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August  2015, 35(8): 3721-3743. doi: 10.3934/dcds.2015.35.3721

Simultaneous controllability of some uncoupled semilinear wave equations

1. 

Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received  July 2014 Revised  December 2014 Published  February 2015

We consider the exact controllability problem for some uncoupled semilinear wave equations with proportional, but different principal operators in a bounded domain. The control is locally distributed, and its support satisfies the geometric control condition of Bardos-Lebeau-Rauch. First, we examine the case of a nonlinearity that is asymptotically linear; using a combination of the Bardos-Lebeau-Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we solve the underlying linear control problem. The linear controllability result thus established, generalizes to higher space dimensions an earlier result of Haraux established in the one-dimensional setting. Then, applying a fixed point argument, we derive the controllability of the nonlinear problem. Afterwards, we use an iterative approach to prove a local controllability result when the nonlinearity is super-linear. Finally, we discuss some extensions of our results and some open problems.
Citation: Louis Tebou. Simultaneous controllability of some uncoupled semilinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721
References:
[1]

F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42 (2003), 871-906. doi: 10.1137/S0363012902402608.

[2]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.

[3]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Math. Acad. Sci. Paris, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.

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S. A. Avdonin and S. A. Ivanov, Families of Exponentials, The method of moments in controllability problems for distributed parameter systems. Translated from the Russian and revised by the authors. Cambridge University Press, Cambridge, 1995.

[5]

S. A. Avdonin, S. A. Ivanov and D. L. Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 947-970. doi: 10.1017/S0308210500000512.

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[7]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[8]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Ser. I Math., 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[9]

P. Cannarsa, V. Komornik and P. Loreti, One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms, Discrete Contin. Dyn. Syst., 8 (2002), 745-756. doi: 10.3934/dcds.2002.8.747.

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G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81. doi: 10.1137/0317007.

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G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-273.

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G. Chen, Control and stabilization for the wave equation in a bounded domain. II, SIAM J. Control Optim., 19 (1981), 114-122. doi: 10.1137/0319009.

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W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control Optim., 14 (1976), 19-25. doi: 10.1137/0314002.

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T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability for parabolic and hyperbolic systems with potentials, Ann. I.H.Poincaré-AN, 25 (2008), 1-41. doi: 10.1016/j.anihpc.2006.07.005.

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X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614. doi: 10.1137/040610222.

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A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1996.

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K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region, SIAM J. Control, 13 (1975), 174-196. doi: 10.1137/0313011.

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A. Haraux, Contrôlabilité exacte d'une membrane rectangulaire au moyen d'une fonctionnelle analytique localisée, C. R. Acad. Sci. Paris Ser. I Math., 306 (1988), 125-128.

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A. Haraux, On a completion problem in the theory of distributed control of wave equations. Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. X (Paris, 1987-1988), 241-271, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

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A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258.

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A. Haraux, A generalized internal control for the wave equation in a rectangle, J. Math. Anal. Appl., 153 (1990), 190-216. doi: 10.1016/0022-247X(90)90273-I.

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A. Haraux, An alternative functional approach to exact controllability of reversible systems, Port. Math. (N.S.), 61 (2004), 399-437.

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L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446.

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L. F. Ho, Exact controllability of second order hyperbolic systems with control in the Dirichlet boundary conditions, J. Math. Pures Appl., 66 (1987), 363-368.

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L. Hormander, Linear Partial Differential Operators, Springer-Verlag, New-York, 1976.

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O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations, Asympt. Anal., 32 (2002), 185-220.

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V. Komornik, Exact controllability in short time for the wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153-164.

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V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994.

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V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.

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J. Lagnese, Control of wave processes with distributed control supported on a subregion, S.I.A.M J. Control and Opt., 21 (1983), 68-85. doi: 10.1137/0321004.

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I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201.

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I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154. doi: 10.1007/BF01442394.

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I. Lasiecka and R. Triggiani, Roberto Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument, Discrete Contin. Dyn. Syst., (2005), suppl., 556-565.

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I. Lasiecka, R. Triggiani and P. F. Yao, An observability estimate in $L_2(\Omega)\times H^{-1}(\Omega)$ for second-order hyperbolic equations with variable coefficients, Control of distributed parameter and stochastic systems (Hangzhou, 1998), 71-78, Kluwer Acad. Publ., Boston, MA, 1999.

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I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57. doi: 10.1006/jmaa.1999.6348.

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I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), 227-325, Contemp. Math., 268, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04315.

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L. Li and X. Zhang, Exact Controllability for Semilinear wave equations, JMAA, 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998.

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W-J. Liu, Exact distributed controllability for the semilinear wave equation, Portugal. Math., 57 (2000), 493-508.

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K. Liu, M. Yamamoto and X. Zhang, Observability inequalities by internal observation and their applications, J. Optim. Theory Appl., 116 (2003), 621-645. doi: 10.1023/A:1023069420681.

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show all references

References:
[1]

F. Alabau-Boussouira, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42 (2003), 871-906. doi: 10.1137/S0363012902402608.

[2]

F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems, SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.

[3]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled systems under geometric conditions, C. R. Math. Acad. Sci. Paris, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.

[4]

S. A. Avdonin and S. A. Ivanov, Families of Exponentials, The method of moments in controllability problems for distributed parameter systems. Translated from the Russian and revised by the authors. Cambridge University Press, Cambridge, 1995.

[5]

S. A. Avdonin, S. A. Ivanov and D. L. Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 947-970. doi: 10.1017/S0308210500000512.

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[7]

N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers, Asymptot. Anal., 14 (1997), 157-191.

