The stability of grey discrete systems (Q2725102)

The authors consider the asymptotic stability of the grey linear discrete system NEWLINE\[NEWLINEx(k+ 1)= A(\otimes) x(k),\quad k\in Z^+= \{0,1,2,\dots\},\quad x(0)= x_0,\tag{1}NEWLINE\]NEWLINE where \(A(\otimes)\) is an \(n\times n\) real matrix with uncertain entries \(\otimes_{ij}\) satisfying \(p_{ij}\leq\otimes_{ij}\leq q_{ij}\), \(i,j= 1,\dots, n\). Let \(N= \{1,2,\dots, n\}\), \(P= (p_{ij})\), \(Q= (q_{ij})\), \(N[P, Q]= \{(a_{ij})\in \mathbb{R}^{n\times n}: p_{ij}\leq a_{ij}\leq q_{ij}, i,j\in N\}\), here \(p_{ij}\), \(q_{ij}\) are some real numbers.NEWLINENEWLINENEWLINEThe grey linear system (1) is called asymptotically stable if \(\rho(A)\) (spectral radius of \(A\)) \(<1\) holds for every matrix \(A= N[P, Q]\). Let \(m_{ij}= \max\{|p_{ij}|,|q_{ij}|\}\), \(M= (m_{ij})\in \mathbb{R}^{n\times n}\), and \(N_1\), \(N_2\) be any subsets of \(N\) satisfying \(N_1\cap N_2= \emptyset\), \(N_1\cup N_2= N\). Denote \(a_i= \sum_j m_{ij}\), \(\alpha_i= \sum_{j\in N_1} m_{ij}\), \(\beta_i= \sum_j m_{ij}\).NEWLINENEWLINENEWLINEThe sufficient conditions for asymptotic stability of system (1) given by \textit{X. L. Peng} et al. [Chin. Bull. Sci. 36, 1273-1274 (1991)] and \textit{J. F. Chen} [Acta Math. Appl. Sin. 18, 123-128 (1995; Zbl 0864.58031)] are: 1) \(\sum_{i,j}m_{ij}\leq 1\), or 2) \(\max_i({1\over r_i})\sum^n_{j=1} r_j m_{ij}< 1\) holds for some natural numbers \(r_1,r_2,\dots, r_n\).NEWLINENEWLINENEWLINEIn the present paper, by a matrix analysis method, several sufficient conditions for asymptotic stability of system (1) are obtained which are applicable in the case when \(\|M\|_\infty> 1\) or \(\|M\|_1> 1\). The main results embodied in Theorem 1 and Theorem 2 are: 1) \(a_i< 1\), \(i\in N\); or 2) if \(a_i\geq 1\) \((i\in N_1\neq\emptyset)\), \(a_j< 1\) \((j\in N_2\neq\emptyset)\), and one of the following relations holds NEWLINE\[NEWLINE\alpha_i(1- \beta_j)- [1- \beta_j] \sum_{t\in N_1} m_{it} a_t+ \beta_j \sum_{t\in N_1} m_{jt} a_t> 0\quad (i\in N_1,\;j\in N_2),NEWLINE\]NEWLINE NEWLINE\[NEWLINE1- \alpha_i- \sum_{t\in N_2} m_{it} a_t> 0\quad (i\in N_1).NEWLINE\]