On the well-posedness and regularity of the wave equation with variable coefficients

Guo, Bao-Zhu; Zhang, Zhi-Xiong

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.

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

[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 | Numdam | MR | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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

[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

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

[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 | Zbl

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

[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

[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

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

[19] R. Triggiani, Exact boundary controllability on $L^2\left(\Omega \right)\times H^{-1}\left(\Omega \right)$ 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

[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

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

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

[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

[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