Fractional Optimal Control Problem of Parabolic Bilinear Systems with Bounded Controls

Abstract

The purpose of this paper is to study a fractional distributed optimal control for a class of infinite-dimensional parabolic bilinear systems evolving on a spatial domain Ω by distributed controls depending on the control operator. Using the Fréchet differentiability, we prove the existence of an optimal control depending on both time and space, that minimizes a quadratic functional which leads into account, the deviation between the desired state and the reached one. Then, we show characterizations of an optimal distributed control for different admissible controls set. Moreover, we developed an algorithm and give simulations that successfully illustrate the theoretically obtained results.

1. Introduction

In engineering and mathematics, control theory deals with the behavior of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system, As an example: the control of vibration which is becoming more and more important for many industries. This generally has to be achieved without additional cost, and thus, detailed knowledge of structural dynamics is required together with familiarity of standard vibration control techniques. We also cited the following works on what concerns the vibration control [1, 2, 3].

The bilinear system involves the product of state and control, linear in state and linear in control but not jointly linear in state and control. The interest of these systems lies in the fact that many natural and industrial processes have intrinsically bilinear structures, This is the case of furnaces for heating metal slabs or heat exchangers, aircraft and robot arms, or energy transmission lines.

Let $\mathrm{\Omega }$ be an open bounded domain of ${\mathrm{ℛ}}^n$, $n\ge 1$, with regular boundary $\mathrm{\partial \Omega }$, and consider a bilinear system described by the equation (see [4])

$$\left\{\begin{array}{ll}\frac{\partial z}{\partial t}\left(x,t\right)=\mathit{Az}\left(x,t\right)+u\left(x,t\right)\mathit{Bz}\left(x,t\right)& Q=\mathrm{\Omega }\times ]0,T[,\\ z\left(x,t\right)=0& \mathrm{\Gamma }=\mathrm{\partial \Omega }\times ]0,T[,\\ z\left(x,0\right)=z_0\left(x\right)& \mathrm{\Omega },\end{array}\right.$$E1

where, $A=\mathrm{\Delta }$ of the domain $\mathcal{D}\left(A\right)=H_0^1\left(\mathrm{\Omega }\right)\cap H^2\left(\mathrm{\Omega }\right)$, $u$ is a control assumed to belong to the set of controls

$$U=\left\{u\in L^2\left(Q\right)/-m\le u\le M\right\}\left(\mathit{\text{with}}M\ge m>0\right).$$E2

$B$ is a bounded control operator on $L^2\left(\mathrm{\Omega }\right)$. For $z_0\in H_0^1\left(\mathrm{\Omega }\right)$ and $u\in U$, system (2) has a unique solution $z\in W=\left\{z\in L^2\left(0,T;H_0^1\left(\mathrm{\Omega }\right)\right)|\frac{\partial z}{\partial t}\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)\right\}$.

Let us consider the fractional quadratic control problem:

$$J\left(u^{\ast }\right)=\underset{u\in U}{min}J\left(u\right),$$E3

with

$$J\left(u\right)=\frac{1}{2}\parallel D_x^{\alpha }z-z_d{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2+\frac{\beta }{2}\parallel u{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2,$$E4

where, $D_x^{\alpha }$ denotes the fractional spacial derivative of order $\alpha \in ]0,1[$, $z$ is a solution of system (2), $z_d\in L^2\left(\mathrm{\Omega }\right)$ is a desired derivative and $\beta$ is a positive constant.

Ractional calculus has emerged as a powerful and efficient mathematical instrument during the past six decades, mainly due to its demonstrated applications in numerous, seemingly diverse, and widespread fields of science and engineering. As an example, The theory of fractional differential equations has received much attention, as they are important for describing the natural models as in diffusion processes, stochastic processes, economics, and hydrology. Moreover, the fractional optimal control has been studied in many works, such as Frederico et al. have studied a fractional optimal control problem in Caputo’s sense. Agrawal [5] have presented an extended approach to a class of distributed system whose dynamics are defined in the sense of Caputo. In [6], they considered the fractional optimal control problem for variable inequalities. In [7], Bahaa studied the fractional optimal control problem for different systems. When $\alpha =0$, problem (2) was considered in many works.: Bradley and Lenhart [8] have shown the existence of such an optimal control and given characterization of such control using necessary optimality conditions. Then, an optimal distributed control for a Kirchhoff plate equation acting on the state position. Also, they collaborated with Yong [9] on the same equation by temporal controls acting on the speed state and with special optimal control in Bradley and Lenhart [10]. For parabolic systems, we have mentioned the work in [11], which established an optimal control of a parabolic equation, modeling one-dimensional fluid through a soil-packed tube in which a contaminant is initially distributed, taking a functional criterion as a combination of the final amount of contaminant and the energy. In the same way, Addou and Benbrik [12] studied a fourth-order parabolic distributed parameter system and derived the existence and uniqueness of temporal bilinear optimal control. Then, Zerrik and El Kabouss [13] extended this problem to a more general class of systems governed by a fourth-order parabolic operator and excited by bounded and unbounded controls. A wide literature has also been considered for infinite hyperbolic systems, especially, by Liang [14] who analyzed an optimal control problem for a wave equation with internal bilinear control, and has given an optimal control that allows minimizing a functional cost which contains the difference between the solution’s position and a desired one. In the case of boundary bilinear controls: Lenhart and Wilson [15] have studied the problem of controlling the solution of the heat equation with the convective boundary condition, such as, that the bilinear control represents a heat transfer coefficient. The used approach consists in finding a unique optimal control in terms of the solution of an optimality system.

