Abstract
In this chapter, we prove the Pontryagin maximum principle. The proof we present follows arguments by Hector Sussmann [244, 247, 248], but in a smooth setting. It is somewhat technical, but provides a uniform treatment of first- and high-order variations. As a result, we not only prove Theorem 2.2.1, but obtain a general high-order version of the maximum principle (e.g., see [140]) from which we then derive the high-order necessary conditions for optimality that were introduced in Sect. 2.8.
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Notes
- 1.
A σ-additive set function ν is said to be absolutely continuous with respect to some measure μ if whenever E is a measurable set for which μ(E) = 0, then also ν(E) = 0.
- 2.
It will always be assumed that M is second countable and Hausdorff. A manifold M is second countable if its topology has a countable basis, and it is Hausdorff if for any two points p and q in M there exist open neighborhoods U of p and V of q that are disjoint. The interested reader is referred to [81] or any other textbook on topology for this background material.
- 3.
It obviously suffices for the vector fields to be of class C r with r large enough that all derivatives exist that need to be taken.
- 4.
This expansion is in terms of exponentials of elements in a Philip Hall basis [52] of the Lie subalgebra \(L(\mathcal{X})\) of the free associative algebra generated by a family of noncommutative indeterminates \(\mathcal{X} =\{ {X}_{1},\ldots,{X}_{r}\}\). We refer the interested reader to [234] for a proof of these algebraic constructions and to [52] for a definition of a Hall basis. We do not need these in our text.
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Schättler, H., Ledzewicz, U. (2012). The High-Order Maximum Principle: From Approximations of Reachable Sets to High-Order Necessary Conditions for Optimality. In: Geometric Optimal Control. Interdisciplinary Applied Mathematics, vol 38. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3834-2_4
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