A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space

Abstract

We consider the linear control system $\dot x = Ax + Bu$. Here A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators $T(t)$ on a Hilbert space E, and B is a bounded linear operator from a Hilbert space H to E. We give sufficient conditions for the existence of a bounded linear operator K from E to H so that the control system with feedback control law $u(t) = Kx(t)$ has the zero solution asymptotically stable. In particular, we study the relationship between the concept of complete controllability and the existence of K.