Control of a nonlinear wave equation with a dynamic boundary condition

Existence, uniqueness, energy decay, and approximate numerical solution for the nonlinear wave equation with dynamic control at the boundary is being studied in this work. The theoretical analysis of the problem will be conducted using the Faedo-Galerkin method and compactness results. To obtain the approximate numerical solution, a combined approach of the finite element method and a finite difference method will be employed, known as the linearized Crank-Nicolson Galerkin method. This method optimizes the calculations and preserves the quadratic order of convergence in time. Finally, numerical experiments are performed, and tables and graphs are presented to illustrate the theoretical convergence rates and demonstrate the consistency between theoretical and numerical results.