Pastravanu, Octavian; Voicu, Mihail Interval matrix systems – flow invariance and componentwise asymptotic stability. (English) Zbl 1028.34047 Differ. Integral Equ. 15, No. 11, 1377-1394 (2002). The authors consider continuous-time and discrete-time interval matrix systems of the form \[ x'(t)=A^Ix(t), t\in{\mathbb T}.\tag{*} \] Here, \(x'\) is the forward shift \(x'(t):=x(t+1)\) for \({\mathbb T}={\mathbb Z}_+\) and the usual derivative for \({\mathbb T}={\mathbb R_+}\). \(A^I\) denotes an interval matrix defined as the family \(A^I:=\{A\in{\mathbb R}^{n\times n}: A^-\leq A\leq A^+\}\) of real matrices, where the order relation is understood componentwise. The first main topic of this paper is to characterize the flow invariance of time-dependent rectangular sets of the form \([-h_1(t),h_1(t)]\times\ldots\times[-h_n(t),h_n(t)]\) in terms of the vector inequality \(\bar{A}h(t)\leq h'(t)\), where the matrix \(\bar{A}\) is constructed from \(A^I\), as well as in terms of a condition on the state-transition matrix of \((\ast)\). In case \(h_i\), \(i=1,\ldots,n\), are exponential functions, these criteria can be simplified using the dominant eigenvalue of the matrix \(\bar{A}\). Another important feature of the paper is to provide necessary and sufficient conditions for the componentwise asymptotic stability. Here, \((\ast)\) is denoted as componentwise asymptotically stable, if there exists a flow-invariant rectangular set satisfying \(\lim_{t\to\infty}h_i(t)=0\) for \(i=1,\ldots,n\). This is true, if and only if \(\bar{A}\) is a stable matrix in the sense of Schur (discrete-time case), or of Hurwitz (continuous-time case), respectively. Moreover, the authors prove that this is equivalent to the concept of componentwise exponential asymptotic stability, and characterize this property by means of certain conditions on \(\bar{A}\). Reviewer: Christian Pötzsche (Augsburg) Cited in 1 Document MSC: 34D05 Asymptotic properties of solutions to ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34C14 Symmetries, invariants of ordinary differential equations 34A30 Linear ordinary differential equations and systems 93D20 Asymptotic stability in control theory 93C05 Linear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations 93C55 Discrete-time control/observation systems 93C41 Control/observation systems with incomplete information Keywords:interval matrix; flow-invariance; componentwise asymptotic stability × Cite Format Result Cite Review PDF