Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result

Abstract

In this paper we study an optimal control problem (OCP) associated to a linear elliptic equation on a bounded domain Ω. The matrixvalued coefficients A of such systems is our control in Ω and will be taken in L 2 (Ω; R N ×N ) which in particular may comprises the case of unboundedness. Concerning the boundary value problems associated to the equations of this type, one may exhibit non-uniqueness of weak solutions-namely, approximable solutions as well as another type of weak solutions that can not be obtained through the L ∞ -approximation of matrix A. Following the direct method in the calculus of variations, we show that the given OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem may have a singular character in the above sense. In view of this we indicate two types of optimal solutions to the above problem: the so-called variational and non-variational solutions, and show that some of that optimal solutions can not be attainable through the L ∞ -approximation of the original problem.

On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems

“…, then the differential operator in the left-hand side of equation 2is neither monotone nor coercive, so two definitions of weak solutions, given by (18) and (22) are not equivalent [27]. Besides, the mapping u → y(u, f ) can be multivalued (see [13] for the details).…”

“…The characteristic feature of such optimal control problem is the unboundedness of skew-symmetric matrix A skew ∈ L q (Ω; R N ×N ). As it was indicated in [12,13,14,17,37], this circumstance can lead to the existence of elements y ∈ W p 0 (Ω) such that y ∈ L ∞ (Ω), Ω ∇ϕ n , A skew ∇ϕ n R N dx = 0 ∀ n ∈ N, and lim…”

Про Необмежені Оптимальні Керування В Коефіцієнтах Для Погано Обумовлених Еліптичних Крайових Задач

“…then the element y = (y Thus, the mapping A → y(A, f ) can be multivalued, in general (see [7] for the details).…”

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

“…Several results for optimal control problems related to elliptic PDE's with unbounded coefficients have been also obtained in [26,21,22,27,28,9,10] and the recent papers [20], [30] with the reference therein).…”