Maximum Principle for Some Optimal Control Problems Governed by 2D Nonlocal Cahn–Hillard–Navier–Stokes Equations

• Published in: Journal of mathematical fluid mechanics Vol. 22; no. 3
• Main Authors: Biswas, Tania, Dharmatti, Sheetal, Mohan, Manil T.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2020; Springer Nature B.V
• Subjects: Classical and Continuum Physics; Computational fluid dynamics; Fluid flow; Fluid mechanics; Fluid- and Aerodynamics; Incompressible flow; Mathematical Methods in Physics; Maximum principle; Meteorology; Navier-Stokes equations; Optimal control; Optimization; Physics; Physics and Astronomy; Theoretical mathematics; Two dimensional models; Velocity distribution; 49J20; Optimal control; 35Q35; Ekeland variational principle; 76D03; Nonlocal Cahn–Hilliard–Navier–Stokes systems; Data assimilation problem; The Pontryagin maximum principle
• ISSN: 1422-6928; 1422-6952
• DOI: 10.1007/s00021-020-00493-8

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.