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Continuity of numeric characteristics for asymptotic stability of solutions to linear difference equations with periodic coefficients. (English) Zbl 1016.39010

The authors study a linear perturbed system of difference equations of the form \[ y(n+1) = (A(n) + B(n))y(n),\quad n \geq 0, \] where \(A(n)\), \(B(n)\) are \(N\times N\) time-periodic matrices. It is supposed that the zero solution to the system \[ x(n+1) = A(n)x(n),\quad n \geq 0, \] is asymptotically stable.
The main aim is to obtain conditions on a perturbation matrix \(B(n)\) that guarantee asymptotic stability of the zero solution to the system under consideration and prove continuity of the series \[ F = \sum_{k=0}^{\infty}(X^*(T))^k(X(T))^k, \] where \[ X(T) = A(T-1)\cdots A(1)A(0). \] In particular, the authors are interested in the following numerical characteristics for asymptotic stability of the zero solution: \[ \omega_1(A,T) = \|F\|, \] where \(\|F\|\) is the spectral norm of the corresponding matrix series, and the following result holds: Let \[ Y(T) = (A(T-1) + B(T-1))\cdots (A(1) + B(1))(A(0) + B(0)) \] be the monodromy matrix of the system under study and let \[ \|Y(T) - X(T)\|\leq \sqrt{\|X(T)\|^2 + \frac{1}{\omega_1(A,T)}} - \|X(T)\|; \] then the zero solution to the system is asymptotically stable.

MSC:

39A11 Stability of difference equations (MSC2000)
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