General theory for 1st order matrix ODE "Z'(t) = A(t)Z(t)"?

Question

I have an ODE of the form $$Z'(t) = A(t)\cdot Z(t), \\ Z(0) = z_0,$$ where $A\in C([0;T];\mathbb R^{2\times 2})$ and $z_0 \in \mathbb R^2$ are given and solutions $Z\in C^1([0;T];\mathbb R^2)$ are being searched. Eventually, I want to be able to make statements of the form "if $A_1$ and $A_2$ are close to each other, then the corresponding solutions $Z_1$ and $Z_2$ must be close to each other". Is there any general theory about this form of ODEs and can you recommend good references? Thank you!

Answer 1

Existence and uniqueness of solutions follow almost trivially from Picard-Lindelöf for all linear ODE with continuous coefficients. This should be contained in any textbook on ODE.

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You can use the Grönwall lemma on $d(t)=\|Z_1(t)-Z_2(t)\|$ where $$ d'\le \|Z_1'-Z_2'\|\le \|A_1\|\,\|Z_1-Z_2\|+\|A_1-A_2\|\,\|Z_2\|= \|A_1\|\,d+\|A_1-A_2\|\,\|Z_2\|. $$ so that with $a(t)=\int_0^t\|A_1(s)\|ds$ $$ d(t)\le e^{a(t)}d(0)+e^{a(t)}\int_0^t e^{-a(s)}\|A_1(s)-A_2(s)\|\,\|Z_2(s)\|\,ds. $$

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Let $\|A_k(t)\|\le L$. Then the Picard iterations $P_k$ have contraction constant $\frac12$ in the weighted supremum norm $\|Z\|=\sup_{t\in[0,T]}e^{-2Lt}\|Z(t)\|$. By the a-priori error estimate of the Banach fixed point theorem $$ \|Z_2-Z_1\|\le 2\|P_1(Z_2)-Z_2\|=2\|P_1(Z_2)-P_2(Z_2)\| $$ which again can be reduced to the difference $\|A_2-A_1\|$.