Analysis of stability for singular discrete linear systems (Q2732024)

The authors consider stability problems of the perturbed discrete general linear systems of the form NEWLINE\[NEWLINEEx(t+ 1)= (A+\Delta A) x(t),\tag{\(*\)}NEWLINE\]NEWLINE where \(x(t)\in \mathbb{R}^n\), \(E\), \(A\) and \(\Delta A\) are \(n\times n\) real constant matrices, for which there exists a complex number \(s\in \mathbb{C}\) such that \((sE-A)\) is invertible and \(\deg[\text{det}(sE- A)]= \text{rank }E\) holds. Denote by \((**)\) the unperturbed system of \((*)\) without the term \(\Delta A\). The following asymptotic stability criterion and two alternative statements of it are proved for system \((**)\) by means of equivalent transformations:NEWLINENEWLINENEWLINETheorem 1. A necessary and sufficient condition for the asymptotic stability of the zero solution \(x(t)\equiv 0\) of system \((**)\) is: there exist non-singular matrices \(P\), \(Q\) such that \(QEP= \left[\begin{smallmatrix} I_1 & 0\\ 0 & N\end{smallmatrix}\right]\) and \(QAP= \left[\begin{smallmatrix} A_1 & 0\\ 0 &I_2\end{smallmatrix}\right]\), and every eigenvalue \(\lambda\in \sigma(A_1)\) is less than one, where \(A_1\) is an \(r\times r\) matrix, \(N\) is a nilpotent matrix, and \(I_i\) are \(n_i\times n_i\) matrices, \(i= 1,2\), \(n_1= r,n_2= n-r\).NEWLINENEWLINENEWLINEBased on the last theorem, a sufficient condition for the asymptotic stability of the zero solution \(x(t)\equiv 0\) of the perturbed system \((*)\) is also derived. An illustrating example is indicated.