伴随向量映射原理

伴随向量映射原理（covector mapping principle）是泛函分析的基礎定理里斯表示定理中的一個特例。名稱是由Ross（英语：I. Michael Ross）和其工作夥伴所命名^[1]^[2]^[3]^[4]^[5]^[6]。伴随向量映射原理提供了運算型最优控制中，可以將离散化和對偶性（dualization）交換順序的條件。

說明

假設要將庞特里亚金最大化原理應用在問題 $B$ ，會從給定的最佳控制問題產生一個边值问题。依照Ross的論點，此边值问题是庞特里亚金提昇（Pontryagin lift），表示為問題 $B^{\lambda }$ 。

現在要離散化問題 $B^{\lambda }$ ，這會產生問題 $B^{\lambda N}$ ，其中 $N$ 表示離散化的點數。為了方便起見，有需要證明下式成立：

N\to \infty ,\quad {\text{Problem }}B^{\lambda N}\to {\text{Problem }}B^{\lambda }

在1960年代Kalman等人^[7]就已證明要求解 $B^{\lambda N}$ 會非常的困難。此困難性稱之為「複雜度咒詛」（curse of complexity）^[8]，是「維度咒詛」（dimensionality）的互補。

在1990年代開始的一系列論文中，Ross和Fahroo證明有更簡單求解問題 $B^{\lambda }$ （因此也包括問題 $B$ ）的方法，作法是先進行離散化（問題 $B^{N}$ ）再進行對偶（問題 $B^{N\lambda }$ ）。此作法需要很小心的進行，以確保解的一致性及收斂。伴随向量映射原理確保可以找到一個伴随向量的映射律，將問題 $B^{N\lambda }$ 的解映射到問題 $B^{\lambda N}$ 的解。

參考資料

^ 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.
^ 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.

[R1-1] 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.

[2] 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

[R2-3] 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.

[R3-4] 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

[R4-5] 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.

[6] 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.

[7] Bryson, A.E. and Ho, Y.C. Applied optimal control. Hemisphere, Washington, DC, 1969.

[8] Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control. Collegiate Publishers. Carmel, CA, 2009. ISBN 978-0-9843571-0-9.

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說明

相關條目

參考資料