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

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

Theodore Tachim-Medjo 1,

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].
Keywords: state-constrained, Two-phase flow model, Allen-Cahn., maximum principle, Navier-stokes.
Mathematics Subject Classification: 35Q93, 49J21, 35A01, 35A02.
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
References:
[1]

H. Abels, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[2]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.  Google Scholar

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. doi: 10.1007/BF00271794.  Google Scholar

[4]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.  Google Scholar

[5]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar

[6]

C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

[7]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.  Google Scholar

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C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[9]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.  Google Scholar

[10]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.  Google Scholar

[11]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluid and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341.  Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar

[13]

S. Li, Optimal controls of Boussinesq equations with state constraints, Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.  Google Scholar

[14]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[16]

A. Onuki, Phase transition of fluids in shear flow, Phase Transition Dynamics, 11 (2009), 641-709. doi: 10.1017/CBO9780511534874.012.  Google Scholar

[17]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

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

[19]

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

[20]

G. Wang, Pontryagin's maximum principle for optimal control of the stationary Navier-Stokes equations, Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.  Google Scholar

[21]

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

show all references

References:
[1]

H. Abels, On a diffuse interface model for a two-phase flow of compressible viscous fluids, Indiana Univ. Math. J., 57 (2008), 659-698. doi: 10.1512/iumj.2008.57.3391.  Google Scholar

[2]

H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2.  Google Scholar

[3]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynam., 1 (1990), 303-325. doi: 10.1007/BF00271794.  Google Scholar

[4]

T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Pysica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307.  Google Scholar

[5]

G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar

[6]

C. Cao and C. G. Gal, Global solutions for the 2D NS-CH model for a two-phase flow of viscous, incompressible fluids with mixed partial viscosity and mobility, Nonlinearity, 25 (2012), 3211-3234. doi: 10.1088/0951-7715/25/11/3211.  Google Scholar

[7]

E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544.  Google Scholar

[8]

C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013.  Google Scholar

[9]

C. G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1.  Google Scholar

[10]

C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6.  Google Scholar

[11]

M. E. Gurtin, D. Polignone and J. Vinals, Two-phase binary fluid and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341.  Google Scholar

[12]

P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar

[13]

S. Li, Optimal controls of Boussinesq equations with state constraints, Nonlinear Anal., 60 (2005), 1485-1508. doi: 10.1016/j.na.2004.11.010.  Google Scholar

[14]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[15]

J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar

[16]

A. Onuki, Phase transition of fluids in shear flow, Phase Transition Dynamics, 11 (2009), 641-709. doi: 10.1017/CBO9780511534874.012.  Google Scholar

[17]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, volume 68. Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[18]

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

[19]

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

[20]

G. Wang, Pontryagin's maximum principle for optimal control of the stationary Navier-Stokes equations, Nonlinear Anal., 52 (2003), 1853-1866. doi: 10.1016/S0362-546X(02)00161-X.  Google Scholar

[21]

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

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