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Gradient Optimal Control of the Bilinear Reaction–Diffusion Equation

By El Hassan Zerrik and Abderrahman Ait Aadi

Submitted: September 9th 2020Reviewed: January 15th 2021Published: May 4th 2021

DOI: 10.5772/intechopen.96041

Downloaded: 78

Abstract

In this chapter, we study a problem of gradient optimal control for a bilinear reaction–diffusion equation evolving in a spatial domain Ω⊂Rn using distributed and bounded controls. Then, we minimize a functional constituted of the deviation between the desired gradient and the reached one and the energy term. We prove the existence of an optimal control solution of the minimization problem. Then this control is characterized as solution to an optimality system. Moreover, we discuss two special cases of controls: the ones are time dependent, and the others are space dependent. A numerical approach is given and successfully illustrated by simulations.

Keywords

  • distributed bilinear systems
  • reaction–diffusion equation
  • controllability
  • optimal control

1. Introduction

The controllability of distributed bilinear systems governed by partial differential equations has been studied by many authors: in [1], the authors developed the weak controllability of the beam and rod equations in the mono-dimensional case. In [2], the author considered the controllability of semilinear parabolic and hyperbolic systemse using distributed controls. In [3], the author studied the exact controllability of the semilinear wave equations in one space dimension. The optimal control problem for a class of infinite dimensional bilinear systems have been consedered in many works. In [4], the author proved the existence and characterization of an optimal control of a bilinear convective-diffusive fluid model using bounded controls. In [5], the author developed optimal control problem of a bilinear heat equation with distributed bounded control. In [6], the authors studied optimal control for a class of bilinear systems using unbounded control. In [7], the authors considered the optimal control problem of the wave equation using bounded boundary control. In [8], the authors considered the optimal control problem of the Kirchhoff plate equation with distributed bounded controls. In [9], the author proved the optimal control of the bilinear wave equation using distributed and bounded controls. The regional optimal control problem of a class of infinite dimensional bilinear systems with unbounded controls was developed in [10], then the authors studied the existence and characterization of an optimal control. In [11], the authors studied the constrained regional optimal control of a bilinear plate equation using distributed and bounded controls. The notion of gradient controllability is very important, since its close to real applications and there exist systems that cannot be controllable but gradient of the state is controllable. Then in [12], the authors proved the regional controllability of parabolic systems using HUM method.

In the present work, we study the gradient optimal control problem of the bilinear reaction–diffusion equation using distributed and bounded controls. Then, we examine the existence and we give characterization of an optimal control. Also, an algorithm and simulations are given. Let Ωbe an open bounded domain of Rn,n1with a C2boundary ∂Ω, we denote by Q=Ω×0Tand Σ=∂Ω×0T, and we consider the bilinear reaction–diffusion equation

ytxtΔyxt=uxtyxtinQyx0=y0xinΩyxt=0onΣ,E1

where uUρuLQρuρa.e.inQis a scalar control function, and ρis a positive constant.

Let us consider the following state space

HL20TH01Ω.

For all y0H01Ωand uUρ, the system (1) has a unique weak solution yH(see for example [13, 14]).

Define the operator

:H01ΩL2Ωn
yy=yx1yxn,

and its adjoint.

Let us recall that the system (1) is weakly gradient controllable if for all ydL2Ωnand ε>0, there exist a control uUρsuch that

y.Tyd.L2Ωnε,

where yd=y1dyndis the gradient of the desired state in L2Ωn.

Our problem consists in finding a control uthat steers the gradient of state close to yd, over the time interval 0Twith a reasonable amount of energy. This may be stated as the following minimization problem

minuUρJu,E2

where

Ju=120Ty.tyd.L2Ωn2dt+β2Qu2xtdQ,E3

with β>0.

The rest of the paper is organized as follows: in section 2, we study the existence of an optimal control solution of (2). In section 3, we give a characterization of an optimal control solution of the problem (2), and we discuss two special cases of an optimal control solution of such problem. Finally, in section 4, we present an algorithm and simulations.

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2. Existence of an optimal control

The main result of the existence of an optimal control solution of (2) is given by the following theorem.

Theorem 1. There exists an optimal control uUρ, solution of (2).

