Maximum principle for some optimal control problems governed by 2D nonlocal Cahn-Hillard-Navier-Stokes equations. (English) Zbl 1440.49003
Summary: This work concerns some optimal control problems associated with the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The Cahn-Hilliard-Navier-Stokes model consists of a Navier-Stokes equation governing the fluid velocity field coupled with a convective Cahn-Hilliard equation for the relative concentration of one of the fluids. A distributed optimal control problem is formulated as the minimization of a cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We establish the first-order necessary conditions of optimality by proving the Pontryagin maximum principle for optimal control of such system via the seminal Ekeland variational principle. The optimal control is characterized using the adjoint variable. We also study an another optimal control problem of finding the unknown optimal initial data. Optimal data initialization problem is also known as the data assimilation problems in meteorology, where determining the correct initial condition is very crucial for the future predictions.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 49S05 | Variational principles of physics |
| 58E30 | Variational principles in infinite-dimensional spaces |
Keywords:
optimal control; nonlocal Cahn-Hilliard-Navier-Stokes systems; Ekeland variational principle; the Pontryagin maximum principle; data assimilation problemReferences:
| [1] | Abergel, F.; Temam, R., On some control problems in fluid mechanics, Theoret. Comput. Fluid Dyn., 1, 303-325 (1990) · Zbl 0708.76106 · doi:10.1007/BF00271794 |
| [2] | Agmon, S., Lectures on Elliptic Boundary Value Problems (1965), Providence, RI: AMS Chelsea Publishing, Providence, RI · Zbl 0151.20203 |
| [3] | Aubin, J-P; Ekeland, I., Applied Nonlinear Analysis (1984), New York: Dover Publications, New York · Zbl 0641.47066 |
| [4] | Ball, JM, Strongly continuous semigroups, weak solutions, and variational constants formula, Proc. Am. Math. Soc., 63, 2, 370-373 (1977) · Zbl 0353.47017 |
| [5] | Biswas, T., Dharmatti, S., Mohan, M.T.: Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn-Hillard-Navier-Stokes equations. arXiv:1802.08413 · Zbl 1488.49004 |
| [6] | Boyer, F., Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptot. Anal., 20, 2, 175-212 (1999) · Zbl 0937.35123 |
| [7] | Colli, P.; Frigeri, S.; Grasselli, M., Global existence of weak solutions to a nonlocal Cahn-Hilliard Navier-Stokes system, J. Math. Anal. Appl., 386, 1, 428-444 (2012) · Zbl 1241.35155 · doi:10.1016/j.jmaa.2011.08.008 |
| [8] | Colli, P.; Sprekels, J.; Colli, P.; Favini, A.; Rocca, E.; Schimperna, G.; Sprekels, J., Optimal boundary control of a nonstandard Cahn-Hilliard system with dynamic boundary condition and double obstacle inclusions, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, 151-182 (2017), Cham: Springer, Cham · Zbl 1382.49004 |
| [9] | Colli, P.; Gilardi, G.; Rocca, E.; Sprekels, J., Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30, 2518-2546 (2017) · Zbl 1378.35175 · doi:10.1088/1361-6544/aa6e5f |
| [10] | Colli, P.; Farshbaf-Shaker, MH; Gilardi, G.; Sprekels, J., Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim., 53, 2, 696-721 (2015) |
| [11] | DiBenedetto, E., Degenerate Parabolic Equations (1993), New York: Springer, New York · Zbl 0794.35090 |
| [12] | Doboszczak, S., Mohan, M.T., Sritharan, S.S.: Necessary conditions for distributed optimal control of linearized compressible Navier-Stokes equations, Submitted, arXiv:1910.11244 · Zbl 1486.49031 |
| [13] | Edmunds, DE, Optimal control of systems governed by partial differential equations, Bull. London Math. Soc., 4, 2, 236-237 (1972) · doi:10.1112/blms/4.2.236 |
| [14] | Ekeland, I., On the variational principle, J. Math. Anal. Appl., 47, 324-353 (1974) · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0 |