[8]

N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Acad. Sci. Paris Ser. I Math., 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.

[9]

P. Cannarsa, V. Komornik and P. Loreti, One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms, Discrete Contin. Dyn. Syst., 8 (2002), 745-756. doi: 10.3934/dcds.2002.8.747.

[10]

G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control Optim., 17 (1979), 66-81. doi: 10.1137/0317007.

[11]

G. Chen, Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures Appl., 58 (1979), 249-273.

[12]

G. Chen, Control and stabilization for the wave equation in a bounded domain. II, SIAM J. Control Optim., 19 (1981), 114-122. doi: 10.1137/0319009.

[13]

W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control Optim., 14 (1976), 19-25. doi: 10.1137/0314002.

[14]

T. Duyckaerts, X. Zhang and E. Zuazua, On the optimality of the observability for parabolic and hyperbolic systems with potentials, Ann. I.H.Poincaré-AN, 25 (2008), 1-41. doi: 10.1016/j.anihpc.2006.07.005.

[15]

H. O. Fattorini, Local controllability of a nonlinear wave equation, Math. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123.

[16]

X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations, SIAM J. Control Optim., 46 (2007), 1578-1614. doi: 10.1137/040610222.

[17]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes, Vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1996.

[18]

K. D. Graham and D. L. Russell, Boundary value control of the wave equation in a spherical region, SIAM J. Control, 13 (1975), 174-196. doi: 10.1137/0313011.

[19]

A. Haraux, Contrôlabilité exacte d'une membrane rectangulaire au moyen d'une fonctionnelle analytique localisée, C. R. Acad. Sci. Paris Ser. I Math., 306 (1988), 125-128.

[20]

A. Haraux, On a completion problem in the theory of distributed control of wave equations. Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. X (Paris, 1987-1988), 241-271, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.

[21]

A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math., 46 (1989), 245-258.

[22]

A. Haraux, A generalized internal control for the wave equation in a rectangle, J. Math. Anal. Appl., 153 (1990), 190-216. doi: 10.1016/0022-247X(90)90273-I.

[23]

A. Haraux, An alternative functional approach to exact controllability of reversible systems, Port. Math. (N.S.), 61 (2004), 399-437.

[24]

L. F. Ho, Observabilité frontière de l'équation des ondes, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446.

[25]

L. F. Ho, Exact controllability of second order hyperbolic systems with control in the Dirichlet boundary conditions, J. Math. Pures Appl., 66 (1987), 363-368.

[26]

L. Hormander, Linear Partial Differential Operators, Springer-Verlag, New-York, 1976.

[27]

O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations, Asympt. Anal., 32 (2002), 185-220.

[28]

V. Komornik, Exact controllability in short time for the wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153-164.

[29]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM, Masson & John Wiley, Paris, 1994.

[30]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.

[31]

J. Lagnese, Control of wave processes with distributed control supported on a subregion, S.I.A.M J. Control and Opt., 21 (1983), 68-85. doi: 10.1137/0321004.

[32]

I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201.

[33]

I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Appl. Math. Optim., 23 (1991), 109-154. doi: 10.1007/BF01442394.

[34]

I. Lasiecka and R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled, nonconservative second-order hyperbolic equations. Partial differential equation methods in control and shape analysis (Pisa), 215-243, Lecture Notes in Pure and Appl. Math., 188, Dekker, New York, 1997.

[35]

I. Lasiecka and R. Triggiani, Roberto Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument, Discrete Contin. Dyn. Syst., (2005), suppl., 556-565.

[36]

I. Lasiecka, R. Triggiani and P. F. Yao, An observability estimate in $L_2(\Omega)\times H^{-1}(\Omega)$ for second-order hyperbolic equations with variable coefficients, Control of distributed parameter and stochastic systems (Hangzhou, 1998), 71-78, Kluwer Acad. Publ., Boston, MA, 1999.

[37]

I. Lasiecka, R. Triggiani and P. F. Yao, Inverse/observability estimates for second-order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235 (1999), 13-57. doi: 10.1006/jmaa.1999.6348.

[38]

I. Lasiecka, R. Triggiani and X. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999), 227-325, Contemp. Math., 268, Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/conm/268/04315.

[39]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.

[40]

L. Li and X. Zhang, Exact Controllability for Semilinear wave equations, JMAA, 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998.

[41]

J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.

[42]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 1, RMA 8, Masson, Paris, 1988.

[43]

J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués, Vol. 2, RMA 9, Masson, Paris, 1988.

[44]

J. L. Lions, An introduction to the methods based on uniqueness for exact controllability of distributed systems, Control of partial differential equations (Santiago de Compostela, 1987), 35-46, Lecture Notes in Control and Inform. Sci., 114, Springer, Berlin, 1989. doi: 10.1007/BFb0002578.

[45]

K. Liu, Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928.

[46]

W-J. Liu, Exact distributed controllability for the semilinear wave equation, Portugal. Math., 57 (2000), 493-508.

[47]

K. Liu, M. Yamamoto and X. Zhang, Observability inequalities by internal observation and their applications, J. Optim. Theory Appl., 116 (2003), 621-645. doi: 10.1023/A:1023069420681.

[48]

L. Markus, Controllability of nonlinear processes, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 78-90. doi: 10.1137/0303008.

[49]

P. Martinez and J. Vancostenoble, Exact controllability in "arbitrarily short time" of the semilinear wave equation, Discrete Contin. Dyn. Syst., 9 (2003), 901-924. doi: 10.3934/dcds.2003.9.901.

[50]

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