The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input $\to$ state” map in $L_2$-norms is established. A structure of the reachable sets for arbitrary $T>0$ is studied. In general case, only the first component $u(·,T)$ of the complete state $\{u(·,T),u_t(·,T)\}$ may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation. If $T_0$ exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input $\to$ output” map) $R^{2T_0}$ uniquely determines $R^T$ for any $T>0$. A procedure recovering $R^{\infty }$ via $R^{2T_0}$ is also described.

[1] S.A. Avdonin, M.I. Belishev and S.A. Ivanov, The controllability in the filled domain for the higher dimensional wave equation with the singular boundary control. Zapiski Nauch. Semin. POMI 210 (1994) 7-21. English translation: J. Math. Sci. 83 (1997). | MR 1334739 | Zbl 0870.93004

[2] C. Bardos, T. Masrour and F. Tatout, Observation and control of Elastic waves. IMA Vol. in Math. Appl. Singularities and Oscillations 191 (1996) 1-16. | MR 1601265 | Zbl 0879.35094

[3] M.I. Belishev, Canonical model of a dynamical system with boundary control in the inverse problem of heat conductivity. St-Petersburg Math. J. 7 (1996) 869-890. | MR 1381977 | Zbl 0866.35134

[4] M.I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC-method). Inv. Prob. 13 (1997) R1-R45. | Zbl 0990.35135

[5] M.I. Belishev, On relations between spectral and dynamical inverse data. J. Inv. Ill-Posed Problems 9 (2001) 547-565. | MR 1881562 | Zbl 0992.35114

[6] M.I. Belishev, Dynamical systems with boundary control: Models and characterization of inverse data. Inv. Prob. 17 (2001) 659-682. | MR 1861475 | Zbl 0988.35164

[7] M.I. Belishev and A.K. Glasman, Boundary control of the Maxwell dynamical system: Lack of controllability by topological reasons. ESAIM: COCV 5 (2000) 207-217. | Numdam | MR 1750615 | Zbl 1121.93307

[8] M.S. Birman and M.Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space. D. Reidel Publishing Comp. (1987). | MR 1192782 | Zbl 0744.47017

[9] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for maxwell's and elasticity systems, in Nonlinear PDE, College de France Seminar J.-L. Lions. Series in Appl. Math. 7 (2002). | Zbl 1038.35159

[10] V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, New-York (1998). | Zbl 0908.35134

[11] F. John, On linear partial differential equations with analytic coefficients. Unique continuation of data. Comm. Pure Appl. Math. 2 (1948) 209-253. | MR 36930 | Zbl 0035.34601

[12] M.G. Krein, On the problem of extension of the Hermitian positive continuous functions. Dokl. Akad. Nauk SSSR 26 (1940) 17-21. | Zbl 0022.35302

[13] I. Lasiecka, J.-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65 (1986) 149-192. | MR 867669 | Zbl 0631.35051

[14] I. Lasiecka, Uniform decay rates for full von Karman system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation. Comm. on PDE's 24 (1999) 1801-1849. | Zbl 0934.35195

[15] I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2$ boundary input hyperbolic equations. Appl. Math. Optim. 7 (1981) 35-93. | MR 600559 | Zbl 0473.35022

[16] I. Lasiecka and R. Triggiani, A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations. Proc. AMS 104 (1988) 745-755. | MR 964851 | Zbl 0699.47034

[17] R. Leis, Initial boundary value problems in Mathematical Physics. John Wiley - Sons LTD and B.G. Teubner, Stuttgart (1986). | Zbl 0599.35001

[18] D.L. Russell, Boundary value control theory of the higher dimensional wave equation. SIAM J. Control 9 (1971) 29-42. | MR 274917 | Zbl 0204.46201

[19] M. Sova, Cosine Operator Functions. Rozprawy matematyczne XLIX (1966). | MR 193525 | Zbl 0156.15404

[20] D. Tataru, Unique continuation for solutions of PDE's: Between Hormander's and Holmgren theorem. Comm. PDE 20 (1995) 855-894. | Zbl 0846.35021

[21] N. Weck, Aussenraumaufgaben in der Theorie stationärer Schwingungen inhomogener elastischer Körper. Math. Z. 111 (1969) 387-398. | MR 263295 | Zbl 0176.09202