TY - JOUR

T1 - Uncertain discrete systems: Stability, instability, and an attractor

AU - Zuber, I.E.

AU - Gelig, A.K.

PY - 2015

Y1 - 2015

N2 - © 2015, Allerton Press, Inc.The system xk + 1 = A(k)xk is considered, where A(k) є ℝn × ℝn is a matrix coefficients aij(k) of which are nonanticipating functionals of any nature. It is assumed that the elements of the first to (p + 1)st columns above the main diagonal and the elements of the (p + 1)st to (n–1)st columns below the main diagonal satisfy the condition $$\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant a.$$ For the other elements, the estimate $$\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant \delta $$ holds. An upper bound for the values of the parameter δ at which system (1) is globally exponentially stable if $$\mathop {\sup }\limits_k |{a_{ii}}(k)| <1$$$$|(i \in \overline {1,n}),$$ and unstable if there exists a j such that $$\mathop {\inf }\limits_k {a_{jj}}(k)| > 1$$ is obtained. In the case where the elements on the main diagonal are functions of xk and satisfy the conditions $$\mathop {\sup }\limits_{|{x_k}| > R} |{a_{ii}

AB - © 2015, Allerton Press, Inc.The system xk + 1 = A(k)xk is considered, where A(k) є ℝn × ℝn is a matrix coefficients aij(k) of which are nonanticipating functionals of any nature. It is assumed that the elements of the first to (p + 1)st columns above the main diagonal and the elements of the (p + 1)st to (n–1)st columns below the main diagonal satisfy the condition $$\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant a.$$ For the other elements, the estimate $$\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant \delta $$ holds. An upper bound for the values of the parameter δ at which system (1) is globally exponentially stable if $$\mathop {\sup }\limits_k |{a_{ii}}(k)| <1$$$$|(i \in \overline {1,n}),$$ and unstable if there exists a j such that $$\mathop {\inf }\limits_k {a_{jj}}(k)| > 1$$ is obtained. In the case where the elements on the main diagonal are functions of xk and satisfy the conditions $$\mathop {\sup }\limits_{|{x_k}| > R} |{a_{ii}

U2 - 10.3103/S1063454115020132

DO - 10.3103/S1063454115020132

M3 - Article

SP - 61

EP - 65

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -