ by χω∗y(x)={y(x),x∈ω0,x∈Ω∖ω  By differentiating the cost functional $$ u \rightarrow Q_{\varepsilon} ( u ) $$ with respect to the control, we have the following lemma: Lemma 3.2 If $$u_{ \varepsilon } \in C_{b}$$ is an optimal control, $$\psi $$ is the solution of (10) and $$p$$ is the solution of (13), then limβ⟶0Qε(uε+βh)−Qε(uε)β =∫ωχω∗χω[∫0T∂2p∂t2ψdt−∫0Tp∂2ψ∂t2dt]dx  +∫0T2εhuεdt. (14) Proof. limβ⟶0Qε(uε+βh)−Qε(uε)β = limβ⟶012∫ω(χωyβ−y1d)2−(χωy−y1d)2βdx  +limβ⟶012∫ω(χω∂yβ∂t−y2d)2−(χω∂y∂t−y2d)2βdx  +limβ⟶0ε∫0T(uε+βh)2−uε2β(t)dt. (15) Therefore limβ⟶0Qε(uε+βh)−Qε(uε)β =limβ⟶012∫ωχω(yβ−y)β(χωyβ+χωy−2y1d)dx  +∫0T(2εhuε+βεh2)dt  +limβ⟶012∫ωχω(∂y∂tβ−∂y∂t)β(χω∂y∂tβ+χω∂y∂t−2y2d)dx =∫ωχωψ(x,T)χω(y(x,T)−χω∗y1d)dx+∫0T2εhuεdt  +∫ωχω∂ψ∂t(x,T)χω(∂y∂t(x,T)−χω∗y2d)dx. (16) Using (10) and (13), we obtain limβ⟶0Qε(uε+βh)−Qε(uε)β  =∫ωχωψ(x,T)χω∂p∂t(x,T)dx−∫ωχω∂ψ∂t(x,T)χωp(x,T)dx+∫0T2εhuεdt  =∫ωχω∗χω[∫0T∂2p∂t2ψdt−∫0Tp∂2ψ∂t2dt]dx+∫0T2εhuεdt. (17) □ The following result characterises and expresses the optimal control, solution of problem (5), in terms of the solution of the adjoint system (13). The result also shows that the optimal control lies in $$C_{b}$$. Theorem 3.3 For an optimal control $$u_{\varepsilon}$$ in $$C_{b}$$, and $$w_{ \varepsilon } = w ( u _{ \varepsilon} )=(y, \displaystyle{\frac{\partial y}{\partial t}} )( u _{ \varepsilon} ) $$ its corresponding solution of Equation (1), then uε(t)=max(−b,min(12ε⟨χωy(x,t);χωp(t)⟩L2(ω),b)), (18) is solution of the problem (5), where $$ \tilde{p} = ( p , p t ) \in C ( [ 0 , T ] ; H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}}) ) $$ is the unique solution of the adjoint system (13). Proof. Let $$ h\in L^{\infty}(0,T)$$ such that $$ u_{\varepsilon} + \beta h \in C_{b}$$ for $$\beta>0$$ . The minimum of $$Q_{\varepsilon}$$ is achieved at $$ u_{\varepsilon}$$, then 0≤limβ⟶0Qε(uε+βh)−Qε(uε)β. (19) From system (10) replacing $$\displaystyle \frac{\partial \psi}{\partial t}$$ in the formula (14), we have 0 ≤limβ⟶0Qε(uε+βh)−Qε(uε)β =∫ωχω∗χω[∫0Tψ∂2p∂t2dt−∫0T(−Δ2ψ+u(t)ψ+h(t)y)pdt]dx+∫0T2εhuεdt. (20) And using the system (13) we obtain 0 ≤∫ωχω∗χω[∫0Tψ(∂2p∂t2+Δ2p−u(t)p)−h(t)ypdt]dx+∫0T2εhuεdt =∫0T2εhuεdt−∫0Th(t)⟨χωy;χωp⟩L2(ω) =∫0Th(t)[2εuε(t)−⟨χωy(x,t);χωp(t)⟩L2(ω)]dt. (21) Note that $$h = h(t)$$ is an arbitrary function with $$u_{\varepsilon}+\beta h \in C_{b}$$ for all small $$\beta$$, by a standard control argument involving the sign of the variation $$h$$ depending on the size of $$u_{\varepsilon}$$, we obtain the desired characterization of $$u_{\varepsilon}$$, namely, uε(t)=max(−b,min(12ε⟨χωy(x,t);χωp(t)⟩L2(ω),b)). (22) The proof of existence of a solution to the adjoint equation (13) is similar to the proof of existence of solution to the state equation since (y¯(T)−χω∗y1d) and (∂y¯∂t(T)−χω∗y2d) in C([0,T],L2(Ω)). see Brezis, 1983; Bradly et al., 1999 The bounded solutions to the optimality system (13) are unique if the control time $$T$$ is sufficiently small. See Beauchard (2011) and Bradly & Lenhart (1997) for similar results. □ 4. Regional control problem We now use a sequence of optimal controls $$u_{\varepsilon}(t)$$ defined by (22) as $$\varepsilon\rightarrow0$$ to identify $$u^{\star}$$ solution of the regional control problem (4) from observations $$w^d=(y_1^d,y_2^d)$$. Propositions 4.1 1. $$(Q_{\epsilon}(u_{\epsilon}))_{\epsilon > 0}$$ is a decreasing sequence as $$\epsilon \rightarrow 0$$. 2. $$\left(\int^{T}_{0} \; u_{\epsilon}^{2}(t)dt \right)_{\epsilon > 0}$$ is an increasing sequence as $$\epsilon \rightarrow 0$$. 3. $$(\|p_{\omega}w_{\epsilon}(T) - w_d\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is a decreasing sequence as $$\epsilon \rightarrow 0$$, and $$\forall \ \epsilon >0 $$ ‖pωwϵ(T)−wd‖(L2(ω))2≤‖pωS(T)w0−wd‖(L2(ω))2. In particular, on subsequence $$(p_{\omega}w_{\epsilon}(T)- w_d)_{\epsilon > 0}$$ converges weakly in $$(L^2(\omega))^{2}$$. Proof. 1. Let $$\epsilon_{1}$$, $$\epsilon_{2}$$ such that $$\epsilon_{2} > \epsilon_{1}$$, using the optimality of $$u_{\epsilon_{1}}$$ for $$Q_{\epsilon_{1}}$$ and the optimality of $$u_{\epsilon_{2}}$$ for $$Q_{\epsilon_{2}}$$, we have: Qϵ1(uϵ1) = ‖pωwϵ1(T)−wd‖(L2(ω))22+ϵ1∫0Tuϵ12(t)dt ≤ ‖pωwϵ2(T)−wd‖(L2(ω))22+ϵ1∫0Tuϵ22(t)dt ≤ ‖pωwϵ2(T)−wd‖(L2(ω))22+ϵ2∫0Tuϵ22(t)dt. (23) which gives Qϵ1(uϵ1)≤Qϵ2(uϵ2). (24) Then the sequence $$(Q_{\epsilon}(u_{\epsilon}))_{\epsilon > 0}$$ is a decreasing. 