Pontryagin's maximum principle and second order optimality condition for optimal control problems for the nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions

TL;DR

This paper develops Pontryagin's maximum principle and second order optimality conditions for controlling two-fluid flow systems modeled by nonlocal Cahn-Hilliard-Navier-Stokes equations in two dimensions.

Contribution

It introduces first and second order optimality conditions for a complex fluid dynamics control problem involving nonlocal PDEs.

Findings

Derivation of Pontryagin's maximum principle for the system

Establishment of second order necessary and sufficient optimality conditions

Application to two-dimensional two-fluid flow control problem

Abstract

In this work, we address some optimal control problems related to the evolution of two isothermal, incompressible, immisible fluids in a two dimensional bounded domain. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We describe the first order necessary conditions of optimality via Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.