Local exact controllability to positive trajectory for parabolic system of chemotaxis

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March 2016, 6(1): 143-165. doi: 10.3934/mcrf.2016.6.143

Local exact controllability to positive trajectory for parabolic system of chemotaxis

Bao-Zhu Guo ^1, and Liang Zhang ^2,

1.	Key Laboratory of System and Control, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China
2.	Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China

Received March 2015 Revised June 2015 Published January 2016

Abstract
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In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani's fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimate for parabolic equations.

Keywords: Kakutani's fixed point theorem., Carleman inequality, chemotaxis system, Local exact controllability.

Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 37C2.

Citation: Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control and Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143

References:

[1]	W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in Handbook of Differential Equations: Evolutionary Equations, Elsevier, 1 (2004), 1-85.
[2]	V. Barbu, Controllability of parabolic and Navier-Stokes equations, Sci. Math. Japon., 56 (2002), 143-211.
[3]	V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, 1993.
[4]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[5]	J. -M. Coron, Control and Nonlinearity, AMS, Providence, RI, 2007.
[6]	E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare(C) Non Linear Analysis, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.
[7]	A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996.
[8]	H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nathr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[10]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa CI. Sci., 24 (1997), 633-683.
[11]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[12]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
[13]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[14]	O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274. doi: 10.2977/prims/1145476103.
[15]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[16]	O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, RI, 1968.
[17]	D. Lamberton, Equations d'évolution liné aires associées à les semi-groupes de contractions dans les espaces $L^p$, J. Funct. Anal., 72 (1987), 252-262. doi: 10.1016/0022-1236(87)90088-7.
[18]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[19]	M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766. doi: 10.1137/110843964.
[20]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekva., 44 (2001), 441-469.
[21]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, LNM 1072, Springer-Verlag, 1984.
[22]	S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66. doi: 10.1006/jmaa.2000.7254.
[23]	A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265.
[24]	G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().
[25]	G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system, J. Optim. Theory Appl., 131 (2006), 453-467. doi: 10.1007/s10957-006-9161-1.
[26]	Z.-A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601.

show all references

References:

[1]	W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in Handbook of Differential Equations: Evolutionary Equations, Elsevier, 1 (2004), 1-85.
[2]	V. Barbu, Controllability of parabolic and Navier-Stokes equations, Sci. Math. Japon., 56 (2002), 143-211.
[3]	V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, 1993.
[4]	P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.
[5]	J. -M. Coron, Control and Nonlinearity, AMS, Providence, RI, 2007.
[6]	E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Annales de l'Institut Henri Poincare(C) Non Linear Analysis, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.
[7]	A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, 1996.
[8]	H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nathr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.
[9]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin, 1981.
[10]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa CI. Sci., 24 (1997), 633-683.
[11]	T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.
[12]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
[13]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.
[14]	O. Yu. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Inst. Math. Sci., 39 (2003), 227-274. doi: 10.2977/prims/1145476103.
[15]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[16]	O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, RI, 1968.
[17]	D. Lamberton, Equations d'évolution liné aires associées à les semi-groupes de contractions dans les espaces $L^p$, J. Funct. Anal., 72 (1987), 252-262. doi: 10.1016/0022-1236(87)90088-7.
[18]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.
[19]	M. Ma, C. Ou and Z. A. Wang, Stationary solutions of a volume filling chemotaxis model with logistic growth and their stability, SIAM J. Appl. Math., 72 (2012), 740-766. doi: 10.1137/110843964.
[20]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekva., 44 (2001), 441-469.
[21]	F. Rothe, Global Solutions of Reaction-Diffusion Systems, LNM 1072, Springer-Verlag, 1984.
[22]	S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 (2001), 45-66. doi: 10.1006/jmaa.2000.7254.
[23]	A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis, Math. Japon., 45 (1997), 241-265.
[24]	G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().
[25]	G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system, J. Optim. Theory Appl., 131 (2006), 453-467. doi: 10.1007/s10957-006-9161-1.
[26]	Z.-A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601.

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