# zbMATH — the first resource for mathematics

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
Full Text: