Low Regularity Lyapunov Stability Theory
Luke Hamilton, supervised by Prof. John Stalker
Abstract ries, given initial conditions; will the forward orbit of any point, in a small enough neighborhood around The classical method of demonstrating the stability an equilibrium, stay within a neighbourhood of it? of an ordinary differential equation is the so-called Consider the autonomous differential equation Lyapunov Direct Method. Traditionally, this re- quires smoothness of the right-hand side of the equa- x˙ = F(x(t)), tion; while this guarantees uniqueness of solutions, x(0) = x0, (∗) we show that this constraint is unneccessary, and that only continuity is needed to demonstrate stability, re- with an equilibrium at x∗, that is, F(x∗) = 0. If F gardless of multiplicity of solutions. is continuously differentiable in x, then the classical method of demonstrating stability in said system is the so-called Lyapunov direct method; for if there 1 Introduction exists a Lyapunov function V , with V continuously differentiable in an -ball about x∗, such that unstable i V (x∗) = 0,
ii V (x) > V (x∗) for all x 6= x∗ in the -ball, x(t) iii ∇V (x) · F(x) ≤ 0, stable then the equilibrium is Lyapunov stable: solutions ε starting sufficiently close to the equilibrium x∗ re- δ main close forever. If 0 x0 iv ∇V (x) · F(x) < 0, then the equilibrium is asymptotically stable: solu- tions starting sufficiently close to the equilibrium x∗ converge to x∗. Notice, however, that despite the hypothesis of a ∂F continuously differentiable F, the derivative ∂x never appears explicitly in the conditions i, ii, iii on V ; in fact, almost all if not every treatment of Lyapunov Figure 1 theory in the literature assumes a sufficiently smooth F; see e.g. [4], [2]. This guarantees local existence In the field of differential equations, stability the- and uniqueness, but if one assumes only continuity of ory addresses the aysymptotic behaviour of trajecto- F, there is no uniqueness result.
1 V (x) For proofs see [1]. For smooth F, we can apply V (x0) Picard–Lindel¨ofto find a unique solution to the Ivp. Otherwise we apply Peano and assume, in the worst case, that for every x0 we can find more than one solution branching from this point. Henceforth we assume that our equilibrium is x∗ = 0 and our local solution is in some neighbourhood about the origin.
V (x∗) = 0 2.2 Global Extension Figure 2 In order to prove asymptotic stability, our solution curves must exist for all t > 0. Given a solution curve x(t) local in space, we must find a continuation global However, in spite of multiple solutions, the condi- in time t > 0. We accomplish this by a bootstrap- tion ∇V (x) · F(x) ≤ 0 implies type lemma. d Lemma 2.3. Assuming i, ii, iii, there exists ρ, τ, v > V (x(t)) = ∇V (x(t)) · F(x(t)) ≤ 0, dt 0 such that when kx0k < δ and V (x0) < v, there exists a solution x : [0, τ] → Rd to (∗), with x(0) = for every solution x(t) of (∗). So intuitively the Lya- x0, kx(t)k < ρ, and V (x(t)) < v for 0 < t < τ. punov direct method should hold a fortiori. ∂F At each stage of the induction we extend the solu- Motivated by the curious absence of ∂x in the Lya- punov conditions above, the object of this project is tion by a further τ, using x(τ) as our new x0, until + to provide a Lyapunov-style result for the case of F we encompass all of R . continuous, but not continuously differentiable, that is, strengthening the Lyapunov direct method to 2.3 Simple Stability give: a local existence result for the Initial Value In the presence of multiple solutions, stability is de- Problem (Ivp) given by (∗); a global extension of this solution for t ≥ 0; and an asymptotic stability fined as follows. result for x near x∗. Ωt−τ (x) ⊂ Ωt(x0)
x ∈ Ωτ (x0) 2 Results x0 We provide proofs in the appendices.
