COLLOCATION VERSUS DIFFERENTIAL INCLUSION IN DIRECT OPTIMIZATION

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 probl em then becomes a parameter optimization problem, The system-governing equations may be satisfied by explicit numerical integration or impli citly, by including nonlinear constraints, which are in bet quadrature rules. A method termed differential inclusion has been recommended fo r the solution of certain classes of such problems because it re duces the size of the parameter optimization problem. It does this by remov ing bounded control variables in favor of bounds on attainable time ra tes 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, becau se it requires the use of an implicit quadrature rule with the lowest possible order of accuracy, i.e., Euler's rule, yields larger rather t han smaller nonlinear programming problems than direct methods, which retain the control variables but use much more sophisticated implicit quadrature rules.