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

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second-order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti-damping. The hybrid system is first reduced to a first order port-Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central-difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter-Kato Theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.