Feedback Stabilization of Semilinear System With Distributed Delay

This work is concerned with the feedback stabilization of a class of semilinear system with distributed delay. We consider an observation condition in terms of the semigroup solution of the linear part of the considered system and parameterized by a positive constant. Some sufficient conditions are given with respect to this parameter and the bounded feedback control to guarantee the feedback stabilization of the semilinear system with distributed delay. Moreover, when this parameter is greater than or equal to 2, an explicit polynomial decay rate of the stabilization state is estimated in the strong stabilization case. In the bilinear case with distributed delay and by using the decomposition method of the state space, we investigate the feedback stabilization of the considered system using some suitable conditions like observability assumption. In the case of strong stabilization, we obtain the same explicit decay estimate of the stabilized state. Furthermore, when the parameter equal to 2, we show that the normalized feedback control exponentially stabilizes the semilinear system. The obtained results are illustrated by three examples and numerical simulations for wave equation with distributed delay.