Finite-dimensional boundary control for 1-D semilinear wave equation

We design a finite-dimensional state-feedback controller for the semilinear wave equation with in-domain damping under Neumann actuation. We employ the dynamic extension to shift control from the boundary to the main equation and design a controller that stabilizes a finite number of dominating modes. To address potential instability caused by the remaining modes (spillover), we develop and compare two methods: the L 2 residue separation method, resulting in an algebraic Riccati equation, and a method based on Young’s inequality, leading to linear matrix inequalities (LMIs). Numerical results show that the L 2 residue separation method provides less conservative conditions, allowing larger Lipschitz constants compared to Young’s inequality approach.