For a system evolving on a spatial domain $\mathrm{\Omega }$, regional controllability concerns the extension of the classical notion of controllability (controllability on the whole domain $\mathrm{\Omega }$) to the controllability only on a subregion $\omega$ of $\mathrm{\Omega }$. This notion is interesting for many reasons: it is close to real applications. For instance, the physical problem that concerns a tunnel furnace where one has to maintain a prescribed temperature only in a subregion of the furnace and may be of great help for systems that are non-controllable on the whole domain but controllable on some subregions, and controlling a system on a subregion $\omega \subset \mathrm{\Omega }$ is cheaper than controlling it in the whole domain. Zerrik and El Kabouss [16] have studied a regional optimal bilinear control of wave equation, taking a functional cost as the sum of the energy and the difference between the solution of the wave equation and the desired state for bounded and unbounded controls. Recently, Zerrik and El Kabouss [17] established an output optimal control problem with a bounded control set. In other words, they considered a problem of controlling only an output of the solution of a parabolic system. In [18], they have studied an optimal control problem for the heat equation in order to give control that leads to a state as the class as possible to the desired state, only on a subregion of the domain of evolution, under constrained controls sets.

In this paper, we consider $0<\alpha <1$, which is very important for modeling many real processes. We study a fractional optimal control problem of parabolic bilinear systems. Using the Frechet differentiability, we prove the existence and give the expression of an optimal control solution of (2). Then we discuss particular cases of admissible controls set.

2. Existence of an optimal control

This section discusses the existence of a solution of the problem (2).

First, let us recall the notion of the weak solution of the system (2).

Definition 1.1.

Let $T>0$, a continuous function $z\in \left[0,T\right]\to L^2\left(\mathrm{\Omega }\right)$ is a weak solution of system (3) on $\left[0,T\right]$, if it satisfies the following integral equation

$$z_u\left(t\right)=S\left(t\right)z_0+{\int }_0^{T}S\left(t-s\right)u\left(.,s\right)z\left(s\right)\mathit{ds},\text{for}\mathrm{all}t\in \left[0,T\right]$$E5

where $S\left(t\right)$ denotes the $C_0$ semi-group generated by $A$ in $L^2\left(\mathrm{\Omega }\right)$.

For fractional Riemann Louiville derivatives, we recall the following definition.

Definition 1.2.

Let $0<\alpha <1$ and $T>0$, the fractional spatial Riemann Liouville derivatives of order $\alpha$ is defined by:

$$D_x^{\alpha }:H_0^1\left(\mathrm{\Omega }\right)\to L^2\left(\mathrm{\Omega }\right)$$E6

$$z\to D_x^{\alpha }z=\frac{d}{\mathit{dx}}{I}_0^{1-\alpha }z,$$E7

where $I_0^{1-\alpha }$ is the Riemann-Liouville integral of $\left(1-\alpha \right)$ order defined by:

$$I_0^{1-\alpha }z\left(x,t\right)=\frac{1}{\mathrm{\Gamma }\left(1-\alpha \right)}{\int }_0^x{\left(x-\tau \right)}^{-\alpha }z\left(\tau ,t\right)\mathit{d\tau }$$E8

with $\mathrm{\Gamma }\left(1-\alpha \right)={\int }_0^{+\infty }{\tau }^{-\alpha }{e}^{-\tau }\mathit{d\tau }.$

In the following, we show the existence of optimal control, solution of problem (3).

Theorem 1.3.

