POD-based mixed-integer optimal control of evolution systems

Abstract

In this chapter the authors consider the numerical treatment of a mixed- integer optimal control problem governed by linear convection-diffusion equations and binary control variables. Using relaxation techniques (introduced by [31] for ordinary differential equations) the original mixed-integer optimal control problem is transferred into a relaxed optimal control problem with no integrality constraints. After an optimal solution to the relaxed problem has been computed, binary admis- sible controls are constructed by a sum-up rounding technique. This allows us to construct – in an iterative process – binary admissible controls such that the cor- responding optimal state and the optimal cost value approximate the original ones with arbitrary accuracy. However, using finite element (FE) methods to discretize the state and adjoint equations yield often to extensive systems which make the frequently calculations time-consuming. Therefore, a model-order reduction based on the proper orthogonal decomposition (POD) method is applied. Compared to the FE case, the POD approach yields to a significant acceleration of the CPU times while the error stays sufficiently small.