Stability of linear positive systems (Q2754811)

The author considers the linear system NEWLINE\[NEWLINE\dot H+MH=G(t),\quad t\geq 0,\tag{1}NEWLINE\]NEWLINE where \(M:\mathcal{E} \mapsto \mathcal{E}\) is a bounded operator in a Banach space \(\mathcal{E}\), which has the structure of a partially ordered space with respect to a fixed cone \(\mathcal{K}\subset\mathcal{E}\). On the basis of the Krein-Bonsall-Karlin theorem on the spectral radius of a monotone operator (see the book by \textit{M. A. Krasnosel'skij, Je. A. Lifshits} and \textit{A. V. Sobolev} [Positive linear systems. The method of positive operators. Sigma Series in Applied Mathematics, 5. Berlin: Heldermann-Verlag (1989; Zbl 0674.47036)]) the author establishes criteria for asymptotic stability of system (1) whose evolution operator has the monotonicity property with respect to \(\mathcal{K}\). E.g., it is shown that if \(M=L-P\), where the operator \(P\) is monotone, \(L\) is monotone invertible, \( P\mathcal{K}\subset\mathcal{K}\subset L\mathcal{K}\), and \(e^{-Lt}\) is monotone with respect to \(\mathcal{K}\), then the following statements are equivalent: (a) system (1) is asymptotically stable; (b) the operator \(M\) is monotone invertible; (c) there exist \(X>0\) and \(Y>0\) such that \(MX=Y\); (d) the spectral radius of the operator \(\lambda L-P\) is less than 1 for any \(\lambda \in \sigma(M)\).NEWLINENEWLINENEWLINESome results on the structure of monotone and monotone invertible operators are also obtained.