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

References:
[1]

SIAM J. Control Optim., 42 (2003), 871-906. doi: 10.1137/S0363012902402608.  Google Scholar

[2]

SIAM J. Control Optim., 37 (1999), 521-542. doi: 10.1137/S0363012996313835.  Google Scholar

[3]

C. R. Math. Acad. Sci. Paris, 349 (2011), 395-400. doi: 10.1016/j.crma.2011.02.004.  Google Scholar

[4]

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.  Google Scholar

[5]

Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 947-970. doi: 10.1017/S0308210500000512.  Google Scholar

[6]

SIAM J. Control and Opt., 30 (1992), 1024-1065. doi: 10.1137/0330055.  Google Scholar

[7]

Asymptot. Anal., 14 (1997), 157-191.  Google Scholar

[8]

C. R. Acad. Sci. Paris Ser. I Math., 325 (1997), 749-752. doi: 10.1016/S0764-4442(97)80053-5.  Google Scholar

[9]

Discrete Contin. Dyn. Syst., 8 (2002), 745-756. doi: 10.3934/dcds.2002.8.747.  Google Scholar

[10]

SIAM J. Control Optim., 17 (1979), 66-81. doi: 10.1137/0317007.  Google Scholar

[11]

J. Math. Pures Appl., 58 (1979), 249-273.  Google Scholar

[12]

SIAM J. Control Optim., 19 (1981), 114-122. doi: 10.1137/0319009.  Google Scholar

[13]

SIAM J. Control Optim., 14 (1976), 19-25. doi: 10.1137/0314002.  Google Scholar

[14]

Ann. I.H.Poincaré-AN, 25 (2008), 1-41. doi: 10.1016/j.anihpc.2006.07.005.  Google Scholar

[15]

Math. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123.  Google Scholar

[16]

SIAM J. Control Optim., 46 (2007), 1578-1614. doi: 10.1137/040610222.  Google Scholar

[17]

Lecture Notes, Vol. 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[18]

SIAM J. Control, 13 (1975), 174-196. doi: 10.1137/0313011.  Google Scholar

[19]

C. R. Acad. Sci. Paris Ser. I Math., 306 (1988), 125-128.  Google Scholar

[20]

Collège de France Seminar, Vol. X (Paris, 1987-1988), 241-271, Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991.  Google Scholar

[21]

Portugal. Math., 46 (1989), 245-258.  Google Scholar

[22]

J. Math. Anal. Appl., 153 (1990), 190-216. doi: 10.1016/0022-247X(90)90273-I.  Google Scholar

[23]

Port. Math. (N.S.), 61 (2004), 399-437.  Google Scholar

[24]

C. R. Acad. Sci. Paris Sér. I Math., 302 (1986), 443-446.  Google Scholar

[25]

J. Math. Pures Appl., 66 (1987), 363-368.  Google Scholar

[26]

Springer-Verlag, New-York, 1976.  Google Scholar

[27]

Asympt. Anal., 32 (2002), 185-220.  Google Scholar

[28]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 153-164.  Google Scholar

[29]

RAM, Masson & John Wiley, Paris, 1994.  Google Scholar

[30]

Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.  Google Scholar

[31]

S.I.A.M J. Control and Opt., 21 (1983), 68-85. doi: 10.1137/0321004.  Google Scholar

[32]

Appl. Math. Optim., 19 (1989), 243-290. doi: 10.1007/BF01448201.  Google Scholar

[33]

Appl. Math. Optim., 23 (1991), 109-154. doi: 10.1007/BF01442394.  Google Scholar

[34]

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.  Google Scholar

[35]

Discrete Contin. Dyn. Syst., (2005), suppl., 556-565.  Google Scholar

[36]

Control of distributed parameter and stochastic systems (Hangzhou, 1998), 71-78, Kluwer Acad. Publ., Boston, MA, 1999.  Google Scholar

[37]

J. Math. Anal. Appl., 235 (1999), 13-57. doi: 10.1006/jmaa.1999.6348.  Google Scholar

[38]

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.  Google Scholar

[39]

Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.  Google Scholar

[40]

