POD-Based Mixed-Integer Optimal Control of the Heat Equation

In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To deal with the discrete controls a variant of the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous optimization variables. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason tailored second-order methods as well as model-order reduction are efficiently combined to speed-up the branch-and-bound method while still ensuring a desired accuracy. In this work the method of proper orthogonal decomposition (POD) is used for the model-order reduction. A posteriori error estimation in each node of the branch-and-bound method guarantees that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.