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# Exponential stability of density-velocity systems with boundary conditions and source term for the $H^2$ norm

3 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : In this paper, we address the problem of the exponential stability of density-velocity systems with boundary conditions. Density-velocity systems are omnipresent in physics as they encompass all systems that consist in a flux conservation and a momentum equation. In this paper we show that any such system can be stabilized exponentially quickly in the $H^2$ norm using simple local feedbacks, provided a condition on the source term which holds for most physical systems, even when it is not dissipative. Besides, the feedback laws obtained only depends on the target values at the boundaries, which implies that they do not depend on the expression of the source term or the force applied on the system and makes them very easy to implement in practice and robust to model errors. For instance, for a river modeled by Saint-Venant equations this means that the feedback laws do not require any information on the friction model, the slope or the shape of the channel considered. This feat is obtained by showing the existence of a basic $H^2$ Lyapunov functions and we apply it to numerous systems: the general Saint-Venant equations, the isentropic Euler equations, the motion of water in rigid-pipe, the osmosis phenomenon, etc.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-02190778
Contributor : Amaury Hayat Connect in order to contact the contributor
Submitted on : Friday, May 7, 2021 - 9:45:33 PM
Last modification on : Wednesday, June 8, 2022 - 12:50:08 PM

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### Citation

Amaury Hayat, Peipei Shang. Exponential stability of density-velocity systems with boundary conditions and source term for the $H^2$ norm. Journal de Mathématiques Pures et Appliquées, Elsevier, 2021, 153, pp.187-212. ⟨10.1016/j.matpur.2021.07.001⟩. ⟨hal-02190778v2⟩

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