Abstract
We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of \(2\)-coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger \(2\)-coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).
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References
Alabau-Boussouira F (2001) Indirect boundary observability of a weakly coupled wave system. C R Acad Sci Paris 333 (Sér I):645–650
Alabau-Boussouira F (2003) A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems. Siam J Control Opt 42:871–906
Alabau-Boussouira F, Léautaud M (2011) Indirect controllability of locally coupled systems under geometric conditions. C R Acad Sci Paris 349 (Sér I):395–400
Alabau-Boussouira F, Léautaud M (2013) Indirect controllability of locally coupled wave-type systems and applications. J Math Pures Appl 99:544–576
Alabau-Boussouira F (2012) Controllability of cascade coupled systems of multi-dimensional evolution PDE’s by a reduced number of controls. C R Acad Sci Paris 350 (Sér I):577–582 (see also arXiv:1109.6645v2)
Alabau-Boussouira F, Ammari K (2011) Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system. J Funct Anal 260:2424–2450
Alabau-Boussouira F (2012) A hierarchic multi-levels energy method for the control of bi-diagonal and mixed n-coupled cascade systems of PDEs by a reduced number of controls (Submitted)
Ammar-Khodja F, Benabdallah A, Dupaix C (2006) Null controllability of some reaction-diffusion systems with one control force. J Math Anal Appl 320:928–943
Ammar-Khodja F, Benabdallah A, Dupaix C, González-Burgos M (2009) A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems. J Evol Equ 9:267–291
Ammar-Khodja F, Benabdallah A, González-Burgos M, de Teresa L (2011) Recent results on the controllability of linear coupled parabolic problems: a survey. Math Control Relat Fields 1:267–306
Bardos C, Lebeau G, Rauch J (1992) Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J Control Opt 30:1024–1065
Bodart O, Fabre C (1995) Controls insensitizing the norm of the solution of a semilinear heat equation. J Math Anal Appl 195:658–683
Bodart O, González-Burgos M, Pérez-García R (2004) Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity. Comm Partial Differ Equ 29(2004):1017–1050
Bodart O, González-Burgos M, Pérez-García R (2004) A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions. SIAM J Control Opt 43:95–969
Burq N (1997) Contrôlabilité exacte des ondes dans des ouverts peu réguliers. Asymptot Anal 14:157–191
Burq N, Gérard P (1997) Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C R Acad Sci Paris Sér I Math 325:749–752
Coron J-M (2007) Control and nonlinearity. In: Mathematical surveys and monographs, vol 136. American Mathematical Society, Providence
Coron J-M, Guerrero S, Rosier L (2010) Null controllability of a parabolic system with a cubic coupling term. SIAM J Control Opt 48:5629–5653
Dáger R (2006) Insensitizing controls for the 1-D wave equation. Siam J Control Opt 45:1758–1768
Dehman B, Le Rousseau J, Léautaud M (2012) Controllability of two coupled wave equations on a compact manifold. HAL hal-00686967 (Preprint)
Ervedoza S, Zuazua E (2011) Observability of heat processes by transmutation without geometric restrictions. Math Control Relat Fields 1:177–187
Ervedoza S, Zuazua E (2011) Sharp observability estimates for heat equations. Arch Rational Mech Anal 202:975–1017
Fernández-Cara E, González-Burgos M, de Teresa L (2010) Boundary controllability of parabolic coupled equations. J Funct Anal 259:1720–1758
González-Burgos M, de Teresa L (2010) Controllability results for cascade systems of \(m\) coupled parabolic PDEs by one control force. Port Math 67:91–113
Guerrero S (2007) Controllability of systems of Stokes equations with one control force: existence of insensitizing controls. Annales de l’Institut Henri Poincaré Analyse Non Linéaire 24:1029–1054
Gueye M (2012) Uniqueness results for Stokes cascade systems and application to insensitizing controls. C R Acad Sci Paris 350 (Sér I):831–835
Gueye M (2012) Insensitizing controls for the Navier-Stokes equations. Accepté pour publication aux Annales IHP. ArXiv:1207.3255
Jaffard S (1990) Contrôle interne exact des vibrations d’une plaque rectangulaire. Port Math 47:423–429
Kavian O, de Teresa L (2010) Unique continuation principle for systems of parabolic equations. ESAIM COCV 16:247–274
Komornik V (1997) Rapid boundary stabilization of linear distributed systems. Siam J Control Opt 35:1591–1613
Lasiecka I (2002) Mathematical control theory of coupled PDEs. In: CBMS-NSF regional conference series in applied mathematics, vol 75. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
Léautaud M (2010) Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems. J Funct Anal 258:2739–2778
Léautaud M (2011) Quelques problèmes de contrôle d’équations aux dérivées partielles : inégalités spectrales, systèmes couplés et limites singulières. Thèse de doctorat de l’université de Paris 6
Lions JL (1988) Contrôlabilité exacte et stabilisation de systèmes distribués. vol 1, 2. Masson, Paris
Lions JL (1989) Remarques préliminaires sur le contrôle des systèmes à données incomplètes. Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54
Mauffrey K (2013) On the null controllability of a 3 \(x\) 3 parabolic system with non constant coefficients by one or two control forces. J Math Pures Appl 99:187–210
Miller L (2004) Geometric bounds on the growth rate of the null-controllability cost for the heat equation in small time. J Differ Equ 204:202–226
Miller L (2005) Controllability cost of conservative systems: resolvent condition and transmutation. J Funct Anal 218:425–444
Miller L (2006) The control transmutation method and the cost of fast controls. Siam J Cont Opt 45:762–772
Olive G (2012) Null-controllability for some linear parabolic systems with controls acting on different parts of the domain and its boundary. Math Control Signals Syst 23:257–280
Phung K-D (2001) Observability and control of Schrödinger equations. Siam J Cont Opt 40:211–230
Rosier L, de Teresa L (2011) Exact controllability of a cascade system of conservative equations. C R Acad Sci Paris 349(Ser I):291–296
Seidman T (1984) Two results on exact boundary control of parabolic equations. Appl Math Opt 11:145–152
Tebou L (2008) Locally distributed desensitizing controls for the wave equation. C R Acad Sci Paris 346(Sér I):407–412
Tebou L (2011) Some results on the controllability of coupled semilinear wave equations: the desensitizing control case. Siam J Control Opt 49:1221–1238
Tenenbaum G, Tucsnak M (2009) Fast and strongly localized observation for the schrödinger equation. Trans AMS 351:951–977
de Teresa L (2000) Insensitizing controls for a semilinear heat equation. CPDE 25:39–72
de Teresa L, Zuazua E (2009) Identification of the class of initial data for the insensitizing control of the heat equation. CPAA 8:457–471
Acknowledgments
I would like to thank Luc Miller for helpful discussions on the sharpness of the Geometric Control Condition and on the transmutation method. I am also very grateful to the referees for their careful reading, and valuable comments and suggestions.
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Alabau-Boussouira, F. Insensitizing exact controls for the scalar wave equation and exact controllability of \(2\)-coupled cascade systems of PDE’s by a single control. Math. Control Signals Syst. 26, 1–46 (2014). https://doi.org/10.1007/s00498-013-0112-8
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DOI: https://doi.org/10.1007/s00498-013-0112-8