Proof:Let unbe a minimizing sequence in Uρ, such that

liminfn+Jun=infuUρJu.E4

Then, according to the nature of the cost function J, we can deduce that unis uniformly bounded in Uρ.

So, we can extract from una subsequence also denoted by unsuch that unuweakly in Uρ.

In other hand, using the weak form of system (1), we deduce that

12ddtynL2Ω2+Ωynyndx=Ωunyn2dx.E5

By integration with respect to time and using the function unis uniformly bounded in LQ, we have

ynL2Ω2+0tynH01Ωdsc10tynL2Ω2ds,E6

where c1is a positive constant.

Using Gronwall’s Lemma, we deduce that ynuniformly bounded in L0TL2Ω, and then ynuniformly bounded in L20TH01Ω.

Using the previous result and system (1), we obtain that ytnis uniformly bounded in L20TH1Ω, and then ynis uniformly bounded in H.

Using the above bounds, we can extract a subsequence satisfying the following convergence properties

ynyweaklyinL20TH01ΩE7
ynystronglyinL2QE8
unuweaklyinL2Q.E9

Since Uρis a closed and convex subset of LQL2Q, Uρis weakly closed in L2Q. Then uUρL2Q. On the other hand, since ρunρfor all n, unuweaklyin LQ, and hence unuweakly in L2Q. By the uniqueness of the weak limit, we obtain u=uand uUρLQ.

Now, we show that unynuyweakly in L2Q.

Since unynuy=unyny+unuy, and using (7), (8) and (9), we obtain unynuyweakly in L2Q.

Thus y=yuis the solution of system (1) with control u.

Since the functional Jis lower semi-continuous with respect to weak convergence (basically Fatou’s lemma), we obtain

Ju12liminfn+0Tyn.tyd.L2Ωn2dt+β2liminfn+Qun2xtdxdtliminfn+Jun=infuUρJu.

Finally, we conclude that uis an optimal control.

3. Characterization of an optimal control

This section is devoted to characterization of an optimal control solution of the problem (2).

3.1 Time and space control dependent

In this part, we give characterization of an optimal control that depend on time and space.

The following result give the differentiability of the mapping uyu.

Lemma 1The mapping uUρyuHis differentiable in the following sense

yu+εhyuεϕweakly  inHasε0,foranyu,u+εhUρ

Moreover, ϕ=ϕyhsatisfies the following system

ϕtxtΔϕxt=uxtϕxt+hxtyxtonQϕx0=0inΩϕxt=0inΣ.E10

Proof:Consider yε=yu+εhand y=yu. Then yεyεis a weak solution of

yεyεtΔyεyε=uyεyε+hyεonQyεyεx0=0inΩyεyεxt=0inΣ.

Using the result (6), it follows that

yεyεHC,

where Cdepends on the Lbound on h, but is independent of ε. Hence on a subsequence, by weak compactness, we have

yεyεϕweaklyinL0TH01Ω
yεyεtϕtweaklyinL0TH1Ω.

By the definition of weak solution, we have

yεyεtψΩyεyεψdx=Ωuyεyεψdx+Ωhyεψdx,E11

for any ψH01Ω, and a.e 0tT.

Letting ε0in (11), we conclude that ϕis the weak solution of system (10).

Now, we give characterization of an optimal control that depend on time and space.

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

uxt=maxρmin1βi=1nyxtxipixtρ,E12

where pC0THis the weak solution of the adjoint system

pitxtΔpixt=uxtpixtonQpixT=yTxiyidinΩpixt=0inΣ.E13

Proof:Let uUρand y=yube the corresponding weak solution, and let u+εhUρ, for ε>0and yε=yu+εh.

Since Jreaches its minimum at u, then

0limε0+Ju+εhJuε=limε0+i=1n12Ω0Tϕxipitdt+0TΔϕxi+uϕxi+hyxipidtdx+limε0+β2Q2hu+εh2dQ.

Then

0QβhudQ+i=1nQhyxipidQ=Qhβu+i=1nQhyxipidQ.

Using a standard control argument based on the choices for the variation hxt, an optimal control is given by

uxt=maxρmin1βi=1nyxtxipixtρ.

3.2 Time or space control dependent

In this subsection, we study two cases of controls: the first ones are time dependent ut, and the others are space dependent ux.

