Collocation Versus Differential Inclusion in Direct Optimization

Abstract

In the so-called direct method of solution of optimal control problems, either the state variable time history or the control variable time history, or both, of the continuous problem are discretized. The problem then becomes a parameter optimization problem. The system-governing equations may be satisŽ ed by explicit numerical integration or implicitly, by including nonlinear constraints, which are in fact quadrature rules. A method termed differential inclusion has been recommended for the solution of certain classes of such problems because it reduces the size of the parameter optimization problem. It does this by removing bounded control variables in favor of bounds on attainable time rates of change of the states. The smaller problem is then in principle solved more quickly and reliably. We demonstrate analytically and with several computed problem solutions that differential inclusion, because it requires the use of an implicit quadrature rule with the lowest possible order of accuracy, i.e., Euler’s rule, yields larger rather than smaller nonlinear programming problems than direct methods, which retain the control variables but use much more sophisticated implicit quadrature rules.