Math 6410 • ODE

[25] Let \(B(t)\) be a continuous real \(d\times d\) matrix function defined for
\(t\in[0,\infty)\) such that
\[
\int_0^\infty \|B(s)\|\,ds<\infty.
\]
Let \(A\) be a real \(d\times d\) matrix such that \(\operatorname{Re}\lambda<0\)
for all eigenvalues of \(A\). Show that the initial value problem

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

(1)

\[
x(0)=x_0
\]


has a bounded solution on \([0,\infty)\). [Provide the complete proof. Do not just quote a theorem.
Hint: Use the variation of parameters formula and Gronwall's Inequality. Gronwall's
Inequality says that if \(c_0\ge0\) and if \(f(t)\), \(\phi(t)\) are nonnegative
continuous functions on the interval \([0,T]\) which satisfy
\[
f(t)\le c_0+\int_0^t \phi(s)f(s)\,ds
\]
for all \(t\in[0,T]\), then
\[
f(t)\le c_0\exp\left(\int_0^t \phi(s)\,ds\right)
\]
for all \(t\in[0,T]\).]