Exact controllability for nonlocal semilinear differential inclusions

Abstract

In this paper, we investigate the controllability of a class of semilinear differential inclusions in Hilbert spaces. Assuming the exact controllability of the associated linear problem, we establish sufficient conditions for achieving the exact controllability of the nonlinear problem. In infinite-dimensional spaces, the compactness of the evolution operator and the linear controllability condition are often incompatible. To address this, we avoid the compactness assumption on the semigroup by employing two distinct approaches: one based on weak topology, and the other on the concept of Gelfand triples. Furthermore, the problem we consider is that of nonlocal controllability, where the solution satisfies a nonlocal initial condition that depends on the behaviour of the solution over the entire time interval.