Control of the Stefan problem in a periodic box

Abstract

In this paper, we consider the one-phase Stefan problem with surface tension, set in a two-dimensional strip-like geometry, with periodic boundary conditions respect to the horizontal direction [Formula: see text]. We prove that the system is locally null-controllable in any positive time, by means of a control supported within an arbitrary open and non-empty subset. We proceed by a linear test and duality, but quickly find that the linearized system is not symmetric and the adjoint has a dynamic coupling between the two states through the (fixed) boundary. Hence, motivated by a Fourier decomposition with respect to [Formula: see text], we consider a family of one-dimensional systems and prove observability results which are uniform with respect to the Fourier frequency parameter. The latter results are also novel, as we compute the full spectrum of the underlying operator for the nonzero Fourier modes. The zeroth mode system, on the other hand, is seen as a controllability problem for the linear heat equation with a finite-dimensional constraint. The complete observability of the adjoint is derived by using a Lebeau–Robbiano strategy, and the local controllability of the nonlinear system is then shown by combining an adaptation of the source term method introduced in [Y. Liu, T. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid–structure interaction model, ESAIM Control Optim. Calc. Var. 19 (2013) 20–42] and a Banach fixed-point argument. Numerical experiments motivate several challenging open problems, foraying even beyond the specific setting we deal with herein.