Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the $L^{2}$ norm

TL;DR

This paper establishes conditions for the global exponential stability and Input-to-State Stability of semilinear hyperbolic systems in the L^2 norm, including nonlocal sources and disturbances, advancing control theory for PDEs.

Contribution

It provides new sufficient conditions for stability of semilinear hyperbolic systems, extending to nonlocal sources and robustness against disturbances.

Findings

Derived internal and boundary stability conditions.

Extended stability results to nonlocal source terms.

Proved robustness of stability under disturbances.

Abstract

In this paper we study the global exponential stability in the $L^{2}$ norm of semilinear $1$-$d$ hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the $L^{2}$ norm with respect to both interior and boundary disturbances.