## Optimal control problems with weakly converging input operators in a nonreflexive framework.(English)Zbl 0957.49010

The paper deals with the recent results on the variational convergence of optimal control problems of the form: $\min \{J_h(u,y): A_h(y) \cap B_h(u) \neq \emptyset , (u,y) \in U \times Y \}, \tag{1}$ where the operators $$A_h: Y \mapsto {\mathcal P}(V) , B_h: U \mapsto {\mathcal P}(V)$$ correspond to the state equation or inclusion, $$B_h$$ being the input operators and $$J$$ is a cost functional. The asymptotic behaviour of such problems reduces to the problem of identification of the $$\Gamma$$-limit of functionals $F_h(u,y) = J_h(u,y) + {\chi}_{A_h(y) \cap B_h(u)}$ ($$\chi$$ denotes the indicator function). This problem in turn can be splitted into subproblems (with auxiliary variable $$v$$) of identification of the $$G$$-limit of inclusions $$v \in A_h(Y)$$ and calculation of the $$\Gamma$$-limit of functionals $$G_h(u,v,y) = J_h(u,y) + {\chi}_{v \in B_h(u)}$$. Two cases are distinguished. If the input operators $$B_h$$ satisfy the strong assumption of sequential Kuratowski continuous convergence, then the limit problem is of the same type as in (1). On the contrary, if a such assumption is dropped the limit problem takes a different form. After abstract setting in the case of nonreflexive space $$V$$ the author considers a measurable framework. Two cases are discussed. In the first case $$B_h$$ are local, possibly nonlinear multivalued operators defined on $$L^p$$ and with values in $$V=L^1$$, while in the second one they are linear, single valued but nonlocal operators. Several examples illustrate the theory developed in the paper.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation
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