Local exponential H^2 stabilization of a 2X2 quasilinear hyperbolic system using backstepping

TL;DR

This paper develops a boundary feedback control law for a quasilinear hyperbolic PDE system, achieving exponential stability using backstepping and Lyapunov methods, with explicit kernel computation.

Contribution

The work introduces a novel backstepping-based control design for quasilinear hyperbolic systems, providing explicit kernel solutions and stability proof.

Findings

Achieves H^2 exponential stability with boundary actuation

Explicit kernel solutions via Goursat-type PDEs

Proves well-posedness using characteristics and approximations

Abstract

In this work, we consider the problem of boundary stabilization for a quasilinear 2X2 system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves H^2 exponential stability of the closedloop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type 4X4 system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.