Coupled wave equations with monotone nonlinearity: existence of solution and controllability results

In this manuscript, we investigate a coupled semilinear system characterised by two wave equations in which the nonlinear function satisfies the monotonicity condition. Initially, the existence of a solution is established through the application of Schauder's fixed point theorem. Thereafter, the results pertaining to the exact controllability are demonstrated for two different cases by establishing a relation between the reachable set of the semilinear system and its linear counterpart. Additionally, an illustrative example is provided to showcase the practical relevance of our theoretical findings.