We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.

Classification: 35K35, 35B37, 49J15, 49J20

Keywords: control problem, time dependent wave equation, damping

@article{COCV_2002__8__375_0, author = {Chambolle, Antonin and Santosa, Fadil}, title = {Control of the wave equation by time-dependent coefficient}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, pages = {375-392}, doi = {10.1051/cocv:2002029}, zbl = {1073.35032}, mrnumber = {1932956}, language = {en}, url = {http://www.numdam.org/item/COCV_2002__8__375_0} }

Chambolle, Antonin; Santosa, Fadil. Control of the wave equation by time-dependent coefficient. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 375-392. doi : 10.1051/cocv:2002029. http://www.numdam.org/item/COCV_2002__8__375_0/

[1] Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108 (1992) 247-262. | Zbl 0785.35067

and ,[2] Smart materials and flexible structures. Control Cybernet. 26 (1997) 161-205. | MR 1472842 | Zbl 0884.73043

and ,[3] Smart Structures and Materials. ASME, New York, ASME, AD 24 (1991).

and ,[4] Adaptronics and Smart Structures. Springer, New York (1999).

,[5] Control in the coefficients of linear hyperbolic equations via spatio-temporal components, in Homogenization. World Science Publishing, River Ridge, NJ, Ser. Adv. Math. Appl. Sci. 50 (1999) 285-315. | MR 1792692 | Zbl 1035.78021

,[6] On a class of quasilinear hyperbolic equations. Math. USSR Sbornik 25 (1975) 145-158. | Zbl 0328.35060

,[7] Magnetostrictive materials and devices, in Encyclopedia of Applied Physics, Vol. 9. VCH Publishers (1994).

,