Exponential stability of second order time deleay semilinear systems with bilinear control

In this paper, we discuss the stabilization of a class of delayed bilinear hyperbolic system. We focus on two cases: the case where the delay occurs in the bilinear control and the case where the delay acts on the state. First, we show existence results under certain assumptions for the considered systems. Second, we design for each system a bilinear control guaranteeing exponential stability under Lipschitz nonlinear perturbation. For the first system, the proof of stability is based on Lyapunov’s abstract functional, while for the second system we prove an equivalence between the stabilization of the delayed system and the observability of the corresponding undamped system. Examples that fall within our abstract framework are presented.