  • Case 1: u=ut.

Here, we consider the admissible controls set

Uρ=uL0T:ρuρa.ein0TE14

and we take the functional cost

Ju=120Ty.tyd.L2Ωn2dt+β20Tu2tdt.E15

Corollary 1Under conditions (14) and (15), an optimal control is given by the formula

ut=maxρmin1βΩi=1nyxtxipixtdxρ,E16

where yis the weak solution of the equation

ytxtΔyxt=utyxtonQyx0=y0x,inΩyxt=0inΣ,

and piis the weak solution of the adjoint equation

pitxtΔpixt=utpixtonQpixT=yTxiyidinΩpixt=0inΣ.

Proof:Using the same steps as in the proof of Theorem 2, let h=htbe an arbitrary function with u+εhUρfor small ε.

We have

0ThtΩi=1nyxtxipixtdx+βutdt0.

By using a standard control argument concerning the sign of the variation h, we obtain

ut=maxρmin1βΩi=1nyxtxipixtdxρ.

  • Case 2:u=ux.

We consider the admissible controls set

Uρ=uLΩ:ρuρa.einΩE17

and we take the functional cost

Ju=120Ty.tyd.L2Ωn2dt+β2Ωu2xdx.E18

Corollary 2Under conditions (17) and (18), an optimal control satisfies

ux=maxρmin1β0Ti=1nyxtxipixtdtρ,E19

where yis the solution of system

ytxtΔyxt=uxyxtonQyx0=y0x,inΩyxt=0inΣ,

and piis the solution of system

pitxtΔpixt=uxtpixtonQpixT=yTxiyidinΩpixt=0inΣ.

Proof:Using the same notations as in the proof of Theorem 2, let h=hxbe an arbitrary function with u+εhUρfor small ε.

We have

Ωhx0Ti=1nyxtxipixtdt+βuxdx0.

A standard control argument gives

ux=maxρmin1β0Ti=1nyxtxipixtdtρ.

4. Algorithm and simulations

We have the solution of the problem (2) is given by the formula

uxt=maxρmin1βi=1nyxtxipixtρ,

where yis the weak solution of the Eq. (1) and piis the weak solution of the adjoint Eq. (13).

The computation of an optimal control solution the problem (2) can be realized by

un+1xt=maxρmin1βi=1nynxtxipinxtρ,u0=0,E20

where ynis the solution of the Eq. (1) associated to unand pnis the solution of the adjoint Eq. (13). Then, we consider the following algorithm

Step1:InitializationInitial statey0,u0andyd.Threshold accuracyεand the final timeT.Step2:Solving the system1givesyn.Solving the system13givespn.Calculateun+1bythe formula20..Untilun+1unLQεstop,elsen=n+1goto step2.Step3:The controlunis optimal.

4.1 Simulations

On Ω=]0,1[, we consider the following equation

ytxtΔyxt=utyxtonQyx0=x1x1+x,inΩyxt=0inΣ,E21

and consider problem (2) with the control set

Uρ=uL0T:ρuρa.ein0T.

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

ut=maxρmin1β01i=1nyxtxipixtdxρ,

where yis solution of the Eq. (21) associated to the control uand pis the solution of the following adjoint system

pitxtΔpixt=utpixtonQpixT=yTxiyidinΩpixt=0inΣ.

We take T=1, ρ=1, β=0.1, y0x=x1x1+x, and ydx=0. Applying the previous algorithm, with ε=104we obtain.

Figure 1 shows that the gradient state is very close to the desired one with error yT=5.33×105and the evolution of control is given by Figure 2.

Figure 1.

The gradient of the state on]0,1[.

Figure 2.

Evolution of the control function.

5. Conclusion

Gradient optimal control problem of the bilinear diffusion equation with distributed and bounded controls is considered. The existence and characterized of an optimal control are proved. The obtained results are tested by numerical examples. Questions are still open, as is the case of gradient optimal control problem of the semilinear reaction–diffusion equation.

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El Hassan Zerrik and Abderrahman Ait Aadi (May 4th 2021). Gradient Optimal Control of the Bilinear Reaction–Diffusion Equation [Online First], IntechOpen, DOI: 10.5772/intechopen.96041. Available from:

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