An Existence Theorem for Control Problems with Unbounded Control Domain

The question of the existence of an optimal solution of a given programming problem can often be answered only under strong assumptions, as for example convexity. To answer this question for optimal control problems the sets of feasible solutions were enlarged and the problems generalized (cf., e.g., [9, 14, 15, 171), so that the existence of an optimal generalized process couldbe proved under weak assumptions; where the importance of the generalized processes exceeds this effect and lies also in their applications [19]. Most of the existing literature refers to bounded control sets; a few papers consider the existence of optimal generalized processes in case of an unbounded control domain but most in case of one-dimensional t-variable [3,9, 11, 13, 161. The existence of an optimal generalized process has . also been proved for problems of Dieudonné-Rashevsky type involving multiple integrals, in which there is now a multidimensional t-variable. However a bounded control domain is always assumed. Our paper shows the existence of optimal generalized processes for control problems of Dieudonnd-Rashevsky type for multiple integrals and with unbounded control domains, where our assumptions are similar to those of [9]. Let us consider the following generalized control problem: