Research article
Issue: № 6 (144), 2024


This article is concerned with an optimal control problem governed by a Laplace equation. Initially, the optimal control problem, governed by a system of partial differential elliptic equations of the second order, is considered. The case of a system, that is singular according to Lions, is considered. In this system a given control may give rise to either no of any state or, on the contrary, the infinitely many ones or that to a single but unstable state

. In this situation, the application of the classic optimal control theory is either very difficult or impossible. Special methods applicable to the control problems, governed by singular distributed systems, are developed in the works of Zh. L. Lions, I. Ekland, P. Marselini, G. Mossino, P. Rivera, and of many other authors. But it should be noted that in most of these works the simplest problem statement is discussed. It is defined by the fact that the set of admissible processes, i.e., the processes, among which we seek the minimum of certain functional, is described by a differential equation and the connected with it, boundary conditions only. In the present work a more general and complex case is considered, namely, the case that in the description of the above-mentioned set there are so-called state constraints. This implies that the phase vector of a system does not leave the given set. In such a statement the optimal control problem, governed by a distributed singular system, is, undoubtedly, of substantial interest. Next it will be shown that the optimal process in this problem is generated by a nonlinear optimal controller and its equation will be obtained.

1. Introduction

Recent decades have witnessed a sustained interest to the maximum principle of Pontryagin’s type for optimal control problems governed by partial differential equations; see e.g.,

and the literature therein. The introducing of state constraints into the problem formulation has earned the additional difficulties and increasing complexity, whereas such constraints are relevant to many applications
. The optimal control theory of PDE’s offers an extremely rich variety of problems. Among them, there are optimal control problems with state constraints for plants described by Laplace equation. In this article, the optimal control problem governed by a Laplace equation is considered. This control system is the singular, according to Lions

2. The optimal control problem governed by a system of elliptic equations. The case of phase constraints

Let Ω be an open bounded subset of ℝl with the Lipshitz boundary Γ, K⊂ℝm be a nonempty set, andimg. Consider the following system:


where img is a state, u(∙) is a control, and A is an elliptic differential operator of the second order: Ay(∙)=p(∙), where img, img and img for a certain Λ>0.

Here img is a space of all continuous in img functions satisfying the Hoelder condition: img. We consider controls img and will seek solutions of problem (1) in a class img. Recall that img is a closure of set img for x∉M, where M⊂int Ω is compact set} in img. The norm in img is defined by the equality img. Suppose that functions img and img are given. Consider an optimal control problem:


on the set img is valid and img

We also assume the following:

1. For almost all x∈Ω a function L(x,y,u) is continuous in (y,u) together with derivative ∂L/∂y. For all (y,u) the function L(x,y,u) is measurable by x and for any r>0 for a certain αr(⋅)∈L( Ω→ℝ) an estimate |L(x,y,u)|+|(∂L(x,y,u))/∂y|≤αr(x) holds for almost all x∈Ω and all y,u∈K such that |y|≤r, |u|≤r.

2. For almost all x∈Ω a function f(x,y,u) is continuous in (y,u) together with derivative ∂f/∂y. For any (y,u) a function f(x,y,u) is measurable by x. There exists s>l/(l-1) such that for any r for a certain βr(⋅)∈Ls (Ω→ℝ) the estimate |f(x,y,u)|+|(∂f(x,y,u))/∂y|≤βr(x) is valid for almost all x∈Ω and any y,u∈K such that |y|≤r, |u|≤r.

3. Functions gi(x,y) are continuous in (x,y) together with derivative ∂gi/∂y and gi(x,0)<0 ∀x∈Γ, i=1,…,q.

Denote by M(Ω) a space of all real regular Borel charges in Ω. It can be identified with the dual to C0(Ω) space

, where C0(Ω)={φ(∙)∈C(Ω) ∶ φ(x)=0 ∀x∈Γ}. Denote by a symbol W01,σ (Ω→ℝh), σ∈[1,∞), a closure of space C0 (Ω→ℝh) in img. In W01,σ (Ω→ℝh) a norm |φ(∙)|≔(|φ(∙)|22+∑_i=1l|∂φ(∙)/∂xi|σσ)1/σ is considered.

