Simultaneous controllability of some uncoupled semilinear wave equations

Louis Tebou

doi:10.3934/dcds.2015.35.3721

Article Contents

2015, Volume 35, Issue 8: 3721-3743. Doi: 10.3934/dcds.2015.35.3721

This issue Previous Article Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities Next Article Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems

Simultaneous controllability of some uncoupled semilinear wave equations

Louis Tebou^1,

1.
Department of Mathematics & Statistics, Florida International University, Miami, FL 33199

Received: June 30, 2014

Revised: November 30, 2014

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 93C20; Secondary: 35K51, 35L53, 35L71, 93B05, 93B07.

Citation:

Related Papers

Cited by

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]	L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.doi: 10.1016/j.jde.2004.05.007.
[51]	A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control, SIAM J. Control Optim., 40 (2001), 777-800.doi: 10.1137/S0363012998345615.
[52]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.doi: 10.1007/978-1-4612-5561-1.
[53]	L. Rosier and L. de Teresa, Exact controllability of a cascade system of conservative equations, C. R. Math. Acad. Sci. Paris, 349 (2011), 291-296.doi: 10.1016/j.crma.2011.01.014.
[54]	H. L. Royden, Real Analysis, Third edition. Prentice-Hall, New Jersey, 1988.
[55]	A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl., 71 (1992), 455-467.
[56]	D. L. Russell, Nonharmonic Fourier series in the control theory of distributed parameter systems, J. Math. Anal. Appl., 18 (1967), 542-560.doi: 10.1016/0022-247X(67)90045-5.
[57]	D. L. Russell, Boundary value control of the higher-dimensional wave equation, SIAM J. Control, 9 (1971), 29-42.doi: 10.1137/0309004.
[58]	D. L. Russell, Boundary value control theory of the higher-dimensional wave equation. II, SIAM J. Control, 9 (1971), 401-419.doi: 10.1137/0309030.
[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., Math. Res. Center, Naval Res. Lab., Washington, D. C., (1971), 241-263. Academic Press, New York, 1972.
[60]	D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Studies in Appl. Math., 52 (1973), 189-211.
[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., Univ. Rhode Island, Kingston, R.I., 1973), pp. 291-319. Lecture Notes in Pure Appl. Math., Vol. 10, Dekker, New York, 1974.
[62]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev., 20 (1978), 639-739.doi: 10.1137/1020095.
[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), 199-229.doi: 10.1137/0324012.
[64]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., (4) 146 (1987), 65-96.doi: 10.1007/BF01762360.
[65]	L. Tebou, Locally distributed desensitizing controls for the wave equation, C.R.A.S. Paris, Série I, 346 (2008), 407-412.doi: 10.1016/j.crma.2008.02.019.
[66]	L. Tebou, Some results on the controllability of coupled semilinear wave equations: The desensitizing control case, SIAM J. Control Optim., 49 (2011), 1221-1238.doi: 10.1137/100803080.
[67]	L. Tebou, Simultaneous observability and stabilization of some uncoupled wave equations, C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 57-62.doi: 10.1016/j.crma.2011.12.001.
[68]	L. Tebou, Sharp observability estimates for a system of two coupled nonconservative hyperbolic equations, Appl. Math. Optim., 66 (2012), 175-207.doi: 10.1007/s00245-012-9168-y.
[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, suppl., 28 (1996), 453-504 (1997).
[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), 331-375.doi: 10.1007/s00245-002-0751-5.
[71]	J. Vancostenoble, Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532.doi: 10.1137/080731396.
[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), 3382-3387.doi: 10.1063/1.435285.
[73]	P. F. Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37 (1999), 1568-1599.doi: 10.1137/S0363012997331482.
[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), 1101-1115.doi: 10.1098/rspa.2000.0553.
[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), 812-834.doi: 10.1137/S0363012999350298.
[76]	E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl., 69 (1990), 1-31.
[77]	E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and Their Applications, H. Brezis and J.-L. Lions, Eds., Pitman, London, 220 (1991), 357-391.
[78]	E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Ann. Inst. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.