Existence theorems for abstract multidimensional control problems

Abstract

In the present paper, the author discusses an abstract formulation of control problems involving general operators ℒ : S → V , ℳ : S → Y from a Banach space S into space V and Y of vector functions in a fixed domain with components in L p , p ⩾1. For this general formulation, the author states closure theorems, lower closure theorems, and existence theorems for an optimal solution. It is then shown that the problems of control involving Dieudonné-Rashevski partial differential equations previously considered by the author are particular cases of the present formulation. Finally, it is shown by examples that problems of control involving usual partial differential equations, linear or not, as well as other functional relations, can be framed in the present formulation. The present work concerns problems with distributed controls . Work concerning problems with distributed as well as boundary controls is forthcoming.