Covector mapping principle
Introduced by Ross and his co-workers,  the covector mapping principle is a fundamental result in computational optimal control. It address the Ross–Fahroo lemma by providing the favorable conditions under which dualization can be commuted with discretization.
An application of Pontryagin's minimum principle to Problem , a given optimal control problem generates a boundary value problem. According to Ross, this boundary value problem is a Pontryagin lift and is represented as Problem .
Now suppose one discretizes Problem . This generates Problem where represents the number of discrete pooints. For convergence, it is necessary to prove that as
In a series of papers starting in the late 1990s, Ross and Fahroo showed that one could arrive at a solution to Problem (and hence Problem ) more easily by discretizing first (Problem ) and dualizing afterwards (Problem ). The sequence of operations must be done carefully to ensure consistency and convergence. The covector mapping principle asserts that a covector mapping theorem can be discovered to map the solutions of Problem to Problem thus completing the circuit.
The covector mapping principle was first implemented in flight by NASA to verify the optimality of the pseudospectral solution generated by DIDO. The first flight implementation was on November 5, 2006, when NASA used the Legendre pseudospectral method to maneuver the International Space Station to perform the Zero Propellant Maneuver. The Zero Propellant Maneuver was discovered by Nazereth Bedrossian using DIDO.
- Ross, I. M., “A Historical Introduction to the Covector Mapping Principle,” Proceedings of the 2005 AAS/AIAA Astrodynamics Specialist Conference, August 7–11, 2005 Lake Tahoe, CA. AAS 05-332.
- Q. Gong, I. M. Ross, W. Kang, F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Computational Optimization and Applications, Vol. 41, pp. 307–335, 2008
- Ross, I. M. and Fahroo, F., “Legendre Pseudospectral Approximations of Optimal Control Problems,” Lecture Notes in Control and Information Sciences, Vol. 295, Springer-Verlag, New York, 2003, pp 327–342.
- Ross, I. M. and Fahroo, F., “Discrete Verification of Necessary Conditions for Switched Nonlinear Optimal Control Systems,” Proceedings of the American Control Conference, June 2004, Boston, MA
- Ross, I. M. and Fahroo, F., “A Pseudospectral Transformation of the Covectors of Optimal Control Systems,” Proceedings of the First IFAC Symposium on System Structure and Control, Prague, Czech Republic, 29–31 August 2001.
- W. Kang, I. M. Ross, Q. Gong, Pseudospectral optimal control and its convergence theorems, Analysis and Design of Nonlinear Control Systems, Springer, pp.109–124, 2008.
- I. M. Ross and F. Fahroo, A Perspective on Methods for Trajectory Optimization, Proceedings of the AIAA/AAS Astrodynamics Conference, Monterey, CA, August 2002. Invited Paper No. AIAA 2002-4727.
- Bryson, A.E. and Ho, Y.C. Applied optimal control. Hemisphere, Washington, DC, 1969.
- Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control. Collegiate Publishers. Carmel, CA, 2009. ISBN 978-0-9843571-0-9.
- W. Kang and N. Bedrossian. Pseudospectral Optimal Control Theory Makes Debut Flight, Saves nasa $1m in Under Three Hours. SIAM News, 40, 2007.