Approximate controllability of semilinear reaction diffusion equations

Previous Article
The simplest semilinear parabolic equation of normal type
MCRF Home
This Issue
Next Article
Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs

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
Related Papers

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 & 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., (). Google Scholar
[2]	S. Axler, P. Bourdon and W. Ramey, "Harmonic Function Theory,", Graduate Texts in Math., 137 (1992). Google Scholar
[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. doi: 10.1093/imamci/dni029. Google Scholar
[4]	D. Barcenas, H. Leiva and W. Urbina, Controllability of the Ornstein-Uhlenbeck equation,, IMA J. Math. Control Inform., 23 (2006), 1. Google Scholar
[5]	D. Barcenas, H. Leiva, Y. Quintana and W. Urbina, Controllability of Laguerre and Jacobi equations,, International Journal of Control, 80 (2007), 1307. doi: 10.1080/00207170701294581. Google Scholar
[6]	R. F. Curtain and A. J. Pritchard, "Infinite Dimensional Linear Systems,", Lecture Notes in Control and Information Sciences, 8 (1978). Google Scholar
[7]	R. F. Curtain and H. J. Zwart, "An Introduction to Infinite Dimensional Linear Systems Theory,", Text in Applied Mathematics, 21 (1995). Google Scholar
[8]	C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of semilinear heat equation,, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31. doi: 10.1017/S0308210500030742. Google Scholar
[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. doi: 10.1007/s002459900069. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	L. Hormander, "Linear Partial Differential Equations,", Springer Verlag, (1969). Google Scholar
[13]	H. Leiva, N. Merentes and J. L. Sanchez, Interior controllability of the $nD$ semilinear heat equation,, African Diaspora Journal of Mathematics, 12 (2011), 1. Google Scholar
[14]	H. Leiva and Y. Quintana, Interior controllability of a broad class of reaction diffusion equations,, Mathematical Problems in Engineering, 2009 (7085). doi: 10.1155/2009/708516. Google Scholar
[15]	K. Naito, Controllability of semilinear control systems dominated by the linear part,, SIAM J. Control Optim., 25 (1987), 715. doi: 10.1137/0325040. Google Scholar
[16]	K. Naito, Approximate controllability for trajectories of semilinear control systems,, J. of Optimization Theory and Appl., 60 (1989), 57. doi: 10.1007/BF00938799. Google Scholar
[17]	M. H. Protter, Unique continuation for elliptic equations,, Transaction of the American Mathematical Society, 95 (1960), 81. doi: 10.1090/S0002-9947-1960-0113030-3. Google Scholar
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 636. doi: 10.1137/1020095. Google Scholar
[19]	L. De Teresa, Approximate controllability of semilinear heat equation in $\mathbbR^N$,, SIAM J. Control Optim., 36 (1998), 2128. doi: 10.1137/S036012997322042. Google Scholar
[20]	L. De Teresa and E. Zuazua, Approximate controllability of semilinear heat equation in unbounded domains,, Nonlinear Anal., 8 (1999). Google Scholar
[21]	Xu Zhang, A remark on null exact controllability of the heat equation,, SIAM J. Control Optim., 40 (2001), 39. doi: 10.1137/S0363012900371691. Google Scholar
[22]	E. Zuazua, Controllability of a system of linear thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar
[23]	E. Zuazua, Control of partial differential equations and its semi-discrete approximation,, Discrete and Continuous Dynamical Systems, 8 (2002), 469. Google Scholar

