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Pointwise Tracking Optimal Control Problem for Cahn Hilliard Navier Stokes system

Sheetal Dharmatti, Greeshma K

TL;DR

The paper develops a rigorous framework for pointwise tracking optimal control of the 2D local Cahn–Hilliard–Navier–Stokes system, where the control enters as a mass source and the objectives are enforced at a finite set of spatial points. It proves well-posedness for the state equations, existence of optimal controls, and differentiability of the control-to-state map via a linearized system; adjoint problems are handled through a transposition method to manage measure-valued sources, enabling first-order optimality conditions. The work extends to a terminal-time tracking variant and to cases with singular potentials by exploiting a separation property that yields higher regularity. The results provide a solid theoretical foundation for designing and analyzing pointwise sensor-based feedback controls in two-phase flows with strong nonlinear coupling, with potential extensions to nonlocal potentials and more general settings.

Abstract

We study a pointwise tracking optimal control problem for the two-dimensional local Cahn Hilliard Navier Stokes system, which models the evolution of two immiscible, incompressible fluids. The source term in the Cahn Hilliard equation acts as a control, and the cost functional measures the deviation of the phase variable from desired values at a finite set of spatial points over time. This setting reflects realistic applications where only a limited number of sensors are available. We also study a variant of the above pointwise tracking control problem where the cost is incorporated with a terminal time pointwise tracking term. The main mathematical difficulty arises from the low regularity of the cost functional due to the pointwise evaluation of the state variables. We prove the existence of strong solutions, establish the existence of an optimal control, and the differentiability of the control to state mapping. We define the adjoint system using a transposition method to characterise optimal control. Moreover, a first-order necessary optimality condition is derived in terms of the adjoint for both problems. Furthermore, we prove that our analysis can be extended to the case of singular potentials.

Pointwise Tracking Optimal Control Problem for Cahn Hilliard Navier Stokes system

TL;DR

The paper develops a rigorous framework for pointwise tracking optimal control of the 2D local Cahn–Hilliard–Navier–Stokes system, where the control enters as a mass source and the objectives are enforced at a finite set of spatial points. It proves well-posedness for the state equations, existence of optimal controls, and differentiability of the control-to-state map via a linearized system; adjoint problems are handled through a transposition method to manage measure-valued sources, enabling first-order optimality conditions. The work extends to a terminal-time tracking variant and to cases with singular potentials by exploiting a separation property that yields higher regularity. The results provide a solid theoretical foundation for designing and analyzing pointwise sensor-based feedback controls in two-phase flows with strong nonlinear coupling, with potential extensions to nonlocal potentials and more general settings.

Abstract

We study a pointwise tracking optimal control problem for the two-dimensional local Cahn Hilliard Navier Stokes system, which models the evolution of two immiscible, incompressible fluids. The source term in the Cahn Hilliard equation acts as a control, and the cost functional measures the deviation of the phase variable from desired values at a finite set of spatial points over time. This setting reflects realistic applications where only a limited number of sensors are available. We also study a variant of the above pointwise tracking control problem where the cost is incorporated with a terminal time pointwise tracking term. The main mathematical difficulty arises from the low regularity of the cost functional due to the pointwise evaluation of the state variables. We prove the existence of strong solutions, establish the existence of an optimal control, and the differentiability of the control to state mapping. We define the adjoint system using a transposition method to characterise optimal control. Moreover, a first-order necessary optimality condition is derived in terms of the adjoint for both problems. Furthermore, we prove that our analysis can be extended to the case of singular potentials.
Paper Structure (14 sections, 18 theorems, 201 equations)

This paper contains 14 sections, 18 theorems, 201 equations.

Table of Contents

  1. Notations and Preliminary results
  2. Pointwise Tracking Optimal Control
  3. Existence of an optimal control
  4. Differentiability of the control to state operator
  5. Differentiability of the control to state operator
  6. First order necessary optimality condition
  7. Characterisation of an optimal control
  8. The Adjoint system
  9. Terminal Time Pointwise Tracking Optimal Control
  10. Differentiability of the control to state operator
  11. First order necessary optimality condition
  12. Characterisation of an optimal control
  13. The Adjoint system
  14. Local CHNS system with a singular potential

Key Result

Proposition 2.1

Let $d\in \{ 2,3 \}$, $\varphi_0 \in V$ and $\textbf{u}_0 \in \mathbb{G}_{div}$. The external source term, $U\in L^2(0,T;H)$. Assume [A1],[A2] holds. Then there exist a weak solution $(\varphi, \textbf{u})$ to the system eq8-eq14 that satisfies, And a weak formulation, for all $\psi \in V$ and $\textbf{v}\in \mathbb{V}_{div}$. Furthermore,

Theorems & Definitions (37)

  • Proposition 2.1: Existence of a weak solution to the local CHNS system
  • Remark 1
  • Proposition 2.2: Existence of a Strong Solution to the Local CHNS system
  • Proposition 2.3: Continuous dependence of the strong solution on initial data and mass source term
  • proof
  • Theorem 3.1: Existence of an optimal control
  • proof
  • Theorem 3.2: Existence of weak solution to the linearized system
  • proof
  • Remark 2
  • ...and 27 more