Control of the Stefan problem in a periodic box

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