Wave equation with hyperbolic boundary condition: a frequency domain approach

Nicolas Vanspranghe (1)

1 GIPSA-INFINITY - GIPSA - Infinite Dimensional Dynamics

In this paper, we investigate the stability of the linear wave equation where one part of the boundary, which is seen as a lower-dimensional Riemannian manifold, is governed by a coupled wave equation, while the other part is subject to a dissipative Robin velocity feedback. We prove that the closed-loop equations generate a semi-uniformly stable semigroup of linear contractions on a suitable energy space. Furthermore, under multiplier-related geometrical conditions, we establish a polynomial decay rate for strong solutions. This is achieved by estimating the growth of the resolvent operator on the imaginary axis.

Keywords: Infinite-dimensional systems; control of partial differential equations; control and estimation of wave equations and systems of elasticity; semigroup and operator theory

Domains: Analysis of PDEs [math.AP]; Optimization and Control [math.OC]

Nicolas Vanspranghe. Wave equation with hyperbolic boundary condition: a frequency domain approach. CPDE - 4th IFAC Workshop on Control of Systems Governed by Partial Differential Equations, Sep 2022, Kiel, Germany. ⟨10.1016/j.ifacol.2022.10.386⟩. ⟨hal-03781188⟩