Problem (3) has at least one solution.

Proof: For $u\in U$, the associated solution of system (3) is one of the equation

$$z_u\left(x,t\right)=S\left(t\right)z_0\left(x\right)+{\int }_0^{T}S\left(t-s\right)u\left(x,s\right)\mathit{Bz}\left(x,s\right)\mathit{ds}.$$E9

Using the bound of the semi-group ${\left(S\left(t\right)\right)}_{t\ge 0}$ over $\left[0,T\right]$, we have

$$\parallel z_u\left(t\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C\parallel z_0{\parallel }_{L^2\left(\mathrm{\Omega }\right)}+C\parallel B{\parallel }_{L^2\left(\mathrm{\Omega }\right)}{\int }_0^T\parallel u\left(s\right)z\left(s\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}\mathit{ds}.$$E10

It follows

$$\parallel z_u\left(t\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C\parallel z_0{\parallel }_{L^2\left(\mathrm{\Omega }\right)}+\mathit{CM}\parallel B{\parallel }_{L^2\left(\mathrm{\Omega }\right)}{\int }_0^T\parallel z\left(s\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}\mathit{ds}.$$

Using the Gronwal inequality, we get

$$\parallel z_u\left(t\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}\le C_1\mathit{exp}\left(\mathit{CM}\parallel B{\parallel }_{L^2\left(\mathrm{\Omega }\right)}T\right).$$E11

with $C_1=C\parallel z_0{\parallel }_{L^2\left(\mathrm{\Omega }\right)}$.

On the other hand, the set $\left\{J\left(u\right)|u\in U\right\}$ is non-empty and is bounded from below by 0.

Let ${\left(u_k\right)}_{k\in \mathbb{N}}$ be a minimizing sequence in $U$ such that $\underset{k\to \infty }{lim}J\left(u_k\right)=\underset{h\in U}{inf}J\left(h\right).$_.

Then ${\left(J\left(u_k\right)\right)}_{k\in \mathbb{N}}$ is bounded. Since $\parallel u_k{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}\le \frac{2}{\beta }J\left(u_k\right)$ thus, ${\left(u_k\right)}_{k\in \mathbb{N}}$ is bounded.

Thus, there exists a subsequence still denoted ${\left(u_k\right)}_{k\in \mathbb{N}}$ that weakly converges to a limit $u^{\ast }\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$.

Since $U$ is closed and convex, $u^{\ast }\in U$.

Let $z_{u_k}$, $z_{u^{\ast }}$ be the corresponding solutions of system (2) to $u_k$ and $u^{\ast }$, we have

$$z_{u_k}\left(t\right)-z_{u^{\ast }}\left(t\right)={\int }_0^{T}S\left(t-s\right)\left[u_k\left(s\right){\mathit{Bz}}_{u^{\ast }}\left(s\right)-u^{\ast }\left(s\right){\mathit{Bz}}_{u^{\ast }}\left(s\right)\right]\mathit{ds},$$E12

$$={\int }_0^{T}S\left(t-s\right)\left[\left(u_k-u^{\ast }\right)\left(s\right){\mathit{Bz}}_{u^{\ast }}\left(s\right)-u_k\left(s\right)\left({\mathit{Bz}}_{u^{\ast }}-{\mathit{Bz}}_{u_k}\right)\left(s\right)\right]\mathit{ds},$$E13

This implies,

$$\mid z_{u_k}-z_{u^{\ast }}\mid \le \mid {\int }_0^{T}S\left(t-s\right)\left(u_k-u^{\ast }\right)\left(s\right){\mathit{Bz}}_{u^{\ast }}\left(s\right)\mathit{ds}\mid e^{{\int }_0^t\parallel S\left(t-s\right)\parallel \parallel u_k\parallel \parallel B\parallel \mathit{ds}}$$E14

Using the boudness of semigroup we get

$$\mid z_{u_k}-z_{u^{\ast }}\mid \le C\mid {\int }_0^{T}S\left(t-s\right)\left(u_k-u^{\ast }\right)\left(s\right){\mathit{Bz}}_{u^{\ast }}\left(s\right)\mathit{ds}\mid .$$E15

By theorem 3.9. in [4] the weak convergence $u_k\rightharpoonup u^{\ast }$ gives $u_k{\mathit{Bz}}_{u^{\ast }}\left(.\right)\rightharpoonup u^{\ast }{\mathit{Bz}}_{u^{\ast }}\left(.\right)$ weakly in $L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right).$_.