2. Using (24), we obtain Qϵ2(uϵ2)−Qϵ1(uϵ2)≤Qϵ2(uϵ1)−Qϵ1(uϵ1), therefore ∫0Tuϵ22(t)dt≤∫0Tuϵ12(t)dt, That allows us to conclude the increasing of the sequence $$\left(\int^{T}_{0} \; u_{\epsilon}^{2}(t)dt\right)_{\epsilon > 0}$$, 3. From the second item, we have ‖pωwϵ1(T)−wd‖(L2(ω))22≤‖pωwϵ2(T)−wd‖(L2(ω))22. which shows that $$(\big\|p_{\omega}w_{\epsilon}(T) - w_d\big\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is decreasing. Now for $$u = 0$$, we have $$w_u(T) = S(T)w_0 \; $$, and ∀ϵ>0,‖pωwϵ(T)−wd‖(L2(ω))22+ϵ∫0Tuϵ(t)2dt≤‖pωS(T)w0−wd‖(L2(ω))22. Then 0≤‖pωwϵ(T)−wd‖(L2(ω))22≤‖pωS(t)w0−wd‖(L2(ω))22,∀ϵ>0. Consequently, $$(\big\|p_{\omega}w_{\epsilon}(T) - w_d \big\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is bounded and the subsequence $$(p_{\omega}w_{\epsilon}(T) - w_{d})_{\epsilon > 0}$$ converges weakly in $$(L^2(\omega))^{2}.$$ □ The following proposition connect the regional penalized problem (5) to the regional control problem (4) as $$\varepsilon \rightarrow 0$$. Precisely, using control method we show that when $$\varepsilon \rightarrow 0$$, the optimal control solution of (5) converges to a feasible solution of (4). Propositions 4.2 1. While the control $$(u_{\varepsilon})_{\varepsilon}\in C_{b}$$ is bounded, the set $$C(\omega)$$ is nonempty. 2. The sequence $$(u_{\epsilon})_{\varepsilon>0}$$ solution of (5) converges strongly to $$ u^{*}$$ in $$ C_{b}$$. 3. $$ u^{*}$$ is the solution of the regional control problem (4). Proof. 1. While the control $$(u_{\varepsilon})_{\varepsilon}\in C_{b}$$ is bounded, we can extract a subsequence $$(u_{\epsilon})_{\epsilon > 0}$$ such that $$u_{\epsilon} \rightarrow u^{*}$$ weakly in $$C_{b}$$ and Ball et al. (1982) gives $$w_{\epsilon} \rightarrow w_{ u^{*}}$$ strongly in $$C([0, T] ; (L^2(\omega))^{2})$$. By the lower semi continuity of the norm see Brezis (1983), we have limϵ→0inf‖uϵ(t)‖L2(0,T)2≥‖u∗(t)‖L2(0,T)2.limϵ→0infQϵ(uϵ)≥‖pωwu∗(T)−wd‖(L2(ω))22. (25) The optimality of $$(u_{\epsilon})_{\epsilon > 0}$$ for the problem (5) gives $$ Q_{\epsilon}(u_{\epsilon}) \leq Q_{\epsilon}(u)\;\; \forall u \in C_{b} $$, and using the upper semicontinuity of the norm, we have limϵ→0supQϵ(uϵ)≤‖pωwu(T)−wd‖(L2(ω))22∀u∈Cb, thus limϵ→0supQϵ(uϵ)≤‖pωwu∗(T)−wd‖(L2(ω))22≤limϵ→0infQϵ(uϵ). Consequently, limϵ→0Qϵ(uϵ)=limϵ→0‖pωwϵ(T)−wd‖(L2(ω))22=‖pωwu∗(T)−wd‖(L2(ω))22. (26) Thereby, limϵ→0‖pωwϵ(T)−wd‖(L2(ω))22=minu∈Cb‖pωwu−wd‖(L2(ω))22=‖pωwu∗(T)−wd‖(L2(ω))22. We conclude that $$u^{*} \in C(\omega)$$ and the set $$C(\omega)$$ is nonempty. 2. To prove the second statement, we have ‖pωwϵ(T)−wd‖(L2(ω))22+ϵ‖uϵ(t)‖L2(0,T)2≤‖pωwu∗(T)−wd‖(L2(ω))22+ϵ‖u∗(t)‖L2(0,T)2, and using (26), we deduce that ‖uϵ(t)‖L2(0,T)2≤‖u∗(t)‖L2(0,T)2,∀ϵ>0. (27) From (25) and (27), we have ‖u∗(t)‖L2(0,T)2≤limϵ→0inf‖uϵ(t)‖L2(0,T)2≤‖u∗(t)‖L2(0,T)2, (28) and ‖uϵ(t)‖L2(0,T)2→‖u∗(t)‖L2(0,T)2asϵ→0. This result with the weak convergence of $$(u_{\epsilon})_{\epsilon > 0}$$ to $$u^{*}$$ in the close set $$C_{b},$$ give the strong convergence. 3. The sequence $$u_{\epsilon}$$ is a solution to (5), then $$\forall u\in C_{b}$$ ‖p ωwϵ(T)−wd‖(L2(ω))22+ϵ∫0Tuϵ2(t)dt≤‖p ωwu(T)−wd‖(L2(ω))22+ϵ∫0Tu2(t)dt. Then by taking the limit, we deduce that ‖u∗(t)‖L2(0,T)2≤‖u(t)‖L2(0,T)2∀u∈C(ω), We conclude that $$u^{*}$$ is a solution to the problem (4). □ Remark 4.1 1. The results of this paper can be extend easily to system (1) with Neumann boundary conditions. 