• Previous Article
    Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities
  • DCDS Home
  • This Issue
  • Next Article
    Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems
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:
[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. 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. Control Optim., 24 (1986), 199. doi: 10.1137/0324012. Google Scholar

[64]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[65]

L. Tebou, Locally distributed desensitizing controls for the wave equation,, C.R.A.S. Paris, 346 (2008), 407. doi: 10.1016/j.crma.2008.02.019. Google Scholar

[66]

L. Tebou, Some results on the controllability of coupled semilinear wave equations: The desensitizing control case,, SIAM J. Control Optim., 49 (2011), 1221. doi: 10.1137/100803080. Google Scholar

[67]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Acad. Sci. Paris, 350 (2012), 57. doi: 10.1016/j.crma.2011.12.001. Google Scholar

[68]

L. Tebou, Sharp observability estimates for a system of two coupled nonconservative hyperbolic equations,, Appl. Math. Optim., 66 (2012), 175. doi: 10.1007/s00245-012-9168-y. Google Scholar

[69]

R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled non-conservative Schrödinger equations. Dedicated to the memory of Pierre Grisvard,, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 453. Google Scholar

[70]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot. Special issue dedicated to the memory of Jacques-Louis Lions,, Appl. Math. Optim., 46 (2002), 331. doi: 10.1007/s00245-002-0751-5. Google Scholar

[71]

J. Vancostenoble, Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials,, SIAM J. Math. Anal., 41 (2009), 1508. doi: 10.1137/080731396. Google Scholar

[72]

H. E. Wilhelm and T. J. van der Werff, Nonlinear wave equations for chemical reactions with diffusion in multicomponent systems,, J. Chem. Phys., 67 (1977), 3382. doi: 10.1063/1.435285. Google Scholar

[73]

P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients,, SIAM J. Control Optim., 37 (1999), 1568. doi: 10.1137/S0363012997331482. Google Scholar

[74]

X. Zhang, Explicit observability estimate for the wave equation with potential and its application,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1101. doi: 10.1098/rspa.2000.0553. Google Scholar

[75]

X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812. doi: 10.1137/S0363012999350298. Google Scholar

[76]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 1. Google Scholar

[77]

E. Zuazua, Exact boundary controllability for the semilinear wave equation,, in Nonlinear Partial Differential Equations and Their Applications, 220 (1991), 357. Google Scholar

[78]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension,, Ann. Inst. Poincaré Anal. Non Linéaire, 10 (1993), 109. Google Scholar

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. 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. 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. Control Optim., 24 (1986), 199. doi: 10.1137/0324012. Google Scholar

[64]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[65]

L. Tebou, Locally distributed desensitizing controls for the wave equation,, C.R.A.S. Paris, 346 (2008), 407. doi: 10.1016/j.crma.2008.02.019. Google Scholar

[66]

L. Tebou, Some results on the controllability of coupled semilinear wave equations: The desensitizing control case,, SIAM J. Control Optim., 49 (2011), 1221. doi: 10.1137/100803080. Google Scholar

[67]

L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations,, C. R. Acad. Sci. Paris, 350 (2012), 57. doi: 10.1016/j.crma.2011.12.001. Google Scholar

[68]

L. Tebou, Sharp observability estimates for a system of two coupled nonconservative hyperbolic equations,, Appl. Math. Optim., 66 (2012), 175. doi: 10.1007/s00245-012-9168-y. Google Scholar

[69]

R. Triggiani, Carleman estimates and exact boundary controllability for a system of coupled non-conservative Schrödinger equations. Dedicated to the memory of Pierre Grisvard,, Rend. Istit. Mat. Univ. Trieste, 28 (1996), 453. Google Scholar

[70]

R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot. Special issue dedicated to the memory of Jacques-Louis Lions,, Appl. Math. Optim., 46 (2002), 331. doi: 10.1007/s00245-002-0751-5. Google Scholar

[71]

J. Vancostenoble, Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials,, SIAM J. Math. Anal., 41 (2009), 1508. doi: 10.1137/080731396. Google Scholar

[72]

H. E. Wilhelm and T. J. van der Werff, Nonlinear wave equations for chemical reactions with diffusion in multicomponent systems,, J. Chem. Phys., 67 (1977), 3382. doi: 10.1063/1.435285. Google Scholar

[73]

P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients,, SIAM J. Control Optim., 37 (1999), 1568. doi: 10.1137/S0363012997331482. Google Scholar

[74]

X. Zhang, Explicit observability estimate for the wave equation with potential and its application,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1101. doi: 10.1098/rspa.2000.0553. Google Scholar

[75]

X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities,, SIAM J. Control Optim., 39 (2000), 812. doi: 10.1137/S0363012999350298. Google Scholar

[76]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 1. Google Scholar

[77]

E. Zuazua, Exact boundary controllability for the semilinear wave equation,, in Nonlinear Partial Differential Equations and Their Applications, 220 (1991), 357. Google Scholar

[78]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension,, Ann. Inst. Poincaré Anal. Non Linéaire, 10 (1993), 109. Google Scholar

[1]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556

[2]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639

[3]

Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317

[4]

Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations & Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019

[5]

Abdelaziz Bennour, Farid Ammar Khodja, Djamel Teniou. Exact and approximate controllability of coupled one-dimensional hyperbolic equations. Evolution Equations & Control Theory, 2017, 6 (4) : 487-516. doi: 10.3934/eect.2017025

[6]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[7]

Ping Lin. Feedback controllability for blowup points of semilinear heat equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1425-1434. doi: 10.3934/dcdsb.2017068

[8]

Víctor Hernández-Santamaría, Luz de Teresa. Robust Stackelberg controllability for linear and semilinear heat equations. Evolution Equations & Control Theory, 2018, 7 (2) : 247-273. doi: 10.3934/eect.2018012

[9]

Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171

[10]

Dario Pighin, Enrique Zuazua. Controllability under positivity constraints of semilinear heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 935-964. doi: 10.3934/mcrf.2018041

[11]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[12]

Tatsien Li, Zhiqiang Wang. A note on the exact controllability for nonautonomous hyperbolic systems. Communications on Pure & Applied Analysis, 2007, 6 (1) : 229-235. doi: 10.3934/cpaa.2007.6.229

[13]

Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011

[14]

Piermarco Cannarsa, Vilmos Komornik, Paola Loreti. One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 745-756. doi: 10.3934/dcds.2002.8.747

[15]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039

[16]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

[17]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699

[18]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429

[19]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control & Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006

[20]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]