Since ${\left(S\left(t\right)\right)}_{t\ge 0}$ is compact, we have

$$\underset{n\to \infty }{lim}\underset{0\le t\le T}{sup}\mid S\left(t-s\right)\left(u_k\left(s\right)-u^{\ast }\left(s\right)\right)\mathit{Bz}\left(s\right)\mathit{ds}\mid =0$$E16

It follows that $z_{u_k}\to z^{\ast }$ strongly in $L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right).$_.

Since for $\alpha \in ]0,1[$, $D_x^{\alpha }$ is continuous from $H_0^1\left(\mathrm{\Omega }\right)\to L^2\left(\mathrm{\Omega }\right)$, then

$$\underset{k\to \infty }{lim}{\int }_0^T\parallel D_x^{\alpha }{z}_{u_k}\left(t\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}={\int }_0^T\parallel D_x^{\alpha }{z}_{u^{\ast }}\left(t\right)-z_d{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

and as $J$ is lower, semi-continuous with respect to weak convergence, we have

$$J\left(u^{\ast }\right)\le \underset{k\to \infty }{lim}infJ\left(u_k\right),$$E17

leading to $J\left(u^{\ast }\right)=\underset{u\in U}{inf}J\left(u_k\right).$.

Remark 1.

If we consider the system (2) with a source $\mathit{\text{termf}}\in L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$

$$\frac{\partial z}{\partial t}=\mathit{Az}+u\left(t\right)\mathit{Bz}+f\mathrm{on}Q$$E18

the same well-posedness and regularity results as hold, but the constant $C_1$ in Eq. (7) takes the form as follows:

$$C_1=C\left({‖z_0‖}_{L^2\left(\mathrm{\Omega }\right)}+\parallel f{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}\right).$$

3. Characterization

We now derive necessary conditions that an optimal control must satisfy. To derive these necessary conditions, we differentiate the cost functional. The differentiation result provides a characterization of the unique optimal control in terms of the optimality system.

In the next, we consider problem (2) and we discuses special cases of the set of admissible controls $U$.

Proposition 1.4.

Let consider the adjoint system given by:

$$\left\{\begin{array}{ll}\frac{\partial p}{\partial t}\left(x,t\right)=-A^{\ast }p\left(x,t\right)+B^{\ast }\left(\mathit{up}\right)\left(x,t\right)+{\left(D_x^{\alpha }\right)}^{\ast }{z}_d\left(x\right)-{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }z\left(x,t\right)& Q\\ p\left(x,t\right)=0& \mathrm{\Gamma },\\ p\left(x,T\right)=0& \mathrm{\Omega }.\end{array}\right.$$E19

where $z_u$ solution of system (2) and ${\left(D_x^{\alpha }\right)}^{\ast }$ is the adjoint operator of $D_x^{\alpha }$.

Then the Frechet derivative of $J$ at $u\in U$ is given by:

$$J^{\prime }\left(u\right)\left(t\right)=p\left(t\right){\mathit{Bz}}_u\left(t\right)+\mathit{\epsilon u}\left(t\right).$$E20

Proof:

The system (12) has a weak solution $p\in L^2\left(O,T;L^2\left(\mathrm{\Omega }\right)\right)$ see [8], that satisfies:

$$p\left(t\right)={\int }_t^T{S}^{\ast }\left(T-s\right)\left[B^{\ast }\left(\mathit{up}\right)\left(s\right)+{\left(D_x^{\alpha }\right)}^{\ast }{z}_d-{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }{z}_u\left(s\right)\right]\mathit{ds},$$E21

where ${\left(S^{\ast }\left(t\right)\right)}_{t\ge 0}$ denotes the C${}_0$ semi-group of generator $-A^{\ast }$, and $B^{\ast }$ the adjoint operator of $B$.

Let consider the following system:

$$\left\{\begin{array}{ll}\frac{\partial y}{\partial t}\left(x,t\right)=\mathit{Ay}\left(x,t\right)+u\left(x,t\right)\mathit{By}\left(x,t\right)+h\left(x,t\right){\mathit{Bz}}_u\left(x,t\right)& Q,\\ y\left(x,t\right)=0& \mathrm{\Gamma },\\ y\left(x,0\right)=0& \mathrm{\Omega },\end{array}\right.$$E22

Let show that the mapping $\mathrm{\Psi }:u\to z$ from $U\to L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)$ is Frechet differentiable, and $y={\mathrm{\Psi }}^{\prime }\left(u\right).h$ is solution of system (15).

The operator $L:h\to y$ from $U$ to $W$ is linear.