2. We have not exclusively use the special case $$y\longrightarrow u(t)y$$ of the damping. The same results hold for system (1) with other types of damping. Thus, all the results of this paper extend to bi-linear wave equations with control acts as a multiplier velocity like {∂2y∂t2+Δ2y=u(t)∂y∂t+f(x,t)Qy(x,0)=y0(x),∂y∂t(x,0)=y1(x)Ωy=0Σ  (29) where $$f\in L^{2}(Q)$$ and we have the following corollary. Corollary 4.2 Let $$u_{\varepsilon}$$ in $$C_{b}$$, and $$ (y, \displaystyle{\frac{\partial y}{\partial t}} )( u _{ \varepsilon} ) $$ its corresponding state solution of (29). There exists a unique weak solution p~=(p,pt)∈C([0,T];H01(Ω)×L2(Ω)), to the adjoint system {∂2p∂t2+Δ2p=u(t)∂y(x,t)∂tQp(x,T)=−(∂y∂t(T)−χω∗y2d)Ω∂p∂t(x,T)=(y(T)−χω∗y1d)Ωp=0Σ  (30) Moreover, uε(t)=max(−b,min(12ε⟨χω∂y(x,t)∂t;χωp(t)⟩L2(ω),b)), (31) is the solution of the problem (5) and the results of propositions (4.2.) hold. 5. Open problem Let consider the fractional diffusion system {D+αy−Δy=u(t)yQI+1−αy(0+)=y0(x)Ωy=0Σ  (32) where $$0< \alpha < 1$$, the control $$u\in L^{2}(0, T)$$. The fractional integral $$I^{1-\alpha}_{+}$$ and derivative $$D^{\alpha}_{+}$$ are understood here in the Riemann Liouville sense, $$I^{1-\alpha}_{+} y(0^{+})=\displaystyle \lim_{t\rightarrow 0^{+}}I^{1-\alpha}_{+}y(t)$$ see Mophou (2011). The question is to solve the regional control problem minu ∈CbJε(u). (33) where the non penalized cost functional $$J_{\varepsilon}$$ is defined by Jε(u)=12‖χωy(T)−yd‖L2(ω)2+ε‖u(t)‖L2([0,T])2. (34) The goal is to give an extension to the classical optimal control theory to a fractional diffusion equation type bi-linear in a bounded domain. This question, among others is still open. Acknowledgements This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project 2015/01/3734. Many thanks to the anonymous referees for valuable comments and suggestions which have been included in the final version of this manuscript. References Ball J. , Marsden J. E. & Slemrod M. ( 1982 ) Controllability for distributed bi-linear systems. SIAM J. , 40 , 575 – 597 . Banks H. T. & Wang Y. ( 1994 ) Damage detection and characterization in smart material structures. Int. Ser. Numer. Math. , 118 , 21 – 43 . Beauchard K. ( 2011 ) Local controllability and non-controllability for a 1D wave equation with bi-linear control. J. Differ. Equ. , 250 , 2064 – 2098 . Google Scholar CrossRef Search ADS Bradly M. E. & Lenhart S. ( 1997 ) Bi-linear optimal control of a Kirchhoff plate to a desired profile. J. Optimal Control Appl. Methods , 18 , 217 – 226 . Google Scholar CrossRef Search ADS Bradly M. E. , Lenhart S. & Yong J. Bi-linear Optimal control of the Velocity term in a Kirchhoff plate equation. J. Math. Anal. Appl. , 238 , 451 – 467 . CrossRef Search ADS Brezis H. ( 1983 ) Analyse fonctionnelle: théorie et application . Paris : Masson . El Jai A. , Pritchard A. J. , Simon M. C. & Zerrik E. ( 1995 ) Regional controllability of distributed systems. Int. J. Control , 62 , 1351 – 1365 . Google Scholar CrossRef Search ADS Khapalov A. Y. ( 2010 ) Controllability of Partial Differential Equations Governed by Multiplicative Controls . Berlin, Heidelberg : Springer . Lasiecka I. & Triggiani R. ( 2008 ) Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation. Control Cybern. , 37 , 935 – 966 . Lenhart S. , Liang M. & Protopopescu V. ( 1999 ) Identification problem for a wave equation via optimal control. Control Distrib. Parameter Stochastic Syst. , 13 , 79 – 84 . Lenhart S. & Liang M. ( 2000 ) Bi-linear optimal control for a wave equation with viscous damping. J. Math. , 26 , 575 – 595 . Liang M. ( 1999 ) Bi-linear optimal control for a wave equation. Math. Models Methods Appl. Sci. , 9 , 45 – 68 . Google Scholar CrossRef Search ADS Lions J. L. ( 1988 ) Contrôlabilité Exacte, Perturbations et Stabilisation des Systèmes Distribués , Tome. 1. Contrôlabilité Exacte . Paris : Masson . Lions J. L. & Magenes E. ( 1968 ) Problèmes aux limites non homogènes et applications , vol. 1 et 2 . Paris : Dunod . Mophou G. M. ( 2011 ) Optimal control of fractional diffusion equation. Comput. Math. Appl. , 61 , 68 – 78 . Google Scholar CrossRef Search ADS Ouzahra M. ( 2014 ) Controllability of the wave equation with bi-linear controls. Eur. J. Control , 20 , 57 – 63 . Google Scholar CrossRef Search ADS Pazy A. ( 1983 ) Semigroups of Linear Operators and Applications to Partial Differential Equations . New York : Springer . Zerrik E. & Ould Sidi M. ( 2010a ) An output controllability of bi-linear distributed system. Int. Rev. Autom. Control , 3 , 466 – 473 . Zerrik E. & Ould Sidi M. ( 2010b ) Regional controllability of linear and semi linear hyperbolic systems. Int. J. Math. Anal. , 4 , 2167 – 2198 . Zerrik E. & Ould Sidi M. ( 2011 ) Regional controllability for infinite dimensional distributed bi-linear systems. Int. J. control , 84 , 2108 – 2116 . Google Scholar CrossRef Search ADS Zerrik E. & Ould Sidi M. ( 2013 ) Constrained regional control problem for distributed bi-linear systems. IET Cont. Theory Appl. , 7 , 1914 – 1921 . Google Scholar CrossRef Search ADS Zuazua E. ( 1990 ) Exponential decay for the semilinear wave equation with localized damping. Commun. Partial Differ. Equ. , 15 , 205 – 235 . Google Scholar CrossRef Search ADS © The authors 2017. 