El Hassan ZERRIK,Nihale EL BOUKHARI

MACS team,Faculty of Sciences,Moulay Ismail University,Meknes,Morocco

Abstract The aim of this paper is to investigate a regional constrained optimal control problem for a class of semilinear distributed systems,which are linear in the control but nonlinear in the state.For a quadratic cost functional and a closed convex set of admissible controls,the existence of an optimal control is proven,and then this is characterized for three cases of constraints.A useful algorithm is developed,and the approach is illustrated through simulations for a heat equation.

Keywords:Semilinear distributed systems,regional optimal control,constraints,heat equation

1 Introduction and problem statement

Many physical problems in fields such as engineering,economics,and life sciences may be modeled by distributed systems evolving on a spatial domain Ω.These are often linear with nonlinear perturbations,and thus take the form of semilinear distributed systems.Such systems are usually subject to constraints on the control,owing to the nature of the system or limited resources.For instance,if the system models the commercial activity of a firm,and the control u models the amount of merchandise bought,then the system might be subject to the constraint m(t)≤ u(t)≤ M(t),where the bounds m(t)and M(t)are often variable in time,depending on supply and demand,which vary following high and low seasons.In addition,if v(t)models the unitary price of that merchandise at time t,then the system might be subject to the constraintwhere M is the trade budget for the period[t1,t2].To describe another scenario,if the control u(t)models the electric current in a heating system,then the consumed power is proportional to u2(t),and hence the constraintcould represent a bound on the consumed energy over[t1,t2].

Previous works dealing with semilinear optimal control problems primarily characterized the optimal control,which minimizes a cost functional,using the Hamilton-Jacobi equation or the generalized Pontrya-gin minimum principle.For instance,control problems with box constraints were investigated in[1]for a class of elliptic and parabolic semilinear equations,using the generalized maximum principle,while in[2],the Mayer problem of semilinear systems was studied using a value function satisfying the Hamilton-Jacobi equation.In[3],the existence of an optimal control was proven for semilinear systems with a compact semigroup,and for second order partial differential equations.Then,an infinite dimensional version of the maximum principle was established.In addition,[4]considered an optimal control problem governed by semilinear parabolic equations with distributed and boundary controls,using Pontryagin’s principle.Later,the authors of[5]proved the existence of an optimal control for a class of semilinear elliptic and parabolic equations,derived first-and second order optimality conditions,and showed that the optimal controls satisfy Pontryagin’s maximum principle.A control problem of semilinear systems was studied using the theory of set-valued mapping in[6].

For a distributed parameter system evolving within a domain Ω,the concept of regional analysis involves approaching a desired state or optimizing the system performance only in a region ω ? Ω.In addition to generalizing the global approach,the regional one is relevant for many real-world applications in thermodynamics,fluid mechanics,and demography,where only a target zone needs to be controlled.This is because it is cheaper with respect to energy,costs,computations,and other such factors,to control the region ω than to control the whole domain Ω.The regional approach was first introduced for linear systems in[7].Then,the authors in[8]studied regional boundary controllability for linear hyperbolic systems,while[9]considered a linear parabolic system and characterized the regional optimal control using the Lagrangian approach.Subsequently,this approach was extended to bilinear systems.In[10],the authors considered a minimum energy regional control problem for bilinear systems,and characterized the optimal control by deriving a corresponding Riccati equation.A similar problem with bounded controls was investigated in[11].

For semilinear systems,[12,13]studied regional controllability for a class of parabolic and hyperbolic semilinear systems,where the nonlinear perturbation does not depend on the control.

This paper extends the regional approach to a wider class of semilinear systems,for which the control term is nonlinear.Optimality conditions are derived for a closed convex set of admissible controls,with an emphasis on three cases of constraints,for which the optimal control is characterized using geometric tools.

More precisely,on a bounded domain Ω?Rnwith a regular boundary?Ω,we consider the following system:

where A is the infinitesimal generator of a strongly continuous semigroup of linear operators(S(t))t≥0on the state space L2(Ω),endowed with its natural inner product〈·,·〉and the associated norm|·|.

The control function is u∈Uad,where Uadis the set of admissible controls,which is assumed to be a nonempty closed and convex subset of L2(0,T).

The inner product and associated norm of L2(0,T)are respectively denoted by 〈·,·〉and ‖·‖.

B ∶L2(Ω)→ L2(Ω)is a nonlinear operator,such that the following hold:

?B is k-Lipschitz,i.e.,

?B is everywhere Fréchet-differentiable.For y∈L2(Ω),B′(y)denotes the Fréchet derivative of B at y.We assume that there exists a constantk≥0 such that the mapping y→B′(y)isk-Lipschitz,i.e.,

In what follows,A＊and[B′(y)]＊denote the respective adjoint operators of A and B′(y).

