Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

TL;DR

This paper develops feedback control strategies to achieve finite-time stabilization of homogeneous quasilinear hyperbolic systems in one dimension, optimizing the stabilization time close to the linearized system's null controllability limit.

Contribution

It introduces time-independent feedback laws for finite-time stabilization in nonlinear hyperbolic systems, extending controllability results to nonlinear boundary conditions.

Findings

Achieved finite-time stabilization with feedback controls

Established local well-posedness for systems with nonlinear boundary conditions

Demonstrated stabilization in any time exceeding the linearized system's optimal control time

Abstract

We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.