The properties of differential-algebraic equations representing optimal control problems

A procedure is described for transforming a general optimal control problem to a system of Differential-Algebraic Equations (DAEs). The Kuhn–Tucker conditions consist of differential equations, complementarity conditions and corresponding inequalities. The latter are converted to equalities by adding a new variable combining the slack variable and the corresponding Lagrange multiplier.We investigate the properties of the resulting DAEs. The index of a system of DAEs determines the well-conditioning of the problem. The concept of the tractability index is used to investigate the index in a systematic way, and during this process, it indicates which components of the system of equations must be differentiated to reduce the index. For an index-3 problem, the index is reduced without increasing the number of equations, and a numerical procedure is used to determine the index.In the examples used here, the DAEs can be solved analytically. The examples are tested by the numerical determination of the index, and the results confirm the previously known properties of these examples.The reformulation proposed here, as well as the index determination, might be used in the future, to develop a methodology to solve optimal control problems.