Necessary and sufficient conditions for the approximate controllability of fractional linear systems via C−semigroups

Abstract

The approximate controllability of fractional linear evolution systems is considered in this paper. Firstly, the definitions of the mild solution and the approximate controllability of fractional linear evolution systems are obtained by using the theory of C?semigroups. Secondly, a new set of necessary and sufficient conditions are established to examine that linear system is approximately controllable with the help of symmetric operator. Moreover, the restricted condition of the state space is weakened by means of the dual mapping. Finally, as applications, the approximate controllability of nonlinear evolution systems are derived under the assumption that the corresponding linear system is approximately controllable. Our work essentially improves and generalized the corresponding results which are based on strongly continuous semigroups.