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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 & Continuous Dynamical Systems - A, 2015, 35 (8) : 3721-3743. doi: 10.3934/dcds.2015.35.3721
##### References:
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Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation,, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 947. doi: 10.1017/S0308210500000512. Google Scholar [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. doi: 10.1137/0330055. Google Scholar [7] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar [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. doi: 10.1016/S0764-4442(97)80053-5. Google Scholar [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. doi: 10.3934/dcds.2002.8.747. Google Scholar [10] G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66. doi: 10.1137/0317007. Google Scholar [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. Google Scholar [12] G. Chen, Control and stabilization for the wave equation in a bounded domain. II,, SIAM J. Control Optim., 19 (1981), 114. doi: 10.1137/0319009. Google Scholar [13] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables,, SIAM J. Control Optim., 14 (1976), 19. doi: 10.1137/0314002. Google Scholar [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. doi: 10.1016/j.anihpc.2006.07.005. Google Scholar [15] H. O. Fattorini, Local controllability of a nonlinear wave equation,, Math. Systems Theory, 9 (1975), 30. doi: 10.1007/BF01698123. Google Scholar [16] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations,, SIAM J. Control Optim., 46 (2007), 1578. doi: 10.1137/040610222. Google Scholar [17] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture Notes, (1996). Google Scholar [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. doi: 10.1137/0313011. Google Scholar [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. Google Scholar [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, (1991), 1987. Google Scholar [21] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245. Google Scholar [22] A. Haraux, A generalized internal control for the wave equation in a rectangle,, J. Math. Anal. Appl., 153 (1990), 190. doi: 10.1016/0022-247X(90)90273-I. Google Scholar [23] A. Haraux, An alternative functional approach to exact controllability of reversible systems,, Port. Math. (N.S.), 61 (2004), 399. Google Scholar [24] L. F. Ho, Observabilité frontière de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443. Google Scholar [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. Google Scholar [26] L. Hormander, Linear Partial Differential Operators,, Springer-Verlag, (1976). Google Scholar [27] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations,, Asympt. Anal., 32 (2002), 185. Google Scholar [28] V. Komornik, Exact controllability in short time for the wave equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153. Google Scholar [29] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994). Google Scholar [30] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer Monographs in Mathematics. Springer-Verlag, (2005). Google Scholar [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. doi: 10.1137/0321004. Google Scholar [32] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Appl. Math. Optim., 19 (1989), 243. doi: 10.1007/BF01448201. Google Scholar [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. doi: 10.1007/BF01442394. Google Scholar [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), (1997), 215. Google Scholar [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), 556. Google Scholar [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. Google Scholar [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. doi: 10.1006/jmaa.1999.6348. Google Scholar [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, (1999), 227. doi: 10.1090/conm/268/04315. Google Scholar [39] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. doi: 10.1080/03605309508821097. Google Scholar [40] L. Li and X. Zhang, Exact Controllability for Semilinear wave equations,, JMAA, 250 (2000), 589. doi: 10.1006/jmaa.2000.6998. Google Scholar [41] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [42] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988). Google Scholar [43] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 2, (1988). Google Scholar [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. doi: 10.1007/BFb0002578. Google Scholar [45] K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt., 35 (1997), 1574. doi: 10.1137/S0363012995284928. Google Scholar [46] W-J. Liu, Exact distributed controllability for the semilinear wave equation,, Portugal. Math., 57 (2000), 493. Google Scholar [47] K. Liu, M. Yamamoto and X. Zhang, Observability inequalities by internal observation and their applications,, J. Optim. Theory Appl., 116 (2003), 621. doi: 10.1023/A:1023069420681. Google Scholar [48] L. Markus, Controllability of nonlinear processes,, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 78. doi: 10.1137/0303008. Google Scholar [49] P. Martinez and J. Vancostenoble, Exact controllability in "arbitrarily short time" of the semilinear wave equation,, Discrete Contin. Dyn. Syst., 9 (2003), 901. doi: 10.3934/dcds.2003.9.901. Google Scholar [50] L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time,, J. Differential Equations, 204 (2004), 202. doi: 10.1016/j.jde.2004.05.007. Google Scholar [51] A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control,, SIAM J. Control Optim., 40 (2001), 777. doi: 10.1137/S0363012998345615. Google Scholar [52] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [53] L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. Sci. Paris, 349 (2011), 291. doi: 10.1016/j.crma.2011.01.014. Google Scholar [54] H. L. Royden, Real Analysis, Third edition. Prentice-Hall,, New Jersey, (1988). Google Scholar [55] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential,, J. Math. Pures Appl., 71 (1992), 455. Google Scholar [56] D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems,, J. Math. Anal. Appl., 18 (1967), 542. doi: 10.1016/0022-247X(67)90045-5. Google Scholar [57] D. L. Russell, Boundary value control of the higher-dimensional wave equation,, SIAM J. Control, 9 (1971), 29. doi: 10.1137/0309004. Google Scholar [58] D. L. Russell, Boundary value control theory of the higher-dimensional wave equation. II,, SIAM J. Control, 9 (1971), 401. doi: 10.1137/0309030. Google Scholar [59] D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory (an outline), Ordinary differential equations,, Proc. Conf., (1971), 241. Google Scholar [60] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,, Studies in Appl. Math., 52 (1973), 189. Google Scholar [61] D. L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions,, Differential games and control theory (Proc. NSF-CBMS Regional Res. Conf., (1973), 291. Google Scholar [62] D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions,, SIAM Rev., 20 (1978), 639. doi: 10.1137/1020095. Google Scholar [63] D. L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region,, SIAM J. 