We consider the linear control system 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.
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Marshall Slemrod (1974) studied this question.
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