Numerical solutions to optimal control problems by finite elements in time with adaptive error control.

摘要

In this research effort, the finite element method in time was applied to boundary value problems with non-linear system dynamics, including optimal control problems. The use of high-order shape functions was compared to using zeroth-order shape functions and shown to reduce the amount of CPU time necessary to achieve a given level of accuracy for a variety of problems.;An adaptive solution refinement methodology was also developed utilizing an a posteriori error estimator and adaptive mesh refinement in order to find the optimal combination of mesh parameters for maximizing accuracy for a given number of parameters. The error estimator includes a bound on the errors in meeting the differential equations at the element level plus a stability factor to gauge how errors propagate through the time interval. It requires only information available from the approximate solution to the problem.;A demonstration of the h-version of the finite element method using this adaptive error control methodology is given for a missile guidance problem in which the final velocity is maximized in a 7-state, 2-control model with non-linear system dynamics over three stages. In one pass through the adaptive error control algorithm, error bounds were at the specified tolerance with significantly fewer elements than the corresponding mesh needed to meet the tolerance through uniform mesh refinement. The number of parameters in the problem necessary to reach a given level of accuracy has been demonstrated to be reduced by as much as two-thirds with a corresponding reduction in CPU time of as much as 60 percent.