0 τ t 2.1 Existence Figure 3 The classical existence theorems are: Definition 2.4. Let Ω (x ) be the set of endpoints Theorem 2.1. (Peano) If F is continuous, there ex- t 0 of the integral curves of (∗) at time t, starting at ists a local solution to (∗). x(0) = x0. Then
and, iΩ t(x0) 6= ∅ for t ≥ 0,
Theorem 2.2. (Picard–Lindel¨of) If F is Lipschitz- iiΩ t(x0) −→ x0 as t −→ 0, that is, ∀ ε > 0, ∃ τ > 0 continuous, there exists a unique local solution to (∗). such that Ωt(x0) ⊂ Oε(x0) for t > τ, and
2 iii for 0 < τ < t, S Ω (x) = Ω (x ). 2.4 Asymptotic Stability x∈Ωτ (x0) t−τ t 0 Note that if we can show the largest solution in We add another hypothesis to show that x(t) −→ 0 as t −→ ∞. Ωt(x0) is stable, then the other solutions in Ωt(x0) are also stable. We now reformulate stability in these Theorem 2.9. A solution 0 to (∗) is asymptotically terms. stable if and only if, in addition to conditions i, ii, iii above, we have Definition 2.5. A solution x∗ to (∗) is stable if iv ∃ δ > 0 such that supx∈Ω (x ) V (x) −→ 0 as t −→ ∀ ε > 0, ∃ δ > 0 such that kx∗ − x0k < δ implies t 0 ∞ for kx0k < δ. supx∈Ωt(x0) kx∗ − xk < ε for t ≥ 0. If we weaken V to be smooth, then the following Definition 2.6. A solution x∗ to (∗) is asymptot- is sufficient to show asymptotic stability. ically stable if it is stable and ∃ δ > 0 such that IV ∃ δ > 0 such that V (x(t)) −→ 0 as t −→ ∞ for kx∗ − x0k < δ implies sup kx∗ − xk −→ 0 x∈Ωt(x0) kx k < δ, as t −→ ∞. 0 Or, alternatively, From this we can find both necessary and sufficient IV ∇V (x) · F(x) < 0, so that V (x(t)) is decreasing, condition for stability, namely: for any solution x(t) to (∗), Theorem 2.7. A solution 0 to (∗) is stable if and Interestingly, the latter is strong enough to apply only if there exists V (x) such that the so-called annulus argument, see e.g. [3], which we present in the appendices in addition to Theorem i ∀ λ > 0, ∃ ξ > 0 such that V (x) > ξ on kxk > λ, 2.9.
ii V (x) −→ 0 as kxk −→ 0, and V is continuous at this point, and, Acknowledgements
The author would like to thank Prof. John Stalker iii supx∈Ω (x ) V (x) is non-increasing for t ≥ 0. t 0 for his guidance throughout the research. Note that if 0 is stable, V need not be smooth or We also acknowledge the support of the Hamilton even continuous. Trust and Prof. Kirk Soodhalter. If we weaken V to be smooth, then we can formu- late the above as: References
Theorem 2.8. A solution 0 to (∗) is stable if there [1] R.P. Agarwal and V. Lakshmikantham. Unique- exists a smooth V (x) such that ness and Nonuniqueness Criteria for Ordinary Differential Equations. Vol. 6. World Scientific, I U(kxk) ≤ V (x) for some increasing, continuous 1993. U, [2] V.I. Arnold and R. Cooke. Ordinary Differ- II V (x) −→ 0 as kxk −→ 0, and, ential Equations. Universitext (Berlin. Print). Springer, 2006. III ∇V (x) · F(x) ≤ 0, so that V (x(t)) is non- [3] Ted A Burton. Volterra integral and differential increasing for any solution x(t) to (∗) equations. Vol. 202. Elsevier, 2005. [4] Hassan K Khalil. Nonlinear systems. Prentice As desired, these are the standard conditions of the Hall, 2002. Lyapunov direct method for x∗ = 0.
3 r (⇐=) Take ε < and λ = inf r V (x). By A Proof of Main Results 2 ε
i ∀ λ > 0, ∃ ξ > 0 such that V (x) > ξ on kxk > λ, sup V (x) < λ. x∈Ωt(x0) ii V (x) −→ 0 as kxk −→ 0, and V is continuous at
this point, and, Let kx0k < δ, and suppose ∃ τ > 0 such
that supx∈Ωt(x0) kxk ≥ ε for t ≥ τ. Then iii supx∈Ωt(x0) V (x) is non-increasing for t ≥ 0. supx∈Ωτ (x0) V (x) > λ. Contradiction. Therefore sup kxk < ε, so 0 is stable. Proof. (=⇒) Assume 0 is stable, that is, ∀ ε > x∈Ωt(x0) 0, ∃ δ > 0 such that kx0k < δ =⇒ sup kxk < x∈Ωt(x0) Theorem 2.9. A solution 0 to (∗) is asymptotically ε for t ≥ 0. Define stable if and only if, in addition to conditions i, ii, iii above, we have V (y) = sup sup kxk, (†) t≥0 x∈Ωt(y) iv ∃ δ > 0 such that supx∈Ωt(x0) V (x) −→ 0 as t −→ ∞ for kx k < δ. By stability, ∀ ε > 0, ∃ λ > 0 such that V (x0) < ξ 0 for kx0k < λ. Thus V (x) −→ 0 as kxk −→ 0 and is Proof. (⇐=) Let conditions i through iv be satisfied. continuous there, giving ii. There are two cases Furthermore V (y) ≥ kyk, giving i. Finally, by virtue of property iii of Ω, for 0 < τ ≤ t, I supx∈Ωt(x0) kxk −→ 0 as t −→ ∞, and we have Ωt−τ (x) ⊂ Ωt(x0), for all x ∈ Ωτ (x0). So
II ∃ γx0 > 0 : supx∈Ωt(x0) kxk > γx0 > 0 for all sup kyk ≤ sup kyk, t ≥ 0. y∈Ωt−τ (x) y∈Ωt(x0) For if I does not hold, then clearly II holds. for all x ∈ Ωτ (x0). Subsequently, taking sup on both As i, ii, iii are satisfied, 0 is stable, that is, ∀ ε > sides, 0, ∃ λ > 0 such that kxk < λ =⇒ supy∈Ωt(x) kyk < ε for t ≥ 0. Now assume II does not hold, that is, V (x) = sup sup kyk ≤ sup sup kyk = V (x0), ∀ λ > 0, ∃ τ > 0 such that sup kxk < λ for t≥τ t≥0 x∈Ωt(x0) y∈Ωt−τ (x) y∈Ωt(x0) t ≥ τ. Thus, which gives iii. sup kyk < ε, y∈Ω (x) Note that the construction given for V does not t−τ guarantee continuity of V over all of U. V will, in gen- for all x ∈ Ωτ (x0) and t ≥ τ. So, by property iii of eral, jump discontinuously at the branching points of Ω, the integral curves (consider the behaviour of solu- sup kxk < ε, tions a(t) and b(t) of Figure 4, one involving a loop, x∈Ωt(x0) the other not.) for all t ≥ τ and arbitrary ε > 0. So I holds. Now we show that II is in contradiction with con- V (a(t)) V (b(t)) dition iv.
