STABILIZABILITY FOR A QUASILINEAR KLEIN-GORDON-WAVE SYSTEM WITH VARIABLE COEFFICIENTS*

This paper concerns with the stabilizability for a quasilinear Klein-Gordon-wave system with variable coefficients in Rn. The stabilizability of linear wave-type equations with the Kelvin-Voigt damping has been considered by Liu-Zhang [SIAM J. Control Optim., 54 (2016), pp. 1859--1871] for an elastic string system, and by Yu-Han [SIAM J. Control Optim., 59 (2021), pp. 1973--1988] for the linear wave system in a cuboidal domain, respectively. In this paper, we establish that there exists a linear feedback control law such that the quasilinear Klein-Gordon-wave system is exponentially stable under certain smallness conditions, where the feedback control is a strong Kelvin-Voigt damping.

Automated summary

This paper investigates the stabilizability of a quasilinear Klein-Gordon-wave system with variable coefficients in Rn. The stabilizability of linear wave-type equations with Kelvin-Voigt damping has been considered by Liu-Zhang and Yu-Han for different systems. In this paper, it is shown that there exists a linear feedback control law that can exponentially stabilize the quasilinear Klein-Gordon-wave system under certain smallness conditions, with the feedback control being a strong Kelvin-Voigt damping.