The properties of differential-algebraic equations representing optimal control problems

Roland England , Susana Go'mez , René Lamour

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

49K15 Problems involving ordinary differential equations

65L80 Methods for differential-algebraic equations

Abstract
This paper outlines a procedure 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. These latter are converted to equalities by the addition of a new variable combining the slack variable and the corresponding Lagrange multipliers. The sign of this variable indicates whether the constraint is active or not.\\ The concept of the tractability index is introduced as a general purpose tool for determining the index of a system of DAEs by checking for the nonsingularity of the elements of the matrix chain. This is helpful in determining the well-conditioning of the problem, and an appropriate method for solving it numerically.\\ In the examples used here, the solution of all the differential equations could be performed analytically. The given examples are tested by the numerical determination of the tractability index chain, and the results confirm the previously known properties of the examples.