SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1656-1679, January 2021.
In this paper, we investigate the $BV$ exponential stability of general systems of scalar conservation laws with positive velocities and under dissipative boundary conditions. The paper is divided into two parts, the first one focusing on linear controls and the last one dealing with saturated laws. For the linear case, the global exponential $BV$ stability is proved. For the saturated case, it is discussed that we cannot expect to have a basin of attraction larger than the region of linearity in a $BV$ context. We rather prove an $L^\infty$ local stability result. An explicit estimate of the basin of attraction is given. The Lyapunov functional is inspired from Glimm's seminal work [J. Glimm, Comm. Pure Appl. Math., 18 (1965), pp. 697--715] reconsidered in [J. M. Coron et al., J. Differential Equations, 262 (2017), pp. 1--30].

中文翻译：

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使用饱和控制的标量守恒定律系统的BV指数稳定性

SIAM控制与优化杂志，第59卷，第2期，第1656-1679页，2021年1月。
在本文中，我们研究了耗散边界条件下具有正速度的标量守恒定律的一般系统的$ BV $指数稳定性。本文分为两部分，第一部分着重于线性控制，最后一部分涉及饱和定律。对于线性情况，证明了全局指数$ BV $的稳定性。对于饱和情况，要讨论的是，我们不能期望在$ BV $上下文中吸引池大于线性区域。我们宁可证明$ L ^ \ infty $局部稳定性结果。给出了对吸引盆地的明确估计。Lyapunov函数的灵感来自Glimm的开创性工作[J. Glimm，通讯。纯应用 [JM Coron et al。，J.Differential Equations，262（2017），pp.1--30]中重新考虑了Math.18（1965），697--715页。