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., (). Google Scholar
[2]	S. Axler, P. Bourdon and W. Ramey, "Harmonic Function Theory,", Graduate Texts in Math., 137 (1992). Google Scholar
[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. doi: 10.1093/imamci/dni029. Google Scholar
[4]	D. Barcenas, H. Leiva and W. Urbina, Controllability of the Ornstein-Uhlenbeck equation,, IMA J. Math. Control Inform., 23 (2006), 1. Google Scholar
[5]	D. Barcenas, H. Leiva, Y. Quintana and W. Urbina, Controllability of Laguerre and Jacobi equations,, International Journal of Control, 80 (2007), 1307. doi: 10.1080/00207170701294581. Google Scholar
[6]	R. F. Curtain and A. J. Pritchard, "Infinite Dimensional Linear Systems,", Lecture Notes in Control and Information Sciences, 8 (1978). Google Scholar
[7]	R. F. Curtain and H. J. Zwart, "An Introduction to Infinite Dimensional Linear Systems Theory,", Text in Applied Mathematics, 21 (1995). Google Scholar
[8]	C. Fabre, J. P. Puel and E. Zuazua, Approximate controllability of semilinear heat equation,, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 31. doi: 10.1017/S0308210500030742. Google Scholar
[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. doi: 10.1007/s002459900069. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	L. Hormander, "Linear Partial Differential Equations,", Springer Verlag, (1969). Google Scholar
[13]	H. Leiva, N. Merentes and J. L. Sanchez, Interior controllability of the $nD$ semilinear heat equation,, African Diaspora Journal of Mathematics, 12 (2011), 1. Google Scholar
[14]	H. Leiva and Y. Quintana, Interior controllability of a broad class of reaction diffusion equations,, Mathematical Problems in Engineering, 2009 (7085). doi: 10.1155/2009/708516. Google Scholar
[15]	K. Naito, Controllability of semilinear control systems dominated by the linear part,, SIAM J. Control Optim., 25 (1987), 715. doi: 10.1137/0325040. Google Scholar
[16]	K. Naito, Approximate controllability for trajectories of semilinear control systems,, J. of Optimization Theory and Appl., 60 (1989), 57. doi: 10.1007/BF00938799. Google Scholar
[17]	M. H. Protter, Unique continuation for elliptic equations,, Transaction of the American Mathematical Society, 95 (1960), 81. doi: 10.1090/S0002-9947-1960-0113030-3. Google Scholar
[18]	D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions,, SIAM Rev., 20 (1978), 636. doi: 10.1137/1020095. Google Scholar
[19]	L. De Teresa, Approximate controllability of semilinear heat equation in $\mathbbR^N$,, SIAM J. Control Optim., 36 (1998), 2128. doi: 10.1137/S036012997322042. Google Scholar
[20]	L. De Teresa and E. Zuazua, Approximate controllability of semilinear heat equation in unbounded domains,, Nonlinear Anal., 8 (1999). Google Scholar
[21]	Xu Zhang, A remark on null exact controllability of the heat equation,, SIAM J. Control Optim., 40 (2001), 39. doi: 10.1137/S0363012900371691. Google Scholar
[22]	E. Zuazua, Controllability of a system of linear thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar
[23]	E. Zuazua, Control of partial differential equations and its semi-discrete approximation,, Discrete and Continuous Dynamical Systems, 8 (2002), 469. Google Scholar

[1]	Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271
[2]	Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[3]	Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[4]	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
[5]	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
[6]	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
[7]	Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026
[8]	Van Tien Nguyen. On the blow-up results for a class of strongly perturbed semilinear heat equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3585-3626. doi: 10.3934/dcds.2015.35.3585
[9]	Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847
[10]	Luc Miller. A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1465-1485. doi: 10.3934/dcdsb.2010.14.1465
[11]	Luisa Malaguti, Cristina Marcelli, Serena Matucci. Continuous dependence in front propagation of convective reaction-diffusion equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1083-1098. doi: 10.3934/cpaa.2010.9.1083
[12]	Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223
[13]	Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
[14]	Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595
[15]	Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks & Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487
[16]	Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[17]	José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269
[18]	Min He. On continuity in parameters of integrated semigroups. Conference Publications, 2003, 2003 (Special) : 403-412. doi: 10.3934/proc.2003.2003.403
[19]	Alastair Fletcher. Quasiregular semigroups with examples. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2157-2172. doi: 10.3934/dcds.2019090
[20]	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

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences