Boundedness and exponential stability for periodic time dependent systems

C. Buse, Department of Mathematics, West University of Timisoara, Timisoara, Romania
A. Zada, Government College University, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore, Pakistan

E. J. Qualitative Theory of Diff. Equ., No. 37. (2009), pp. 1-9.

Abstract: The time dependent $2$-periodic system
\begin{equation*}
\dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\eqno{(A(t))}
\end{equation*}
is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem
\begin{equation*}
\left\{\begin{split}
\dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\
y(0) &= 0
\end{split}\right.
\end{equation*}
is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries.

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