Abstract
Optimal control theory is deeply rooted in the classical mathematical subject referred to as the calculus of variations; the name of which seems to go back to the famous mathematician Leonhard Euler (1707–1783). Calculus of variations deals with minimization of expressions of the form.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Newton’s minimal resistance problem can also be seen as a calculus of variations problem, and it predates the brachistochrone problem by almost 10 years.
- 2.
For more on the history of the brachistochrone problem and subsequent developments see H.J. Sussmann and J.C. Willems. 300 years of optimal control: from the brachistochrone to the maximum principle. IEEE Control Systems Magazine, 17: 32–44, 1997.
- 3.
Leibniz’ integral rule says that \(\frac{\text {d}}{\text {d}\alpha }\int G(\alpha ,t)\text {d}t=\int \frac{\partial G(\alpha ,t)}{\partial \alpha }\text {d}t\) if \(G(\alpha ,t)\) and \(\frac{\partial G(\alpha ,t)}{\partial \alpha }\) are continuous in t and \(\alpha \). Here they are continuous because F and \(\delta _x\) are assumed \(C^1\).
- 4.
The integration by parts rule holds if \(\frac{\partial }{\partial \dot{x}^\textsc {t}}F(t,{{} \texttt {x}}_*(t),\dot{{{} \texttt {x}}}_*(t))\) and \(\delta _x(t)\) are \(C^1\) with respect to time. This holds if \(F,{{} \texttt {x}}_*,\delta _x\) are \(C^2\) in all their arguments.
- 5.
A little-o function \({{\,\mathrm{\mathfrak {o}}\,}}:\mathbb {R}^m\rightarrow \mathbb {R}^k\) is any function with the property that \(\lim _{y\rightarrow 0} \frac{\Vert {{\,\mathrm{\mathfrak {o}}\,}}(y)\Vert }{\Vert y\Vert }=0\).
- 6.
Quick derivation: since the cotangent \(\cos (\phi /2)/\sin (\phi /2)\) for \(\phi \in [0,2\pi ]\) ranges over all real numbers once (including \(\pm \infty \)) it follows that any \(\text {d}y/\text {d}x\) can uniquely be written as \(\text {d}y/\text {d}x =\cos (\phi /2)/\sin (\phi /2)\) with \(\phi \in [0,2\pi ]\). Then (1.28) implies that \({{} \texttt {y}}(\phi )=c^2/(1+\cos ^2(\phi /2)/\sin ^2(\phi /2)) =c^2\sin ^2(\phi /2) =c^2(1-\cos (\phi ))/2\) and then \(\text {d}x/\text {d}\phi = (\text {d}y/\text {d}\phi )/(\text {d}y/\text {d}x) = [c^2\sin (\phi /2)\cos (\phi /2)]/[\cos (\phi /2)/\sin (\phi /2)] = c^2\sin ^2(\phi /2) = c^2(1-\cos (\phi ))/2\). Integrating this expression shows that \({{} \texttt {x}}(\phi )=c^2(\phi -\sin (\phi ))/2+d\) where d is some integration constant. This d equals zero because \((x,y){{\,\mathrm{:=}\,}}(0,0)\) is on the curve. (See Exercise 1.4 for more details.)
- 7.
This hyperbolic cosine solution can be derived using separation of variables (see Appendix A.3). However, there is a technicality in this derivation that is often overlooked, see Exercise 1.6, but we need not worry about that now.
- 8.
The relation between positive semi-definite Hessians and convexity is explained in Appendix A.7.
- 9.
Lagrange multipliers are usually denoted as \(\lambda \). We use \(\mu \) in order to avoid a confusion in the next chapter.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Meinsma, G., van der Schaft, A. (2023). Calculus of Variations. In: A Course on Optimal Control. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-031-36655-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-031-36655-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-36654-3
Online ISBN: 978-3-031-36655-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)