伴随向量映射原理

伴随向量映射原理（covector mapping principle）是泛函分析的基础定理里斯表示定理中的一个特例。名称是由Ross（英语：I. Michael Ross）和其工作伙伴所命名^[1]^[2]^[3]^[4]^[5]^[6]。伴随向量映射原理提供了运算型最优控制中，可以将离散化和对偶性（dualization）交换顺序的条件。

说明

假设要将庞特里亚金最大化原理应用在问题 $B$ ，会从给定的最佳控制问题产生一个边值问题。依照Ross的论点，此边值问题是庞特里亚金提升（Pontryagin lift），表示为问题 $B^{\lambda }$ 。

现在要离散化问题 $B^{\lambda }$ ，这会产生问题 $B^{\lambda N}$ ，其中 $N$ 表示离散化的点数。为了方便起见，有需要证明下式成立：

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

在1960年代Kalman等人^[7]就已证明要求解 $B^{\lambda N}$ 会非常的困难。此困难性称之为“复杂度咒诅”（curse of complexity）^[8]，是“维度咒诅”（dimensionality）的互补。

在1990年代开始的一系列论文中，Ross和Fahroo证明有更简单求解问题 $B^{\lambda }$ （因此也包括问题 $B$ ）的方法，作法是先进行离散化（问题 $B^{N}$ ）再进行对偶（问题 $B^{N\lambda }$ ）。此作法需要很小心的进行，以确保解的一致性及收敛。伴随向量映射原理确保可以找到一个伴随向量的映射律，将问题 $B^{N\lambda }$ 的解映射到问题 $B^{\lambda N}$ 的解。

参考资料

^ 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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说明

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参考资料