Interval matrix systems -- flow invariance and componentwise asymptotic stability (Q1850184)

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 \(\bar{A}\).