Lyapunov function and smooth periodic solutions to quasilinear 1D hyperbolic systems

TL;DR

This paper develops a Lyapunov-based framework to establish existence, uniqueness, and stability of small smooth periodic solutions in 1D quasilinear hyperbolic systems with complex boundary conditions, including perturbation robustness.

Contribution

It introduces a Lyapunov function approach to analyze periodic solutions and stability for nonlinear hyperbolic systems with nonlocal boundary conditions, extending previous methods.

Findings

Established conditions for exponential stability of linearized boundary value problems.

Proved existence and uniqueness of smooth periodic solutions under non-resonance conditions.

Demonstrated robustness of solutions under small perturbations of system coefficients.

Abstract

We apply a Lyapunov function to obtain conditions for the existence and uniqueness of small classical time-periodic solutions to first order quasilinear 1D hyperbolic systems with (nonlinear) nonlocal boundary conditions in a strip. The boundary conditions cover different types of reflections from the boundary as well as integral operators with delays. In the first step we use a Lyapunov approach to derive sufficient conditions for the robust exponential stability of the boundary value problems for a linear(ized) homogeneous problem. Under those conditions and a number of non-resonance conditions, in the second step we prove the existence and uniqueness of smooth time-periodic solutions to the corresponding linear nonhomogeneous problems. In the third step, we prove a perturbation theorem stating that the periodic solutions survive under small perturbations of all coefficients of the hyperbolic system. In the last step, we apply the linear results to construct small and smooth time-periodic solutions to the quasilinear problems.