For a given u∈L2(0,T)and y0∈L2(Ω),we say that y is a mild solution(or solution in the sense of semigroups)to system(1)if y ∈ C([0,T]；L2(Ω))and y is solution to the following integral equation:

The existence and uniqueness of a mild solution to system(1)is proven in[14].Indeed,because B is k-Lipschitz,it follows that

Hence,according to Theorem 2.5.in[14],equation(2)has a unique solution in C([0,T],L2(Ω)).

Let ω?Ω be a region of positive Lebesgue measure,and let χω∶L2(Ω)→ L2(ω)be the restriction operator given by

We consider the following quadratic cost functional:

The regional optimal control problem consists of finding a control that steers the state close to ydwithin the subregion ω,with a reasonable amount of energy.In practice,this may be stated as a minimization of the cost functional(4).Thus,this study is concerned with solving the following problem:

The remainder of this paper is organized as follows.In Section 2,the existence of an optimal control is proven,and necessary optimality conditions are formulated,which lead to the characterization of the optimal control for three cases of constraints.Next,a sufficient condition for the uniqueness of the optimal control is formulated,and a useful algorithm is developed.In Section 3,the theoretical results are illustrated through simulations for a heat equation.

2 Optimal control problem

We first prove the existence of an optimal control solving problem(5).

Proposition 1There exists a solution u＊∈ Uadto problem(5).

To prove Proposition 1,we need the following lemma.

Lemma 1[15,Page 250] Let Λ be a continuous linear operator,mapping a normed space X into a reflexive separable Banach space Y.A necessary and sufficient condition forΛto be compact is that the adjoint operator Λ＊maps every sequence(zn)that is weakly convergent to zero in Y＊to a sequence(Λ＊zn)that converges to zero for the norm in X＊.

Proof of Proposition 1The set{J(u)|u∈Uad}is nonempty and nonnegative,and thus it has a nonnegative infimum.Let(un)n∈Nbe a minimizing sequence in Uad.Becauseit follows that(un)n∈Nis bounded.Then,there exists a subsequence,still denoted by(un)n∈N,that converges weakly to a limit u＊.Because Uadis closed and convex,it is closed for the weak topology,which implies that u＊∈ Uad.

Let ynand y＊be the unique solutions of system(1)associated to unand u＊,respectively.

Here,ynand y＊can be written in the form(2),which gives

so that

Using Gronwall’s lemma,we obtain that

There exist constants M ≥ 1 and ρ∈R such that‖S(t)‖≤ M eρt.Then,the above inequality yields

Let us prove that Λtis compact for any t∈ [0,T].

To this end,let(zm)be a sequence in L2(Ω)such that zm?0 weakly.Without loss of generality,we can assume that|zm|≤ 1,?m ∈ N.Then,

Because zm?0 weakly,it follow that0 a.e.on[0,T].Furthermore,for every s∈[0,T]we have that

Then,by applying the dominated convergence theorem we obtain

Hence,by virtue of Lemma 1,the operator Λtis compact.

It follows from the weak convergence(un? u＊)? 0 that

Therefore,by the inequality(6)we obtain

Hence,the continuity of the operator χωyields that

and by Fatou’s lemma we obtain

Because norms are lower semi-continuous for the weak topology,it follows that the weak convergence of(un)nyields that

Now,we give the necessary optimality conditions by characterizing the derivative of the cost functional(4).

Proposition 2Let u＊be an optimal control solving problem(5).Then,u＊satisfies

where J′(u＊)is the Fréchet derivative of J at u＊,which is given by

Here,y is the solution of system(1)associated to u＊,and p is the solution of the following adjoint equation:

ProofAccording to[14],the mapping u■→yu,where yuis the mild solution of(1)associated to u,is Fréchet differentiable.Let Duy denote the derivative of u ■→ yuat u ∈ Uadand zh=Duy ·h,for a given h∈L2(0,T).Then,zhsatisfies the following integral equation:

where A＊generates the semigroup S＊(t)=(S(t))＊,?t ∈R+.Furthermore,because B is k-Lipschitz,it follows that‖（B′(y(t))）＊‖≤ k,?t∈]0,T[.

Then,for any u∈Uad,it follows using a similar proof to that of Theorem 2.5.in[14]that equation(9)has a unique mild solution in C([0,T],L2(Ω)),which is the solution of the associated integral equation,given by

Let u,u+h∈Uad.Then,

Using calculations similar to those given above,we obtain

and it is easy to see that

Then,J is Fréchet-differentiable over Uad,and its derivative at u is given by

From the expression(11),we have that

Applying Fubini’s theorem yields

By equation(11),we have that

Then,

It is easy to check that

and applying Fubini’s theorem leads to

It follows that

Therefore,the derivative of J can be written as

By identifying J′(u)with its representative in L2(0,T),we obtain(8).

