References

The indirect approach to the continuous-time optimal control problem (OCP) formulates the necessary conditions of optimality as a two-point boundary value problem (TP-BVP), which generally requires numerical methods. The direct approach to the continuous-time OCP relies heavily on numerical methods too, namely the methods for solving nonlinear programs (NLP) and methods for solving ordinary differential equations (ODE). Numerical methods for both approaches share a lot of common principles and tools, and these are collectively presented in the literature as called numerical optimal control. A recommendable (and freely online available) introduction to these methods is [1]. Shorter version of this is in chapter 8 of [2], which is also available online. A more comprehensive treatment is in [3].

Some survey papers such as [4] and [5] can also be useful, although now primarily as historical accounts. Similarly with the classics [6] and [7], which cover the indirect approach only.

Another name under which the numerical methods for the direct approach are presented is trajectory optimization. There are quite a few tutorials and surveys such as [8] and [9].

References

[1]

S. Gros and M. Diehl, “Numerical Optimal Control (Draft).” Systems Control; Optimization Laboratory IMTEK, Faculty of Engineering, University of Freiburg, Apr. 2022. Available: https://www.syscop.de/files/2020ss/NOC/book-NOCSE.pdf

[2]

J. B. Rawlings, D. Q. Mayne, and M. M. Diehl, Model Predictive Control: Theory, Computation, and Design, 2nd ed. Madison, Wisconsin: Nob Hill Publishing, LLC, 2017. Available: http://www.nobhillpublishing.com/mpc-paperback/index-mpc.html

[3]

J. T. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, 3rd ed. in Advances in Design and Control. Society for Industrial and Applied Mathematics, 2020. doi: 10.1137/1.9781611976199.

[4]

A. V. Rao, “A survey of numerical methods for optimal control,” Advances in the Astronautical Sciences, vol. 135, no. 1, pp. 497–528, 2009, Accessed: Jun. 09, 2016. [Online]. Available: http://vdol.mae.ufl.edu/ConferencePublications/trajectorySurveyAAS.pdf

[5]

O. von Stryk and R. Bulirsch, “Direct and indirect methods for trajectory optimization,” Annals of Operations Research, vol. 37, no. 1, pp. 357–373, Dec. 1992, doi: 10.1007/BF02071065.

[6]

D. E. Kirk, Optimal Control Theory: An Introduction, Reprint of the 1970 edition. Dover Publications, 2004.

[7]

A. E. Bryson Jr. and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation and Control, Revised edition. CRC Press, 1975.

[8]

M. Kelly, “An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation,” SIAM Review, vol. 59, no. 4, pp. 849–904, Jan. 2017, doi: 10.1137/16M1062569.

[9]

M. P. Kelly, “Transcription Methods for Trajectory Optimization: A beginners tutorial,” arXiv:1707.00284 [math], Jul. 2017, Accessed: Apr. 06, 2021. [Online]. Available: http://arxiv.org/abs/1707.00284