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Regional optimal control problem with minimum energy for a class of bilinear distributed systems

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Abstract

Abstract The aim of this paper is to propose a method for solving a regional control problem with minimum energy for a system governed by a wave equation via bi-linear bounded control. Using optimal control techniques, we show how to bring the solution for a two-dimensional wave equation with control act as multiplier of the state, close to a desired profile only on a subregion of the system domain. 1. Introduction Let the system described by the wave bi-linear equation {∂2y∂t2+Δ2y=u(t)yQy(x,0)=y0(x),∂y∂t(x,0)=y1(x)Ωy=0Σ  (1) where $${\it{\Omega}}$$ is an open bounded domain of $$I\!\!R^2$$ with $$C^2$$ boundary, $$ Q = {\it{\Omega}} \times (0,T)$$ for $$T> 0$$ and $$ {\it{\Sigma}}= \partial{\it{\Omega}} \times (0,T)$$. The control $$ u\in C_b= \{ u\in L^{\infty} (0,T); -b\leq u(t) \leq b\}, $$ for a constant positive $$b$$. We rewrite the system (1) in the parabolic form. {∂w(x,t)∂t=Aw(x,t)+Bw(x,t)Qw(x,0)=w0(x)Ω  (2) where w=(yyt),A=(0I−Δ20),Bw=(0u(t)y)andw0=(y0y1). $${\mathcal A}$$ generates a $${\mathcal C}^{0}$$ semigroup $$S(t)_{t\geq 0 } $$ of isometries on $$H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$. Consequently, there exists a $$K>0$$ such that $$\big\|S(t)\big\| \leq K$$, $$\forall t\in [0,T]$$ see Lasiecka & Triggiani (2008) and Pazy (1983). For $$w_{0}\in H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$, the mild solution of (2) is written as wu(x,t)=S(t)w0(x)+∫0tS(t−s)Bwu(x,s)ds. (3) The existence of a unique solution $$w_u(x,t)$$ in $${\mathcal C}([0,T]; H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}}))$$ satisfying (3) follows from standard results in Brezis (1983) and Lions (1988). Let $$w_{u}(T)=(y_{u}(T) ,\displaystyle{\frac{\partial y_{u}}{\partial t}}(T) )$$ the solution of (3) at time $$T$$, for $$\omega \subset{\it{\Omega}}$$, $$w^d=(y_1^d,y_2^d)$$ is the desired observations on $$L^2(\omega)\times L^2(\omega)$$ and $$\chi_{_\omega}: L^2({\it{\Omega}}) \longrightarrow L^2(\omega)$$ is the restriction operator to $$\omega $$. Given $$\tilde{w}=(\tilde{w}_1^d$$, $$\tilde{w}_2^d)$$, we define the scale product on the space product $$L^2(\omega)\times L^2(\omega)$$ by ⟨wd,w~⟩(L2(ω))2=(w1d,w~1d)L2(ω)+(w2d,w~2d)L2(ω), and denoted by $$\big\|\cdot \big\|_{(L^2(\omega))^{2}}$$ its corresponding norm. We have ‖pωw(T)−wd‖(L2(ω))22 = ⟨pωw(T)−wd,pωw(T)−wd⟩(L2(ω))2 = ⟨χωyu(T)−y1d,χωyu(T)−y1d⟩L2(ω)  +⟨χω(∂yu(T)∂t)−y2d,χω(∂yu(T)∂t)−y2d⟩L2(ω) = ‖χωyu(T)−y1d‖L2(ω)2+‖χω(∂yu(T)∂t)−y2d‖L2(ω)2, where $$p_{\omega}$$ is the restriction map pω:L2(Ω)×L2(Ω)⟶L2(ω)×L2(ω)(y1,y2)⟶(χωy1,χωy2). Our main objective is to solve the regional with minimum energy control problem governed by the bi-linear distributed wave equation (1) {Minimize ‖u‖L2(0,T)2subject to u∈C(ω).  (4) where C(ω)={u∈Cb:‖pωwu(T)−wd‖L2(ω)×L2(ω)2 is minimum }. $$C(\omega)$$ is the set of bounded control with corresponding state $$w_u(T)$$ in a neighborhood of the desired state $$w^d$$. In terms of applications, a problem like this arises, e.g., in the context of “smart materials”, whose properties can be altered by applying various external factors such as temperature, electrical current or magnetic field (Banks & Wang, 1994; Khapalov, 2010). For the background on controllability of bi-linear distributed systems, Ball et al. (1982) have proved for the first time the global approximate controllability in $$H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$ of bi-linear beam