Optimal control of parabolic equations by spectral decomposition

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations. It consists in finding the initial datum that minimises a cost functional, which comprises an energy term on the control and a regulation term given by a distributed cost on the state, and such that the final state lies within a prescribed distance to a given target. Beyond the particularities of the chosen problem, our aim here is to propose a new methodology −based on a spectral decomposition of the operator governing the evolution of the system−, describe its implementation for this kind of problem, and finally perform some preliminary numerical experiments to assess its behavior.