Optimal feedback control for a class of infinite dimensional semilinear systems with distributed delay

This paper focuses on the problem of weak stabilization, strong stabilization and optimal control for a class of semilinear systems with distributed delay in a Hilbert space. We propose new family of feedback controls to discuss the weak and strong stabilization of this class of equations. Sufficient conditions for weak and strong stabilization are investigated as well. Moreover, we show that one of the used family of feedback controls is the unique solution of an appropriate minimization problem. Furthermore, some results in finite dimension are given. The obtained results are illustrated by an examples with numerical simulations for physical systems with distributed delay.