Using remark (1) we have

$$\parallel y{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}\le C\parallel {\mathit{hBz}}_u{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}\le C_3\parallel z_u{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)},$$

It follows that $L$ is continuous.

Now to show that $\mathrm{\Psi }$ is Frechet differentiable, it suffices to prove that

$$\underset{\parallel h{\parallel }_U\to 0}{lim}\frac{\parallel \mathrm{\Psi }\left(u+h\right)-\mathrm{\Psi }\left(u\right)-L\left(h\right){\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}}{\parallel h{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}}=0.$$

Setting $z_h=\mathrm{\Theta }\left(u+h\right)$, $\psi =z_h-z_u.$ and $\mathrm{\Phi }=\psi -y$, then $\psi$ and $\mathrm{\Phi }$ are solutions of the following systems

$$\left\{\begin{array}{ll}\frac{\partial \psi }{\partial t}\left(x,t\right)=\mathit{A\psi }\left(x,t\right)+u\left(x,t\right)\mathit{B\psi }\left(x,t\right)+h\left(x,t\right){\mathit{Bz}}_h\left(x,t\right)& Q,\\ \psi =0& \mathrm{\Gamma },\\ \psi \left(x,0\right)=0& \mathrm{\Omega },\end{array}\right.$$E23

and

$$\left\{\begin{array}{ll}\frac{\mathrm{\partial \Phi }}{\partial t}\left(x,t\right)=A\mathrm{\Phi }\left(x,t\right)u\left(x,t\right)B\mathrm{\Phi }\left(x,t\right)+h\left(x,t\right)\mathit{B\psi }\left(x,t\right)& Q,\\ \mathrm{\Phi }\left(x,t\right)==0& \mathrm{\Gamma },\\ \mathrm{\Phi }\left(x,0\right)=0& \mathrm{\Omega },\end{array}\right.$$E24

It follows that

$$\parallel \psi {\parallel }_{L^2\left(0,T;H^1\left(\mathrm{\Omega }\right)\right)}\le C\parallel h{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}.$$

and

$$\parallel \mathrm{\Phi }{\parallel }_{L^2\left(0,T;H^1\left(\mathrm{\Omega }\right)\right)}\le \parallel \mathit{hB\psi }{\parallel }_{L^2\left(\mathrm{\Gamma }\right)}\le C\parallel h{\parallel }_U\parallel \psi {\parallel }_{L^2\left(0,T;H^1\left(\mathrm{\Omega }\right)\right)}.$$

Then $\parallel \mathrm{\Phi }{\parallel }_{L^2\left(0,T;H^1\left(\mathrm{\Omega }\right)\right)}\le C\parallel h{\parallel }_U^2.$

It means that

$$\parallel \mathrm{\Theta }\left(u+h\right)-\mathrm{\Theta }\left(u\right)-{\mathrm{\Theta }}^{\prime }\left(u\right).h{\parallel }_{L^2\left(0,T;H^1\left(\mathrm{\Omega }\right)\right)}\le C\parallel h{\parallel }_U^2.$$

We conclude that $\mathrm{\Theta }$ is Fréchet differentiable.

Let consider $u,u+h\in U$, then

$$\frac{1}{2}\parallel D_x^{\alpha }{z}_u-z_d{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2-\frac{1}{2}\parallel D_x^{\alpha }{z}_{u+h}-z_d{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2$$E25

$$={\int }_0^T<D_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right),D_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-2z_d{>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}$$E26

$$={\int }_0^T<z_{u+h}\left(t\right)-z_u\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-2{\left(D_x^{\alpha }\right)}^{\ast }{z}_d{>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}$$E27

$$={\int }_0^T<y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d{>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}+o\parallel h\parallel ,$$E28

and

$$\frac{\beta }{2}\left(\parallel u+h{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2-\parallel u{\parallel }_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}^2\right)=\beta <u,h{>}_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}+o\parallel h\parallel .$$

Then $J$ is Fréchet differentiable, and its derivative is given by:

$$\begin{array}{l}{J}^{\prime }\left(u\right).h={\int }_0^T<y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d{>}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}+\beta <u,\\ h{>}_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right)}+o‖h‖\end{array}$$

Using the system (15), we have

$${⟨y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}$$E29

$$={⟨{\int }_0^{T}S\left(t-s\right)\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right)\mathit{ds},{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}$$E30

