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Finite-time comparison of bounded vector fields

Statement

Let $n \geq 1$ , $T \geq 0$ , $U \subseteq R^{n}$ be open, and $K \subseteq U$ be compact and convex. Let $F_{ϵ}, F_{0} : [0, T] \times U \to R^{n}$ be continuous. Suppose that on $[0, T] \times K$ both fields are bounded in the Euclidean norm by $B \geq 0$ , and that there is $L \geq 0$ such that

$∥ F_{0} (t, z) - F_{0} (t, w) ∥_{2} \leq L ∥ z - w ∥_{2} (t \in [0, T], z, w \in K) .$

Also suppose

$∥ F_{ϵ} (t, z) - F_{0} (t, z) ∥_{2} \leq η (t \in [0, T], z \in K)$

for some $η \geq 0$ . Let $z_{ϵ}, z_{0} \in C^{1} ([0, T], R^{n})$ solve ${\dot{z}}_{ϵ} = F_{ϵ} (t, z_{ϵ})$ and ${\dot{z}}_{0} = F_{0} (t, z_{0})$ , with both trajectories contained in $K$ . Then, for every $t \in [0, T]$ ,

$∥ z_{ϵ} (t) - z_{0} (t) ∥_{2} \leq (∥ z_{ϵ} (0) - z_{0} (0) ∥_{2} + η t) e^{L t} .$

In particular, equal initial states give $\sup_{0 \leq t \leq T} ∥ z_{ϵ} (t) - z_{0} (t) ∥_{2} \leq η T e^{L T}$ . The field bound $B$ alone is not a stability hypothesis; the comparison-field Lipschitz bound $L$ and uniform field error $η$ are part of the assumptions. No existence through $T$ follows unless the trajectories are known to exist there.

Facts & Assumptions

Given: The interval, open state domain, compact convex set, bounded continuous fields, the comparison-field state-Lipschitz constant, uniform field-error bound, and the two existing $C^{1}$ solution curves in the Statement.

[L1]

On a nondegenerate compact interval $[a, b]$ with $a < b$ , a differentiable vector-valued curve with integrable derivative has derivative integral equal to the endpoint difference (If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz).

[L2]

For an integrable vector-valued function, $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ when $a \leq b$ (For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

[L3]

If $u, a, b$ are continuous on $[0, T]$ , $u, b \geq 0$ , $u (t) \leq a (t) + \int_{0}^{t} b (s) u (s) d s$ , and $a$ is nondecreasing, then $u (t) \leq a (t) \exp (\int_{0}^{t} b (s) d s)$ (Gronwall's integral inequality with variable and constant coefficients).

[L4]

Each $C^{1}$ solution has a continuous vector derivative; vector continuity and integrability are componentwise, and every continuous real component on $[0, T]$ is Riemann integrable (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

Proof

technique · subtract the integral equations and apply Gronwall's inequality

1.1givenL1L2L4algebra

Write $d (t) : = ∥ z_{ϵ} (t) - z_{0} (t) ∥_{2}$ . At $t = 0$ the claimed estimate is equality; if $T = 0$ , this proves the result. Assume $T > 0$ and fix $0 < t \leq T$ . By [L4] each solution derivative is vector-integrable on the nondegenerate interval $[0, t]$ , so [L1] applies there. Subtracting the resulting integral equations, adding and subtracting $F_{0} (s, z_{ϵ} (s))$ , and applying [L2] gives $d (t) \leq d (0) + \int_{0}^{t} (∥ F_{ϵ} (s, z_{ϵ} (s)) - F_{0} (s, z_{ϵ} (s)) ∥_{2} + ∥ F_{0} (s, z_{ϵ} (s)) - F_{0} (s, z_{0} (s)) ∥_{2}) d s \leq d (0) + η t + L \int_{0}^{t} d (s) d s$ , because both states lie in $K$ , the uniform field-error bound controls the first integrand term by $η$ , and the Lipschitz bound for $F_{0}$ controls the second by $L d (s)$ . At $t = 0$ this integral inequality also holds as equality, since the zero-length integral is zero by The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral componentwise. Thus it holds on all of $[0, T]$ .

2.1step 1.1L3algebra∎

Set $a (t) = d (0) + η t$ and $b (t) = L$ on $[0, T]$ . These functions are continuous, $a$ is nondecreasing, and $b, d \geq 0$ , so [L3] yields $d (t) \leq (d (0) + η t) e^{L t}$ for every $t$ . If $d (0) = 0$ , then $η t \leq η T$ and $e^{L t} \leq e^{L T}$ on $[0, T]$ , which gives the stated supremum estimate.

Remarks

The trajectories are assumed to exist on the common finite interval and remain in the stated set. Local existence and continuation are separate obligations; compact containment cannot be replaced by pointwise boundedness of the vector field on a larger, unspecified domain.

Depends on

Dependency tree · 2 levels

Sources

Gautam Iyer, Brief Notes on Introduction to Differential Equations, complete ODE notes