Abstract
Is it always satisfactory to consider only smooth solutions of the basic problem, as we have done in the previous chapter? By the middle of the 19th century, the need to go beyond continuously differentiable functions was becoming increasingly apparent. In fact, the calculus of variations may have been the first subject to acknowledge a need to consider nonsmooth functions. We begin by exhibiting a simple problem that has a natural solution in the class of piecewise-smooth functions, but has no solution in that of smooth functions. This shows that the very existence of solutions is an impetus for admitting nonsmooth arcs. The need also became apparent in physical applications (soap bubbles, for example, generally have corners and creases). Spurred by these considerations, the theory of the basic problem was extended to the context of piecewise-smooth functions. In this chapter, we develop this theory, but within the more general class of Lipschitz functions x. We extend the setting of the basic problem in one more way, by allowing x to be vector-valued.
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Notes
- 1.
The calculus of variations seems to have been the first subject to acknowledge a need to consider nonsmooth functions.
- 2.
This is known classically as the first Erdmann condition, whereas the conclusion of Prop. 14.4 is the second Erdmann condition. There does not seem to be a third.
- 3.
This is but one example showing that our revered ancestors did not know about convex functions and their useful properties.
- 4.
When θ f = π, the final point is antipodal to the initial point, and there is another (equally good) possibility: k =−1. When θ f = 0, the geodesic reduces to a vertical segment.
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© 2013 Springer-Verlag London
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Clarke, F. (2013). Nonsmooth extremals. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_15
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DOI: https://doi.org/10.1007/978-1-4471-4820-3_15
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Publisher Name: Springer, London
Print ISBN: 978-1-4471-4819-7
Online ISBN: 978-1-4471-4820-3
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