American Institute of Mathematical Sciences

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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

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 & 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, 1 (2004), 1.   Google Scholar

[2]

V. Barbu, Controllability of parabolic and Navier-Stokes equations,, Sci. Math. Japon., 56 (2002), 143.   Google Scholar

[3]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems,, Academic Press, (1993).   Google Scholar

[4]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.   Google Scholar

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J. -M. Coron, Control and Nonlinearity,, AMS, (2007).   Google Scholar

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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.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

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A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations,, Lecture Notes Series 34, (1996).   Google Scholar

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H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nathr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[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.   Google Scholar

[11]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

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D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.   Google Scholar

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D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[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.  doi: 10.2977/prims/1145476103.  Google Scholar

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E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[16]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, AMS, (1968).   Google Scholar

[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.  doi: 10.1016/0022-1236(87)90088-7.  Google Scholar

[18]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).  doi: 10.1142/3302.  Google Scholar

[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.  doi: 10.1137/110843964.  Google Scholar

[20]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekva., 44 (2001), 441.   Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, LNM 1072, (1072).   Google Scholar

[22]

S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations,, J. Math. Anal. Appl., 256 (2001), 45.  doi: 10.1006/jmaa.2000.7254.  Google Scholar

[23]

A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, Math. Japon., 45 (1997), 241.   Google Scholar

[24]

G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().   Google Scholar

[25]

G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system,, J. Optim. Theory Appl., 131 (2006), 453.  doi: 10.1007/s10957-006-9161-1.  Google Scholar

[26]

Z.-A. Wang, Mathematics of traveling waves in chemotaxis,, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

show all references

References:
[1]

W. Arendt, Semigroups and evolution equations: functional calculus, regularity and kernel estimates, in, Handbook of Differential Equations: Evolutionary Equations, 1 (2004), 1.   Google Scholar

[2]

V. Barbu, Controllability of parabolic and Navier-Stokes equations,, Sci. Math. Japon., 56 (2002), 143.   Google Scholar

[3]

V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems,, Academic Press, (1993).   Google Scholar

[4]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.   Google Scholar

[5]

J. -M. Coron, Control and Nonlinearity,, AMS, (2007).   Google Scholar

[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.  doi: 10.1016/S0294-1449(00)00117-7.  Google Scholar

[7]

A. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations,, Lecture Notes Series 34, (1996).   Google Scholar

[8]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis,, Math. Nathr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[9]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer-Verlag, (1981).   Google Scholar

[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.   Google Scholar

[11]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[12]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103.   Google Scholar

[13]

D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.  doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[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.  doi: 10.2977/prims/1145476103.  Google Scholar

[15]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[16]

O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type,, AMS, (1968).   Google Scholar

[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.  doi: 10.1016/0022-1236(87)90088-7.  Google Scholar

[18]

G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996).  doi: 10.1142/3302.  Google Scholar

[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.  doi: 10.1137/110843964.  Google Scholar

[20]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekva., 44 (2001), 441.   Google Scholar

[21]

F. Rothe, Global Solutions of Reaction-Diffusion Systems,, LNM 1072, (1072).   Google Scholar

[22]

S.-U. Ryu and A. Yagi, Optimal control of Keller-Segel equations,, J. Math. Anal. Appl., 256 (2001), 45.  doi: 10.1006/jmaa.2000.7254.  Google Scholar

[23]

A. Yagi, Norm behavior of solutions to a parabolic system of chemotaxis,, Math. Japon., 45 (1997), 241.   Google Scholar

[24]

G. Wang and C. Zhang, Observability estimate from measurable sets in time for some evolution equations,, , ().   Google Scholar

[25]

G. Wang and L. Zhang, Exact local controllability of a one-control reaction-diffusion system,, J. Optim. Theory Appl., 131 (2006), 453.  doi: 10.1007/s10957-006-9161-1.  Google Scholar

[26]

Z.-A. Wang, Mathematics of traveling waves in chemotaxis,, Discrete Contin. Dyn. Syst-Series B., 18 (2013), 601.  doi: 10.3934/dcdsb.2013.18.601.  Google Scholar

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