Emerging Principles in Fast Trajectory Optimization


The confluence of major breakthroughs in optimal control theory and new algorithms has made possible the real-time computation of optimal trajectories. This implies that mission analysis can be carried out rapidly with the only limitation being the designer‘s imagination. This course will introduce the student to the major advancements that have taken place over the last decade in both theory and algorithms for fast trajectory optimization. Students will acquire a broad perspective on recent developments in the mathematical foundations of trajectory optimization; “old hats” will also acquire a new perspective to some old ideas. The overall objective of this course is to outline the new foundations related to convergence of solutions that have emerged in recent years and the accompanying breakthroughs in general techniques for problem solving. These techniques are intended to enhance, not replace, special techniques that are in common use. Anyone involved in aerospace research will benefit from this course.

Key Topics:

  • Emerging perspectives in mathematical analysis and trajectory optimization.
  • Review of optimal control theory.
    • Covector Mapping Principle for computational optimal control.
  • Introduction to pseudospectral optimal control.
  • New techniques in verification and validation.
  • Real-time optimal control.

Who Should Attend:

The purpose of this course is to introduce the student to new general techniques in trajectory optimization and optimal control that have emerged in recent years. These techniques are intended to enhance, not replace, special techniques that are in common use. Thus, anyone involved in trajectory optimization, control of complex nonlinear systems, and aerospace research will benefit from this course.

Course Information:

Type of Course: Instructor-Led Short Course
Course Level: Intermediate/Advanced
Course Length: 2 days
AIAA CEU's available: Yes



I. Emerging perspectives in mathematical analysis and trajectory optimization
II. What kinds of trajectory optimization problems are solvable today and why
III. A Modern perspective on classical optimality conditions
IV. Modern terminologies; illustrative examples
V. Pontryagin’s Principle and the Curse of Complexity
VI. Bellman’s Principle and the Curse of Dimensionality
VII. The Covector Mapping Principle
VIII. Quick review of the “forgotten” methods of Bernoulli and Euler
IX. Sequences, limits and convergences
X. Illustrative counter examples
XI. How to commute discretization with dualization and why
XII. Introduction to pseudospectral optimal control
XIII. Pseudospectral methods on arbitrary grids
XIV. How to choose discretization grids
XV. Analysis on computational properties
XVI. Putting it all together: CMP, PMP and HJB
XVII. Optimality test through Covector Mapping Principle
XVIII. Quantifying suboptimality via Bellman’s Principle
XIX. Anti-aliasing in trajectory optimization
XX. Bellman methods to improve feasibility
XXI. Real-time optimal control
XXII. Closed-form solution vs. closed-loop solution
XXIII. What is real time?
XXIV. Closed-loop optimal control structures and examples




I. Michael Ross is a Professor and Program Director of Control and Optimization at the Naval Postgraduate School in Monterey, California. He leads a group of faculty, research associates, postdoctoral fellows, GS staff and graduate students towards advancing and applying efficient methods for trajectory optimization. A Fellow of the AAS, Ross received the 2010 AIAA Mechanics and Control of Flight Award for “… fundamentally changing the landscape of flight mechanics …”. He is the author of DIDO, the first object-oriented software for solving optimal control problems. From 1999 to 2001, he was a Visiting Associate Professor at Draper Labs during which time he introduced pseudospectral methods for solving nonlinear trajectory design and control problems in the areas of launch and entry guidance, attitude control and inertial navigation. The Founding Book Review Editor for the Journal of Guidance, Control and Dynamics, Ross has served on several AIAA committees and has taught short courses at Draper Laboratory, NASA and various DoD organizations.

Dr. Gong is an assistant professor at the Dept. of Applied Mathematics and Statistics, University of California, Santa Cruz. Before that, he was a research scientist at the Department of Electrical and Computer Engineering at the University of Texas at San Antonio and Department of Mechanical and Astronautical Engineering at Naval Postgraduate School. He received his Ph.D. degree in Electrical Engineering and Computer Science from Case Western Reserve University in 2004. Dr. Gong's research interests include computational optimal control, trajectory optimization, robust and adaptive control of nonlinear systems, real-time optimal control and their industry applications. Dr. Gong received Research Associateship Award from National Research Council in 2004.




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