Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

Florian Monteghetti (1) , Ghislain Haine (2) , Denis Matignon (2)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 DISC - Département d'Ingénierie des Systèmes Complexes

This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)).

Keywords: Wave equation; Stability; Acoustic boundary condition; memory damping; diffusive representation; fractional kernels; completely monotone kernels; delay feedbacks.

Domains: Mathematics [math]; General Mathematics [math.GM]

Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions. Mathematical Control and Related Fields, 2019, 9 (4), pp.759-791. ⟨10.3934/mcrf.2019049⟩. ⟨hal-02362852⟩