Criterion For The Exponential Stability of Discrete Evolution Family Over Banach Spaces
Résumé
Let $T(1)$ be the algebraic generator of the discrete semigroup $\textbf{T}=\{T(n)\}_{n\geq 0}$. We prove that
the system $x_{n+1}=T(1)x_n$ is uniformly exponentially stable if and only if for each real number $\mu$ and each $q$-periodic sequence $z(n)$ with $z(0)=0$ the unique solution of the Cauchy Problem
\begin{equation*}
\left\{
\begin{split}
% \nonumber to remove numbering (before each equation)
x_{n+1} &= T(1)x_{n}+e^{i\mu(n+1)}z(n+1), \\
x_0&= 0
\end{split}
\right.\eqno{(T(1), \mu,0)}
\end{equation*}
is bounded.
We also extend the above result to $q$-periodic system $y_{n+1} = A_{n}y_{n}$ i.e. we proved that the system $y_{n+1} = A_{n}y_{n}$
is uniformly exponentially stable if and only if for each real number $\mu$ and each $q$-periodic sequence $z(n)$, with $z(0)=0$ the unique solution of the Cauchy Problem
\begin{equation*}
\left\{
\begin{split}
% \nonumber to remove numbering (before each equation)
y_{n+1} &= A_{n}y_{n}+e^{i\mu(n+1)}z(n+1), \\
y_0&= 0
\end{split}
\right.\eqno{(A_{n}, \mu,0)}
\end{equation*}
is bounded. Here, $A_{n}$ is a sequence of bounded linear operators on Banach space $\mathcal{X}$.