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Stability of a time discrete perturbed dynamical systems with delay. (English) Zbl 0966.39009
The authors investigate the absolute stability of the system of delay difference equations \[ u_{k+1}=Au_k+Bu_{k-\tau}+F_k(u_k,u_{k-\tau}), \tag{*} \] where \(\tau\) is a positive integer, \(A,B\in \mathbb{C}^{n\times n}\), \(u_k\in \mathbb{C}^{n}\), and the nonlinearity \(F\) satisfies the restriction \(\|F_k(x,y)\|\leq p\|x\|+q\|y\|\), \(k=0,1,2,\dots\), where \(p,q\) are positive real constant and \(\|\cdot\|\) is the usual norm in \(\mathbb{C}^n\) . Recall that the zero solution of (*) is said to be absolutely \(l^2\)-stable, if there exists a constant \(\Gamma >0\) (depending only on the numbers \(p,q\) in the restriction for \(F\)) such that for every solution \(\{u_k\}_{k=-\tau}^\infty\) to (*) satisfies \[ \|u\|_{l^2}:=\left(\sum_{k=0}^\infty \|u_k\|^2\right)^{1\over 2} \leq \Gamma \sum_{k=-\tau}^0 \|u_k\|. \] A typical result is the following statement.
Theorem. Suppose that all zeros of \(\det(zI-A-z^{-\tau}B)\) have modulus less than 1 and \((p+q)M<1\), where \(M=\max_{|z|=1}\|(zI-A-z^{-\tau}B)^{-1}\|\). Then the zero solution to (*) is absolutely \(l^2\)-stable.

MSC:
39A12 Discrete version of topics in analysis
39A11 Stability of difference equations (MSC2000)
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
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