Existence and Uniqueness for Linear Systems

The existence and uniqueness theorem for linear systems guarantees that solutions exist for all time and depend continuously on initial conditions.

Statement

Theorem5.6Existence and uniqueness for linear systems

Consider the initial value problem $\mathbf{x}' = A(t)\mathbf{x} + \mathbf{g}(t)$ , $\mathbf{x}(t_0) = \mathbf{x}_0$ , where $A(t)$ and $\mathbf{g}(t)$ are continuous on an open interval $I$ containing $t_0$ . Then there exists a unique solution $\mathbf{x}(t)$ defined on all of $I$ .

RemarkGlobal existence

Unlike the nonlinear case, where solutions may blow up in finite time, linear systems always have solutions defined on the entire interval where the coefficients are continuous. This is because linear growth in $\mathbf{x}$ is controlled by Gronwall's inequality.

Proof Sketch

Proof

The proof uses the Picard iteration method adapted to systems.

Step 1. Rewrite as an integral equation: $\mathbf{x}(t) = \mathbf{x}_0 + \int_{t_0}^t [A(s)\mathbf{x}(s) + \mathbf{g}(s)]\,ds$ .

Step 2. Define Picard iterates: $\mathbf{x}_0(t) = \mathbf{x}_0$ and $\mathbf{x}_{n+1}(t) = \mathbf{x}_0 + \int_{t_0}^t [A(s)\mathbf{x}_n(s) + \mathbf{g}(s)]\,ds$ .

Step 3. On any compact subinterval $[t_0-\delta, t_0+\delta] \subset I$ , let $M = \max\|A(t)\|$ and $K = \max\|\mathbf{g}(t)\|$ . Then $\|\mathbf{x}_{n+1}(t) - \mathbf{x}_n(t)\| \leq \frac{(M|t-t_0|)^n}{n!}C$ for some constant $C$ , showing the series $\sum(\mathbf{x}_{n+1}-\mathbf{x}_n)$ converges uniformly.

Step 4. The limit $\mathbf{x}(t) = \lim_{n\to\infty}\mathbf{x}_n(t)$ satisfies the integral equation. Uniqueness follows from Gronwall's inequality: if $\mathbf{x}_1, \mathbf{x}_2$ are two solutions, $\|\mathbf{x}_1 - \mathbf{x}_2\| \leq M\int_{t_0}^t\|\mathbf{x}_1-\mathbf{x}_2\|\,ds$ , implying $\mathbf{x}_1 = \mathbf{x}_2$ . $\blacksquare$

■

Consequences

Theorem5.7Structure of the solution space

The homogeneous system $\mathbf{x}' = A(t)\mathbf{x}$ has a solution space of dimension $n$ . Any $n$ linearly independent solutions form a fundamental set, and the general solution of the nonhomogeneous system is

$\mathbf{x}(t) = \Phi(t)\mathbf{c} + \mathbf{x}_p(t)$

where $\Phi(t)$ is a fundamental matrix and $\mathbf{x}_p$ is any particular solution.

ExampleVerifying the structure theorem

For $\mathbf{x}' = \begin{pmatrix}0&1\\-1&0\end{pmatrix}\mathbf{x}$ , the solutions $\mathbf{x}_1 = (\cos t, -\sin t)^T$ and $\mathbf{x}_2 = (\sin t, \cos t)^T$ are linearly independent (their Wronskian is $1$ ). Every solution is $c_1\mathbf{x}_1 + c_2\mathbf{x}_2$ , a $2$ -dimensional space.

RemarkContinuous dependence on parameters

The solution $\mathbf{x}(t; t_0, \mathbf{x}_0)$ depends continuously on $t_0$ , $\mathbf{x}_0$ , and the coefficients $A(t), \mathbf{g}(t)$ . This stability is crucial for numerical methods and perturbation theory.