We show that a Kalman rank condition is necessary and sufficient for the uniqueness of solution to a system of wave equations associated with incomplete internal observation without any restriction neither on the controlled subregion nor on the coupling matrices. The obtained result can be applied to the approximate internal controllability of the corresponding system.
Accepted:
Published online:
DOI: 10.5802/crmath.341
Tatsien Li 1; Bopeng Rao 2
@article{CRMATH_2022__360_G6_729_0, author = {Tatsien Li and Bopeng Rao}, title = {Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability}, journal = {Comptes Rendus. Math\'ematique}, pages = {729--737}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.341}, zbl = {07547271}, language = {en}, }
TY - JOUR AU - Tatsien Li AU - Bopeng Rao TI - Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability JO - Comptes Rendus. Mathématique PY - 2022 SP - 729 EP - 737 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.341 LA - en ID - CRMATH_2022__360_G6_729_0 ER -
%0 Journal Article %A Tatsien Li %A Bopeng Rao %T Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability %J Comptes Rendus. Mathématique %D 2022 %P 729-737 %V 360 %I Académie des sciences, Paris %R 10.5802/crmath.341 %G en %F CRMATH_2022__360_G6_729_0
Tatsien Li; Bopeng Rao. Uniqueness theorem for a coupled system of wave equations with incomplete internal observation and application to approximate controllability. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 729-737. doi : 10.5802/crmath.341. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.341/
[1] Leader-follower synchronization and ISS analysis for a network of boundary-controlled wave PDEs, IEEE Control Sys. Lett., Volume 5 (2021) no. 2, pp. 683-688 | DOI | MR
[2] A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., Volume 42 (2003) no. 3, pp. 871-903 | DOI | MR | Zbl
[3] A hierarchic multi-level energy method for the control of bidiagonal and mixed -coupled cascade systems of PDE’s by a reduced number of controls, Adv. Differ. Equ., Volume 18 (2013) no. 11-12, pp. 1005-1073 | MR | Zbl
[4] A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., Volume 1 (2009) no. 3, pp. 427-457 | MR
[5] A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., Volume 9 (2009) no. 2, pp. 267-291 | DOI | Zbl
[6] Applied Functional Analysis, Applications of Mathematics, 3, Springer, 1976
[7] An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, Clarendon Press, 1998
[8] Controllability of two coupled wave equations on a compact manifold, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 1, pp. 113-187 | DOI | MR | Zbl
[9] Optimisation and adaptation of synchronisation controllers for networked second-order infinite-dimensional systems, Int. J. Control, Volume 92 (2019) no. 1, pp. 112-131 | DOI | MR | Zbl
[10] A generalized internal control for the wave equation in a rectangle, J. Math. Anal. Appl., Volume 153 (1990) no. 1, pp. 190-216 | DOI | MR | Zbl
[11] Contributions to the theory of optimal control, Bol. Soc. Mat. Mex., II. Ser., Volume 5 (1960), pp. 102-119 | MR | Zbl
[12] Control of wave processes with distributed controls on a subregion, SIAM J. Control Optim., Volume 21 (1983), pp. 68-85 | DOI | MR | Zbl
[13] Approximate internal controllability and synchronization of a coupled system of wave equations (in course)
[14] Criteria of Kalman type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls, SIAM J. Control Optim., Volume 54 (2016) no. 1, pp. 49-73 | MR | Zbl
[15] Kalman criterion on the uniqueness of continuation for the nilpotent system of wave equations, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 11-12, pp. 1188-1192 | MR | Zbl
[16] Boundary synchronization for hyperbolic systems, Progress in Nonlinear Differential Equations and their Applications, 94, Birkhäuser, 2019
[17] Uniqueness of solution to systems of elliptic operators and application to asymptotic synchronization of linear dissipative systems, ESAIM, Control Optim. Calc. Var., Volume 26 (2020), 117, 26 pages | DOI | MR | Zbl
[18] Approximate boundary synchronization by groups for a couples system of wave equationswith coupled Robin boundary conditions, ESAIM, Control Optim. Calc. Var., Volume 27 (2021), 10, 30 pages | Zbl
[19] Quelques méthodes de résolution des problèmes aux limites non linéaires, Études mathématiques, Gauthier-Villars, 1969 | Numdam
[20] Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1: Contrôlabilité exacte, Recherches en Mathématiques Appliquées, 8, Masson, 1988
[21] Internal controllability for parabolic systems involving analytic non-local terms, Chin. Ann. Math., Ser. B, Volume 39 (2018) no. 2, pp. 281-296 | DOI | MR | Zbl
[22] Internal observability for coupled systems of linear partial differential equations, SIAM J. Control Optim., Volume 57 (2019) no. 2, pp. 832-853 | DOI | MR | Zbl
[23] A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin. Dyn. Syst., Volume 23 (2009) no. 1-2, pp. 399-413 | MR | Zbl
[24] Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, 44, Springer, 1982
[25] Exact controllability of a cascade system of conservative equations, C. R. Math. Acad. Sci. Paris, Volume 349 (2011) no. 5-6, pp. 291-295 | DOI | MR | Zbl
[26] Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., Volume 18 (1967), pp. 542-560 | DOI | MR | Zbl
[27] Optimal control problem for exact synchronization of parabolic system, Math. Control Relat. Fields, Volume 9 (2019) no. 3, pp. 411-424 | DOI | MR | Zbl
[28] Sufficiency of Kalman rank condition for the approximate boundary controllability on spherical domain, Math. Methods Appl. Sci., Volume 44 (2021) no. 17, pp. 13509-13525 | MR | Zbl
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