Using the Gronwall lemma, we get

$$\begin{array}{l}{\int }_0^T{⟨y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}\\ =\begin{array}{l}⟨{\int }_0^T{\int }_s^{T}S\left(t-s\right)\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right)\mathit{\text{dtds,}}{\left(D_x^{\alpha }\right)}^{\ast }\\ {\times D_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}\end{array}\\ =\begin{array}{l}⟨{\int }_0^T\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right)\mathit{dt},{\int }_s^T{S}^{\ast }\left(t-s\right){\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)\\ {-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d\mathit{dt}⟩}_{L^2\left(\mathrm{\Omega }\right)}\end{array}\end{array}$$

A variational formulation of system (12) leads to:

$${\int }_s^T{S}^{\ast }\left(t-s\right)\left[{\left(D_x^{\alpha }\right)}^{\ast }{z}_d-{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }{z}_u\left(s\right)\right]\mathit{ds}=p\left(s\right)-{\int }_s^T{S}^{\ast }\left(T-s\right)B^{\ast }\left(\mathit{up}\right)\left(t\right)\mathit{dt},$$E31

It means that

$$\begin{array}{l}{\int }_0^T{⟨y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}\\ ={\int }_0^T{⟨\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right),p\left(s\right)-{\int }_s^T{S}^{\ast }\left(t-s\right)B^{\ast }\left(\mathit{up}\right)\left(t\right)\mathit{dt},⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{ds}\mathbb{N}\mathit{\text{onumber}}\end{array}$$

Using the Gronwall lemma once more gives

$$\begin{array}{l}{\int }_0^T{⟨\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right),{\int }_s^T{S}^{\ast }\left(t-s\right)B^{\ast }\left(\mathit{up}\right)\left(s\right)\mathit{dt}⟩}_{L^2\left(\mathrm{\Omega }\right)}\\ ={\int }_0^T{\int }_0^t{⟨\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right),S^{\ast }\left(t-s\right)B^{\ast }\left(\mathit{up}\right)\left(s\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{\text{dsdt}}\\ ={\int }_0^T{\int }_0^t{⟨S\left(t-s\right)\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right),B^{\ast }\left(\mathit{up}\right)\left(s\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{\text{dsdt}}\\ ={\int }_0^T{⟨{\int }_0^{t}S\left(t-s\right)\left(u\left(s\right)\mathit{By}\left(s\right)+h\left(s\right)\mathit{Bz}\left(s\right)\right)\mathit{ds},B^{\ast }\left(\mathit{up}\right)\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}\\ ={\int }_0^T{⟨y\left(t\right),B^{\ast }\left(\mathit{up}\right)\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}={\int }_0^T{⟨u\left(t\right)\mathit{By}\left(t\right),p\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}\end{array}$$

Then inequality (3) becomes

$${\int }_0^T{⟨y_h\left(t\right),{\left(D_x^{\alpha }\right)}^{\ast }{D}_x^{\alpha }\left(z_{u+h}\left(t\right)-z_u\left(t\right)\right)-{\left(D_x^{\alpha }\right)}^{\ast }{z}_d⟩}_{L^2\left(\mathrm{\Omega }\right)}={\int }_0^T{⟨h\left(t\right)\mathit{Bz}\left(t\right),p\left(t\right),⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}$$

Then the Frechet derivative of $J$ is given by:

$$J^{\prime }\left(u\right).h={\int }_0^T{⟨h\left(t\right),\mathit{Bz}\left(t\right)p\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}+\beta {⟨u\left(t\right),h\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

The following results characterize and give an expression of an optimal control solution of problem (2) in several cases of admissible controls sets.

Proposition 1.5.

An optimal control solution of problem (2) is given by

$$u^{\ast }\left(x,t\right)=max\left(m,min\left(-\frac{1}{\beta }\mathit{Bz}\left(x,t\right)p(x,t),M\right)\right)$$E32

Proof:

The Frechet differential of $J$ is given by

$$J^{\prime }\left(u\right).h={\int }_0^T{⟨h\left(t\right)\mathit{Bz}\left(t\right),p\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}+\beta {⟨u\left(t\right),h\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

Since $J$ achieves its minimum at $u^{\ast }$, we have

$$0\le {\int }_0^T{⟨h\left(t\right)\mathit{Bz}\left(t\right),p\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}+\beta {⟨u\left(t\right),h\left(t\right)⟩}_{L^2\left(\mathrm{\Omega }\right)}\mathit{dt}.$$

Taking $h=max\left(m,min\left(-\frac{1}{\beta }\mathit{Bz}\left(x,t\right)p(x,t),M\right)\right)-u^{\ast }$, we show that $h\left(u^{\ast }+\frac{1}{\beta }\mathit{Bzp}\right)$ is negative and then

$$\left(max\left(m,min\left(-\frac{1}{\beta }\mathit{Bz}\left(x,t\right)p(x,t),M\right)\right)-u^{\ast }\right)\left(u^{\ast }+\frac{1}{\beta }\mathit{Bzp}\right)=0.$$

If $M\le -\frac{1}{\beta }\mathit{Bzp}$ we have $\left(M-u^{\ast }\right)\left(u^{\ast }+\frac{1}{\beta }\mathit{Bzp}\right)=0$, thus $u^{\ast }=M$.

