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September  2019, 9(3): 411-424. doi: 10.3934/mcrf.2019019

Optimal control problem for exact synchronization of parabolic system

Lijuan Wang 1, and  Qishu Yan 2,,

1. 

School of Mathematics and Statistics, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China
2. 

School of Science, Hebei University of Technology, Tianjin, 300400, China

* Corresponding author: Qishu Yan

Received  December 2016 Revised  September 2017 Published  April 2019

Fund Project: The first author is supported by National Natural Science Foundation of China under grants 11371285 and 11771344. The second author is supported by National Natural Science Foundation of China under grant 11701138.

In this paper, we consider the exact synchronization of a kind of parabolic system and obtain Pontryagin's maximum principle for a related optimal control problem. The method relies on the properties of the null controllability for parabolic system and an observability estimate for a linear parabolic system.

Keywords: Exact synchronization, Pontryagin's maximum principle, parabolic system.
Mathematics Subject Classification: Primary: 35K40, 49K20; Secondary: 35B50, 35B99.
Citation: Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, Pitman, Boston, 1984.  Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.  Google Scholar

[3]

M. A. Demetriou, Synchronization and consensus controllers for a class of parabolic distributed parameter systems, Systems and Control Letters, 62 (2013), 70-76.  doi: 10.1016/j.sysconle.2012.10.010.  Google Scholar

[4]

Ch. Huygens, Oeuvres Complètes, Vol.15, Swets & Zeitlinger B.V., Amsterdam, 1967. Google Scholar

[5]

F. A. Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differential Equations and Applications, 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[6]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[7]

T-T. Li and B. P. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chinese Annals of Mathematics - B, 34 (2013), 139-160.  doi: 10.1007/s11401-012-0754-8.  Google Scholar

[8]

T-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asymptotic Analysis, 86 (2014), 199-226.   Google Scholar

[9]

T-T. Li and B. P. Rao, On the state of exact synchronization of a coupled system of wave equations, Comptes Rendus Mathématique-Académie des Sciencs-Paris, 352 (2014), 823-829.  doi: 10.1016/j.crma.2014.08.007.  Google Scholar

[10]

T-T. Li, B. P. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1-D wave equations, ESAIM: Control, Optimisation and Calculus of Variations, 20 (2014), 339-361.  doi: 10.1051/cocv/2013066.  Google Scholar

[11]

T-T. Li and B. P. Rao, Kalman-type criteria for the approximate controllability and approximate synchronization of a coupled system of wave equations, Comptes Rendus Mathématique-Académie des Sciencs-Paris, 353 (2015), 63-68.  doi: 10.1016/j.crma.2014.10.023.  Google Scholar

[12]

T-T. Li and B. P. Rao, On the exactly synchronizable state to a coupled system of wave equations, Portugaliae Mathematica, 72 (2015), 83-100.  doi: 10.4171/PM/1958.  Google Scholar

[13]

T-T. Li and B. P. Rao, Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls, SIAM Journal on Control and Optimization, 54 (2016), 49-72.  doi: 10.1137/140989807.  Google Scholar

[14]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

H. W. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 975-994.  doi: 10.1051/cocv/2010034.  Google Scholar

[16]

S. Strogatz, Sync: The Emerging Science of Spontaneous Order, THEIA, New York, 2003.  Google Scholar

[17]

G. S. Wang and L. J. Wang, State-constrained optimal control governed by non-well-posed parabolic differential equations, SIAM Journal on Control and Optimization, 40 (2002), 1517-1539.  doi: 10.1137/S0363012900377006.  Google Scholar

[18]

N. Wiener, Cybernetics, or Control and Communication in the Animal and the Machine, MIT Press, Cambridge, 1961.  Google Scholar

[19]

C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific, Singapore, 2007. doi: 10.1142/6570.  Google Scholar

show all references

References:
[1]

V. Barbu, Optimal Control of Variational Inequalities, Pitman, Boston, 1984.  Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, New York, 1993.  Google Scholar

[3]

M. A. Demetriou, Synchronization and consensus controllers for a class of parabolic distributed parameter systems, Systems and Control Letters, 62 (2013), 70-76.  doi: 10.1016/j.sysconle.2012.10.010.  Google Scholar

[4]

Ch. Huygens, Oeuvres Complètes, Vol.15, Swets & Zeitlinger B.V., Amsterdam, 1967. Google Scholar

[5]

F. A. Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differential Equations and Applications, 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[6]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[7]

T-T. Li and B. P. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chinese Annals of Mathematics - B, 34 (2013), 139-160.  doi: 10.1007/s11401-012-0754-8.  Google Scholar

[8]

T-T. Li and B. P. Rao, Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asymptotic Analysis, 86 (2014), 199-226.   Google Scholar

[9]

T-T. Li and B. P. Rao, On the state of exact synchronization of a coupled system of wave equations, Comptes Rendus Mathématique-Académie des Sciencs-Paris, 352 (2014), 823-829.  doi: 10.1016/j.crma.2014.08.007.  Google Scholar

[10]

T-T. Li, B. P. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1-D wave equations, ESAIM: Control, Optimisation and Calculus of Variations, 20 (2014), 339-361.  doi: 10.1051/cocv/2013066.  Google Scholar

[11]

T-T. Li and B. P. Rao, Kalman-type criteria for the approximate controllability and approximate synchronization of a coupled system of wave equations, Comptes Rendus Mathématique-Académie des Sciencs-Paris, 353 (2015), 63-68.  doi: 10.1016/j.crma.2014.10.023.  Google Scholar

[12]

T-T. Li and B. P. Rao, On the exactly synchronizable state to a coupled system of wave equations, Portugaliae Mathematica, 72 (2015), 83-100.  doi: 10.4171/PM/1958.  Google Scholar

[13]

T-T. Li and B. P. Rao, Criteria of Kalman's type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls, SIAM Journal on Control and Optimization, 54 (2016), 49-72.  doi: 10.1137/140989807.  Google Scholar

[14]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

H. W. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 975-994.  doi: 10.1051/cocv/2010034.  Google Scholar

[16]

S. Strogatz, Sync: The Emerging Science of Spontaneous Order, THEIA, New York, 2003.  Google Scholar

[17]

G. S. Wang and L. J. Wang, State-constrained optimal control governed by non-well-posed parabolic differential equations, SIAM Journal on Control and Optimization, 40 (2002), 1517-1539.  doi: 10.1137/S0363012900377006.  Google Scholar

[18]

N. Wiener, Cybernetics, or Control and Communication in the Animal and the Machine, MIT Press, Cambridge, 1961.  Google Scholar

[19]

C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific, Singapore, 2007. doi: 10.1142/6570.  Google Scholar

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