Expressing optimal control problems as differential algebraic equations

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.