Math 6410 • ODE

Let \(B(t)\) be a \(C^1\) real \(d\times d\) matrix function defined for
\(t\in[0,\infty)\) such that

\[
\lim_{t\to\infty}B(t)=0.
\]


Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname{Re}\lambda<0\)
for all eigenvalues of \(A\). Show that there is \(\delta>0\) such that if
\(|x_0|\le\delta\), then the initial value problem

\[
\dot{x}=Ax+B(t)x
\]

(2)

\[
x(0)=x_0
\]


has a solution such that \(x(t)\to0\) as \(t\to\infty\). [Provide the complete proof. Do not just
quote a theorem.]

[Hint: Use the variation of parameters formula and Gronwall's Inequality, which says that
for \(f(t)\), \(\phi(t)\) nonnegative continuous functions on the interval
\([0,T]\) and for \(c_0\ge0\), if
\[
f(t)\le c_0+\int_0^t \phi(s)f(s)\,ds
\]
for all \(t\in[0,T]\), then
\[
f(t)\le c_0\exp\left(\int_0^t \phi(s)\,ds\right)
\]
for all \(t\in[0,T]\). You might first show that there is a constant \(k\) such
that if \(\epsilon\) is small enough and \(|x_0|<\epsilon\) then the solution
remains bounded \(|x(t)|\le k|x_0|\) for all \(t\in[0,T]\).]