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The '''covector mapping principle''' is a special case of [[Riesz_representation_theorem|Riesz' representation theorem]], which is a fundamental theorem in functional analysis. | |||
The name was coined by by [[I. Michael Ross|Ross]] and co-workers,<ref name = "R1">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.</ref><ref> | |||
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</ref> | |||
<ref name ="R2">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.</ref><ref name = "R3">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</ref><ref name = "R4">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.</ref><ref>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.</ref> | |||
It provides conditions under which dualization can be commuted with [[discretization]] in the case of computational [[optimal control]]. | |||
==Description== | |||
An application of [[Pontryagin's minimum principle]] to Problem <math> B </math>, 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 <math>B^\lambda</math>. [[Image:CMP-OptimalControl.png|thumb|300px|center|Illustration of the Covector Mapping Principle (adapted from Ross and Fahroo | |||
.<ref>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.</ref>]] Now suppose one discretizes Problem <math>B^\lambda</math>. This generates Problem<math>B^{\lambda N}</math> where <math>N</math> represents the number of discrete pooints. For convergence, it is necessary to prove that as | |||
:<math> N \to \infty, \quad \text{Problem } B^{\lambda N} \to \text{Problem } B^\lambda </math> | |||
In the 1960s Kalman and others <ref>Bryson, A.E. and Ho, Y.C. Applied optimal control. Hemisphere, Washington, DC, 1969.</ref> showed that solving Problem <math> B^{\lambda N}</math> is extremely difficult. This difficulty, known as the curse of complexity,<ref>Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control. Collegiate Publishers. Carmel, CA, 2009. ISBN 978-0-9843571-0-9.</ref> is complementary to the curse of dimensionality. | |||
In a series of papers starting in the late 1990s, Ross and Fahroo showed that one could arrive at a solution to Problem <math> B^{\lambda}</math> (and hence Problem <math> B </math>) more easily by discretizing first (Problem <math> B^{N}</math>) and dualizing afterwards (Problem <math> B^{N \lambda}</math>). 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 <math> B^{N \lambda}</math> to Problem <math> B^{\lambda N}</math> thus completing the circuit. | |||
==See also== | |||
*[[Legendre pseudospectral method]] | |||
*[[Ross–Fahroo pseudospectral method]]s | |||
*[[Ross–Fahroo lemma]] | |||
==References== | |||
{{Reflist}} | |||
[[Category:Optimal control]] | |||
Revision as of 20:17, 26 January 2014
The covector mapping principle is a special case of Riesz' representation theorem, which is a fundamental theorem in functional analysis. The name was coined by by Ross and co-workers,[1][2] [3][4][5][6] It provides conditions under which dualization can be commuted with discretization in the case of computational optimal control.
Description
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 the 1960s Kalman and others [8] showed that solving Problem is extremely difficult. This difficulty, known as the curse of complexity,[9] is complementary to the curse of dimensionality.
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.
See also
References
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- ↑ 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.