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$C^{1}$ dependence of solutions on initial data

Statement

Let $F : D \subseteq R \times R^{n} \to R^{n}$ be continuous and $C^{1}$ in the state variable. Fix data $(t_{0}, x_{0}) \in D$ and a compact time interval $I = [t_{0} - h, t_{0} + h]$ on which the corresponding solutions through nearby initial states all exist. Then the solution map

$Φ : I \times U_{0} \to R^{n}, Φ (t, y) = x (t; y),$

is $C^{1}$ in the initial-state variable $y$ on some neighbourhood $U_{0}$ of $x_{0}$ . For each $y \in U_{0}$ , the derivative matrix $D_{y} Φ (t, y)$ is the solution of the variational equation along $t \mapsto x (t; y)$ .

Facts & Assumptions

Given: The field $F$ , the compact interval $I$ , and the common local family $Φ (t, y) = x (t; y)$ of solutions through nearby initial states.

[F1]

The variational equation along a solution is $A^{'} (t) = D_{x} F (t, x (t)) A (t)$ with initial condition $A (t_{0}) = I_{n}$ (The variational equation along an ODE solution).

[F2]

Solutions satisfy the corresponding Volterra integral equation (A first-order initial value problem is equivalent to its Volterra integral equation).

[L1]

Nearby initial data and parameters give uniformly close solutions on one common compact time interval (Continuous dependence of ODE solutions on initial data and parameters).

[L2]

Linear matrix ODEs on a compact interval have unique solutions (Linear matrix ODEs have unique global solutions on a fixed interval).

[L3]

The norm of a vector-valued integral is at most the integral of the norm (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).

[L4]

Gronwall's integral inequality controls difference equations of Volterra type (Gronwall's integral inequality with variable and constant coefficients).

Proof

technique · direct

1.1F1L1L2

By [L1], after shrinking $U_{0}$ if needed, every solution graph $(t, Φ (t, y))$ with $y \in U_{0}$ lies in one compact cylinder $K \subseteq D$ . Fix $y \in U_{0}$ , and let $A_{y} : I \to M_{n} (R)$ be the unique solution of the variational equation below.

$A_{y}^{'} (t) = D_{x} F (t, Φ (t, y)) A_{y} (t), A_{y} (t_{0}) = I_{n},$

whose existence on $I$ is given by [L2]. Because $D_{x} F$ is continuous on the compact set $K$ , it is bounded and uniformly continuous there.

2.1F2L1step 1.1algebra

Fix $y \in U_{0}$ and an increment $u \in R^{n}$ with $y + u \in U_{0}$ . By [F2], the difference $z_{u} (t) : = Φ (t, y + u) - Φ (t, y)$ satisfies the Volterra equation below, and the state-variable mean-value formula gives the matrix field $B_{u}$ .

$z_{u} (t) = u + \int_{t_{0}}^{t} (F (s, Φ (s, y + u)) - F (s, Φ (s, y))) d s .$

For each $s \in I$ , the one-variable mean-value formula in the state variable gives

$F (s, Φ (s, y + u)) - F (s, Φ (s, y)) = B_{u} (s) z_{u} (s),$

where

$B_{u} (s) : = \int_{0}^{1} D_{x} F (s, Φ (s, y) + θ z_{u} (s)) d θ .$

By [L1], $z_{u} \to 0$ uniformly on $I$ as $u \to 0$ , so the uniform continuity of $D_{x} F$ on $K$ gives

$ε (u) : = \sup_{s \in I} ∥ B_{u} (s) - D_{x} F (s, Φ (s, y)) ∥ ⟶ 0.$

3.1L3L4step 1.1step 2.1algebra

Put $ρ_{u} (t) : = z_{u} (t) - A_{y} (t) u$ . Subtracting the Volterra equations for $z_{u}$ and $A_{y} u$ gives the identity below.

$ρ_{u} (t) = \int_{t_{0}}^{t} B_{u} (s) ρ_{u} (s) d s + \int_{t_{0}}^{t} (B_{u} (s) - D_{x} F (s, Φ (s, y))) A_{y} (s) u d s .$

Let $M : = \sup_{K} ∥ D_{x} F ∥$ and $C_{y} : = \sup_{s \in I} ∥ A_{y} (s) ∥$ . Then $∥ B_{u} (s) ∥ \leq M$ , and [L3] gives, for $t \geq t_{0}$ ,

$∥ ρ_{u} (t) ∥_{2} \leq M \int_{t_{0}}^{t} ∥ ρ_{u} (s) ∥_{2} d s + C_{y} ε (u) ∥ u ∥ (t - t_{0}) .$

Applying [L4] on $[t_{0}, t]$ and its time-reflected form on $[t, t_{0}]$ yields a constant $C_{y}^{'}$ independent of $u$ such that $\sup_{t \in I} ∥ ρ_{u} (t) ∥_{2} \leq C_{y}^{'} ε (u) ∥ u ∥$ . Since $ε (u) \to 0$ , this proves

$\sup_{t \in I} \frac{∥ Φ (t, y + u) - Φ (t, y) - A_{y} (t) u ∥_{2}}{∥ u ∥} ⟶ 0.$

Therefore $A_{y} (t) = D_{y} Φ (t, y)$ for every $t \in I$ .

4.1L1L3L4step 1.1step 3.1algebra

For $y, y^{'} \in U_{0}$ , write $E (t) : = A_{y^{'}} (t) - A_{y} (t)$ and $C_{y} (t) : = D_{x} F (t, Φ (t, y))$ . Then the continuity estimate below, together with the same Gronwall argument as in step 3.1, proves continuity of the derivative matrix.

$E (t) = \int_{t_{0}}^{t} C_{y^{'}} (s) E (s) d s + \int_{t_{0}}^{t} (C_{y^{'}} (s) - C_{y} (s)) A_{y} (s) d s .$

By [L1], $Φ (\cdot, y^{'}) \to Φ (\cdot, y)$ uniformly on $I$ as $y^{'} \to y$ , so the uniform continuity of $D_{x} F$ on $K$ gives $\sup_{s \in I} ∥ C_{y^{'}} (s) - C_{y} (s) ∥ \to 0$ . Applying [L3] and [L4] exactly as in step 3.1 shows $\sup_{t \in I} ∥ E (t) ∥_{2} \to 0$ . Hence $y \mapsto D_{y} Φ (t, y) = A_{y} (t)$ is continuous for each $t$ , and the derivative matrix is exactly the variational-equation solution.

5.1step 3.1step 4.1∎

Steps 3.1 and 4.1 prove that $Φ$ is $C^{1}$ in the initial-state variable and that its derivative matrix is the solution of the variational equation.

Depends on

Used by

Continuous dependence does not by itself imply differentiable dependence False statement
Local fully nonlinear Charpit graph construction Theorem
Smooth dependence of ODE solutions on parameters Theorem
Smooth dependence of solutions on initial data Theorem

Dependency tree · two levels

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Sources

Nigel Hitchin, Differentiable Manifolds, Appendix §10.3, Lemma 10.6 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)

C1C1 dependence of solutions on initial data