Approximate controllability of semilinear reaction diffusion equations

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June 2012, 2(2): 171-182. doi: 10.3934/mcrf.2012.2.171

Approximate controllability of semilinear reaction diffusion equations

Hugo Leiva ^1, , Nelson Merentes ^2, and José L. Sánchez ^3,

1.	Universidad de los Andes, Facultad de Ciencias, Departamento de Matemática, Mérida 5101, Venezuela
2.	Universidad Central de Venezuela, Facultad de Ciencias, Departamento de Matemática, Caracas 1051, Venezuela
3.	Universidad Central de Venezuela, Facultad de Ciencias, Departamento de Matem, Caracas 1051, Venezuela

Received November 2011 Revised January 2012 Published May 2012

Abstract
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In this paper we prove the approximate controllability of the a broad class of semilinear reaction diffusion equation in a Hilbert space, with application to the semilinear $n$D heat equation, the Ornstein-Uhlenbeck equation, amount others.

Keywords: Interior controllability, strongly continuous semigroups, semilinear reaction diffusion equations, heat equation, compact semigroups..

Mathematics Subject Classification: Primary: 93B05; Secondary: 93C1.

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

References:

[1]	J. Appell, H. Leiva, N. Merentes and A. Vignoli, Un espectro de compresión no lineal con aplicaciones a la controlabilidad aproximada de sistemas semilineales,, preprint., ().
[2]	S. Axler, P. Bourdon and W. Ramey, "Harmonic Function Theory," Graduate Texts in Math., 137, Springer Verlag, New York, 1992.
[3]	D. Barcenas, H. Leiva and Z. Sívoli, A broad class of evolution equations are approximately controllable, but never exactly controllable, IMA J. Math. Control Inform., 22 (2005), 310-320. doi: 10.1093/imamci/dni029.
[4]	D. Barcenas, H. Leiva and W. Urbina, Controllability of the Ornstein-Uhlenbeck equation, IMA J. Math. Control Inform., 23 (2006), 1-9.
[5]	D. Barcenas, H. Leiva, Y. Quintana and W. Urbina, Controllability of Laguerre and Jacobi equations, International Journal of Control, 80 (2007), 1307-1315. doi: 10.1080/00207170701294581.
[6]	R. F. Curtain and A. J. Pritchard, "Infinite Dimensional Linear Systems," Lecture Notes in Control and Information Sciences, 8, Springer Verlag, Berlin, 1978.
[7]	R. F. Curtain and H. J. Zwart, "An Introduction to Infinite Dimensional Linear Systems Theory," Text in Applied Mathematics, 21, Springer Verlag, New York, 1995.
[8]	C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61. doi: 10.1017/S0308210500030742.
[9]	J. I. Díaz, J. Henry and A. M. Ramos, On the approximate controllability of some semilinear parabolic boundary-value problems, Appl. Math. Optim., 37 (1998), 71-97. doi: 10.1007/s002459900069.
[10]	E. Fernandez-Cara, Remark on approximate and null controllability of semilinear parabolic equations, ESAIM: Proceeding of Controle et Equations aux Derivees Partielles, 4 (1998), 73-81.
[11]	E. Fernandez-Cara and E. Zuazua, Controllability for blowing up semilinear parabolic equations, C. R. Acad. Sci. Paris Sér I Math., 330 (2000), 199-204.
[12]	L. Hormander, "Linear Partial Differential Equations," Springer Verlag, 1969.
[13]	H. Leiva, N. Merentes and J. L. Sanchez, Interior controllability of the $nD$ semilinear heat equation, African Diaspora Journal of Mathematics, Special Vol. in Honor of Profs. C. Corduneanu, A. Fink and S. Zaidman., 12 (2011), 1-12.
[14]	H. Leiva and Y. Quintana, Interior controllability of a broad class of reaction diffusion equations, Mathematical Problems in Engineering, 2009, Article ID 708516, 8 pp. doi: 10.1155/2009/708516.
[15]	K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25 (1987), 715-722. doi: 10.1137/0325040.
[16]	K. Naito, Approximate controllability for trajectories of semilinear control systems, J. of Optimization Theory and Appl., 60 (1989), 57-65. doi: 10.1007/BF00938799.
[17]	M. H. Protter, Unique continuation for elliptic equations, Transaction of the American Mathematical Society, 95 (1960), 81-91. doi: 10.1090/S0002-9947-1960-0113030-3.
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 636-739. doi: 10.1137/1020095.
[19]	L. De Teresa, Approximate controllability of semilinear heat equation in $\mathbbR^N$, SIAM J. Control Optim., 36 (1998), 2128-2147. doi: 10.1137/S036012997322042.
[20]	L. De Teresa and E. Zuazua, Approximate controllability of semilinear heat equation in unbounded domains, Nonlinear Anal., 8 (1999).
[21]	Xu Zhang, A remark on null exact controllability of the heat equation, SIAM J. Control Optim., 40 (2001), 39-53. doi: 10.1137/S0363012900371691.
[22]	E. Zuazua, Controllability of a system of linear thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.
[23]	E. Zuazua, Control of partial differential equations and its semi-discrete approximation, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.

