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

This paper explores the stabilisation of delayed bilinear hyperbolic systems, examining scenarios with delays in bilinear control and state. Our contributions are twofold: First, we establish existence results under specific conditions for these systems. Second, we design bilinear controls ensuring exponential stability under Lipschitz nonlinear perturbations. For control delay, stability is demonstrated using Lyapunov's abstract function. In contrast, for state delay, we show an equivalence between the stabilisation of the delayed system and the observability of the corresponding undamped system. Our theoretical findings are exemplified through practical models, emphasising their applicability in real-world scenarios.