Contribution
Lecture notes, not a research paper. Chapters 1–2 restate the viability theorems and the viability kernel and capture basin from the author’s earlier work, for ordinary differential equations, as tools for solving Hamilton–Jacobi equations later in the notes. For this record their value is a first-hand, compact statement of the definitions.
Key insight
Fix a set K of states you require, independently of the dynamics. Then ask of each state whether some evolution from it can keep satisfying K. The answer carves K into the viability kernel, which can, and the rest, which cannot and will leave K in finite time however it evolves. Nothing about attractors, equilibria or return to a pattern enters into it.
Assumptions
- Finite-dimensional state space; dynamics x′ = f(x) with f continuous, and with linear growth for the chapter 2 results (the general theory uses set-valued maps; the notes restrict to equations “for the sake of simplicity”, p. 4).
- K locally compact for Nagumo’s theorem, closed for the kernel results.
Key results
- Definitions 1.1.1–1.1.2. x(·) viable in K on [0, T] if x(t) ∈ K for all t. K locally viable if from any x0 ∈ K some solution is viable on some [0, T], T > 0; globally if T = ∞ is always possible. K invariant if every solution from K stays in it. A repeller: all solutions from K leave in finite time.
- Theorem 1.3.1 (Nagumo). K locally viable ⇔ f(x) ∈ T_K(x) for all x ∈ K, equivalently ⟨p, f(x)⟩ ≤ 0 for every normal p ∈ N_K(x).
- Definition 2.1.3. Viab_f(C, T): states of C from which some solution stays in C on [0, T]; Viab_f(C) for T = ∞. Capt_f(C): states from which some solution reaches C in finite time. Capt^K_f(C): from which some solution stays in K until it reaches C.
- Proposition 2.1.5. Viab_f(C) is the largest subset of C viable under f; C \ Viab_f(C) is a repeller.
- Theorem 2.3.1. The viability kernel is the largest closed D ⊂ K with f(x) ∈ T_D(x) for all x ∈ D.
- Proposition 2.3.2. From a compact set outside the kernel, every solution leaves K within a bounded time.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Viability (some evolution stays) and invariance (all do) differ, and coincide when solutions are unique | strong | definitions and remark, p. 7 |
| C2 | Viability of K is decided by the dynamics on K alone | strong | remark, p. 7 |
| C3 | Every closed K has a largest viable subset, and from the rest of K every evolution exits in finite time | strong | Propositions 2.1.5, 2.3.2; Theorem 2.3.1 |
| C4 | These results are “mathematical metaphors” of evolutionary economics, population dynamics and biological evolution | weak | asserted, p. 5; no application in chapters 1–2 |
Concepts
- viable evolution — one that stays in the constraint set K over the horizon considered.
- viability kernel — the states of K from which at least one evolution stays in K for ever.
- capture basin — the states from which at least one evolution reaches a target in finite time.
- repeller — a set from every state of which every evolution leaves in finite time.
- stability (§2.6) — here, whether viability is preserved under limits of sets, not stability of a state or attractor.
Connections
The notes credit Nagumo (1942) for the viability theorem and Frankowska for the use of kernels and capture basins to characterise value functions. They point to Aubin’s monograph (LIT-738) for the general, set-valued theory.
Bearing on the record
Source of THEORY-134. The owner’s essay (§6.1) cites “Aubin’s mathematical theory of viability” for the claim that stability and viability are different questions. The notes support the half of that claim that defines viability: a state space, admissible dynamics, an independently specified constraint set, and the states that can keep satisfying it over a horizon. They do not make the comparison with stability. That step is the essay’s, and it is sound only once the essay says what the constraint set is for the system in question.
The record’s organizational-autonomy accounts (Barandiaran, Di Paolo & Rohde, LIT-566) call an agent’s organization “precarious”. The viability kernel is one way to make that word precise: a system is precarious where its kernel is small or its state is near the kernel’s boundary. That reading is the record’s, not either source’s.
Limitations
- Deterministic single-valued dynamics only in these chapters; the set-valued case, where viability and invariance genuinely diverge, is in the monograph.
- No worked application outside mathematics in the chapters read.
Open questions
How to specify K for a system whose constraints are partly set by the system itself, as with an organism or an organization that changes what it needs. The notes take K as given.