and rod equations in the mono-dimensional case. Positive results on exact and approximative controllability of wave equations are given by Zuazua (1990), Lenhart et al. (1999, 2000), Khapalov (2010) and Ouzahra (2014) and negative results by Ball et al. (1982) and Beauchard (2011) due to the choice of functional spaces. Optimal control problem of wave equation type (1), has been treated in Liang (1999) where a bi-linear control has been used to bring the state solutions close to a desired profile on the whole domain. Then the author established the existence and uniqueness of the solution of the optimality system and thus, determined the unique optimal control in terms of the solution of the optimality system. Bradly & Lenhart (1997) and Bradly et al. (1999) treated $${\it{\Delta}}^{2}$$ type of bi-linear control for the Kirchhoff plate equation. In this paper our goal is to solve the regional control problem (4), which consists in finding a bounded control $$u\in C_{b}$$ with minimum energy such its corresponding solution $$w_{u}(T)$$ at time $$T$$ very close to the observations $$w^d$$ only on a subregion on $$\omega \subset{\it{\Omega}}$$. This can be regarded as an extension of the research work in El Jai et al. (1995) and Zerrik & Ould Sidi (2010a,b, 2011, 2013). Besides, focusing on regional controllability would allow for a reduction in the number of physical actuators, offer the potential to reduce computational requirements in some cases, and also offer the possibility to discuss those systems which are not controllable on the whole domain, etc. The paper is organized as follows: we begin by proving the existence of a solution for the associate regional penalized problem in Section 2. In Section 3, we give necessary conditions for the problem (5). In Section 4, we connect the regional penalized problem (5) to the regional control problem (4) as $$\varepsilon \rightarrow 0$$. Specifically, we show that when $$\varepsilon \rightarrow 0$$, the optimal control solution of (5) converges to a feasible solution of (4). The paper ends with open problem. 2. Regional penalized problem To solve the optimal control problem (4), we propose an approach based on quadratic cost control problem, which involves the minimization of the control norm, the final state and speed errors. For $$\varepsilon>0$$, let associate to (4) the problem: minu ∈CbQε(u). (5) where the non-penalized cost functional $$Q_{\varepsilon}$$ is defined by Qε(u) =12‖pωw(T)−wd‖(L2(ω))22+ε‖u(t)‖L2([0,T])2 =12‖χ ωyu(T)−y1d‖L2(ω)2+12‖χ ω(∂yu(T)∂t0)−y2d‖L2(ω)2+ε‖u(t)‖L2([0,T])2. (6) The next theorem justifies that the minimum of the problem (5) exists, Theorem 2.1 There exists an optimal control $$u_{\varepsilon} \in C_b$$, which minimizes the non-penalized cost functional $$Q_{\varepsilon}(u)$$ over $$u$$ in $$C_b$$. Proof. For given $$ w(0) \in H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}})$$ and $$u\in C_{b}$$ the weak solution of (1) satisfies ‖w‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT‖w(0)‖H01(Ω)×L2(Ω). (7) Using (3), and the bound $$K$$ of the strongly continuous semigroup $$(S(t))_{t\geq 0}$$ in all finite interval of $$[0,T]$$ see Pazy (1983), we have ‖wu(t)‖C([0,T];H01(Ω)×L2(Ω))≤K‖w(0)‖H01(Ω)×L2(Ω)+Kb∫0t‖wu(s)‖H01(Ω)×L2(Ω)ds. Using Gronwall inequality see Ball et al. (1982), we obtain (7). Let $$ ( u )_{ n } \in C_{b} $$ be a minimizing sequence such that limn→∞Qε(un)=infu∈CbQε(u)=Q∗, Equation (7) gives a priori estimate of $$ w_{n}=w(u_{n})$$ with bounds independent of $$n$$. On a subsequence, by weak compactness, there exists $$ w_{n}=w(u_{ n })=\big(y_n, {\frac{\partial y_{n}}{\partial t}} \big)$$ such that un⇀u∗weakly in L2(0,T).yn⇀yweakly in L2([0,T];H01(Ω)).∂yn∂t⇀∂y∂tweakly in L2([0,T];L2(Ω)).