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 (Gros and Diehl 2022). Shorter version of this is in chapter 8 of (Rawlings, Mayne, and Diehl 2017), which is also available online. A more comprehensive treatment is in (Betts 2020).

Some survey papers such as (Rao 2009) and (von Stryk and Bulirsch 1992) can also be useful, although now primarily as historical accounts. Similarly with the classics (Kirk 2004) and (Bryson and Ho 1975), 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 (M. Kelly 2017) and (M. P. Kelly 2017).

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References

Betts, John T. 2020. Practical Methods for Optimal Control Using Nonlinear Programming. 3rd ed. Advances in Design and Control. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611976199.
Bryson, Arthur E., Jr., and Yu-Chi Ho. 1975. Applied Optimal Control: Optimization, Estimation and Control. Revised edition. CRC Press.
Gros, Sebastien, and Moritz Diehl. 2022. “Numerical Optimal Control (Draft).” Systems Control; Optimization Laboratory IMTEK, Faculty of Engineering, University of Freiburg. https://www.syscop.de/files/2020ss/NOC/book-NOCSE.pdf.
Kelly, Matthew. 2017. “An Introduction to Trajectory Optimization: How to Do Your Own Direct Collocation.” SIAM Review 59 (4): 849–904. https://doi.org/10.1137/16M1062569.
Kelly, Matthew P. 2017. “Transcription Methods for Trajectory Optimization: A Beginners Tutorial.” arXiv:1707.00284 [Math], July. http://arxiv.org/abs/1707.00284.
Kirk, Donald E. 2004. Optimal Control Theory: An Introduction. Reprint of the 1970 edition. Dover Publications.
Rao, Anil V. 2009. “A Survey of Numerical Methods for Optimal Control.” Advances in the Astronautical Sciences 135 (1): 497–528. http://vdol.mae.ufl.edu/ConferencePublications/trajectorySurveyAAS.pdf.
Rawlings, James B., David Q. Mayne, and Moritz M. Diehl. 2017. Model Predictive Control: Theory, Computation, and Design. 2nd ed. Madison, Wisconsin: Nob Hill Publishing, LLC. http://www.nobhillpublishing.com/mpc-paperback/index-mpc.html.
von Stryk, O., and R. Bulirsch. 1992. “Direct and Indirect Methods for Trajectory Optimization.” Annals of Operations Research 37 (1): 357–73. https://doi.org/10.1007/BF02071065.