Second Look at Approximating Differential Inclusions

Abstract

In many direct methods for numerically solving optimal control problems, a collocation technique is used. What distinguishes numerous direct collocation schemes is the discretization of the time history and the way the state equations are satisfied at various discrete points. In one of the earliest schemes, cubic splines were used as the interpolating polynomials over the time segments. The state differential equations were imposed at the midpoints by way of a Hermiteﾃ__Simpson implicit integration method. Generalizations of these collocation schemes were employed by Herman and Conway and Conway and Larson in the form of higher-order Gaussﾃ__Lobatto and by Enright and Conway in the form of Rungeﾃ__Kutta-type quadrature rules. The use of higher-order integration rules facilitates a larger step size that results in a smaller number of discretization nodes or optimization variables. Because the efficiency and even convergence of nonlinear programming (NLP)problems improves for a problem of smaller size, finding ways to accurately and ef ficiently discretize optimal control problems is of great interest in this area of research.