∂2yn∂t2⇀∂2y∂t2weakly in L2([0,T];H−1(Ω)). (8) By classical argument see Lions & Magenes (1968) and using the fact that $$w_{ n }(0)= w_{ 0 }$$. Taking the limit in the system (1), associated with $$(u_n, w(u_{ n }))$$ as $$n \longrightarrow \infty$$ from this, we obtain $$w^{*}=w(u^{*})$$ which is the weak solution of (1) with control $$u^*$$. Since the objective functional is lower semi-continuous with respect to weak convergence, we obtain Qε(u∗)≤limn→+∞infQε(un)=Q∗≤Qε(u∗). We conclude that $$u^{*}$$ is an optimal control. □ Remark 2.2 If we consider the state equation with a source term $$f\in L^{\infty}(0,T;L^2({\it{\Omega}}))$$, ∂2y∂t2+Δ2y=u(t)y+f(t,x)on Q. The same well-posedness and regularity results as (1) hold, and we have ‖w‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT(‖w(0)‖H01(Ω)×L2(Ω)+‖f‖L∞(0,T;L2(Ω))). (9) where $$w=\big(y, {\frac{\partial y}{\partial t}}\big)$$ and $$w(0)=(w_0, w_1)$$. 3. Necessary conditions We derive necessary conditions that every optimal control must satisfy by differentiate our functional criterion $$ Q_{\varepsilon}(u) $$ with respect to $$u$$. Firstly, we examine the differentiability of $$u\longrightarrow w(u)$$ with respect to u and the differentiation of $$ Q_{\varepsilon} $$ gives a characterization of optimal controls in terms of the optimality system. Lemma 3.1 The mapping u∈Cb→w(u)∈C([0,T];H01(Ω)×L2(Ω)), is differentiable in the following sense: w(u+εh)−w(u)ε⇀ψ~weakly in L∞([0,T];H01(Ω)×L2(Ω)), as $$ \varepsilon \rightarrow 0 , $$ for any $$ u , u + \varepsilon h \in C_{b}. $$ Moreover, the limit $$ \tilde{\psi} = ( \psi , \psi t ) $$ is a weak state of the wave bi-linear equation {∂2ψ∂t2+Δ2ψ=u(t)ψ+h(t)yQψ(x,0)=0,∂ψ∂t(x,0)=0Ωψ=0Σ  (10) Proof. Let $$ w^{ \varepsilon } = w ( u + \varepsilon h ) $$ and $$ w = w( u )$$. From (1), $$(w ^{ \varepsilon } - w ) / \varepsilon $$ is a weak solution of (yε−yε)tt+Δ2(yε−yε)=u(t)(yε−yε)+hyε in Q. (11) Using Remark 2.2 with source term $$ hy ^{\varepsilon }, $$ we obtain ‖(wε−wε)‖C([0,T];H01(Ω)×L2(Ω))≤KeKbT‖hyε‖L∞(0,T;L2(Ω))≤C3. (12) where $$C_{3}$$ is independent of $$\epsilon$$ since the bound on $$ \big\| y^{ \varepsilon }\big\| _{L^{\infty}(0,T;L^2({\it{\Omega}}))}$$ is independent of $$\epsilon$$, and the weak convergence to $$\psi$$ is obtained. □ Next, consider the optimality system {∂2p∂t2+Δ2p=u(t)pQp(x,T)=−(∂y∂t(T)−χω∗y2d)Ω∂p∂t(x,T)=(y(T)−χω∗y1d)Ωp=0Σ  (13) where $$\chi _{\omega }^{*}$$ is the adjoint of $$\chi_\omega$$ defined from $$ L^2 (\omega ) \longrightarrow L^2 ({\it{\Omega}} )$$ by χω∗y(x)={y(x),x∈ω0,x∈Ω∖ω  By differentiating the cost functional $$ u \rightarrow Q_{\varepsilon} ( u ) $$ with respect to the control, we have the following lemma: Lemma 3.2 If $$u_{ \varepsilon } \in C_{b}$$ is an optimal control, $$\psi $$ is the solution of (10) and $$p$$ is the solution of (13), then limβ⟶0Qε(uε+βh)−Qε(uε)β =∫ωχω∗χω[∫0T∂2p∂t2ψdt−∫0Tp∂2ψ∂t2dt]dx  +∫0T2εhuεdt. (14) Proof. limβ⟶0Qε(uε+βh)−Qε(uε)β = limβ⟶012∫ω(χωyβ−y1d)2−(χωy−y1d)2βdx  +limβ⟶012∫ω(χω∂yβ∂t−y2d)2−(χω∂y∂t−y2d)2βdx  +limβ⟶0ε∫0T(uε+βh)2−uε2β(t)dt. (15) Therefore limβ⟶0Qε(uε+βh)−Qε(uε)β =limβ⟶012∫ωχω(yβ−y)β(χωyβ+χωy−2y1d)dx  +∫0T(2εhuε+βεh2)dt  +limβ⟶012∫ωχω(∂y∂tβ−∂y∂t)β(χω∂y∂tβ+χω∂y∂t−2y2d)dx =∫ωχωψ(x,T)χω(y(x,T)−χω∗y1d)dx+∫0T2εhuεdt  +∫ωχω∂ψ∂t(x,T)χω(∂y∂t(x,T)−χω∗y2d)dx. (16) Using (10) and (13), we obtain limβ⟶0Qε(uε+βh)−Qε(uε)β  =∫ωχωψ(x,T)χω∂p∂t(x,T)dx−∫ωχω∂ψ∂t(x,T)χωp(x,T)dx+∫0T2εhuεdt  =∫ωχω∗χω[∫0T∂2p∂t2ψdt−∫0Tp∂2ψ∂t2dt]dx+∫0T2εhuεdt. (17) □ The following result characterises and expresses the optimal control, solution of problem (5), in terms of the solution of the adjoint system (13). The result also shows that the optimal control lies in $$C_{b}$$. Theorem 3.3 For an optimal control $$u_{\varepsilon}$$ in $$C_{b}$$, and $$w_{ \varepsilon } = w ( u _{ \varepsilon} )=(y, \displaystyle{\frac{\partial y}{\partial t}} )( u _{ \varepsilon} ) $$ its corresponding solution of Equation (1), then uε(t)=max(−b,min(12ε⟨χωy(x,t);χωp(t)⟩L2(ω),b)), (18) is solution of the problem (5), where $$ \tilde{p} = ( p , p t ) \in C ( [ 0 , T ] ; H^{1}_{0}({\it{\Omega}})\times L^{2}({\it{\Omega}}) ) $$ is the unique solution of the adjoint system (13). Proof. Let $$ h\in L^{\infty}(0,T)$$ such that $$ u_{\varepsilon} + \beta h \in C_{b}$$ for $$\beta>0$$ . The minimum of $$Q_{\varepsilon}$$ is achieved at $$ u_{\varepsilon}$$, then 0≤limβ⟶0Qε(uε+βh)−Qε(uε)β. (19) From system (10) replacing $$\displaystyle \frac{\partial \psi}{\partial t}$$ in the formula (14), we have 0 ≤limβ⟶0Qε(uε+βh)−Qε(uε)β =∫ωχω∗χω[∫0Tψ∂2p∂t2dt−∫0T(−Δ2ψ+u(t)ψ+h(t)y)pdt]dx+∫0T2εhuεdt. (20) And