Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints

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March 2012, 2(1): 61-80. doi: 10.3934/mcrf.2012.2.61

Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints

1.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China, China

Received June 2011 Revised November 2011 Published January 2012

Abstract
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This paper deals with the Pontryagin's principle of optimal control problems governed by the 2D Navier-Stokes equations with integral state constraints and coupled integral control--state constraints. As an application, the necessary conditions for the local solution in the sense of $L^r(0,T;L^2(\Omega))$ ($2 < r < \infty$) are also obtained.

Keywords: Pontryagin’s principle, State constraints, Control-state constraints, Optimal control, 2D Navier-Stokes equation, Local optimal solution..

Mathematics Subject Classification: Primary: 35Q30, 49K20; Secondary: 93C2.

Citation: Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

References:

[1]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 2^nd edition, (1979). Google Scholar
[2]	X. Li and J. Yong, "Optimal Control Theory for Infinite, Dimensional System, (1995). Google Scholar
[3]	G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint,, Nonlinear Anal., 51 (2002), 509. doi: 10.1016/S0362-546X(01)00843-4. Google Scholar
[4]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems,, Nonlinear Anal., 52 (2003), 1911. doi: 10.1016/S0362-546X(02)00282-1. Google Scholar
[5]	H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations,, Nonlinear Anal., 73 (2010), 3924. doi: 10.1016/j.na.2010.08.026. Google Scholar
[6]	G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints,, SIAM J. Control Optim., 41 (2002), 583. doi: 10.1137/S0363012901385769. Google Scholar
[7]	E. Casas, J.-P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints,, SIAM J. Control Optim., 39 (2000), 1182. Google Scholar
[8]	J.-P. Raymond and H. Zidani, Pontryagin's principles for state-constrained control problems governed by semilinear parabolic equations with unbounded controls,, SIAM J. Control Optim., 36 (1998), 1853. Google Scholar
[9]	F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267. doi: 10.3934/mcrf.2011.1.267. Google Scholar
[10]	S. W. Hansen and O. Yu Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189. doi: 10.3934/mcrf.2011.1.189. Google Scholar
[11]	V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93. doi: 10.1016/S0167-6911(96)00083-7. Google Scholar
[12]	L. Baudouin, E. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem,, Math. Control Relat. Fields, 1 (2011), 307. doi: 10.3934/mcrf.2011.1.307. Google Scholar
[13]	I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inv. Ill-Posed Problems, 12 (2004), 43. Google Scholar
[14]	M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations,, Math. Control Relat. Fields, 1 (2011), 149. doi: 10.3934/mcrf.2011.1.149. Google Scholar
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions,, Math. Control Relat. Fields, 1 (2011), 177. doi: 10.3934/mcrf.2011.1.177. Google Scholar
[16]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional Conference Series in Applied Mathematics, (1983). Google Scholar
[17]	H. O. Fattorini and S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211. doi: 10.1017/S0308210500028444. Google Scholar
[18]	E. Casas, J.-P. Raymond and H. Zidani, Optimal control problem governed by semilinear elliptic equations with integral control constraints and pointwise state constraints,, in, 126 (1998), 89. Google Scholar
[19]	L. Cesari, "Optimization, Theory and Applications,", Springer-Verlag, (1983). Google Scholar
[20]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs,, Nonlinear Anal., 31 (1998), 15. doi: 10.1016/S0362-546X(96)00306-9. Google Scholar
[21]	X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895. doi: 10.1137/0329049. Google Scholar

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References:

[1]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis,", 2^nd edition, (1979). Google Scholar
[2]	X. Li and J. Yong, "Optimal Control Theory for Infinite, Dimensional System, (1995). Google Scholar
[3]	G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint,, Nonlinear Anal., 51 (2002), 509. doi: 10.1016/S0362-546X(01)00843-4. Google Scholar
[4]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems,, Nonlinear Anal., 52 (2003), 1911. doi: 10.1016/S0362-546X(02)00282-1. Google Scholar
[5]	H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations,, Nonlinear Anal., 73 (2010), 3924. doi: 10.1016/j.na.2010.08.026. Google Scholar
[6]	G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints,, SIAM J. Control Optim., 41 (2002), 583. doi: 10.1137/S0363012901385769. Google Scholar
[7]	E. Casas, J.-P. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints,, SIAM J. Control Optim., 39 (2000), 1182. Google Scholar
[8]	J.-P. Raymond and H. Zidani, Pontryagin's principles for state-constrained control problems governed by semilinear parabolic equations with unbounded controls,, SIAM J. Control Optim., 36 (1998), 1853. Google Scholar
[9]	F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey,, Math. Control Relat. Fields, 1 (2011), 267. doi: 10.3934/mcrf.2011.1.267. Google Scholar
[10]	S. W. Hansen and O. Yu Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189. doi: 10.3934/mcrf.2011.1.189. Google Scholar
[11]	V. Barbu, The time optimal control of Navier-Stokes equations,, Systems Control Lett., 30 (1997), 93. doi: 10.1016/S0167-6911(96)00083-7. Google Scholar
[12]	L. Baudouin, E. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem,, Math. Control Relat. Fields, 1 (2011), 307. doi: 10.3934/mcrf.2011.1.307. Google Scholar
[13]	I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inv. Ill-Posed Problems, 12 (2004), 43. Google Scholar
[14]	M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations,, Math. Control Relat. Fields, 1 (2011), 149. doi: 10.3934/mcrf.2011.1.149. Google Scholar
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions,, Math. Control Relat. Fields, 1 (2011), 177. doi: 10.3934/mcrf.2011.1.177. Google Scholar
[16]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis,", CBMS-NSF Regional Conference Series in Applied Mathematics, (1983). Google Scholar
[17]	H. O. Fattorini and S. Sritharan, Necessary and sufficient conditions for optimal controls in viscous flow problems,, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211. doi: 10.1017/S0308210500028444. Google Scholar
[18]	E. Casas, J.-P. Raymond and H. Zidani, Optimal control problem governed by semilinear elliptic equations with integral control constraints and pointwise state constraints,, in, 126 (1998), 89. Google Scholar
[19]	L. Cesari, "Optimization, Theory and Applications,", Springer-Verlag, (1983). Google Scholar
[20]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs,, Nonlinear Anal., 31 (1998), 15. doi: 10.1016/S0362-546X(96)00306-9. Google Scholar
[21]	X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems,, SIAM J. Control Optim., 29 (1991), 895. doi: 10.1137/0329049. Google Scholar

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