On the well-posedness and regularity of the wave equation with variable coefficients
ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 776-792.

An open-loop system of a multidimensional wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation is considered. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The riemannian geometry method is used in the proof of regularity and the feedthrough operator is explicitly computed.

DOI : https://doi.org/10.1051/cocv:2007040
Classification : 35J50,  93C20,  93C25
Mots clés : wave equation, transfer function, well-posed and regular system, boundary control and observation
@article{COCV_2007__13_4_776_0,
author = {Guo, Bao-Zhu and Zhang, Zhi-Xiong},
title = {On the well-posedness and regularity of the wave equation with variable coefficients},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {776--792},
publisher = {EDP-Sciences},
volume = {13},
number = {4},
year = {2007},
doi = {10.1051/cocv:2007040},
mrnumber = {2351403},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv:2007040/}
}
TY  - JOUR
AU  - Guo, Bao-Zhu
AU  - Zhang, Zhi-Xiong
TI  - On the well-posedness and regularity of the wave equation with variable coefficients
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2007
DA  - 2007///
SP  - 776
EP  - 792
VL  - 13
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007040/
UR  - https://www.ams.org/mathscinet-getitem?mr=2351403
UR  - https://doi.org/10.1051/cocv:2007040
DO  - 10.1051/cocv:2007040
LA  - en
ID  - COCV_2007__13_4_776_0
ER  - 
Guo, Bao-Zhu; Zhang, Zhi-Xiong. On the well-posedness and regularity of the wave equation with variable coefficients. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 4, pp. 776-792. doi : 10.1051/cocv:2007040. http://www.numdam.org/articles/10.1051/cocv:2007040/

[1] K. Ammari, Dirichlet boundary stabilization of the wave equation. Asymptotic Anal. 30 (2002) 117-130. | MR 1919338 | Zbl 1020.35042

[2] K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM: COCV 6 (2001) 361-386. | EuDML 90598 | Numdam | MR 1836048 | Zbl 0992.93039

[3] C.I. Byrnes, D.S. Gilliam, V.I. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation. J. Dyn. Control Syst. 8 (2002) 341-370. | Zbl 1010.93052

[4] A. Cheng and K. Morris, Well-posedness of boundary control systems. SIAM J. Control Optim. 42 (2003) 1244-1265. | MR 2044794 | Zbl 1049.35041

[5] R.F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems: a survey. IMA J. Math. Control Inform. 14 (1997) 207-223. | MR 1470034 | Zbl 0880.93021

[6] R.F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), in Control and Estimation of Distributed Parameter Systems, F. Kappel, K. Kunisch and W. Schappacher Eds., Birkhäuser, Basel 91 (1989) 41-59. | MR 1033051 | Zbl 0686.93049

[7] R. Glowinski, J.W. He and J.L. Lions, On the controllability of wave models with variable coefficients: a numerical investigation. Comput. Appl. Math. 21 (2002) 191-225. | MR 2009952 | Zbl 1125.35309

[8] B.Z. Guo and Y.H. Luo, Controllability and stability of a second order hyperbolic system with collocated sensor/actuator. Syst. Control Lett. 46 (2002) 45-65. | MR 2011071 | Zbl 0994.93021

[9] B.Z. Guo and Z.C. Shao, Regularity of a Schrödinger equation with Dirichlet control and collocated observation. Syst. Control Lett. 54 (2005) 1135-1142. | Zbl 1129.35447

[10] B.Z. Guo and Z.C. Shao, Regularity of an Euler-Bernoulli plate equation with Neumann control and collocated observation. J. Dyn. Control Syst. 12 (2006) 405-418. | Zbl 1111.93033

[11] B.Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and collocated observation. SIAM J. Control Optim. 44 (2005) 1598-1613. | Zbl 1134.35318

[12] L. Hörmander, The Analysis of Linear Partial Differential Operators III. Springer-Verlag, Berlin (1985). | MR 781536 | Zbl 0601.35001

[13] B. Kellogg, Properties of elliptic boundary value problems, in Mathematical Foundations of the Finite Elements Methods. Academic Press, New York (1972) Chapter 3.

[14] V. Komornik, Exact controllability and stabilization: The Multiplier Method. John Wiley and Sons. Ltd., Chichester (1994). | MR 1359765 | Zbl 0937.93003

[15] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with ${L}^{2}\left(0,\infty ;{L}^{2}\left(\Gamma \right)\right)$-feedback control in the Dirichlet boundary conditions. J. Diff. Eqns. 66 (1987) 340-390. | Zbl 0629.93047

[16] I. Lasiecka and R. Triggiani, The operator ${B}^{*}L$ for the wave equation with Dirichlet control. Abstract Appl. Anal. N${}^{\circ }$7 (2004) 625-634. | Zbl 1065.35171

[17] I. Lasiecka, J.L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pure Appl. 65 (1986) 149-192. | Zbl 0631.35051

[18] R.B. Melrose and J. Sjöstrand, Singularities of boundary value problems I. Comm. Pure Appl. Math. 31 (1978) 593-617. | Zbl 0368.35020

[19] R. Triggiani, Exact boundary controllability on ${L}^{2}\left(\Omega \right)×{H}^{-1}\left(\Omega \right)$ of the wave equation with Dirichlet boundary control acting on a portion of the boundary $\partial \Omega$, and related problems. Appl. Math. Optim. 18 (1988) 241-277. | Zbl 0656.93011

[20] M. Tucsnak and G. Weiss, How to get a conservative well-posed linear system out of thin air II, controllability and stability. SIAM J. Control Optim. 42 (2003) 907-935. | Zbl 1125.93383

[21] G. Weiss, Transfer functions of regular linear systems I: characterizations of regularity. Trans. Amer. Math. Soc. 342 (1994) 827-854. | Zbl 0798.93036

[22] G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems. SIAM J. Control Optim. 39 (2000) 1204-1232. | Zbl 0981.93032

[23] G. Weiss, O.J. Staffans and M. Tucsnak, Well-posed linear systems-a survey with emphasis on conservative systems. Int. J. Appl. Math. Comput. Sci. 11 (2001) 7-33. | Zbl 0990.93046

[24] P.F. Yao, On the observability inequalities for exact controllablility of wave equations with variable coefficients. SIAM J. Control Optim. 37 (1999) 1568-1599. | Zbl 0951.35069

Cité par Sources :