using the system (13) we obtain 0 ≤∫ωχω∗χω[∫0Tψ(∂2p∂t2+Δ2p−u(t)p)−h(t)ypdt]dx+∫0T2εhuεdt =∫0T2εhuεdt−∫0Th(t)⟨χωy;χωp⟩L2(ω) =∫0Th(t)[2εuε(t)−⟨χωy(x,t);χωp(t)⟩L2(ω)]dt. (21) Note that $$h = h(t)$$ is an arbitrary function with $$u_{\varepsilon}+\beta h \in C_{b}$$ for all small $$\beta$$, by a standard control argument involving the sign of the variation $$h$$ depending on the size of $$u_{\varepsilon}$$, we obtain the desired characterization of $$u_{\varepsilon}$$, namely, uε(t)=max(−b,min(12ε⟨χωy(x,t);χωp(t)⟩L2(ω),b)). (22) The proof of existence of a solution to the adjoint equation (13) is similar to the proof of existence of solution to the state equation since (y¯(T)−χω∗y1d) and (∂y¯∂t(T)−χω∗y2d) in C([0,T],L2(Ω)). see Brezis, 1983; Bradly et al., 1999 The bounded solutions to the optimality system (13) are unique if the control time $$T$$ is sufficiently small. See Beauchard (2011) and Bradly & Lenhart (1997) for similar results. □ 4. Regional control problem We now use a sequence of optimal controls $$u_{\varepsilon}(t)$$ defined by (22) as $$\varepsilon\rightarrow0$$ to identify $$u^{\star}$$ solution of the regional control problem (4) from observations $$w^d=(y_1^d,y_2^d)$$. Propositions 4.1 1. $$(Q_{\epsilon}(u_{\epsilon}))_{\epsilon > 0}$$ is a decreasing sequence as $$\epsilon \rightarrow 0$$. 2. $$\left(\int^{T}_{0} \; u_{\epsilon}^{2}(t)dt \right)_{\epsilon > 0}$$ is an increasing sequence as $$\epsilon \rightarrow 0$$. 3. $$(\|p_{\omega}w_{\epsilon}(T) - w_d\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is a decreasing sequence as $$\epsilon \rightarrow 0$$, and $$\forall \ \epsilon >0 $$ ‖pωwϵ(T)−wd‖(L2(ω))2≤‖pωS(T)w0−wd‖(L2(ω))2. In particular, on subsequence $$(p_{\omega}w_{\epsilon}(T)- w_d)_{\epsilon > 0}$$ converges weakly in $$(L^2(\omega))^{2}$$. Proof. 1. Let $$\epsilon_{1}$$, $$\epsilon_{2}$$ such that $$\epsilon_{2} > \epsilon_{1}$$, using the optimality of $$u_{\epsilon_{1}}$$ for $$Q_{\epsilon_{1}}$$ and the optimality of $$u_{\epsilon_{2}}$$ for $$Q_{\epsilon_{2}}$$, we have: Qϵ1(uϵ1) = ‖pωwϵ1(T)−wd‖(L2(ω))22+ϵ1∫0Tuϵ12(t)dt ≤ ‖pωwϵ2(T)−wd‖(L2(ω))22+ϵ1∫0Tuϵ22(t)dt ≤ ‖pωwϵ2(T)−wd‖(L2(ω))22+ϵ2∫0Tuϵ22(t)dt. (23) which gives Qϵ1(uϵ1)≤Qϵ2(uϵ2). (24) Then the sequence $$(Q_{\epsilon}(u_{\epsilon}))_{\epsilon > 0}$$ is a decreasing. 2. Using (24), we obtain Qϵ2(uϵ2)−Qϵ1(uϵ2)≤Qϵ2(uϵ1)−Qϵ1(uϵ1), therefore ∫0Tuϵ22(t)dt≤∫0Tuϵ12(t)dt, That allows us to conclude the increasing of the sequence $$\left(\int^{T}_{0} \; u_{\epsilon}^{2}(t)dt\right)_{\epsilon > 0}$$, 3. From the second item, we have ‖pωwϵ1(T)−wd‖(L2(ω))22≤‖pωwϵ2(T)−wd‖(L2(ω))22. which shows that $$(\big\|p_{\omega}w_{\epsilon}(T) - w_d\big\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is decreasing. Now for $$u = 0$$, we have $$w_u(T) = S(T)w_0 \; $$, and ∀ϵ>0,‖pωwϵ(T)−wd‖(L2(ω))22+ϵ∫0Tuϵ(t)2dt≤‖pωS(T)w0−wd‖(L2(ω))22. Then 0≤‖pωwϵ(T)−wd‖(L2(ω))22≤‖pωS(t)w0−wd‖(L2(ω))22,∀ϵ>0. Consequently, $$(\big\|p_{\omega}w_{\epsilon}(T) - w_d \big\|_{(L^2(\omega))^{2}})_{\epsilon> 0}$$ is bounded and the subsequence $$(p_{\omega}w_{\epsilon}(T) - w_{d})_{\epsilon > 0}$$ converges weakly in $$(L^2(\omega))^{2}.$$ □ The following proposition connect the regional penalized problem (5) to the regional control problem (4) as $$\varepsilon \rightarrow 0$$. Precisely, using control method we show that when $$\varepsilon \rightarrow 0$$, the optimal control solution of (5) converges to a feasible solution of (4). Propositions 4.2 1. While the control $$(u_{\varepsilon})_{\varepsilon}\in C_{b}$$ is bounded, the set $$C(\omega)$$ is nonempty. 2. The sequence $$(u_{\epsilon})_{\varepsilon>0}$$ solution of (5) converges strongly to $$ u^{*}$$ in $$ C_{b}$$. 3. $$ u^{*}$$ is the solution of the regional control problem (4). Proof. 