Optimization of an Unsteady System Governed by PDEs using a Multi-Objective Descent Method

The aim of this work is to develop an approach to solve optimization problems in which the functional that has to be minimized is time dependent. In the literature, the most common approach when dealing with unsteady problems, is to consider a time-average criterion. However, this approach is limited since the dynamical nature of the state is neglected. These considerations lead to the alternative idea of building a set of cost functionals by evaluating a single criterion at different sampling times. In this way, the optimization of the unsteady system is defined as a multi-objective optimization problem, that will be solved using an appropriate descent algorithm. Moreover, we also consider a hybrid approach, for which the set of cost functionals is built by doing a time-average operation over multiple intervals. These strategies are illustrated and applied to a non-linear unsteady system governed by a one-dimensional convection-diffusion-reaction partial differential equation