References

Indirect approach to optimal control is based on calculus of variations (and its later extension in the form of Pontryagin’s principle of maximum). Calculus of variations is an advanced mathematical discipline that requires non-trivial foundations and effort to master. In our course, however, we take the liberty of aiming for intuitive understanding rather than mathematical rigor. At roughly the same level, the calculus of variations is introduced in books on optimal control, such as the classic and affordable (Kirk 2004), the popular and online available (Lewis, Vrabie, and Syrmo 2012), or very accessible and also freely available online (Liberzon 2011).

With anticipation, we provide here a reference to the paper (Sussmann and Willems 1997), which shows how the celebrated Pontryagin’s principle of maximum extends the calculus of variations significantly. But we will only discuss this in the next chapter.

For those interested in a having a standard reference for the calculus of variations, the classic (Gelfand and Fomin 2020) is recommendable, the more so that it is fairly slim.

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References

Gelfand, I. M., and S. V. Fomin. 2020. Calculus of Variations. Reprint of the 1963 edition. Mineola, N.Y: Dover Publications.
Kirk, Donald E. 2004. Optimal Control Theory: An Introduction. Reprint of the 1970 edition. Dover Publications.
Lewis, Frank L., Draguna Vrabie, and Vassilis L. Syrmo. 2012. Optimal Control. 3rd ed. John Wiley & Sons. https://lewisgroup.uta.edu/FL%20books/Lewis%20optimal%20control%203rd%20edition%202012.pdf.
Liberzon, Daniel. 2011. Calculus of Variations and Optimal Control Theory: A Concise Introduction. Princeton University Press. http://liberzon.csl.illinois.edu/teaching/cvoc/cvoc.html.
Sussmann, H. J., and J. C. Willems. 1997. “300 Years of Optimal Control: From the Brachystochrone to the Maximum Principle.” IEEE Control Systems 17 (3): 32–44. https://doi.org/10.1109/37.588098.