Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 94, pp. 1-5.
Title: A characterization of dichotomy in terms of boundedness of solutions
for some Cauchy problems
Author: Akbar Zada (Government College Univ., Lahore, Pakistan)
Abstract:
We prove that a quadratic matrix of order $n$ having complex
entries is dichotomic (i.e. its spectrum does not intersect
the imaginary axis) if and only if there exists a projection
$P$ on $ \mathbb{C}^n$ such that $Pe^{tA}=e^{tA}P$ for all
$t\ge 0$ and for each real number $\mu$ and each vector
$b \in \mathbb{C}^n$ the solutions of the following
two Cauchy problems are bounded:
$$\displaylines{
\dot x(t) = A x(t) + e^{i \mu t}Pb,\quad t\geq 0, \cr
x(0) = 0
}$$
and
$$\displaylines{
\dot{y}(t)= -Ay(t) + e^{i\mu t}(I-P)b, \quad t\geq 0, \cr
y(0) = 0\,.
}$$
Submitted May 29, 2008. Published July 05, 2008.
Math Subject Classifications: 47D06, 35B35.
Key Words: Stable and dichotomic matrices; Cauchy problem;
spectral decomposition theorem.