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Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup. (English) Zbl 1157.93006
Summary: We study the controllability problem for a system governed by a semilinear differential inclusion in a Banach space not assuming that the semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows us to apply the topological degree theory for condensing operators and to obtain the controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity. As application we consider the controllability for a system governed by a perturbed wave equation.
MSC:
93B05Controllability
34A60Differential inclusions
34H05ODE in connection with control problems
34G25Evolution inclusions
47H04Set-valued operators
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49J24Optimal control problems with differential inclusions (existence) (MSC2000)