Final-state, Open-loop Control of the Heat equation in Tensorial Domains (full version)

In this paper, a quadratic optimal control problem will be considered for the heat equation in tensorial domains with homogeneous Dirichlet boundary conditions, in which the control function (depending only on time) constitutes a source term. These problems involve choosing a control function (with or without "peak-value" constraints) to approximately steer the solution of the heat equation to a desired function at the end of a prescribed (finite) time-interval. To compute approximations to the desired optimal control functions, semi-discrete, spectral (with eigenfunctions) Galerkin approximations to the corresponding heat equation and the corresponding (approximating) control problems are tackled. Two simple, illustrative examples are presented in the final section.