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The fundamental theorem for nonautonomous smooth ODEs

Statement

Let $D \subseteq R \times R^{n}$ be open and let $F : D \to R^{n}$ be smooth. For every base point $(t_{0}, x_{0}) \in D$ there are $h > 0$ and an open neighbourhood $W$ of $(t_{0}, x_{0})$ such that each initial pair $(s, y) \in W$ has a unique solution on $[s - h, s + h]$ , and the resulting local solution map is smooth in the initial time and initial state.

Facts & Assumptions

Given: A smooth nonautonomous field $F (t, x)$ and a base point $(t_{0}, x_{0}) \in D$ .

[L1]

Autonomous smooth ODEs have unique local smooth flows (The fundamental theorem for autonomous smooth ODEs).

[L2]

Smooth parameter-dependent ODEs have solutions depending smoothly on the parameters (Smooth dependence of ODE solutions on parameters).

Proof

technique · direct

1.1givenL1construct

On the open state domain $D \subseteq R \times R^{n}$ , define the augmented autonomous field $V (s, x) = (1, F (s, x))$ , so its equations are $s^{'} (r) = 1$ and $x^{'} (r) = F (s (r), x (r))$ . This field is smooth on $D$ . Apply [L1] with open state domain $U = D$ at $(t_{0}, x_{0})$ : there are $h > 0$ and an open neighbourhood $W \subseteq D$ of $(t_{0}, x_{0})$ such that every initial pair in $W$ has a unique augmented solution on the common elapsed-time interval $[- h, h]$ , taking values in $D$ .

2.1L2step 1.1∎

The first equation forces $s (t) = s_{0} + t$ when $s (0) = s_{0}$ , so the second equation becomes exactly the original nonautonomous system with initial time $s_{0}$ . Reading the initial value $(s_{0}, x_{0})$ as a parameter, [L2] makes the resulting solution map smooth in $(s_{0}, x_{0})$ . This is precisely the claimed local theorem for nonautonomous smooth ODEs.

The fundamental theorem for nonautonomous smooth ODEs

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources