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arXiv · 1801.02353

Exponential stability of general 1-D quasilinear systems with source terms for the $C^1$ norm under boundary conditions

Amaury Hayat
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Abstract

We address the question of the exponential stability for the $C^{1}$ norm of general 1-D quasilinear systems with source terms under boundary conditions. To reach this aim, we introduce the notion of basic $C^{1}$ Lyapunov functions, a generic kind of exponentially decreasing function whose existence ensures the exponential stability of the system for the $C^{1}$ norm. We show that the existence of a basic $C^{1}$ Lyapunov function is subject to two conditions: an interior condition, intrinsic to the system, and a condition on the boundary controls. We give explicit sufficient interior and boundary conditions such that the system is exponentially stable for the $C^{1}$ norm and we show that the interior condition is also necessary to the existence of a basic $C^{1}$ Lyapunov function. Finally, we show that the results conducted in this article are also true under the same conditions for the exponential stability in the $C^{p}$ norm, for any $p\geq1$.

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Amaury Hayat. 2018-01-08. Exponential stability of general 1-D quasilinear systems with source terms for the $C^1$ norm under boundary conditions. https://arxiv.org/abs/1801.02353

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