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June  2018, 8(2): 397-410. doi: 10.3934/mcrf.2018016

Compact perturbations of controlled systems

1. 

Institut de Mathématique de Marseille, Aix Marseille Université, 39, rue J. Joliot Curie, 13453 Marseille Cedex 13, France

2. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, Cours de la Libération, 33405 Talence, France

* Corresponding author: Guillaume Olive

Received  December 2016 Revised  February 2018 Published  March 2018

In this article we study the controllability properties of general compactly perturbed exactly controlled linear systems with admissible control operators. Firstly, we show that approximate and exact controllability are equivalent properties for such systems. Then, and more importantly, we provide for the perturbed system a complete characterization of the set of reachable states in terms of the Fattorini-Hautus test. The results rely on the Peetre lemma.

Citation: Michel Duprez, Guillaume Olive. Compact perturbations of controlled systems. Mathematical Control & Related Fields, 2018, 8 (2) : 397-410. doi: 10.3934/mcrf.2018016
References:
[1]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.  Google Scholar

[2]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[3]

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields, 4 (2014), 263-287.  doi: 10.3934/mcrf.2014.4.263.  Google Scholar

[4]

N. Cȋndea and M. Tucsnak, Internal exact observability of a perturbed Euler-Bernoulli equation, Ann. Acad. Rom. Sci. Ser. Math. Appl., 2 (2010), 205-221.   Google Scholar

[5]

J.-M. CoronL. Hu and G. Olive, Stabilization and controllability of first-order integro-differential hyperbolic equations, J. Funct. Anal., 271 (2016), 3554-3587.  doi: 10.1016/j.jfa.2016.08.018.  Google Scholar

[6]

K. -L. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[7]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.  Google Scholar

[8]

E. Fernández-CaraQ. Lü and E. Zuazua, Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54 (2016), 2009-2019.  doi: 10.1137/15M1044291.  Google Scholar

[9]

S. Hadd, Unbounded perturbations of C0-semigroups on Banach spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7.  Google Scholar

[10]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.  Google Scholar

[11]

V. Komornik and P. Loreti, Observability of compactly perturbed systems, J. Math. Anal. Appl., 243 (2000), 409-428.  doi: 10.1006/jmaa.1999.6678.  Google Scholar

[12]

I. Lasiecka and R. Triggiani, Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument, Discrete Contin. Dyn. Syst., (2005), 556-565.  doi: 10.1006/jmaa.1999.6678.  Google Scholar

[13]

C. Laurent, Internal control of the Schrödinger equation, Math. Control Relat. Fields, 4 (2014), 161-186.  doi: 10.3934/mcrf.2014.4.161.  Google Scholar

[14]

T. Li and B. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2017), 473-488.  doi: 10.1007/s11401-017-1078-5.  Google Scholar

[15]

J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, vol. 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.  Google Scholar

[16]

M. Mehrenberger, Observability of coupled systems, Acta Math. Hungar., 103 (2004), 321-348.  doi: 10.1023/B:AMHU.0000028832.47891.09.  Google Scholar

[17]

A. F. NevesH. d. S. Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986), 320-344.  doi: 10.1016/0022-1236(86)90029-7.  Google Scholar

[18]

J. Peetre, Another approach to elliptic boundary problems, Comm. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.  Google Scholar

[19]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[20]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.  Google Scholar

[21]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optimization, 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[23]

E. Zuazua, Contrôlabilité exacte d'un modèle de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 173-176.   Google Scholar

[24]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), vol. 220 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1991,357-391.  Google Scholar

show all references

References:
[1]

M. Badra and T. Takahashi, On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems, ESAIM Control Optim. Calc. Var., 20 (2014), 924-956.  doi: 10.1051/cocv/2014002.  Google Scholar

[2]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[3]

F. Boyer and G. Olive, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields, 4 (2014), 263-287.  doi: 10.3934/mcrf.2014.4.263.  Google Scholar

[4]

N. Cȋndea and M. Tucsnak, Internal exact observability of a perturbed Euler-Bernoulli equation, Ann. Acad. Rom. Sci. Ser. Math. Appl., 2 (2010), 205-221.   Google Scholar

[5]

J.-M. CoronL. Hu and G. Olive, Stabilization and controllability of first-order integro-differential hyperbolic equations, J. Funct. Anal., 271 (2016), 3554-3587.  doi: 10.1016/j.jfa.2016.08.018.  Google Scholar

[6]

K. -L. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.  Google Scholar

[7]

H. O. Fattorini, Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.  doi: 10.1137/0304048.  Google Scholar

[8]

E. Fernández-CaraQ. Lü and E. Zuazua, Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54 (2016), 2009-2019.  doi: 10.1137/15M1044291.  Google Scholar

[9]

S. Hadd, Unbounded perturbations of C0-semigroups on Banach spaces and applications, Semigroup Forum, 70 (2005), 451-465.  doi: 10.1007/s00233-004-0172-7.  Google Scholar

[10]

V. Komornik, Exact Controllability and Stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994, The multiplier method.  Google Scholar

[11]

V. Komornik and P. Loreti, Observability of compactly perturbed systems, J. Math. Anal. Appl., 243 (2000), 409-428.  doi: 10.1006/jmaa.1999.6678.  Google Scholar

[12]

I. Lasiecka and R. Triggiani, Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument, Discrete Contin. Dyn. Syst., (2005), 556-565.  doi: 10.1006/jmaa.1999.6678.  Google Scholar

[13]

C. Laurent, Internal control of the Schrödinger equation, Math. Control Relat. Fields, 4 (2014), 161-186.  doi: 10.3934/mcrf.2014.4.161.  Google Scholar

[14]

T. Li and B. Rao, Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls, Chin. Ann. Math. Ser. B, 38 (2017), 473-488.  doi: 10.1007/s11401-017-1078-5.  Google Scholar

[15]

J. -L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, vol. 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988, Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.  Google Scholar

[16]

M. Mehrenberger, Observability of coupled systems, Acta Math. Hungar., 103 (2004), 321-348.  doi: 10.1023/B:AMHU.0000028832.47891.09.  Google Scholar

[17]

A. F. NevesH. d. S. Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986), 320-344.  doi: 10.1016/0022-1236(86)90029-7.  Google Scholar

[18]

J. Peetre, Another approach to elliptic boundary problems, Comm. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.  Google Scholar

[19]

J. Rauch and M. Taylor, Exponential decay of solutions to hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[20]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.  Google Scholar

[21]

R. Triggiani, A note on the lack of exact controllability for mild solutions in Banach spaces, SIAM J. Control Optimization, 15 (1977), 407-411.  doi: 10.1137/0315028.  Google Scholar

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[23]

E. Zuazua, Contrôlabilité exacte d'un modèle de plaques vibrantes en un temps arbitrairement petit, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 173-176.   Google Scholar

[24]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987-1988), vol. 220 of Pitman Res. Notes Math. Ser., Longman Sci. Tech., Harlow, 1991,357-391.  Google Scholar

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