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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
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show all references

References:
[1]

2nd edition, North-Holland, Amsterdam, 1979. Google Scholar

[2]

Dimensional System," Birkhäuser, Boston, 1995. Google Scholar

[3]

Nonlinear Anal., 51 (2002), 509-536. doi: 10.1016/S0362-546X(01)00843-4.  Google Scholar

[4]

Nonlinear Anal., 52 (2003), 1911-1931. doi: 10.1016/S0362-546X(02)00282-1.  Google Scholar

[5]

Nonlinear Anal., 73 (2010), 3924-3939. doi: 10.1016/j.na.2010.08.026.  Google Scholar

[6]

SIAM J. Control Optim., 41 (2002), 583-606. doi: 10.1137/S0363012901385769.  Google Scholar

[7]

SIAM J. Control Optim., 39 (2000), 1182-1203.  Google Scholar

[8]

SIAM J. Control Optim., 36 (1998), 1853-1879. Google Scholar

[9]

Math. Control Relat. Fields, 1 (2011), 267-306. doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[10]

Math. Control Relat. Fields, 1 (2011), 189-230. doi: 10.3934/mcrf.2011.1.189.  Google Scholar

[11]

Systems Control Lett., 30 (1997), 93-100. doi: 10.1016/S0167-6911(96)00083-7.  Google Scholar

[12]

Math. Control Relat. Fields, 1 (2011), 307-330. doi: 10.3934/mcrf.2011.1.307.  Google Scholar

[13]

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]

Math. Control Relat. Fields, 1 (2011), 149-175. doi: 10.3934/mcrf.2011.1.149.  Google Scholar

[15]

Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177.  Google Scholar

[16]

CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983. Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 211-251. doi: 10.1017/S0308210500028444.  Google Scholar

[18]

in "Control and Estimation of Distributed Parameter Systems" (Vorau, 1996), Internat. Ser. Numer. Math., 126, Birkhäuser, Basel, (1998), 89-102.  Google Scholar

[19]

Springer-Verlag, New York, 1983. Google Scholar

[20]

Nonlinear Anal., 31 (1998), 15-31. doi: 10.1016/S0362-546X(96)00306-9.  Google Scholar

[21]

SIAM J. Control Optim., 29 (1991), 895-908. doi: 10.1137/0329049.  Google Scholar

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