[Paper Review] Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach

Pierluigi Colli, Gianni Gilardi+1 more|arXiv (Cornell University)|Sep 11, 2017

Solidification and crystal growth phenomena23 references18 citations

TL;DR

This paper studies optimal velocity control for a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions using a 'deep quench' approach. By leveraging variational methods and regularization techniques, the authors establish the existence of optimal controls and derive necessary optimality conditions, providing a rigorous framework for phase separation processes with constraints on the velocity and phase variables.

ABSTRACT

In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents a difficulty for the analysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by the same authors, the bulk and surface free energies are of double obstacle type, which renders the state constraint nondifferentiable. It is well known that for such cases standard constraint qualifications are not satisfied so that standard methods do not apply to yield the existence of Lagrange multipliers. In this paper, we overcome this difficulty by taking advantage of results established in the quoted paper for logarithmic nonlinearities, using a so-called `deep quench approximation'. We derive results concerning the existence of optimal controls and the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system.

Motivation & Objective

• To address optimal control of a convective Cahn-Hilliard system with double obstacles, modeling phase separation with constraints on the phase variable and its velocity.
• To incorporate dynamic boundary conditions that model surface effects in phase separation processes.
• To develop a 'deep quench' approach to handle the double obstacle potential, enabling the analysis of singular limits and optimal control in non-smooth settings.
• To establish the existence of optimal controls and derive first-order necessary optimality conditions for the control problem.
• To extend existing results on Cahn-Hilliard systems by including velocity control and dynamic boundary conditions

Proposed method

• Employing a 'deep quench' regularization technique to approximate the double obstacle potential, transforming the non-smooth control problem into a sequence of smooth, solvable problems.
• Using variational formulation and weak solutions in a Hilbert space framework to handle the coupled PDE system with dynamic boundary conditions.
• Applying the method of asymptotic analysis to pass to the limit in the regularized problems, recovering solutions to the original non-smooth problem.
• Deriving the first-order optimality system via the Lagrange multiplier rule in the context of the regularized problem, then passing to the limit to obtain the optimality system for the original problem.
• Utilizing the theory of maximal monotone operators and subdifferentials to treat the double obstacle potential and its subdifferential in the weak formulation.
• Establishing compactness and convergence results to justify the passage from regularized to original problem in the limit

Experimental results

Research questions

• RQ1 Does an optimal control exist for the convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions under velocity control?
• RQ2 How can the non-smooth double obstacle potential be regularized to allow for variational analysis and optimality conditions?
• RQ3 What are the necessary optimality conditions for the control problem, and how do they relate to the state and adjoint variables?
• RQ4 Can the 'deep quench' approach be successfully applied to systems with dynamic boundary conditions and velocity control?
• RQ5 What is the limiting behavior of the regularized optimal control problems as the regularization parameter tends to zero?

Key findings

• The existence of an optimal control is proven for the non-smooth Cahn-Hilliard system with double obstacles and dynamic boundary conditions.
• The 'deep quench' approach successfully regularizes the double obstacle potential, enabling the derivation of first-order optimality conditions.
• A necessary optimality system is derived in the limit, involving the state, adjoint, and control variables, with the subdifferential of the double obstacle potential appearing in the adjoint equation.
• The convergence of the regularized optimal solutions to a solution of the original problem is established under suitable assumptions.
• The method allows for the inclusion of velocity control, extending previous results on Cahn-Hilliard systems with only state control.
• The analysis is conducted in a weak formulation framework, ensuring applicability to physically relevant, non-smooth phase transition models.