Control and inverse problems for the heat equation with strong singularities

Abstract

We consider a linear system composed of N + 1 one dimensional heat equations connected by pointmass-like interface conditions. Assume an L 2 Dirichlet boundary control at one end, and Dirichlet boundary condition on the other end. Given any L 2 -type initial temperature distribution, we show that the system is null controllable in arbitrarily small time. The proof uses known results for exact controllability for the associated wave equation. An argument using the Fourier Method reduces the control problem for both the heat equation and the wave equation to certain moment problems. Controllability is then proved by relating minimality properties of the family of exponential functions associated to the wave with the family associated to the heat equation. Based on the controllability result we solve the dynamical inverse problem, i.e. recover unknown parameters of the system from the Dirichlet-to-Neumann map given at a boundary point.