Electron. J. Diff. Eqns., Vol. 2008(2008), No. 94, pp. 1-5.

A characterization of dichotomy in terms of boundedness of solutions for some Cauchy problems

Akbar Zada

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.

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Akbar Zada
Government College University
Abdus Salam School of Mathematical Sciences, (ASSMS)
Lahore, Pakistan
email: zadababo@yahoo.com

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