Sensitivity of Schur stability of monodromy matrix (Q632916)

The authors study the sensitivity of Schur stability of a linear difference equation system with periodic coefficients \[ x\left(n+1\right) =A\left( n\right) x\left( n\right),\quad n\in \mathbb{Z}, \tag{1} \] where \(A\left( n\right) \) is an \(N\times N\) dimensional matrix with a period \(T\). The Schur stability of the monodromy matrix \(X\left( T\right) =A\left( T-1\right) A\left( T-2\right) \dots A\left( 1\right) A\left( 0\right) \) of the system (1) is equivalent to the Schur stability of the system (1). The authors obtain results which replace the constant upper bound provided by Theorem 2 of \textit{K. Aydin, H. Bulgak} and \textit{G. V. Demidenko} [Selçuk J. Appl. Math. 2, No.~2, 5--10 (2001; Zbl 1016.39010)] for \(\left\| Y\left( T\right) -X\left( T\right) \right\| \) with varying and sharper ones, where \(X\left( T\right) \) is the monodromy matrix of the system (1) and \( Y\left( T\right) \) is the monodromy matrix of the perturbed system \[ y\left( n+1\right) =\left( A\left( n\right) +B\left( n\right) \right) y\left( n\right) ,\;n\in \mathbb{Z}. \]