Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space

Jean-Michel Coron (1, 2, 3) , Long Hu (4) , Guillaume Olive (5) , Peipei Shang (6)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 SU - Sorbonne Université
4 School of Mathematics [Shandong]
5 Institute of Mathematics [Krakow]
6 School of Mathematical Sciences [Shanghai]

In this article we are interested in the boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space. We extend the so called ``backstepping method'' by introducing appropriate time-dependent integral transformations in order to map our initial system to a new one which has desired stability properties. The kernels of the integral transformations involved are solutions to non standard multi-dimensional hyperbolic PDEs, where the time dependence introduces several new difficulties in the treatment of their well-posedness. This work generalizes previous results of the literature, where only time-independent systems were considered.

Mots clés: Hyperbolic systems; Boundary stabilization; Non-autonomous systems; Backstepping method

Domaines: Optimisation et contrôle [math.OC]

Jean-Michel Coron, Long Hu, Guillaume Olive, Peipei Shang. Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space. Journal of Differential Equations, 2021, 271, pp.1109-1170. ⟨10.1016/j.jde.2020.09.037⟩. ⟨hal-02553027⟩