Now,let u＊be an optimal control,and w ∈Uad.The convexity of Uadimplies that

Then,J(u＊)≤J(u＊+λ(w?u＊)).Because J is Fréchet differentiable,we then obtain that

Therefore,

Hence,we obtain(7). □

By the above optimality condition,the optimal control can be characterized for special cases of constraints,as shown by the following propositions.

Proposition 3Let

where m,M∈L2(0,T)satisfy m(t)＜M(t)a.e.on[0,T].Then,an optimal control is given by

Proof?If m(t)＜ u＊(t)＜ M(t)over a set I?]0,T[of positive Lebesgue measure:

For h ∈ L∞(0,T)that is sufficiently small and null outside I it holds that u＊+h,u＊?h ∈ Uad.

It follows from applying the condition(7)to both u＊+h and u＊? h that〈J′(u＊),h〉=0.

Hence,from the density of L∞(I)in L2(I),J′(u＊)=0 over I,implying that u＊(t)=

Because m(t)＜ u＊(t)＜ M(t),the above equality is equivalent to(13)over I.

?If u＊(t)=m(t)over a set I ?]0,T[of positive Lebesgue measure:

Let h∈L∞(0,T)be null outside I such that h≥0 on I.

If‖h‖L∞(0,T)is sufficiently small,then u＊+h ∈ Uad.Hence, 〈J′(u＊),h〉≥ 0,implying that J′(u＊) ≥ 0 on I.

Therefore,

which is equivalent to(13)on I.

?The case that u＊(t)=M(t)is similar to the above case. □

In many real applications,such as heating systems or population dynamics,the constraints representing the available energy or maximum cost may not be possible to model by(12),and may instead need to be written as integral inequalities or bounds on norms.The following propositions provide characterizations of the optimal control for such constraints.

Proposition 4Assume that

where M＞0.Then,the optimal control is given by

ProofIf u＊∈ int(Uad),then applying condition(7)to a neighborhood of u＊within Uadimplies that J′(u＊)=0,which yields(15)with ‖J′(u＊)‖=0.

If u＊∈ ?(Uad),then the case J′(u＊)=0 is similar to the above case.We assume that J′(u＊)≠ 0.

Then,the optimality condition(7)can be written as

By the strict convexity of the space L2(0,T),we obtain

Replacing J′(u＊)by its expression leads to

which yields(15). □

Proposition 5Let v∈L2(0,T){0},m＜M,and

Then,the optimal control is given by

where

Proof?If m ＜ 〈v,u＊〉＜ M,then u＊∈ int(Uad),and hence J′(u＊)=0.Expression(8)implies that(17)with δ=0.

?If〈v,u＊〉=M:Let w ∈ (R v)⊥(i.e.,the orthogonal space of R v).Then,u＊?w,u＊+w ∈ Uad,and hence the condition(7)yields that〈J′(u＊),w〉=0.

Therefore,J′(u＊)∈ R v.

which implies(17)with δ=1.

?Similarly to the previous case,if〈v,u＊〉=m,then(16)holds with δ = ?1. □

The next proposition provides a sufficient condition for the uniqueness of the optimal control,solving problem(5).

Proposition 6We assume that Uadis bounded.

There exists a constant η≥0,depending on the parameters of system(1),such that if

holds,then the optimal control,solving problem(5),is unique.

ProofLet u and v be two optimal controls.Then,by the condition(7),we have that

which yields

We know that

Then,the above equality yields

where

Then,by the inequality(19),there exists a constant η＞ 0 that does not depend on the parameters α,β,and ε such that

Remark 1The above proof also shows that if(18)holds,then there exists a unique control that satisfies the necessary optimality condition(7).Hence,(7)also becomes a sufficient condition of optimality.

To search for a control that satisfies the optimality condition(7),we introduce the following algorithm,based on the steepest descent method.

Algorithm

Step 1Choose an initial control u0∈Uad,a threshold accuracy κ,a region ω,and a step length λ.Initialize with n=0.

Step 2Compute ynsolving(1)and pnsolving(9)using the finite difference method.Compute J′(un)by(8).

Step 3Compute un+1by

where PUaddenotes the projection operator on the closed convex set Uadin L2(0,T).

Step 4n=n+1,go to Step 2 and repeat until‖un+1?un‖≤ κ.

The above algorithm converges if the step length λ is appropriately chosen.We refer to chapter XV of[15]for further details on the choice of λ.

Remark 2Let w∈L2(0,T).If Uadis given by(12),then PUad(w)is written as

If Uadis given by(14),then PUad(w)is written as

If Uadis given by(16),then PUad(w)is written as

where δ=max[m,min[M,〈v,w〉]].

