CONTROLLER SYNTHESIS FOR A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM GOVERNED BY A LAPLACE EQUATION

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.,

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

(1)

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

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

(2)

on the set  is valid and 

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

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

(3)

(4)

(5)

(6)

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 , .

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

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

(7)

(8)

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), are given. We choose controls u(∙) in the class L∞(Ω) and look for the state y(∙) in the class , where α∈(0,1). We assume that 

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 , .

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 .

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 are such that:

(9)

(10)

(11)

(12)

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 .

Proof of Lemma 1.

By the definition of generalized solution of the homogeneous Dirichlet problem

(13)

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

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:

(14)

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

Remark 2.

By the theorem 14.1 (

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 , whence in view of (14) . This means that the function h(∙) can be substituted into (12)

(15)

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

(16)

Indeed, if (16) is violated, then according to (10) and (11) v=0 and μ(dx)=0. But then (15) takes the form , 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 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,  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 . 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 .

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

(17)

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

(18)

Here

(19)

Note that formulas (17), (18), (19) uniquely determine the process  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) (

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

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

,

Thus,

Choosing now

we get an estimate of the form

where ϰ>0, C>0.

This estimate means that the coercivity condition (9.2) (

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