| [15] | Evans, LC, Partial Differential Equations (2010), Providence: American Mathematical Society, Providence · Zbl 1194.35001 |
| [16] | Fattorini, HO; Sritharan, SS, Necessary and sufficient conditions for optimal controls in viscous flow problems, Proc. R. Soc. Lond. Ser. A, 124, 211-251 (1994) · Zbl 0800.49047 |
| [17] | Fattorini, HO, Infinite Dimensional Optimization and Control Theory (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 0931.49001 |
| [18] | Frigeri, S.; Gal, CG; Grasselli, M., On nonlocal Cahn-illiard-Navier-Stokes systems in two dimensions, J. Nonlinear Sci., 26, 4, 847-893 (2016) · Zbl 1348.35171 · doi:10.1007/s00332-016-9292-y |
| [19] | Frigeri, S.; Grasselli, M.; Krejci, P., Strong solutions for two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems, J. Differ. Equ., 255, 9, 2587-2614 (2013) · Zbl 1284.35312 · doi:10.1016/j.jde.2013.07.016 |
| [20] | Frigeri, S.; Rocca, E.; Sprekels, J., Optimal distributed control of a nonlocal Cahn-Hilliard Navier-Stokes system in two dimension, SIAM J. Control Optim., 54, 1, 221-250 (2015) · Zbl 1335.49010 · doi:10.1137/140994800 |
| [21] | Frigeri, S.; Grasselli, M.; Sprekels, J., Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential, Appl. Math. Optim. (2018) · Zbl 1440.35235 · doi:10.1007/s00245-018-9524-7 |
| [22] | Fursikov, AV, Optimal Control of Distributed Systems: Theory and Applications (2000), Rhode Island: American Mathematical Society, Rhode Island · Zbl 1027.93500 |
| [23] | Gunzburger, M.D.: Perspectives in Flow Control and Optimization. SIAM’s Advances in Design and Control series, Philadelphia (2003) · Zbl 1088.93001 |
| [24] | Lions, J-L, Optimal Control of Systems Governed by Partial Differential Equations (1971), Berlin: Springer, Berlin · Zbl 0203.09001 |
| [25] | Lions, J-L, Quelques méthodes de résolution des problémes aux limites non linéaires (1969), Paris: (French) Dunod Gauthier-Villars, Paris · Zbl 0189.40603 |
| [26] | Ladyzhenskaya, OA, The Mathematical Theory of Viscous Incompressible Flow (1969), New York: Gordon and Breach, New York · Zbl 0184.52603 |
| [27] | Li, X.; Yong, J., Optimal Control Theory for Infinite Dimensional Systems (1995), Boston: Birkhäuser, Boston |
| [28] | Mejdo, T., hRobust control of a Cahn-Hilliard-Navier-Stokes model, Commun. Pure Appl. Anal., 15, 6, 2075 (2016) · Zbl 1348.93134 · doi:10.3934/cpaa.2016028 |
| [29] | Medjo, T., Optimal control of a Cahn-Hilliard-Navier-Stokes model with state constraints, J. Convex Anal, 22, 1135-1172 (2015) · Zbl 1330.49020 |
| [30] | Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 3, 13, 115-162 (1959) · Zbl 0088.07601 |
| [31] | Simon, J., Compact sets in the space \(\text{L}^p(0, T;\text{ B })\), Annali di Matematica, 146, 65-96 (1986) · Zbl 0629.46031 · doi:10.1007/BF01762360 |
| [32] | Sritharan, S.S.: Optimal control of viscous flow. SIAM Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia (1998) · Zbl 0920.76004 |
| [33] | Temam, R., Navier-Stokes Equations, Theory and Numerical Analysis (1984), Amsterdam: North-Holland, Amsterdam · Zbl 0568.35002 |
| [34] | Wang, G., Pontryagin maximum principle of optimal control governed by fluid dynamic systems with two point boundary state constraint, Nonlinear Anal., 51, 509-536 (2002) · Zbl 1013.49018 · doi:10.1016/S0362-546X(01)00843-4 |
| [35] | Wang, G.; Wang, L., Maximum principle of state-constrained optimal control governed by fluid dynamic systems, Nonlinear Anal., 52, 1911-1931 (2003) · Zbl 1032.49033 · doi:10.1016/S0362-546X(02)00282-1 |
| [36] | Zhao, X.; Liu, C., Optimal control problem for viscous Cahn-Hilliard equation, Nonlinear Anal. Theory Methods Appl., 74, 17, 6348-6357 (2011) · Zbl 1228.49008 · doi:10.1016/j.na.2011.06.015 |
| [37] | Zheng, J.; Wang, Y., Optimal control problem for Cahn-Hilliard equations with state constraint, J. Dyn. Control Syst., 21, 2, 257-272 (2015) · Zbl 1321.49037 · doi:10.1007/s10883-014-9259-y |
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