# American Institute of Mathematical Sciences

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 and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
##### References:
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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. [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. [14] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4. [15] J. L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, New York, 1971. [16] A. Onuki, Phase transition of fluids in shear flow, Phase Transition Dynamics, 11 (2009), 641-709. doi: 10.1017/CBO9780511534874.012. [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. [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. [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. [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. [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.
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