Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface.In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the "optimize-then-discretize" paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional.By means of an example setup of a container with an in-and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. Two-Phase Stefan ProblemWe consider a two-dimensional two-phase Stefan problem as in [2,3] where a domain Ω ∈ R 2 is split into the solid phase Ω s (t) and the liquid phase Ω l (t) as in figure 1. The phases are separated by the moving interface Γ I (t) which we model by the Stefan condition and which we represent as a graph, h : [0, T ] × Γ cool → R, over the lower boundary Γ cool . Further, Γ cool is a cooling boundary, the inlet Γ in is a heat source and Γ out is the outlet. The temperature distribution is characterized by the heat equation, and the system is fully coupled with Navier-Stokes equations which describe the fluid flow in Ω l (t). A detailed description of the system can be found in [1]. We solve the system numerically with finite elements and mesh movement methods with the software FEniCS. The code is available at https://gitlab.mpi-magdeburg.mpg.de/baran/Stefan_Problem_in_FEniCS.git.

The optimal control of moving boundary problems receives growing attention in science and technology. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. The Stefan problem is coupled with incompressible Navier{Stokes equations. We take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state and the control costs. With the formal Lagrange approach and an adjoint system we derive the gradient of the cost functional. The derived formulations can be used to achieve a desired interface position. Among others, we address how to handle the weak discontinuity of the temperature along the interface with mesh movement methods in a finite element framework.

Problems featuring moving interfaces appear in many applications. They can model solidification and melting of pure materials, crystal growth and other multi-phase problems. The control of the moving interface enables to, for example, influence production processes and, thus, the product material quality.We consider the two-phase Stefan problem that models a solid and a liquid phase separated by the moving interface. In the liquid phase, the heat distribution is characterized by a convection-diffusion equation. The fluid flow in the liquid phase is described by the Navier-Stokes equations which introduces a differential algebraic structure to the system. The interface movement is coupled with the temperature through the Stefan condition, which adds additional algebraic constraints. Our formulation uses a sharp interface representation and we define a quadratic tracking-type cost functional as a target of a control input.We compute an open loop optimal control for the Stefan problem using an adjoint system. For a feedback representation, we linearize the system about the trajectory defined by the open loop control. This results in a linear-quadratic regulator problem, for which we formulate the differential Riccati equation with time varying coefficients. This Riccati equation defines the corresponding feedback gain.Further, we present the feedback formulation that takes into account the structure and the differential algebraic components of the problem. Also, we discuss how the complexities that come, for example, with mesh movements, can be handled in a feedback setting.We aim to control and stabilize a coupled nonlinear system. As exemplary coupled nonlinear system we use a two-phase two-dimensional Stefan problem. We compute a reference solution (x,ũ) with an open loop control as from [1]. Since the open loop controlũ is not robust against perturbations and uncertainties, it should be stabilized. We choose a Riccati feedback strategy to achieve the stabilization. The system is linearized around (x,ũ) which leads to the following linear-quadratic control problem:Here, x is the state, u the input, and y the output of the system. Details regarding the linearization and the matrices M, A, B, and C are given in Section 2. We use mesh movement techniques to represent the interface and resolve the discontinuity of the temperature gradient along the interface. The spatial discretization mesh is updated in every time step with the mesh velocity V mesh from equations (2b) and (2c). Thus, not only the functions in equation (1) are time dependent, but also the matrices. We omit the time dependence "(t)" for the sake of brevity in what follows. The optimal control u = −Kx can be computed with the feedback matrix K = B T XM [2]. The matrix X is the solution of the generalized differential Riccati equation with time dependent coefficients: −M TẊ M = C T C + (Ṁ + A) T XM + M T X(Ṁ + A) − M T XBB