Chambolle, Antonin; Santosa, Fadil Control of the wave equation by time-dependent coefficients. (English) Zbl 1073.35032 ESAIM, Control Optim. Calc. Var. 8, 375-392 (2002). Authors’ abstract: 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. Reviewer: Sergiu Aizicovici (Athens/Ohio) Cited in 6 Documents MSC: 35B37 PDE in connection with control problems (MSC2000) 49J15 Existence theories for optimal control problems involving ordinary differential equations 49J20 Existence theories for optimal control problems involving partial differential equations 74M05 Control, switches and devices (“smart materials”) in solid mechanics 93C20 Control/observation systems governed by partial differential equations 35L20 Initial-boundary value problems for second-order hyperbolic equations Keywords:damping; energy decay × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] P. D’Ancona and S. Spagnolo , Global solvability for the degenerate Kirchhoff equation with real analytic data . Invent. Math. 108 ( 1992 ) 247 - 262 . Zbl 0785.35067 · Zbl 0785.35067 · doi:10.1007/BF02100605 [2] P. Destuynder and A. Saidi , Smart materials and flexible structures . Control Cybernet. 26 ( 1997 ) 161 - 205 . MR 1472842 | Zbl 0884.73043 · Zbl 0884.73043 [3] G. Haritos and A. Srinivasan , Smart Structures and Materials . ASME, New York, ASME, AD 24 ( 1991 ). [4] H. Janocha , Adaptronics and Smart Structures . Springer, New York ( 1999 ). [5] K. Lurié , 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 · Zbl 1035.78021 [6] S. Pohozaev , On a class of quasilinear hyperbolic equations . Math. USSR Sbornik 25 ( 1975 ) 145 - 158 . Zbl 0328.35060 · Zbl 0328.35060 · doi:10.1070/SM1975v025n01ABEH002203 [7] J. Restorff , Magnetostrictive materials and devices , in Encyclopedia of Applied Physics, Vol. 9. VCH Publishers ( 1994 ). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.