Theorem 1.

Let hypotheses 1-3 be satisfied and (y0,u0) be an optimal process in problem (2). Then there exists a function ψ(∙)∈W01,σ(Ω→ℝh), where σ<l/(l-1), the charges μi(dx)∈M(Ω),i=1,…,q and a number λ0∈R are such that


Here H[x,y,u]=ψ*(x)f(x,y,u)-λ0L(x,y,u) is a Hamiltonian function and A*ψ(x)=p(x), where img, img.

In (3) all the addends are considered as generalized functions

. This equation is of the elliptic type of the second order with respect to ψ(∙). The inclusion ψ(x)∈W01,σ(Ω→ℝh) involves the validity of the homogeneous Dirichlet boundary condition ψ|Г=0. The equation (3) with measures μi(dx) was studied in
. Relation (1) implies that y0|Г=0 and therefore condition 3 results in gi [x,y0(x)]<0 ∀x∈Γ,i=1,…,q. By inclusion (5) we have supp μi(dx)⊂int Ω ∀i=1,…,q. The proof of Theorem 1 is given in

3. The optimal control problem governed by a Laplace equation. The case of phase constraints

Let n≥2 and Ω⊂ℝn be an open bounded subset. Consider the following optimal control problem:


where y=y(x)∈ℝ is a state, u=u(x)∈ℝ is a control, ∆ – Laplace operator, numbers k>0,0<u_-<u_+ and a continuous function φ(x), img are given. We choose controls u(∙) in the class L(Ω) and look for the state y(∙) in the class img, where α∈(0,1). We assume that img

Let us explain the introduced notation. Lp(Ω) is the space of all functions y(∙), summable with degree p∈[1,∞), defined on the set Ω and having finite norm imgimg.

H0α(Ω) is the Banach space of continuous functions y(∙), defined on Ω, vanished on the boundary ∂Ω of the set Ω and having finite norm 


where img.

Let us now write down for problem (7), (8) the Pontryagin’s maximum principle formulated in Theorem 1.

Let [y(∙),u(∙)] be an optimal process in problem (7), (8). By Theorem 1 there is a function ψ(∙)∈W01,σ(Ω), a number v≥0 and a finite regular Borel measure img are such that:


Equality (12) holds for any function h(∙)∈H0α (Ω)∩W1,2(Ω) for which ∆h(∙)∈L(Ω). The notation z0=arg maxz∈Zf(x) means that the function f(z) reaches its maximum on the set Z at the point z0.

Lemma 1.

The inequality y(x)≥0 is valid img.

Proof of Lemma 1.

By the definition of generalized solution of the homogeneous Dirichlet problem

for the equation from (7) we have:


The equality (13) holds for any function h(∙)∈W01,2(Ω). According to Lemma 12 (

) h(∙)≔y_(x)∈W_01,2(Ω), where y_(x)≔min⁡{y(x),0}. Let Ω_≔{x∈Ω∶y(x)<0} and Ω+≔Ω\Ω_={x∈Ω∶y(x)≥0}. It is easy to verify that y_ (x)=y(x), ∇y_(x)=∇y(x) a.a.x∈Ω_, y_(x)=0,∇y_ (x)=0 a.a.x∈Ω+.

Substituting h(∙)≔y_(∙) into (13), we get


Since u(x)≥u_>0 and y_(x)≤0, then both terms on the right side are non-negative, while the expression on the left side is not positive. Therefore, Ω|∇y_(x)|2 dx=0, and since y_(∙)∈W01,2(Ω), then y_ (x)=0 and that means y(x)≥0 for almost all x. Recalling that y(∙)∈H0α(Ω), we come to the conclusion of the Lemma 1.

Remark 1.

Note that y(x)>0 at least at one point x∈Ω. Indeed, otherwise y(∙)≡0 and according to (7) 0=u(x)≥u_, that is impossible due to inequality u_>0.

