Asymptotic exponential cones of Metzler matrices and their use in the solution of an algebraic problem
Several algebraic problems arising in the context of positive switched systems are still unsolved. In particular, in the investigating reachability property for continuous-time switched systems, the problem has originated under what conditions one can find nonnegative integers \(\tau_1, \tau_2, \dots, \tau_m\) such that a strictly positive vector \({\mathbf v}\) belongs to the boundary of the cone generated by the columns of the matrix \(e^{A_{i_{1}} \tau_1}, e^{A_{i_{2}} \tau_2}, \dots, e^{A_{i_{m}} \tau_m}\), where \(A_{i_{1}}, A_{i_{2}}, \dots, A_{i_{m}}\) are \(m\) Metzler matrices (real square matrices with nonnegative off-diagonal entries). In this paper, the analysis, which was started by the authors in [Linear Algebra Appl. 425, No.~2--3, 283--302 (2007; Zbl 1123.93033)], is developed further by widening and deepening the knowledge about asymptotic exponential cones of a family of Metzler matrices and these results are used to provide necessary and/or sufficient conditions for the solvability of the given algebraic problem. The problem solution is also investigated in the specific case when dealing with at most two exponential matrices.
scientific article; zbMATH DE number 5667902
scientific article