An improved estimate in the method of freezing

Abstract:

Let $\dot x = A(t)x$ and ${\lambda _k}(t)$ be the eigenvalues of the matrix $A(t)$. The main result of the Method of Freezing [1] states that if $\left \| {A(t)} \right \| \leqslant a$. Re ${\lambda _k}(t) \leqslant {\lambda _0}$ and $\left \| {A(t) - A(s)} \right \| \leqslant \delta \left | {t - s} \right |$, then \[ \left \| {x(t)} \right \| \leqslant \left \| {x({t_0})} \right \|{D_\delta }\exp ({\lambda _0} + 2a{\lambda _\delta })(t - {t_0})\quad (t \geqslant {t_0})\] for all solutions of the system, where \[ {\lambda _\delta } = {({C_n} \cdot \delta /4{a^2})^{1/(n + 1)}}\]. The previous best known value. ${C_n} = n(n + 1)/2$, is reduced to the substantially smaller value $2{n^n}{e^{ - n}}/(n - 1)! < \sqrt {2n/\pi }$.