If $m\le -\frac{1}{\beta }\mathit{Bzp}\le M$ we have $\left(-\frac{1}{\beta }\mathit{Bzp}-u^{\ast }\right)\left(u^{\ast }+\frac{1}{\beta }\mathit{Bzp}\right)=0$.

Therefore $u^{\ast }=-\frac{1}{\beta }\mathit{Bzp}.$

Now, if $m\ge -\frac{1}{\beta }\mathit{Bzp}$, we have $\left(m-u^{\ast }\right)\left(u^{\ast }+\frac{1}{\beta }\mathit{Bzp}\right)=0$ and then $u^{\ast }=m$.

We conclude that,

$$u^{\ast }\left(x,t\right)=max\left(m,min\left(-\frac{1}{\beta }\mathit{Bz}\left(x,t\right)p(x,t),M\right)\right).$$

The next proposition shows a necessary optimality condition.

Proposition 1.6.

Let $u^{\ast }\in U$ be an optimal control, then:

$$\forall v\in U,<J^{\prime }\left(u\right),u^{\ast }-v{>}_{L^2\left(O,T;L^2\left(\mathrm{\Omega }\right)\right)}\ge 0.$$

Proof:

If $v=u$_, we get the condition.

If $v$ is different than $u$, and since $U$ is convex we have

$$u^{\ast }+\lambda \left(v-u^{\ast }\right)\in U,\text{for}\mathrm{any}\lambda \in ]0,1[$$

It follows

$$J\left(u^{\ast }\right)\le J\left(u^{\ast }\right)+\lambda \left(v-u^{\ast }\right)$$

which gives

$$J\left(u^{\ast }\right)\le J\left(u^{\ast }\right)+\lambda <J^{\prime }\left(u^{\ast }\right),v-u^{\ast }{>}_{L^2\left(O,T;L^2\left(\mathrm{\Omega }\right)\right)}+o\left(\lambda \left(v-u^{\ast }\right)\right)$$

Then,

$$<J^{\prime }\left(u^{\ast }\right),v-u^{\ast }{>}_{L^2\left(O,T;L^2\left(\mathrm{\Omega }\right)\right)}\ge \frac{1}{\lambda }\left(\lambda \left(v-u^{\ast }\right)\right).$$

Since $o\left(\lambda \left(v-u^{\ast }\right)\right)=\parallel \lambda \left(v-u^{\ast }\right)\parallel \phi \left(\lambda \left(v-u^{\ast }\right)\right)$, with $\underset{\parallel z\parallel \to 0}{lim}\phi \left(z\right)=0$. Then

$$\underset{\lambda \to 0}{lim}\frac{1}{\lambda }o\left(\lambda \left(v-u^{\ast }\right)\right)=\underset{\parallel z\parallel \to 0}{lim}\parallel \lambda \left(v-u^{\ast }\right)\parallel \phi \left(\lambda \left(v-u^{\ast }\right)\right)=\parallel v-u^{\ast }\parallel \underset{\lambda \to 0}{lim}\phi \left(\lambda \left(v-u^{\ast }\right)\right)=0.$$

we conclude that,

$$<J^{\prime }\left(u\right),u^{\ast }-v{>}_{L^2\left(O,T;L^2\left(\mathrm{\Omega }\right)\right)}\ge \underset{\lambda \to 0}{lim}\frac{1}{\lambda }o\left(\lambda \left(v-u^{\ast }\right)\right)=0.$$

Corollary 1.

Let $g\in L^2\left(\mathrm{\Omega }\right)$_, such that $\mid g\mid \mathbb{N}\mathit{eq}0$ and assuming that $U=L^2\left(0,T\right).$_.

