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 [1]. 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., [3], [4], [5] 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 [6], [7], [8]. 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 [1].

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 ℝlMissing Mark : sup with the Lipshitz boundary Γ, K⊂ℝmMissing Mark : sup be a nonempty set, and

`[LATEX_FORMULA]\left\{\begin{array}{c} A y=f[x, y(x), u(x)], u(x) \in K, x \in \Omega \\ \left.y\right|_{\Gamma}=0 \end{array}\right.[/LATEX_FORMULA]`

where

Here

`[LATEX_FORMULA]J(y,u):=\int_\Omega L[x,y(x),u(x)]dx→inf [/LATEX_FORMULA]`

on the set

We also assume the following:

1. For almost all

2. For almost all

3. Functions

Denote by

[9]

Let hypotheses 1-3 be satisfied and

`[LATEX_FORMULA]A^* \psi(x)-\nabla_y \mathrm{H}\left[\mathrm{x}, y^0(x), u^0(x)\right]+\sum_{i=1}^q \mu_i(d x) \nabla_y g_i\left[\mathrm{x}, y^0(x)\right]=0, x \in \Omega[/LATEX_FORMULA]`

`[LATEX_FORMULA]\mathrm{H}\left[\mathrm{x}, y^0(x), u^0(x)\right]=\max _{v \in K} \mathrm{H}\left[\mathrm{x}, y^0(x), \mathrm{v}\right] \text { for almost all } x \in \Omega[/LATEX_FORMULA]`

`[LATEX_FORMULA]\lambda^0 \geq 0, \mu_i(d x) \geq 0, \operatorname{supp} \mu_i(d x) \subset\left\{x: g_i\left[\mathrm{x}, y^0(x)\right]=0\right\} \forall i=1, \ldots, q[/LATEX_FORMULA]`

`[LATEX_FORMULA]\lambda^0+\int_{\Omega}|\psi(x)| d x+\sum_{i=1}^q \mu_i(\Omega)>0[/LATEX_FORMULA]`

Here

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

[10][9][2]

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

Let

`[LATEX_FORMULA]\Delta y=u y^3-k u^2, u_{-} \leq u \leq u_{+}, x \in \Omega[/LATEX_FORMULA]`

`[LATEX_FORMULA]\left.y\right|_{\Gamma}=0, y(x) \leq \varphi(x) \forall x \in \bar{\Omega}, \int_{\Omega} y(x)^2 d x \rightarrow \min[/LATEX_FORMULA]`

where

Let us explain the introduced notation.

where [LATEX_FORMULA]<y(\cdot)>_{x, \Omega}^{(\gamma)}:=\sup _{x^{\prime}, x^{\prime \prime} \in \Omega} \frac{\left|y\left(x^{\prime \prime}\right)-y\left(x^{\prime}\right)\right|}{\left|x^{\prime \prime}-x^{\prime}\right|^\gamma} 0<\gamma<1[/LATEX_FORMULA].

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

Let

`[LATEX_FORMULA]u(x)=\arg \max _{\omega \in\left[u_{-}, u_{+}\right]} \psi(x)\left[w y(x)^3-k w^2\right] \text { a.a. } x \in \Omega[/LATEX_FORMULA]`

`[LATEX_FORMULA]\mu(d x) \geq 0, \operatorname{supp} \mu(d x) \subset\{x \in \bar{\Omega}: y(x)=\varphi(x)\}, v \geq 0[/LATEX_FORMULA]`

`[LATEX_FORMULA]v+\mu(\overline{\Omega})+\int_{\Omega}|\psi(x)| d x>0[/LATEX_FORMULA]`

`[LATEX_FORMULA]\int_{\Omega} \psi(x)\left[\Delta h(x)-3 u(x) y^2(x) h(x)\right] d x+\int_{\Omega} h(x) \mu(d x)+2 v \int_{\Omega} h(x) y(x) d x=0[/LATEX_FORMULA]`

Equality (12) holds for any function

Lemma 1.

The inequality

Proof of Lemma 1.

By the definition of generalized solution of the homogeneous Dirichlet problem [11] for the equation from (7) we have:

`[LATEX_FORMULA]-\int_{\Omega}<\nabla y, \nabla h>d x=\int_{\Omega} u(x) y(x)^3 h(x) d x-k \int_{\Omega} u(x)^2 h(x) d x[/LATEX_FORMULA]`

The equality (13) holds for any function

[12, II, 3]

Substituting

Since

Note that

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

Lemma 2.

Let

`[LATEX_FORMULA]\Delta h(x)=a(x) h(x)+f(x),\left.h\right|_{\Gamma} \equiv 0[/LATEX_FORMULA]`

If

By the theorem 14.1 (

[12, III, 14]

Lemma 2 allows for the following clarification.

Let, under the conditions of Lemma

Then

The statement of Lemma 3 obviously follows from Lemma 2.

Let us establish its important consequence concerning the function

The function

Proof of Lemma 4.

Consider a function

`[LATEX_FORMULA]\int_{\Omega} \psi(x) f(x) d x=-\int_{\Omega} h(x) \mu(d x)-2 v \int_{\Omega} h(x) y(x) d x[/LATEX_FORMULA]`

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

`[LATEX_FORMULA]v+\mu(\overline{\Omega})>0[/LATEX_FORMULA]`

Indeed, if (16) is violated, then according to (10) and (11)

It follows that

where

Thus, the optimal process is generated by a nonlinear controller

`[LATEX_FORMULA]u(x)=\chi\left\{\frac{y^3}{2 k}\right\}[/LATEX_FORMULA]`

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

`[LATEX_FORMULA]\Delta y-a(y)=0,\left.y\right|_{\Gamma} \equiv 0[/LATEX_FORMULA]`

Here

`[LATEX_FORMULA]a(y):=\chi\left\{\frac{y^3}{2 k}\right\} y^3-k \chi^2\left\{\frac{y^3}{2 k}\right\}[/LATEX_FORMULA]`

Note that formulas (17), (18), (19) uniquely determine the process [LATEX_FORMULA][y(\cdot), u(\cdot)] \in\left(W_0^{1,2} \cap L^4\right) \times L^{\infty}[/LATEX_FORMULA] 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) ([12, IV, 9]):

for any two elements y and z from

[12, IV, 9]

To estimate

[12, II, 1]

Let us take here

Thus,

Choosing now

we get an estimate of the form

where

This estimate means that the coercivity condition (9.2) (

[12, IV, 9][12, IV, 9][12, IV, 9][12, IV, 9]

4. Conclusion

The purpose of the paper was to demonstrate the practical application of the theory developed in [2] 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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