Abstract
The main objective of this paper is to study the optimal distributed control for a model of homogeneous incompressible two-phase flows. We apply the well-posedness and regularity results to establish the existence of optimal controls as well as the first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
Recall for definiteness that it is easy to prove that a couple (v, ϕ) belonging to the function spaces and satisfying integral equation indicated above, is the solution of system (2.2) as well.
References
Blesgen T. A generalization of the Navier-Stokes equation to two-phase flow. Physica D: Nonlinear Phenomena 1999;32:1119–1123.
Caginalp G. An analysis of a phase field model of a free boundary. Arch Ration Mech Anal 1986;92(3):205–245.
Chepyzhov V.V., Vishik M.I. 2002. Attractors for equations of mathematical physics, American Mathematical Society, Providence, RI.
Feireisl E., Petzeltová H., Rocca E., Schimperna G. Analysis of a phase-field model for two-phase compressible fluids. Mathematical Models and Methods in Applied Sciences 2010;20(7):1129–1160.
Frigeri S., Rocca E., Sprekels J. Optimal distributed control of a nonlocal Cahn-Hilliard/ Navier-Stokes system in two dimensions. SIAM J CONTROL OPTIM 2016;54(1):221–250.
Fursikov A.V., Gunzburger M.D., Hou L.S. Optimal boundary control for the evolutionary Navier-Stokes system: the three-dimensional case. SIAM J CONTROL OPTIM 2005;43(6):2191–2232.
Gal C.G. Long-time behavior for a model of homogeneous incompressible two-phase flows. Discrete Contin. Dyn. 2010;28(1):1–39.
Gal C.G., Grasselli M. Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D. ANN I H POINCARE-AN 2010a;27(1):401–436.
Gal C.G., Grasselli M. Trajectory attractors for binary fluid mixtures in 3D. Chin. Ann. Math. Ser. B 2010b;31(5):655–678.
Medjo T.T. Pullback attractors for a non-autonomous homogeneous two-phase flow model. Int. J. Differ. Equ. 2012;253:1779–1806.
Sohr H. 2001. The Navier-Stokes equations: an elementary functional analytic approach, Birkh auser Verlag Basel.
Temam R. Infinite dimensional dynamical systems in mechanics and physics. New York: Springer-Verlag; 1997.
Tröltzsch F. Optimal control of partial differential equations: theory, methods and applications. Rhode Island: American Mathematical Society Providence; 2010.
Funding
This work was supported by the National Science Foundation of China Grant (11401459,11801427, 11871389), the Natural Science Foundation of Shaanxi Province (2018JQ1009, 2018JM1012), and the Fundamental Research Funds for the Central Universities (xjj2018088).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Li, F., You, B. Optimal Distributed Control for a Model of Homogeneous Incompressible Two-Phase Flows. J Dyn Control Syst 27, 153–177 (2021). https://doi.org/10.1007/s10883-020-09500-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-020-09500-7
Keywords
- Distributed optimal control
- First-order necessary optimality conditions
- Adjoint state system
- Two-phase flow model.