Exponential stability preserving of two spatially discretized port-Hamiltonian systems

Abstract: For the ideal transmission line governed by telegrapher’s equations, a mixed finite element method, a generalization of popular spatially discretized schemes, has been proposed. This numerical approximation scheme preserves both the Dirac structure and passivity, ensuring that the spatially discretized system retains its port-Hamiltonian nature. In this paper, we apply this method to spatially discretize two infinite-dimensional port-Hamiltonian systems with variable coefficients and boundary controls. We then investigate the preservation of exponential stability in the resulting semi-discretized systems, demonstrating their uniform exponential stability with respect to discretization parameters. For both semi-discretized models, the uniform exponential stabilities are derived through frequency domain analysis. Finally, numerical simulations validate the effectiveness of this semi-discrete scheme.