Asymptotical stability of partial difference equations with variable coefficients
type: Item
instance of: scholarly article
title: Asymptotical stability of partial difference equations with variable coefficients
zbMATH Open document ID: 1016.39009
DOI: 10.1016/S0022-247X(02)00430-4
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The paper is devoted to the linear \(n\)-variate dynamical system with scalar complex input \(b[t]\) and state \(x[t]\) in discrete time \(t:=(t_{1},\ldots,t_{n})\) described by the initial value problem for partial difference equation \[ x[t]=\begin{cases} \sum_{d\in\mathbf{D}}a_{d}[t]x[t-d]+b[t], &t\in\mathbf{T},\\ x_{t}, &t\in\mathbf{T}_{0}, \end{cases} \] \(1\leq |\mathbf{D}|<\infty\), \(\mathbf{D}\subset\mathbb{N}^{n}\), \(\sup_{\tau\in\mathbf{T}_{0}}|x_{\tau}|<\infty\), \(\mathbf{T}_{0}\cap\mathbf{T}=\emptyset\). The time set \(\mathbf{T}\) is the nonnegative orthant of the integer lattice \(\mathbb{Z}^{n}\). The initial set \(\mathbf{T}_{0}(\mathbf{D})\) is chosen so that there is a unique solution \(\{x[t]\}_{t\in\mathbf{T}}\) for any bounded initial sequence \(\{x_{\tau}\}_{\tau\in\mathbf{T}_{0}}\) and input sequence \(\{b[t]\}_{t\in\mathbf{T}}\). The system parameter space \(A:=\mathbb{C}^{\mathbf{D}\times\mathbf{T}}\) is made up of all coefficient sequence vectors \(a:=\{a_{d}[t]\}_{d\in\mathbf{D},t\in\mathbf{T}}\). The main result presents a time-independent part of the exponential stability domain in the parameter space: \(S_{\varepsilon}:=\{a\in A:\sum_{d\in\mathbf{D}}|a_{d}[t]|\leq 1-\varepsilon, t\in\mathbf{T}\}\) for any small \(\varepsilon>0\). Two illustrative examples give a comparison of the sufficient stability criteria with the exact stability set. A different approach in the univariate case is set out in another paper of the author [J. Math. Anal. Appl. 273, 378--408 (2002; Zbl 1018.39001)].
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 1866773
zbMATH Keywords: partial difference equations
zbMATH Keywords: multivariate dynamical systems
zbMATH Keywords: asymptotic stability
zbMATH Keywords: initial value problem
zbMATH Keywords: exponential stability
zbMATH Keywords: variable coefficients
author: F. G. Boese
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
cites work: A necessary and sufficient condition for robust asymptotic stability of time-variant discrete systems
cites work: Stability of polynomials with time-variant coefficients
cites work: Boundary Retention Effects upon Contaminant Dispersion in Secondary Flows
cites work: On the Asymptotical Stability of Multivariate Dynamical Systems
cites work: On ordinary difference equations with variable coefficients
cites work: Computational Aspects of Three-Term Recurrence Relations
cites work: On the theory of discrete systems
cites work: On the solution of a second order linear homogeneous difference equation with variable coefficients
cites work: Solutions of linear difference equations with variable coefficients
cites work: On the Solution of a Linear Homogeneous Difference Equation with Variable Coefficients
cites work: Multidimensional constant linear systems
cites work: One Expression for the Solutions of Second Order Difference Equations
cites work: Q3735790
cites work: Robust stability of multidimensional difference equations with shift- variant coefficients
description: scientific article; zbMATH DE number 1866773