Exact controllability of infinite dimensional systems with controls of minimal norm

Abstract

The paper deals with the exact controllability of a semilinear system in a separable Hilbert space. A bounded linear part is considered and a linear control introduced. The state space is compactly embedded in a Banach space and the nonlinear term is continuous in its state variable in the norm of the Banach space. An infinite sequence of finite dimensional controllability problems is introduced and the solution is obtained by a limiting procedure. To the best of our knowledge, the method is new in controllability theory. An application to an integro-differential system in euclidean spaces completes the discussion.