June  2016, 6(2): 335-362. doi: 10.3934/mcrf.2016006

Optimal control of a two-phase flow model with state constraints

1. 

Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, United States

Received  February 2015 Revised  April 2015 Published  April 2016

We investigate in this article the Pontryagin's maximum principle for a class of control problems associated with a two-phase flow model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity $v, $ coupled with a convective Allen-Cahn model for the order (phase) parameter $\phi. $ The optimal problems involve a state constraint similar to that considered in [18]. We derive the Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier-Stokes and the Allen-Cahn systems makes the analysis of the control problem more involved. In particular, the associated adjoint systems have less regularity than the one derived in [18].
Citation: Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
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show all references

References:
[1]

Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[2]

Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.  Google Scholar

[3]

Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. doi: 10.1007/BF00271794.  Google Scholar

[4]

Pysica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.  Google Scholar

[5]

Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar

[6]

Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

[7]

Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.  Google Scholar

[8]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[9]

Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.  Google Scholar

[10]

Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.  Google Scholar

[11]

Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341.  Google Scholar

[12]

Rev. Modern Phys., 49 (1977), 435-479. Google Scholar

[13]

Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.  Google Scholar

[14]

Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

Springer-Verlag, New York, 1971.  Google Scholar

[16]

Phase Transition Dynamics, 11 (2009), 641-709. doi: 10.1017/CBO9780511534874.012.  Google Scholar

[17]

Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

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

[19]

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

[20]

Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.  Google Scholar

[21]

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

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