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

Optimal control problem for exact synchronization of parabolic system

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.

Citation: Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control and Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019
References:
[1]

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

[2]

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

[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.

[4]

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

[5]

F. A. KhodjaA. BenabdallahC. 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.

[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.

[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.

[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. 

[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.

[10]

T-T. LiB. 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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

show all references

References:
[1]

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

[2]

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

[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.

[4]

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

[5]

F. A. KhodjaA. BenabdallahC. 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.

[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.

[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.

[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. 

[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.

[10]

T-T. LiB. 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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[19]

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

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