Guaranteed satisfaction of inequality state constraints in PDE-constrained optimization

Abstract

Optimal control problems with inequality path constraints (IPCs) are present in several engineering problems described by partial differential equations (PDE). We propose an algorithm to solve PDE-constrained dynamic optimization problems with guaranteed satisfaction of IPCs. The algorithm is based on a solution of a sequence of approximated dynamic optimization problems following the strategy of Fu et al. (Automatica, 2015). For the approximation, the path constraint is imposed only on a set of discrete points and is thus relaxed. At the same time, it is restricted by an adaptive back-off and by the error bound of the PDE solution. The approximation is iteratively improved: at iterations without violation, the restriction is reduced; at iterations with violation, the point of maximal violation is added to the discrete set. Under mild assumptions, we prove finite termination of the algorithm. We test the algorithm on an optimal grade change of a tubular reactor.