Assume II holds, that is, supx∈Ωt(x0) kxk > γx0 for t ≥ 0. By i, ∀γ > 0, ∃ ε > 0 such that V (x) > ε
for kxk > γ. Let γ = γx0 . Taking sup preserves
the inequality, so ∃ ε > 0 : supx∈Ωt(x0) V (x) > ε, for Figure 4 t ≥ 0.
4 Now by iv, we have ∀ ε > 0, ∃ τ > 0 : Bβ sup V (x) < ε for t ≥ τ. Thus, ∃ τ > 0 : x∈Ωt(x0) Ω sup V (x) < ε for t ≥ τ. Contradiction. α x∈Ωt(x0) B Therefore I holds, so 0 is asymptotically stable. δ (=⇒) Let 0 be asymptotically stable. Therefore it 0 is stable, so there exists V such that the conditions i, ii, iii of the preceeding theorem are satisfied. We show that the construction (†) also enjoys iv. By asymptotic stability, ∃ ε > 0 such that kxk <
ε =⇒ ∀ ξ > 0, ∃ τ > 0 such that supz∈Ωt(x) kzk < ξ for t ≥ τ. And by stability, ∀ ε > 0, ∃ δ > 0 such that Figure 5 when kx0k < δ,
sup kxk < ε By continuity of V , there is some δ > 0 such that x∈Ωt(x0) Bδ ⊂ Ωα ⊂ Bβ, see Figure 5. Since V is decreasing, for all t ≥ 0. Thus, x(t) ∈/ Bδ. Let sup kyk < ξ y∈Ωt−τ (x) γ = − max ∇V (x) · F(x) > 0, δ≤kxk≤ for t ≥ τ and x ∈ Ωτ (x0). By the construction (†), which does not depend on choice of solution, hence for any solution x(t) of (∗), V (x) = sup sup kyk = sup sup kyk ≤ ξ t≥0 y∈Ωt(x) t≥τ y∈Ωt−τ (x) Z t d V (x(t)) = V (x(τ)) + V (x(s)) ds τ ds for x ∈ Ωτ (x0). Therefore supx∈Ωt(x0) V (x) ≤ ξ, when t = τ. But this holds a fortiori for t ≥ τ as Z t well, since V is non-increasing. So when kx k < δ, ≤ V (x(τ)) + ∇V (x(s)) · F(x(s)) ds 0 τ ∀ ξ > 0, ∃ τ > 0 such that sup V (x) ≤ ξ for x∈Ωt(x0) ≤ V (x(τ)) + γτ − γt, all t ≥ τ. So V satisfies iv. which implies V (x(t)) < 0 eventually: contradiction. Theorem A.1. (Annulus argument) If Therefore α = 0. i V (x) > 0 for x 6= 0, V (x0) ii V (0) = 0, and, iii ∇V (x) · F(x) < 0, then for every solution x(t) of (∗), we have x(t) −→ 0 V (x(t)) → α V (x) = α as t −→ ∞. Proof. To show that for all there exists a τ such that kx(t)k < for t > τ, start by letting β = x0 ε δ {x : V (x) = α} minkxk= V (x). 0 It is sufficient to show V (x(t)) −→ 0 as t −→ ∞ for every solution to (∗). Let x(t) be one such so- Figure 6 lution. As V is monotone decreasing, and bounded from below by 0, V (x(t)) −→ α ≥ 0. Clearly α < β. Let Ωα = {x : V (x) ≤ α}.
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