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

Freddi, L.

Port. Math. 57, No. 1, 97-126 (2000).

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.

Reviewer: Z.Denkowski (Kraków)

MSC:

49J45

Methods involving semicontinuity and convergence; relaxation

Keywords:

optimal control; \(\Gamma\)-convergence; functionals defined on measures; weak convergence; multivalued operators; inclusions