3 Simulations

On a bounded domainΩwith a regular boundary?Ω,we consider the following heat equation with Neumann boundary conditions:

where c ＞ 0.Denote y(t)=z(·,t)and A=cΔ,with

Then,equation(20)takes the form of system(1),and A generates a strongly continuous semigroup(S(t))t≥0.

The operator B ∶L2(Ω)→ L2(Ω)is nonlinear,and k-Lipschitz and everywhere Fréchet-differentiable.

The purpose of this application is to steer the state y as close as possible to yd=0 at time T,within a target region ω ? Ω.Then,we consider the optimal control problem(5)with the following cost functional:

over the following set of admissible controls

Two cases are considered below:a one-dimensional case,where Ω=]0,1[,and a two-dimensional case with Ω =]0,1[×]0,1[.

3.1 One-dimensional case

We first consider equation(20)on Ω=]0,1[.The semigroup(S(t))t≥0is written as

where φ0=1 and φn(x)Furthermore,B ∶L2(Ω)→ L2(Ω)is 1-Lipschitz,and is given by

where D?Ω is the actuator zone,and has a positive Lebesgue measure.

For y ∈ L2(Ω),B′(y)is self-adjoint,and is given by

Simulations are performed using the above algorithm with the following parameters:

?For ω =]0,0.4[:

The evolution of the optimal control is illustrated in Fig.1,while the associated states at times t=0 and t=T=1 are presented in Fig.2.The final state is significantly close to yd=0 within the region ω,where the error is ‖χωy(T)‖L2(ω)=8.16 × 10?4.

Fig.1 Optimal control for ω=]0,0.4[.

Fig.2 State for ω=]0,0.4[.

?For ω = Ω:

The optimal control and the associated states at times t=0 and t=T=1 are presented in Figs.3 and 4,respectively.The final state is close to yd=0,but the regional case exhibits a better performance.The error is‖χωy(T)‖L2(ω)=5.89 × 10?3.

Fig.3 Optimal control for ω=]0,1[.

Fig.4 State for ω=]0,1[.

The following table shows the evolution of the error ‖χωy(T)‖L2(ω)and the cost J(u＊)with respect to the region ω:

ω]0,0.4[ ]0,0.6[ ]0,0.8[ ]0,1[‖χωy(T)‖× 10?3 0.81 2.18 4.78 5.89 J(u＊)× 10?3 0.18 0.38 1.29 1.90

3.2 Two-dimensional case

Now,we consider equation(20)on Ω =]0,1[×]0,1[.The semigroup(S(t))t≥0is written as

where φn,m(x)=en(x1)em(x2),such that

where D?Ω is the actuator zone,and has positive Lebesgue measure.

For y ∈ L2(Ω),B′(y)is self-adjoint,and is given by

Simulations are performed using the above algorithm,with the following parameters:c=0.2,D=]0.6,1[×]0.6,1[,T=1,M=2,y0(x)=0.05+x1x2(1 ?x1)(1?x2).

?For ω =]0,0.4[×]0,0.4[:

The evolution of the optimal control is illustrated in Fig.5,while the final state y(T)is depicted in Fig.6.The final state is close to yd=0 within the region ω,with an error equal to ‖χωy(T)‖L2(ω)=4.13 × 10?4.

Fig.5 Optimal control for ω =]0,0.4[×]0,0.4[.

Fig.6 State for ω =]0,0.4[×]0,0.4[.

?For ω = Ω:

The optimal control and final state are illustrated in Figs.7 and 8,respectively.Similarly to the one dimensional case,the final state is close to yd=0,where the error is ‖χωy(T)‖L2(ω)=4.10 × 10?3,but the regional case provides a better precision.

The table below shows the evolution of the error‖χωy(T)‖L2(ω)and the cost J(u＊)with respect to the region ω:

‖χωy(T)‖× 10?3 0.41 0.99 2.10 4.10 J(u＊)× 10?4 0.54 0.80 1.25 2.64

As shown by the above simulations,the regional approach provides an improvement in the performance of the system in the target region ω,even when the actuator zone D and ω are disjoint.The smaller the area of the region ω is,the smaller the error‖χωy(T)? yd‖L2(ω)and the cost J(u＊)are,which means that it is cheaper to steer the system to the desired state on a region than on the whole domain.This is crucial to many real-world applications where only a region within the spatial domain has to be controlled,particularly by a distant actuator.

Fig.7 Optimal control for ω =]0,1[×]0,1[.

Fig.8 State for ω =]0,1[×]0,1[.

4 Conclusions

In this paper,we studied a regional quadratic control problem for a class of semilinear distributed systems.We formulated optimality conditions in the generalcase.Next,we derived characterizations of the optimal control for special cases of constraints.Simulations illustrated the obtained results,and demonstrated the relevance of the regional approach.Many issues remain unresolved,such as regional optimal control problems with bound-ary subregions.These issues will be the focus of a future research paper.