Expressing optimal control problems as differential algebraic equations

Abstract

The purpose of this paper is to present an approach to express certain types of optimal control problems in terms of a system of differential algebraic equations (DAEs). This system is obtained using calculus of variations to get the Kuhn–Tucker conditions. The inequalities associated with the complementarity conditions are converted to equalities by the addition of a new variable. Such systems of DAEs are well known in the Chemical Engineering literature, and there are a number of established numerical methods for their solution. Also, we introduce here the concept of the tractability index as a general purpose way of determining the index, by establishing which part of the system of DAEs must be differentiated and how many times. This provides a systematic way of determining the index, without needing to differentiate the whole system. Numerical examples from Chemical Engineering are used to illustrate the methodology.