JMAA, 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998.  Google Scholar

[41]

SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[42]

Vol. 1, RMA 8, Masson, Paris, 1988.  Google Scholar

[43]

Vol. 2, RMA 9, Masson, Paris, 1988.  Google Scholar

[44]

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.  Google Scholar

[45]

S.I.A.M J. Control and Opt., 35 (1997), 1574-1590. doi: 10.1137/S0363012995284928.  Google Scholar

[46]

Portugal. Math., 57 (2000), 493-508.  Google Scholar

[47]

J. Optim. Theory Appl., 116 (2003), 621-645. doi: 10.1023/A:1023069420681.  Google Scholar

[48]

J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 78-90. doi: 10.1137/0303008.  Google Scholar

[49]

Discrete Contin. Dyn. Syst., 9 (2003), 901-924. doi: 10.3934/dcds.2003.9.901.  Google Scholar

[50]

J. Differential Equations, 204 (2004), 202-226. doi: 10.1016/j.jde.2004.05.007.  Google Scholar

[51]

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Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[53]

C. R. Math. Acad. Sci. Paris, 349 (2011), 291-296. doi: 10.1016/j.crma.2011.01.014.  Google Scholar

[54]

New Jersey, 1988.  Google Scholar

[55]

J. Math. Pures Appl., 71 (1992), 455-467.  Google Scholar

[56]

J. Math. Anal. Appl., 18 (1967), 542-560. doi: 10.1016/0022-247X(67)90045-5.  Google Scholar

[57]

SIAM J. Control, 9 (1971), 29-42. doi: 10.1137/0309004.  Google Scholar

[58]

SIAM J. Control, 9 (1971), 401-419. doi: 10.1137/0309030.  Google Scholar

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Proc. Conf., Math. Res. Center, Naval Res. Lab., Washington, D. C., (1971), 241-263. Academic Press, New York, 1972.  Google Scholar

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Studies in Appl. Math., 52 (1973), 189-211.  Google Scholar

[61]

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.  Google Scholar

[62]

SIAM Rev., 20 (1978), 639-739. doi: 10.1137/1020095.  Google Scholar

[63]

SIAM J. Control Optim., 24 (1986), 199-229. doi: 10.1137/0324012.  Google Scholar

[64]

Ann. Mat. Pura Appl., (4) 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[65]

C.R.A.S. Paris, Série I, 346 (2008), 407-412. doi: 10.1016/j.crma.2008.02.019.  Google Scholar

[66]

SIAM J. Control Optim., 49 (2011), 1221-1238. doi: 10.1137/100803080.  Google Scholar

[67]

C. R. Acad. Sci. Paris, Ser. I, 350 (2012), 57-62. doi: 10.1016/j.crma.2011.12.001.  Google Scholar

[68]

Appl. Math. Optim., 66 (2012), 175-207. doi: 10.1007/s00245-012-9168-y.  Google Scholar

[69]

Rend. Istit. Mat. Univ. Trieste, suppl., 28 (1996), 453-504 (1997).  Google Scholar

[70]

Appl. Math. Optim., 46 (2002), 331-375. doi: 10.1007/s00245-002-0751-5.  Google Scholar

[71]

SIAM J. Math. Anal., 41 (2009), 1508-1532. doi: 10.1137/080731396.  Google Scholar

[72]

J. Chem. Phys., 67 (1977), 3382-3387. doi: 10.1063/1.435285.  Google Scholar

[73]

SIAM J. Control Optim., 37 (1999), 1568-1599. doi: 10.1137/S0363012997331482.  Google Scholar

[74]

R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 1101-1115. doi: 10.1098/rspa.2000.0553.  Google Scholar

[75]

SIAM J. Control Optim., 39 (2000), 812-834. doi: 10.1137/S0363012999350298.  Google Scholar

[76]

J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar

[77]

in Nonlinear Partial Differential Equations and Their Applications, H. Brezis and J.-L. Lions, Eds., Pitman, London, 220 (1991), 357-391.  Google Scholar

[78]

Ann. Inst. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  Google Scholar

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