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Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations. (English) Zbl 0796.49008

Summary: We consider differential inclusions related to PDEs of parabolic type and some control problems with integral cost functionals associated to them. Given a sequence of such problems, we investigate first the asymptotic behavior of solution sets (mild solutions, or more precisely selection- trajectory pairs) for differential inclusions, and we get some semicontinuity or continuity results (Kuratowski convergence of solution sets). Then, we prove the \(\Gamma\)-convergence of cost functionals, related to the above Kuratowski convergence of solution sets. Finally, applying the Buttazzo-Dal Maso abstract scheme, based on the sequential \(\Gamma\)-convergence, we obtain results concerning the asymptotic behavior (hence, also stability results) for optimal solutions to control problems as well as the convergence of minimal values.

MSC:

49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
35K55 Nonlinear parabolic equations
49J45 Methods involving semicontinuity and convergence; relaxation
49J20 Existence theories for optimal control problems involving partial differential equations
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