1. While the control $$(u_{\varepsilon})_{\varepsilon}\in C_{b}$$ is bounded, we can extract a subsequence $$(u_{\epsilon})_{\epsilon > 0}$$ such that $$u_{\epsilon} \rightarrow u^{*}$$ weakly in $$C_{b}$$ and Ball et al. (1982) gives $$w_{\epsilon} \rightarrow w_{ u^{*}}$$ strongly in $$C([0, T] ; (L^2(\omega))^{2})$$. By the lower semi continuity of the norm see Brezis (1983), we have limϵ→0inf‖uϵ(t)‖L2(0,T)2≥‖u∗(t)‖L2(0,T)2.limϵ→0infQϵ(uϵ)≥‖pωwu∗(T)−wd‖(L2(ω))22. (25) The optimality of $$(u_{\epsilon})_{\epsilon > 0}$$ for the problem (5) gives $$ Q_{\epsilon}(u_{\epsilon}) \leq Q_{\epsilon}(u)\;\; \forall u \in C_{b} $$, and using the upper semicontinuity of the norm, we have limϵ→0supQϵ(uϵ)≤‖pωwu(T)−wd‖(L2(ω))22∀u∈Cb, thus limϵ→0supQϵ(uϵ)≤‖pωwu∗(T)−wd‖(L2(ω))22≤limϵ→0infQϵ(uϵ). Consequently, limϵ→0Qϵ(uϵ)=limϵ→0‖pωwϵ(T)−wd‖(L2(ω))22=‖pωwu∗(T)−wd‖(L2(ω))22. (26) Thereby, limϵ→0‖pωwϵ(T)−wd‖(L2(ω))22=minu∈Cb‖pωwu−wd‖(L2(ω))22=‖pωwu∗(T)−wd‖(L2(ω))22. We conclude that $$u^{*} \in C(\omega)$$ and the set $$C(\omega)$$ is nonempty. 2. To prove the second statement, we have ‖pωwϵ(T)−wd‖(L2(ω))22+ϵ‖uϵ(t)‖L2(0,T)2≤‖pωwu∗(T)−wd‖(L2(ω))22+ϵ‖u∗(t)‖L2(0,T)2, and using (26), we deduce that ‖uϵ(t)‖L2(0,T)2≤‖u∗(t)‖L2(0,T)2,∀ϵ>0. (27) From (25) and (27), we have ‖u∗(t)‖L2(0,T)2≤limϵ→0inf‖uϵ(t)‖L2(0,T)2≤‖u∗(t)‖L2(0,T)2, (28) and ‖uϵ(t)‖L2(0,T)2→‖u∗(t)‖L2(0,T)2asϵ→0. This result with the weak convergence of $$(u_{\epsilon})_{\epsilon > 0}$$ to $$u^{*}$$ in the close set $$C_{b},$$ give the strong convergence. 3. The sequence $$u_{\epsilon}$$ is a solution to (5), then $$\forall u\in C_{b}$$ ‖p ωwϵ(T)−wd‖(L2(ω))22+ϵ∫0Tuϵ2(t)dt≤‖p ωwu(T)−wd‖(L2(ω))22+ϵ∫0Tu2(t)dt. Then by taking the limit, we deduce that ‖u∗(t)‖L2(0,T)2≤‖u(t)‖L2(0,T)2∀u∈C(ω), We conclude that $$u^{*}$$ is a solution to the problem (4). □ Remark 4.1 1. The results of this paper can be extend easily to system (1) with Neumann boundary conditions. 2. We have not exclusively use the special case $$y\longrightarrow u(t)y$$ of the damping. The same results hold for system (1) with other types of damping. Thus, all the results of this paper extend to bi-linear wave equations with control acts as a multiplier velocity like {∂2y∂t2+Δ2y=u(t)∂y∂t+f(x,t)Qy(x,0)=y0(x),∂y∂t(x,0)=y1(x)Ωy=0Σ  (29) where $$f\in L^{2}(Q)$$ and we have the following corollary. Corollary 4.2 Let $$u_{\varepsilon}$$ in $$C_{b}$$, and $$ (y, \displaystyle{\frac{\partial y}{\partial t}} )( u _{ \varepsilon} ) $$ its corresponding state solution of (29). There exists a unique weak solution p~=(p,pt)∈C([0,T];H01(Ω)×L2(Ω)), to the adjoint system {∂2p∂t2+Δ2p=u(t)∂y(x,t)∂tQp(x,T)=−(∂y∂t(T)−χω∗y2d)Ω∂p∂t(x,T)=(y(T)−χω∗y1d)Ωp=0Σ  (30) Moreover, uε(t)=max(−b,min(12ε⟨χω∂y(x,t)∂t;χωp(t)⟩L2(ω),b)), (31) is the solution of the problem (5) and the results of propositions (4.2.) hold. 5. Open problem Let consider the fractional diffusion system {D+αy−Δy=u(t)yQI+1−αy(0+)=y0(x)Ωy=0Σ  (32) where $$0< \alpha < 1$$, the control $$u\in L^{2}(0, T)$$. The fractional integral $$I^{1-\alpha}_{+}$$ and derivative $$D^{\alpha}_{+}$$ are understood here in the Riemann Liouville sense, $$I^{1-\alpha}_{+} y(0^{+})=\displaystyle \lim_{t\rightarrow 0^{+}}I^{1-\alpha}_{+}y(t)$$ see Mophou (2011). The question is to solve the regional control problem minu ∈CbJε(u). (33) where the non penalized cost functional $$J_{\varepsilon}$$ is defined by Jε(u)=12‖χωy(T)−yd‖L2(ω)2+ε‖u(t)‖L2([0,T])2. (34) The goal is to give an extension to the classical optimal control theory to a fractional diffusion equation type bi-linear in a bounded domain. This question, among others is still open. Acknowledgements This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project 2015/01/3734. Many thanks to the anonymous referees for valuable comments and suggestions which have been included in the final version of this manuscript. References Ball J. , Marsden J. E. & Slemrod M. ( 1982 ) Controllability for distributed bi-linear systems. SIAM J. , 40 , 575 – 597 . Banks H. T. & Wang Y. ( 1994 ) Damage detection and characterization in smart material structures. Int. Ser. Numer. Math. , 118 , 21 – 43 . Beauchard K. ( 2011 ) Local controllability and non-controllability for a 1D wave equation with bi-linear control. J. Differ. Equ. , 250 , 2064 – 2098 . Google Scholar CrossRef Search ADS Bradly M. E. & Lenhart S. 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Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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Published: Apr 21, 2017

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