The following auxiliary fact can be proved in a similar way.

Lemma 2.

Let a(∙),f(∙)∈L(Ω) and h(∙)∈W01,2(Ω) be the solution to the Dirichlet problem:


If a(x)≥0 and f(x)≤0 for almost all x∈Ω, then img

Remark 2.

By the theorem 14.1 (

) img.

Lemma 2 allows for the following clarification.

Lemma 3.

Let, under the conditions of Lemma 2f(x)<0 for almost all x∈Ω.

Then h(x)>0 ∀x∈intΩ.

Proof of Lemma 3.

The statement of Lemma 3 obviously follows from Lemma 2.

Let us establish its important consequence concerning the function ψ(∙) from the maximum principle (9) – (12).

Lemma 4.

The function ψ(x)>0 for almost all x∈Ω.

Proof of Lemma 4.

Consider a function f(∙)∈L(Ω) such that f(x)≤0 for almost all x∈Ω and f(x)<0 for all points x from some set of positive measure. Let us define h(∙)∈W01,2(Ω) as a generalized solution to the Dirichlet problem (14) with a(x)≔3u(x)y2(x). Since a(x)≥u_- y2(x)≥0 due to the second relation from (7), this solution exists and is uniquely determined. According to the Remark img, whence in view of (14) img. This means that the function h(∙) can be substituted into (12)


It follows that the nondegeneracy condition (11) can be refined as follows:


Indeed, if (16) is violated, then according to (10) and (11) v=0 and μ(dx)=0. But then (15) takes the form img, where the non-positive and non-zero function f(∙)∈L(Ω) is arbitrary. But then ψ(∙)=0, which, along with the equalities v=0 and μ(dx)=0, contradicts (11). In (15) h(x)>0 ∀x ∈ intΩ by Lemma 3. Since img by assumption, and y(x)=0 for x∈Γ due to the boundary condition from (7), then according to (10) supp μ(dx)⊂ int Ω and μ(dx)≥0. At the same time, by Lemma 1 and Remark 1, img and maxx∈Ω y(x)>0. From this and from (16) it follows that in (15) the right-hand side is strictly negative and therefore img. Since here f(∙)∈L(Ω) is an arbitrary non-positive and non-zero function, we come to the conclusion of Lemma 4. Taking Lemma 4 into account, relation (9) is transformed to the form img.

It follows that


where χ(v):=v for u_≤v≤u+, χ(v):=u_ for v<u_, χ(v):=u+ for v>u+.

Thus, the optimal process is generated by a nonlinear controller


Substituting this equality into the first equation from (7), taking (8) into account, leads to the relations




Note that formulas (17), (18), (19) uniquely determine the process img which is optimal. Indeed, for this it is enough to verify that the boundary value problem (18) is uniquely solvable. For this purpose, we note that function (19) is strictly monotonic:


Therefore, the first equation from (18) satisfies the condition of strict monotonicity (9.33) (



for any two elements y and z from W01,2(Ω)∩L4(Ω) that are not identically equal to each other. In addition, the coercivity condition (9.2) (

) is satisfied. Indeed,


To estimate Ω|y|dx let’s use Young’s inequality: img (

), where p>1 and img.

Let us take here a:=|y(x)|, b≔1, p≔4, then q≔4/3. We get




Choosing now


we get an estimate of the form


where ϰ>0, C>0.

This estimate means that the coercivity condition (9.2) (

) is actually true for m=2 and q=4. The validity of condition (1) (
) is obvious. In view of the established facts, the existence of a solution to problem (18) is guaranteed by Theorem 9.1 (
), and the uniqueness is guaranteed by the concluding remark 9.1 (

4. Conclusion

The purpose of the paper was to demonstrate the practical application of the theory developed in

to the state-constrained optimal control problem for a Laplace equation. As a result, it was found that the optimal process in this problem is generated by a nonlinear optimal controller and its equation was obtained.

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