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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

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.
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," 2nd edition, North-Holland, Amsterdam, 1979. Google Scholar [2] X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional System," Birkhäuser, Boston, 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), 509-536. 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), 1911-1931. 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), 3924-3939. 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), 583-606. 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), 1182-1203.  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), 1853-1879. 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), 267-306. 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), 189-230. 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), 93-100. 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), 307-330. 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), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 211-251. 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 "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.  Google Scholar [19] L. Cesari, "Optimization, Theory and Applications," Springer-Verlag, New York, 1983. Google Scholar [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.  Google Scholar [21] X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.  Google Scholar

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##### References:
 [1] R. Temam, "Navier-Stokes Equations, Theory and Numerical Analysis," 2nd edition, North-Holland, Amsterdam, 1979. Google Scholar [2] X. Li and J. Yong, "Optimal Control Theory for Infinite Dimensional System," Birkhäuser, Boston, 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), 509-536. 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), 1911-1931. 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), 3924-3939. 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), 583-606. 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), 1182-1203.  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), 1853-1879. 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), 267-306. 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), 189-230. 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), 93-100. 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), 307-330. 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), 43-123; Part II: $L_2(\Omega)$-estimates, J. Inv. Ill-Posed Problems, 12 (2004), 183-231, (MR2061430).  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] R. Temam, "Navier-Stokes Equations and Nonlinear Functional Analysis," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 211-251. 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 "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.  Google Scholar [19] L. Cesari, "Optimization, Theory and Applications," Springer-Verlag, New York, 1983. Google Scholar [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.  Google Scholar [21] X. J. Li and J. Yong, Necessary conditions for optimal control of distributed parameter systems, SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.  Google Scholar
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