Asymptotical stability of partial difference equations with variable coefficients
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)].
scientific article; zbMATH DE number 1866773
scientific article