# University of Minnesota

Aerospace Engineering and Mechanics

**Fall 2002 Seminar Series**

*New, Efficient Methods for the Solution of Optimal Control Problems *

*Abstract*

**The necessary conditions for the solution of a continuous optimal control
problem are well known from the calculus of variations. They (the
Euler-Lagrange equations) constitute a two-point-boundary-value problem.
Unfortunately, for problems of any sophistication this problem is nonlinear and
difficult to solve using conventional methods, for example "shooting"
methods. The method of direct collocation with nonlinear programming (DCNLP)
has become an accepted and useful alternative. In this method the necessary
conditions are not explicitly satisfied. Instead the problem is discretized and
converted into a nonlinear programming problem. **

**There are many forms in which this method may be applied; for example
there are several different implicit integration formulas that may be used,
there are several choices for how the control variables of the system may be
parameterized, and there is a choice of solvers for the nonlinear programming
problem. We have found that best results are obtained when the method is
tailored to the characteristics of the problem. **

**The DCNLP method will be described and examples of "tailoring"
this method to specific problems will be shown. These problems will include
optimal trajectories for Earth orbit to Lunar orbit transfer, for low-thrust
interception of a dangerous asteroid, for LEO to GEO orbit transfer, and for
unpowered landing of the HL-20 lifting body vehicle. It will also be shown how
we have recently extended the method to allow solution of differential games
(two-player) problems, exemplified by the case of optimal strategies for two
F-16 fighters in combat.**

### Friday, October 11, 2002

209 Akerman
Hall

2:30-3:30 p.m.

### Refreshments served after the seminar in
227 Akerman Hall.

Disability accomodations provided upon request.

Contact Kristal Belisle, Senior
Secretary, 625-8000.