Dissipative boundary conditions for 2 × 2 hyperbolic systems of conservation laws for entropy solutions in BV

Abstract

In this article, we investigate the BV stability of 2×2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions. More precisely, we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in BV. Our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose BV norms are controlled through a Lyapunov functional. This Lyapunov functional is inspired by the one proposed in J. Glimm's seminal work [16], modified with some suitable weights in the spirit of the previous works [10,9].

BV Exponential Stability for Systems of Scalar Conservation Laws Using Saturated Controls

“…Also, it should be mentioned that in [8,10], the stability was proved for m = 2 but the argument can be adapted for any integer m > 0. c. For BV ([0, 1]), few results are known. To the authors' knowledge only [9] deals with this case. They take a 2 × 2 system of conservation laws and give a sufficient condition on H to ensure the local BV stability.…”

“…An explicit estimate of the basin of attraction is given. The Lyapunov functional is inspired from Glimm's seminal work [18] reconsidered in [9].…”

Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

“…In [6,7,30], by acting suitably on both sides of the interval, one can steer asymptotically any initial data with sufficiently small total variations to any close constant steady states. Recently, it was shown in [12] the existence and exponential convergence of an entropy BV solution to the null-steady-state with small BV initial conditions. To our knowledge, no results exist concerning the boundary stabilization of steady states with jump discontinuities for nonlinear hyperbolic systems.…”

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

“…The asymptotic stabilization of entropy solutions to systems has been studied in [4,9,18]. It seems difficult to investigate the exact controllability of entropy solutions of scalar conservation laws in several space dimensions using the techniques designed for the one dimensional case.…”