The uniform exponential stability of wave equation with dynamical boundary damping discretized by the order reduction finite difference

Because the uniform exponential stabilities with respect to the discretized parameter play key roles in the computing of optimal control and the inverse problem of observability, they were broadly and intensively discussed. It is well known that, for the continuous wave equation, it is exponentially stable. If the continuous system is discretized in spacial variable by finite difference method, the numerical scheme yields spurious high frequency oscillations which induce the deficiency of the uniform exponential stability. To restore the uniform qualitative behaviors, researchers introduced the methods of vanishing viscosity terms and filtering. However, there are rare results on the uniform exponential stability of the wave equation with dynamical boundary condition. In this note, we shall apply finite difference approach of order reduction to study this question. That is to say, we reduce the order of the wave equation and then discretized spacial variable by finite difference method, the uniform exponential stability is tested by introducing suitable Lyapunov function and without any remedy. © 2020, Editorial Department of Control Theory & Applications South China University of Technology. All right reserved.