The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input state” map in -norms is established. A structure of the reachable sets for arbitrary is studied. In general case, only the first component of the complete state 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 exceeds the time needed for shear waves to fill the entire domain, then the response operator (“input output” map) uniquely determines for any . A procedure recovering via is also described.
Classification : 93C20, 74B05, 35B65, 34K35
Mots clés : isotropic elasticity, dynamical Lame system, regularity of solutions, structure of sets reachable from the boundary in a short time, boundary controllability
@article{COCV_2002__8__143_0, author = {Belishev, M. I. and Lasiecka, Irena}, title = {The dynamical {Lame} system : regularity of solutions, boundary controllability and boundary data continuation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {143--167}, publisher = {EDP-Sciences}, volume = {8}, year = {2002}, doi = {10.1051/cocv:2002058}, zbl = {1064.93019}, mrnumber = {1932948}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2002058/} }
TY - JOUR AU - Belishev, M. I. AU - Lasiecka, Irena TI - The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2002 DA - 2002/// SP - 143 EP - 167 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2002058/ UR - https://zbmath.org/?q=an%3A1064.93019 UR - https://www.ams.org/mathscinet-getitem?mr=1932948 UR - https://doi.org/10.1051/cocv:2002058 DO - 10.1051/cocv:2002058 LA - en ID - COCV_2002__8__143_0 ER -
Belishev, M. I.; Lasiecka, I. The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 143-167. doi : 10.1051/cocv:2002058. http://www.numdam.org/articles/10.1051/cocv:2002058/
[1] 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
, and ,[2] Observation and control of Elastic waves. IMA Vol. in Math. Appl. Singularities and Oscillations 191 (1996) 1-16. | MR 1601265 | Zbl 0879.35094
, and ,[3] 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] Boundary control in reconstruction of manifolds and metrics (the BC-method). Inv. Prob. 13 (1997) R1-R45. | Zbl 0990.35135
,[5] On relations between spectral and dynamical inverse data. J. Inv. Ill-Posed Problems 9 (2001) 547-565. | MR 1881562 | Zbl 0992.35114
,[6] Dynamical systems with boundary control: Models and characterization of inverse data. Inv. Prob. 17 (2001) 659-682. | MR 1861475 | Zbl 0988.35164
,[7] 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
and ,[8] Spectral Theory of Self-Adjoint Operators in Hilbert Space. D. Reidel Publishing Comp. (1987). | MR 1192782 | Zbl 0744.47017
and ,[9] 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
, , and ,[10] Inverse Problems for Partial Differential Equations. Springer-Verlag, New-York (1998). | Zbl 0908.35134
,[11] 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] On the problem of extension of the Hermitian positive continuous functions. Dokl. Akad. Nauk SSSR 26 (1940) 17-21. | Zbl 0022.35302
,[13] Non homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65 (1986) 149-192. | MR 867669 | Zbl 0631.35051
, and ,[14] 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] A cosine operator approach to modeling boundary input hyperbolic equations. Appl. Math. Optim. 7 (1981) 35-93. | MR 600559 | Zbl 0473.35022
and ,[16] 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
and ,[17] Initial boundary value problems in Mathematical Physics. John Wiley - Sons LTD and B.G. Teubner, Stuttgart (1986). | Zbl 0599.35001
,[18] Boundary value control theory of the higher dimensional wave equation. SIAM J. Control 9 (1971) 29-42. | MR 274917 | Zbl 0204.46201
,[19] Cosine Operator Functions. Rozprawy matematyczne XLIX (1966). | MR 193525 | Zbl 0156.15404
,[20] Unique continuation for solutions of PDE's: Between Hormander's and Holmgren theorem. Comm. PDE 20 (1995) 855-894. | Zbl 0846.35021
,[21] Aussenraumaufgaben in der Theorie stationärer Schwingungen inhomogener elastischer Körper. Math. Z. 111 (1969) 387-398. | MR 263295 | Zbl 0176.09202
,Cité par Sources :