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LIT-714: The Method of Characteristics Revisited: A Viability Approach

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StatusActive — read 2026-10-05 (NOTE-576): the introduction and chapters 1–2 of the author's arXiv lecture notes, the parts that define viability. Worth reading as the shortest first-hand statement the record holds of Aubin's viability concepts: a set is viable if from every state at least one evolution stays in it, invariant if all do, and the viability kernel is the largest viable subset of a constraint set. Chapters 3 onward apply these to Hamilton–Jacobi equations and were not read. Stands in for the unread monograph LIT-738.
Version2
Tagsmathematics, complex-systems, control-theory
Cited as “paper” byNOTE-576 — The Method of Characteristics Revisited: A Viability Approach
Cited as “source” byTHEORY-134 — Whether a system can persist is a question about a constraint set fixed independently of its dynamics: a set is viable when from each of its states some evolution can stay in it, and when it is not, its viability kernel is the largest part that is, from the rest of which every evolution leaves in finite time

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Jean-Pierre Aubin (1999), “The Method of Characteristics Revisited: A Viability Approach. A Mini-Course”, lecture notes, Université de Paris-Dauphine — ARXIV-math/9906178

Key takeaways

  • Viability is a property of a set under a dynamics. An evolution is viable in K on [0, T] if it stays in K. K is viable under f if from every initial state in K at least one solution stays in K, and invariant if every solution does (Definitions 1.1.1–1.1.2). Viability “depends only on the behavior of f on K”; invariance depends on f outside K too (remark, p. 7).
  • Nagumo’s theorem. For locally compact K and continuous f, K is locally viable if and only if f(x) lies in the contingent cone to K at every x in K (Theorem 1.3.1): at each state, the velocity must not point out of K.
  • When K is not viable, two questions arise: from which states can an evolution stay in K as long as possible, and from outside K, which can return to it soonest (chapter 2 introduction). The answers are the viability kernel (the states of K from which some solution stays in K for ever; the largest viable subset of K, Proposition 2.1.5) and the capture basin (the states from which some solution reaches a target in finite time). Outside the kernel, every evolution leaves K in finite time (Proposition 2.3.2): C minus its kernel is a repeller.
  • “Stability” in these notes is not Lyapunov stability. §2.6, “Stability Properties”, concerns whether viability survives taking limits of sets. The notes do not compare viability with the stability of an equilibrium or an attractor; that comparison is not made here.

Standing in the record

Filed on 2026-10-05 at the owner’s request, as the readable source for an essay’s appeal to “Aubin’s mathematical theory of viability” (§6.1). The record’s account is THEORY-134.

What the notes give the essay is a precise vocabulary: a state space, dynamics, a constraint set K fixed independently of the dynamics, and the kernel of states that can keep satisfying K. What they do not give it is the comparison with stability. That is the essay’s own step, and it needs the essay to say what K is for an organization.

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