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##### 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. doi: 10.1137/S0363012902402608. Google Scholar [2] F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems,, SIAM J. Control Optim., 37 (1999), 521. doi: 10.1137/S0363012996313835. Google Scholar [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. doi: 10.1016/j.crma.2011.02.004. Google Scholar [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, (1995). Google Scholar [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. doi: 10.1017/S0308210500000512. Google Scholar [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. doi: 10.1137/0330055. Google Scholar [7] N. Burq, Contrôlabilité exacte des ondes dans des ouverts peu réguliers,, Asymptot. Anal., 14 (1997), 157. Google Scholar [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. doi: 10.1016/S0764-4442(97)80053-5. Google Scholar [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. doi: 10.3934/dcds.2002.8.747. Google Scholar [10] G. Chen, Control and stabilization for the wave equation in a bounded domain,, SIAM J. Control Optim., 17 (1979), 66. doi: 10.1137/0317007. Google Scholar [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. Google Scholar [12] G. Chen, Control and stabilization for the wave equation in a bounded domain. II,, SIAM J. Control Optim., 19 (1981), 114. doi: 10.1137/0319009. Google Scholar [13] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables,, SIAM J. Control Optim., 14 (1976), 19. doi: 10.1137/0314002. Google Scholar [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. doi: 10.1016/j.anihpc.2006.07.005. Google Scholar [15] H. O. Fattorini, Local controllability of a nonlinear wave equation,, Math. Systems Theory, 9 (1975), 30. doi: 10.1007/BF01698123. Google Scholar [16] X. Fu, J. Yong and X. Zhang, Exact controllability for multidimensional semilinear hyperbolic equations,, SIAM J. Control Optim., 46 (2007), 1578. doi: 10.1137/040610222. Google Scholar [17] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations,, Lecture Notes, (1996). Google Scholar [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. doi: 10.1137/0313011. Google Scholar [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. Google Scholar [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, (1991), 1987. Google Scholar [21] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps,, Portugal. Math., 46 (1989), 245. Google Scholar [22] A. Haraux, A generalized internal control for the wave equation in a rectangle,, J. Math. Anal. Appl., 153 (1990), 190. doi: 10.1016/0022-247X(90)90273-I. Google Scholar [23] A. Haraux, An alternative functional approach to exact controllability of reversible systems,, Port. Math. (N.S.), 61 (2004), 399. Google Scholar [24] L. F. Ho, Observabilité frontière de l'équation des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443. Google Scholar [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. Google Scholar [26] L. Hormander, Linear Partial Differential Operators,, Springer-Verlag, (1976). Google Scholar [27] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations,, Asympt. Anal., 32 (2002), 185. Google Scholar [28] V. Komornik, Exact controllability in short time for the wave equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153. Google Scholar [29] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM, (1994). Google Scholar [30] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer Monographs in Mathematics. Springer-Verlag, (2005). Google Scholar [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. doi: 10.1137/0321004. Google Scholar [32] I. Lasiecka and R. Triggiani, Exact controllability of the wave equation with Neumann boundary control,, Appl. Math. Optim., 19 (1989), 243. doi: 10.1007/BF01448201. Google Scholar [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. doi: 10.1007/BF01442394. Google Scholar [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), (1997), 215. Google Scholar [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), 556. Google Scholar [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. Google Scholar [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. doi: 10.1006/jmaa.1999.6348. Google Scholar [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, (1999), 227. doi: 10.1090/conm/268/04315. Google Scholar [39] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur,, Comm. Partial Differential Equations, 20 (1995), 335. doi: 10.1080/03605309508821097. Google Scholar [40] L. Li and X. Zhang, Exact Controllability for Semilinear wave equations,, JMAA, 250 (2000), 589. doi: 10.1006/jmaa.2000.6998. Google Scholar [41] J. L. Lions, Exact controllability, stabilization and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [42] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 1, (1988). Google Scholar [43] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation Des Systèmes Distribués,, Vol. 2, (1988). Google Scholar [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. doi: 10.1007/BFb0002578. Google Scholar [45] K. Liu, Locally distributed control and damping for the conservative systems,, S.I.A.M J. Control and Opt., 35 (1997), 1574. doi: 10.1137/S0363012995284928. Google Scholar [46] W-J. Liu, Exact distributed controllability for the semilinear wave equation,, Portugal. Math., 57 (2000), 493. Google Scholar [47] K. Liu, M. Yamamoto and X. Zhang, Observability inequalities by internal observation and their applications,, J. Optim. Theory Appl., 116 (2003), 621. doi: 10.1023/A:1023069420681. Google Scholar [48] L. Markus, Controllability of nonlinear processes,, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 78. doi: 10.1137/0303008. Google Scholar [49] P. Martinez and J. Vancostenoble, Exact controllability in "arbitrarily short time" of the semilinear wave equation,, Discrete Contin. Dyn. Syst., 9 (2003), 901. doi: 10.3934/dcds.2003.9.901. Google Scholar [50] L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time,, J. Differential Equations, 204 (2004), 202. doi: 10.1016/j.jde.2004.05.007. Google Scholar [51] A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control,, SIAM J. Control Optim., 40 (2001), 777. doi: 10.1137/S0363012998345615. Google Scholar [52] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [53] L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations,, C. R. Math. Acad. 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