Exact controllability of infinite dimensional linear systems via bounded control functions

This paper considers the problem of exact controllability of linear control systems with compact semigroup in non-reflexive state spaces and control spaces. The weak convergence of bounded sequence of control functions with respect to the subspace of the dual space is studied and a lemma for the weak convergence is proved. Based on the lemma and the strong separation theorem, a necessary and sufficient condition for exact controllability of compact semigroups with respect to the space consisting all essentially bounded measurable functions is obtained. An example of heat equation is presented to illustrate the utility of the result. Some related problems remaining for further study are addressed.