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 & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
References:

show all references

References:
 [1] Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110 [2] Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 [3] Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [4] Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009 [5] T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 [6] Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198 [7] Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 [8] K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591 [9] Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157 [10] Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011 [11] Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401 [12] Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016 [13] G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010 [14] Zhen Cheng, Wenjun Wang. The Cauchy problem of a two-phase flow model for a mixture of non-interacting compressible fluids. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021151 [15] 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 [16] Eduardo Casas, Fredi Tröltzsch. State-constrained semilinear elliptic optimization problems with unrestricted sparse controls. Mathematical Control & Related Fields, 2020, 10 (3) : 527-546. doi: 10.3934/mcrf.2020009 [17] Lars Grüne, Hasnaa Zidani. Zubov's equation for state-constrained perturbed nonlinear systems. Mathematical Control & Related Fields, 2015, 5 (1) : 55-71. doi: 10.3934/mcrf.2015.5.55 [18] Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090 [19] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3973-3987. doi: 10.3934/dcdss.2020467 [20] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397

2020 Impact Factor: 1.284

Metrics

• HTML views (0)
• Cited by (1)

Other articlesby authors

• on AIMS
• on Google Scholar