Optimal control problems for a semi-linear integro-differential evolution system with infinite delay

We study in this work the optimal control and time optimal control problems for a semi-linear integro-differential evolution system with infinite delay in Hilbert space. We first study the existence and uniqueness result of mild solutions in space (Formula presented.) for the considered system, and the main tools here are the theory of resolvent operators and fractional powers of operators and (Formula presented.) norm. In this way the obtained results are more general and can be applied to the systems involving terms having spatial derivatives. Then we discuss the Lagrange optimal control problem for the system via limit arguments. The time optimal control problem is also proposed and investigated deliberately for it. Finally, an example is provided to show the applications of the obtained results.