Then an optimal control is given by

$$u^{\ast }\left(x,t\right)=v^{\ast }\left(t\right)g\left(x\right)$$E33

with $v^{\ast }\left(t\right)=-\frac{1}{\beta \parallel g{\parallel }_{L^2\left(\mathrm{\Omega }\right)}}{\int }_{\mathrm{\Omega }}\mathit{Bz}\left(x,t\right)p\left(x,t\right)$

Particularly, if $g\left(x\right)={\mathcal{1}}_D\left(x\right)$, with $D\subset \mathrm{\Omega }$ is the actuator location and ${\mathcal{1}}_D$ is the characteristic function such that its measure $\mu \left(D\right)$ is non-zero, then an optimal control $v^{\ast }\left(t\right)$ is given by

$$v^{\ast }\left(t\right)=max\left(m,min\left(-\frac{1}{\mathit{\beta \mu }\left(D\right)}{\int }_{\mathrm{\Omega }}\mathit{Bz}\left(x,t\right)p(x,t)\mathit{dx},M\right)\right).$$E34

Proof:

Let $v\in L^2\left(0,T\right)$, such that $w\left(x,t\right)=v\left(t\right)g\left(x\right)$ it follows from (1.6) that ${⟨J^{\prime }\left(u^{\ast }\right),w⟩}_{L^2\left(0,T;L^2\left(\mathrm{\Omega }\right)\right))}=0$ which gives

$${\int }_0^{T}v\left(t\right){\int }_{\mathrm{\Omega }}g\left(x\right)J^{\prime }\left(u^{\ast }\right)\left(x,t\right)\mathit{\text{dxdt}}=0\forall v\in L^2\left(0,T\right)$$

Hence

$${\int }_{\mathrm{\Omega }}g\left(x\right)J^{\prime }\left(u^{\ast }\right)\left(x,t\right)\mathit{dx}=0\forall t\in ]0,T[$$

Then ${⟨J^{\prime }\left(u^{\ast }\right)\left(t\right),g⟩}_{L^2\left(\mathrm{\Omega }\right)}=0$, it means

$${⟨\mathit{Bz}\left(t\right)p\left(t\right),g⟩}_{L^2\left(\mathrm{\Omega }\right)}+\beta v^{\ast }\left(t\right){⟨g,g⟩}_{L^2\left(\mathrm{\Omega }\right)},\forall t\in ]0,T[$$

which leads to formula (25).

4. Algorithm and simulations

In this section, we give an example to illustrate the usefulness of our main results.

The optimality condition (25) shows that the optimal control $u^{\ast }$ is a function of $z$ and $p$ which themselves are functions of $u^{\ast }.$ Then the control cannot be directly computed. For this reason, we introduce the following algorithm.

• Step 1: Choose an initial control $u_0\in U$ a threshold accuracy $\epsilon >0,$ and initialize with $k=0$;

• Step 2: Compute $z_k$, solution of (2) and $p_k,$ solution of (12) relatively to $v_k$.

• Step 3: Compute

  $$v_{k+1}=max\left(m,min\left(-\frac{1}{\mathit{\beta \mu }\left(D\right)}{\int }_{\mathrm{\Omega }}{\mathit{Bz}}_k\left(x,t\right)p_k(x,t)\mathit{dx},M\right)\right).$$E35

• Step 4: If $‖u_{k+1}-u_k‖>\epsilon ,k=k+1,$ go to step $2.$ Otherwise $u^{\ast }=u_k$

For simulations, we consider a bilinear system described by the equation:

$$\left\{\begin{array}{ll}\frac{\partial z}{\partial t}\left(x,t\right)=\mathrm{\Delta }z\left(x,t\right)+u\left(x,t\right)z\left(x,t\right)& \mathrm{\Omega }\times ]0,1[\\ z\left(0,t\right)=z\left(1,t\right)=0& ]0,1[\\ z\left(x,0\right)=x\left(x-1\right)& \mathrm{\Omega };\end{array}\right.$$E36

We consider problem (2) with $\alpha =0.2$ and $z_d\left(x\right)=0.62x^3+1.7x^2+0.023$.

Applying the above algorithm, we obtain the following figures (Figures 1 and 2):

Figure 1.

Final state.

Figure 2.

The evolution of an optimal control.

The desired state is obtained with error $\parallel D_x^{\alpha }z\left(x,T\right)-z_d\left(x\right){\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2={\mathrm{5.27.10}}^{-4}$ and a cost $J\left(u^{\ast }\right)={\mathrm{2.31.10}}^{-3}$.

5. Conclusion

In this work, we discuss the question of fractional optimal control problem of parabolic bilinear systems with bounded controls, we obtain a distributed control solution, that minimizes a quadratic functional. This work gives an opening to other questions; this is the case of the fractional optimal control problem of hyperbolic systems. This will be the purpose of a future research paper.