Optimal Control Problems for a Semilinear Evolution System with Infinite Delay

In this paper we study the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay. We first establish the results of existence and uniqueness of mild solution and the compactness of the solution operator for the control system. Then, based on these results, we investigate the optimal control problem with integral cost function and the time optimal control problem respectively. Under some conditions we show the existence of optimal controls for the both cases of bounded and unbounded admissible control sets. We also obtain the existence of time optimal control to a target set. In addition, a convergence theorem of time optimal controls to a point target set is proved. Finally, an example is given to show the application of the main results.