Uniform exponential stability approximations of semi-discretization schemes for two hybrid systems

Abstract: This paper deals with the uniform exponential stabilities (UESs) of two hybrid control systems consisting of wave equation and a second-order ordinary differential equation. Linear feedback law and local viscosity, and nonlinear feedback law and  interior anti-damping are considered, respectively. Firstly, the hybrid system is reduced to a first order port-Hamiltonian system with dynamical boundary conditions and the resulting systems  are then discretized by  average central-difference scheme.  Secondly,  the UES of the discrete system is obtained without  prior knowledge on the exponential stability of continuous system.  The frequency domain characterization of UES for a family of  contractive semigroups and discrete multiplier method are utilized to verify main results, respectively. Finally,  the convergence analysis of the numerical approximation scheme is performed by the Trotter-Kato Theorem. Most interestingly,  the exponential stability of the  continuous system is derived by the convergence of energy and UES and this is a new idea to investigate the exponential stability of some complicate systems. The effectiveness of the numerical approximating  scheme is verified by numerical simulation.