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Smooth dependence of solutions on initial data

Statement

Let $F : D \subseteq R \times R^{n} \to R^{n}$ be smooth in the state variable, and suppose nearby initial states share one compact local time interval $I$ . Then the solution map

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

is smooth in the initial-state variable $y$ on some neighbourhood $U_{0}$ of the base point.

Facts & Assumptions

Given: The common local solution map $Φ (t, y) = x (t; y)$ on a compact time interval $I$ .

[L1]

The solution map is $C^{1}$ in the initial data, and the first derivative is given by the variational equation ( $C^{1}$ dependence of solutions on initial data).

[F1]

The variational equation is a linear matrix ODE whose coefficients are the state derivatives of the vector field along the base solution (The variational equation along an ODE solution).

[L2]

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

[F2]

Every solution satisfies the Volterra integral equation (A first-order initial value problem is equivalent to its Volterra integral equation).

Proof

technique · direct

1.1

By [L1], the derivative $D_{y} Φ (t, y)$ exists and is the solution of the variational equation below.

L1F1

$\partial_{t} (D_{y} Φ) = D_{x} F (t, Φ (t, y)) D_{y} Φ, D_{y} Φ (t_{0}, y) = I_{n} .$

Because $F$ is smooth and $Φ$ is continuous, the coefficient $D_{x} F (t, Φ (t, y))$ is as regular in $y$ as $Φ$ is.

2.1

For each $r \geq 1$ , let $E_{r}$ be the finite-dimensional space of $r$ -linear maps $(R^{n})^{r} \to R^{n}$ , with $E_{0} : = R^{n}$ . Repeated differentiation of the ODE or of its Volterra form produces the finite-dimensional jet system below.

F1F2step 1.1algebra

$J_{r}^{'} = G_{r} (t, J_{r})$

for $J_{r} = (X_{0}, \dots, X_{r}) \in E_{0} \times \dots \times E_{r}$ , whose first components are

$X_{0}^{'} = F (t, X_{0}), X_{1}^{'} = D_{x} F (t, X_{0}) X_{1},$

and whose higher components have the form $X_{m}^{'} = D_{x} F (t, X_{0}) X_{m} + P_{m} (t, X_{0}, \dots, X_{m - 1})$ , where $P_{m}$ is a universal polynomial expression in derivatives of $F$ and lower jets. For example, $X_{2}^{'} = D_{x} F (t, X_{0}) X_{2} + D_{x}^{2} F (t, X_{0}) [X_{1}, X_{1}]$ . Because $F$ is smooth, each $G_{r}$ is smooth in its state variables.

3.1

Fix $r \geq 1$ . If $Φ$ is $C^{r}$ in $y$ , then its jet $J_{r} (t, y) = (Φ, D_{y} Φ, \dots, D_{y}^{r} Φ) (t, y)$ is a continuous solution of the system from step 2.1 on the same compact interval $I$ , with initial data $X_{0} (t_{0}) = y$ , $X_{1} (t_{0}) = I_{n}$ , and $X_{m} (t_{0}) = 0$ for $m \geq 2$ . The highest-jet component is linear in $X_{r}$ once the lower jets are fixed, so [L2] supplies its unique evolution on $I$ . Applying [L1] to this enlarged smooth system shows that $J_{r}$ depends $C^{1}$ on the initial value $y$ . In particular its first component $Φ$ is $C^{r + 1}$ in $y$ .

L1L2step 2.1

4.1

Step 1.1 is the base case $r = 1$ . Step 3.1 upgrades $C^{r}$ regularity of $Φ$ to $C^{r + 1}$ for every $r \geq 1$ , so by induction $Φ$ is $C^{m}$ in $y$ for every $m$ . Therefore $Φ$ is smooth in the initial-state variable.

step 1.1step 3.1

5.1

Hence the solution map depends smoothly on initial data.

step 4.1∎

Depends on

Used by

Smooth dependence in an ODE with a parameter Example
Continuous dependence does not by itself imply differentiable dependence False statement
Smooth dependence of ODE solutions on parameters Theorem
The fundamental theorem for autonomous smooth ODEs Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nigel Hitchin, Differentiable Manifolds, Appendix §10.3, Theorem 10.7 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)