On L∞ stabilization of diagonal semilinear hyperbolic systems by saturated boundary control

Mathias Dus,Francesco Ferrante,Christophe Prieur

doi:10.1051/cocv/2019069

Abstract

This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L2([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L∞ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.

Highlights

We focus on feedback boundary control of a diagonal system of semilinear hyperbolic partial differential equations (PDEs)
In [14], the authors prove that two-dimensional quasilinear hyperbolic systems with opposite velocities are stabilizable with bounded C1 boundary control inputs
Thanks to Theorem A.1 given in Appendix, we will show that if the initial data is H2([0, 1]) and satisfies compatibility conditions of order 1 (2.2) the previous system of PDEs has a unique solution in C0([0, T ], H1([0, 1])) ∩ C1([0, T ], L2([0, 1])) for any T > 0

Summary

Introduction

Only a few papers focused on saturated boundary control of systems modeled via PDEs. For example, in [14], the authors prove that two-dimensional quasilinear hyperbolic systems with opposite velocities are stabilizable with bounded C1 boundary control inputs. Inspired by [1], the authors in [17] relies on the theory of nonlinear semigroups to prove well-posedness and global H1 exponential stability for the wave equation, in the presence of distributed or boundary saturated controllers. The main idea consists of using a sector bounded approach inspired by the literature of finite dimensional systems [19], to ensure exponential decay of an H1-Lyapunov functional In this manuscript, as opposed to [14, 17], we directly consider the following system of semilinear hyperbolic PDEs of arbitrary dimension d ∈ N. Note that ([4], Thm. 3.3) was proven for small initial data and for quasilinear systems

Definition of the system and contribution

Main results

Numerical example

Conclusion

Existence and uniqueness of solution with a smoothed saturation