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Nearby critical values force nonminimum escape

Statement

Let $m \geq 1$ , let $f \in C^{2} (U)$ on open $U \subseteq R^{m}$ , and let $z$ be an equilibrium for $x^{'} = - \nabla f (x)$ . Suppose there is a neighborhood $W$ of $z$ on which every critical point $y$ satisfies $f (y) = f (z)$ . If $z$ is not a local minimum, it is not Lyapunov stable.

Here stability is defined for maximal forward solutions in $U$ . Write $x (t; x_{0})$ for the solution with $x (0; x_{0}) = x_{0} \in U$ , and $[0, T_{+} (x_{0}))$ for the forward part of its maximal interval, where $0 < T_{+} (x_{0}) \leq \infty$ . Lyapunov stability means that for every $ε > 0$ there is $δ > 0$ such that $x_{0} \in U$ and $∥ x_{0} - z ∥_{2} < δ$ imply $∥ x (t; x_{0}) - z ∥_{2} < ε$ for every $0 \leq t < T_{+} (x_{0})$ . For a complete flow this is the usual point-stability definition. In fact, the conclusion supplies a fixed $r > 0$ such that every $δ > 0$ admits such an $x_{0}$ and a time $t < T_{+} (x_{0})$ with $∥ x (t; x_{0}) - z ∥_{2} > r$ .

Facts & Assumptions

[F1]

The Euclidean gradient has coordinates $g_{i} = \partial_{i} f$ for $0 \leq i < m$ by The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case. The $C^{2}$ convention is Cᵏ maps and multi-index derivative notation in Euclidean space: $f$ , its first partials and every ordered second partial are continuous on $U$ . Continuously totally differentiable maps use Continuously differentiable maps, local inverses, and local diffeomorphisms. Solutions and maximal intervals use First-order systems, initial value problems, and solutions on intervals and Extensions, maximal solutions, maximal intervals, and global solutions.

[F2]

Local minimality and the complete-flow point-stability definition are supplied by Point stability, attraction and stability of a set. The maximal-forward-solution definition in the Statement uses the same distance quantifiers over each solution's available forward times; it specializes to that definition when a complete flow is given.

[F3]

Continuous partial derivatives imply total differentiability with Jacobian derivative by If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative. A uniform bound on the total derivative on a convex open set gives the corresponding Lipschitz bound by On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂. Closed bounded Euclidean balls are compact by Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, and each continuous real function on a nonempty compact set is bounded by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value.

[F4]

Local Lipschitz continuity in the state variable means a uniform state Lipschitz bound on each compact time-state cylinder, as in Local Lipschitz continuity in the state variable, locally uniform in time and parameters. Under continuity and this condition on an open time-state domain, Picard-Lindelöf local existence and uniqueness for first-order systems gives local existence and uniqueness, and Every Picard–Lindelöf initial value problem has one maximal solution on an open interval gives a unique maximal solution on an open interval containing the initial time.

[F5]

The autonomous clause 2 of Compact continuation for locally Lipschitz ODEs: a solution whose graph stays in a compact subset of the domain near a finite endpoint extends past it extends a solution past a finite forward endpoint when its states stay in a compact subset of $U$ near that endpoint. This supplier has domain mathematics. For a trajectory defined at all forward times and trapped in compact $K \subset U$ , A compactly trapped gradient trajectory has a critical accumulation point gives $y \in K$ with $\nabla f (y) = 0$ and $f (y) \leq f (x (0))$ .

Proof

Given: The data and hypotheses in the Statement.

Goal: Prove instability in the stated maximal-forward-solution sense and the escape property.

1.1F1F3F4givenalgebra

Put $g = \nabla f$ . By the continuous-partials criterion, both $f$ and $g$ are totally differentiable, with continuous derivative entries, so in particular they are continuous and $f \in C^{1} (U)$ in the sense required by compact critical accumulation. On any closed ball $Q = \overline{B} (a, R) \subset U$ with $R > 0$ , compactness and the extreme-value theorem give a common finite $M \geq 0$ bounding every $∣ \partial_{j} g_{i} ∣$ . For $u \in Q$ and $v \in R^{m}$ , the Jacobian formula gives $∣ (D g (u) v)_{i} ∣ \leq M \sum_{j < m} ∣ v_{j} ∣ \leq m M ∥ v ∥_{2}$ , and hence $∥ D g (u) v ∥_{2} \leq m \sqrt{m} M ∥ v ∥_{2}$ . The total-derivative mean-value inequality on $B (a, R)$ gives this Lipschitz constant there, and continuity extends the inequality to $Q$ . Thus the autonomous field $F (t, u) = - g (u)$ on open $D = R \times U$ is continuous and satisfies the uniform state Lipschitz condition on every compact time-state cylinder.

2.1F2F3F4step 1.1givenchoose

Choose $r > 0$ with $K = \overline{B} (z, r) \subset W \cap U$ ; $K$ is compact. Failure of local minimality gives, for every $δ > 0$ , a point $x_{0} \in B (z, \min {δ, r})$ with $f (x_{0}) < f (z)$ . Apply local existence and uniqueness and the maximal-solution theorem to $F$ on $D$ at $(0, x_{0})$ , using step 1.1. Its unique maximal solution has forward interval $[0, T_{+} (x_{0}))$ .

3.1F5step 1.1step 2.1given

Suppose $x (t; x_{0}) \in K$ for every $0 \leq t < T_{+} (x_{0})$ . If $T_{+} (x_{0}) < \infty$ , autonomous compact continuation applies with $K \subset U$ and extends the trajectory past this endpoint, contradicting maximality. Hence $T_{+} (x_{0}) = \infty$ . Compact critical accumulation now applies to $f \in C^{1} (U)$ and this trajectory in $K$ , giving $y \in K$ with $\nabla f (y) = 0$ and $f (y) \leq f (x_{0}) < f (z)$ . Since $K \subset W$ , the common-critical-value hypothesis gives $f (y) = f (z)$ , a contradiction.

4.1F2step 2.1step 3.1∎

Therefore some $t \in [0, T_{+} (x_{0}))$ has $x (t; x_{0}) \notin K$ , so $∥ x (t; x_{0}) - z ∥_{2} > r$ . The radius $r$ is independent of $δ$ , and step 2.1 gives $∥ x_{0} - z ∥_{2} < δ$ for every $δ > 0$ . This is the negation of stability in the Statement, with tolerance $ε = r$ , and proves both the asserted instability and the stated escape property.

Depends on

Used by · 1

Łojasiewicz inequality and analytic point stability

Dependency tree · 2 levels

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