Unique Continuation and Control for the Heat Equation from an Oscillating Lower Dimensional Manifold

Abstract

We consider the linear heat equation with Dirichlet boundary conditions in a bounded domain of $\R^n$, $n\geq 1$, and with a control acting on a lower-dimensional time-dependent manifold of dimension $k\leq n-1$. We analyze the approximate controllability problem. This problem is equivalent to a suitable uniqueness or unique continuation property of solutions of the heat equation without control. More precisely, it consists of proving that the unique solution of the Dirichlet problem vanishing on the time-dependent manifold is identically zero. This uniqueness problem, however, does not fit in the class of classical Cauchy problems and therefore, the existing tools based on power series expansions, Carleman inequalities, and doubling properties do not seem to apply. We give sufficient conditions on the time-dependent manifold for this uniqueness property to hold. The techniques we employ combine the Fourier series representation and the time analyticity of solutions and allow us to reduce the problem to a uniqueness question for the eigenfunctions of the Laplacian. We then apply well-known results on the nodal sets of these eigenfunctions. We also analyze the asymptotic behavior of the control when the time-oscillation of the manifold supporting the control increases. When the frequency of oscillation tends to infinity we prove that the controls converge to an approximate control for the same heat equation but on a manifold of dimension $k+1$ that is independent of time. This is done under suitable time-periodicity assumptions on the original manifold and confirms the fact that increasing time-oscillations of the support of the control increases the efficiency of the control mechanism.