Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

Abstract

This paper deals for the first time with the Dirichlet problem for discrete (P D ), discrete approximation problem on a uniform grid and differential (P C ) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.

Sufficient Conditions of Optimality for Convex Differential Inclusions of Elliptic Type and Duality

“…A lot of problems in economic dynamics, as well as classical problems on optimal control in vibrations, chemical engineering, heat, diffusion processes, differential games, and so on, can be reduced to such investigations with ordinary and partial differential inclusions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. We refer the reader to the survey papers [11,[16][17][18][19][20]. The present paper is organized as follows.…”

“…In Section 2 first are given some suitable definitions, supplementary notions and results considered by author in [18,19]. Then a certain extremal Dirichlet's problem is formulated for so-called elliptic differential (P C ) inclusions with Laplace's operator and with second order elliptic operator in the multidimensional case.…”

“…Then a certain extremal Dirichlet's problem is formulated for so-called elliptic differential (P C ) inclusions with Laplace's operator and with second order elliptic operator in the multidimensional case. In the reviewed results for optimality the arisen adjoint inclusions using the locally adjoint multivalued (LAM) functions are stated in the Euler-Lagrange form [9,18,19]. It turn out that such form of optimality conditions automatically implies the Weierstrass-Pontryagin maximum condition.…”

“…It turn out that such form of optimality conditions automatically implies the Weierstrass-Pontryagin maximum condition. Apparently it happens because the LAM is more applicable apparat in different type of problems governed by differential inclusions [16][17][18][19][20].…”

“…It means that if some pair of feasible solutions (u(. ),u inclusions in [17][18][19]. Some duality relations and optimality conditions for an extremum of different control problems with partial differential inclusions can be found in [18,19].…”