Minimization with constraints described by DAEs

Abstract

The present chapter collects results obtained by means of the projector based approach to DAEs, which are relevant in view of optimization. We do not at all undertake to offer an overview concerning the large field of control and optimization with DAE constraints and do not touch the huge arsenal of direct minimization methods. We address the basic topics of adjoint and self-adjoint DAEs and discuss the Hamiltonian property. We provide a necessary extremal conditions for the case of a nonlinear cost and a nonlinear DAE constraint. Necessary and sufficient extremal conditions are given for linear-quadratic problems. Moreover, an appropriate generalization of the Riccati feedback solution is developed. In each part, we direct particular attention to the properties of the resulting optimality DAE. If one intends to apply indirect optimization, then one should take great care to ensure good properties, such as regularity with index 1, in advance by utilizing the scope of modeling. By providing criteria in terms of the original problem data we intend to assist specialists in modeling.