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

Huaiqiang Yu ^1, and Bin Liu ^1,

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 and 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, North-Holland, Amsterdam, 1979.
[2]	X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional System," Birkhäuser, Boston, 1995.
[3]	G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint, Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.
[4]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.
[5]	H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations, Nonlinear Anal., 73 (2010), 3924-3939. doi: 10.1016/j.na.2010.08.026.
[6]	G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.
[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), 1182-1203.
[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), 1853-1879.
[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), 267-306. doi: 10.3934/mcrf.2011.1.267.
[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), 189-230. doi: 10.3934/mcrf.2011.1.189.
[11]	V. Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett., 30 (1997), 93-100. doi: 10.1016/S0167-6911(96)00083-7.
[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), 307-330. doi: 10.3934/mcrf.2011.1.307.
[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), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).
[14]	M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175. doi: 10.3934/mcrf.2011.1.149.
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions, Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177.
[16]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
[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), 211-251. doi: 10.1017/S0308210500028444.
[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 "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.
[19]	L. Cesari, "Optimization, Theory and Applications," Springer-Verlag, New York, 1983.
[20]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Anal., 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.
[21]	X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.

show all references

References:

[1]	R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," 2^nd edition, North-Holland, Amsterdam, 1979.
[2]	X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional System," Birkhäuser, Boston, 1995.
[3]	G. Wang, Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint, Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.
[4]	G. Wang and L. Wang, Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.
[5]	H. Liu, Optimal control problems with state constraint governed by Navier-Stokes equations, Nonlinear Anal., 73 (2010), 3924-3939. doi: 10.1016/j.na.2010.08.026.
[6]	G. Wang, Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.
[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), 1182-1203.
[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), 1853-1879.
[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), 267-306. doi: 10.3934/mcrf.2011.1.267.
[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), 189-230. doi: 10.3934/mcrf.2011.1.189.
[11]	V. Barbu, The time optimal control of Navier-Stokes equations, Systems Control Lett., 30 (1997), 93-100. doi: 10.1016/S0167-6911(96)00083-7.
[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), 307-330. doi: 10.3934/mcrf.2011.1.307.
[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), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).
[14]	M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175. doi: 10.3934/mcrf.2011.1.149.
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions, Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177.
[16]	R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
[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), 211-251. doi: 10.1017/S0308210500028444.
[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 "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.
[19]	L. Cesari, "Optimization, Theory and Applications," Springer-Verlag, New York, 1983.
[20]	V. Barbu, Optimal control of Navier-Stokes equations with periodic inputs, Nonlinear Anal., 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.
[21]	X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.

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