Optimal control of differential-algebraic equations from an ordinary differential equation perspective

TL;DR

This paper introduces a novel framework for solving optimal control problems for linear differential-algebraic systems by defining the input index and an augmented system, enabling the use of ODE control techniques.

Contribution

The paper presents the concepts of input index and augmented system, facilitating the analysis and numerical solution of DAE-based optimal control problems.

Findings

Characterization of consistent initial values via a Kalman-like matrix

Derivation of an augmented system for DAE optimal control analysis

Provision of checkable conditions for stage cost consistency

Abstract

We study the Optimal Control Problem (OCP) for regular linear differential-algebraic systems (DAEs). To this end, we introduce the input index, which allows, on the one hand, to characterize the space of consistent initial values in terms of a Kalman-like matrix and, on the other hand, the necessary smoothness properties of the control. The latter is essential to make the problem accessible from a numerical point of view. Moreover, we derive an augmented system as the key to analyze the OCP with tools well-known from optimal control of ordinary differential equations. The new concepts of the input index and the augmented system provide easily checkable sufficient conditions which ensure that the stage costs are consistent with the differential-algebraic system.