show all references

References:

[1]	J. Appell, H. Leiva, N. Merentes and A. Vignoli, Un espectro de compresión no lineal con aplicaciones a la controlabilidad aproximada de sistemas semilineales,, preprint., ().
[2]	S. Axler, P. Bourdon and W. Ramey, "Harmonic Function Theory," Graduate Texts in Math., 137, Springer Verlag, New York, 1992.
[3]	D. Barcenas, H. Leiva and Z. Sívoli, A broad class of evolution equations are approximately controllable, but never exactly controllable, IMA J. Math. Control Inform., 22 (2005), 310-320. doi: 10.1093/imamci/dni029.
[4]	D. Barcenas, H. Leiva and W. Urbina, Controllability of the Ornstein-Uhlenbeck equation, IMA J. Math. Control Inform., 23 (2006), 1-9.
[5]	D. Barcenas, H. Leiva, Y. Quintana and W. Urbina, Controllability of Laguerre and Jacobi equations, International Journal of Control, 80 (2007), 1307-1315. doi: 10.1080/00207170701294581.
[6]	R. F. Curtain and A. J. Pritchard, "Infinite Dimensional Linear Systems," Lecture Notes in Control and Information Sciences, 8, Springer Verlag, Berlin, 1978.
[7]	R. F. Curtain and H. J. Zwart, "An Introduction to Infinite Dimensional Linear Systems Theory," Text in Applied Mathematics, 21, Springer Verlag, New York, 1995.
[8]	C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31-61. doi: 10.1017/S0308210500030742.
[9]	J. I. Díaz, J. Henry and A. M. Ramos, On the approximate controllability of some semilinear parabolic boundary-value problems, Appl. Math. Optim., 37 (1998), 71-97. doi: 10.1007/s002459900069.
[10]	E. Fernandez-Cara, Remark on approximate and null controllability of semilinear parabolic equations, ESAIM: Proceeding of Controle et Equations aux Derivees Partielles, 4 (1998), 73-81.
[11]	E. Fernandez-Cara and E. Zuazua, Controllability for blowing up semilinear parabolic equations, C. R. Acad. Sci. Paris Sér I Math., 330 (2000), 199-204.
[12]	L. Hormander, "Linear Partial Differential Equations," Springer Verlag, 1969.
[13]	H. Leiva, N. Merentes and J. L. Sanchez, Interior controllability of the $nD$ semilinear heat equation, African Diaspora Journal of Mathematics, Special Vol. in Honor of Profs. C. Corduneanu, A. Fink and S. Zaidman., 12 (2011), 1-12.
[14]	H. Leiva and Y. Quintana, Interior controllability of a broad class of reaction diffusion equations, Mathematical Problems in Engineering, 2009, Article ID 708516, 8 pp. doi: 10.1155/2009/708516.
[15]	K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25 (1987), 715-722. doi: 10.1137/0325040.
[16]	K. Naito, Approximate controllability for trajectories of semilinear control systems, J. of Optimization Theory and Appl., 60 (1989), 57-65. doi: 10.1007/BF00938799.
[17]	M. H. Protter, Unique continuation for elliptic equations, Transaction of the American Mathematical Society, 95 (1960), 81-91. doi: 10.1090/S0002-9947-1960-0113030-3.
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 636-739. doi: 10.1137/1020095.
[19]	L. De Teresa, Approximate controllability of semilinear heat equation in $\mathbbR^N$, SIAM J. Control Optim., 36 (1998), 2128-2147. doi: 10.1137/S036012997322042.
[20]	L. De Teresa and E. Zuazua, Approximate controllability of semilinear heat equation in unbounded domains, Nonlinear Anal., 8 (1999).
[21]	Xu Zhang, A remark on null exact controllability of the heat equation, SIAM J. Control Optim., 40 (2001), 39-53. doi: 10.1137/S0363012900371691.
[22]	E. Zuazua, Controllability of a system of linear thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.
[23]	E. Zuazua, Control of partial differential equations and its semi-discrete approximation, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.

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