Exponential stability of positive implicit dynamic equations with constant coefficients

Abstract

In this paper, the problem of positivity and stability for linear time-invariant implicit dynamic equations is generally studied. We provide necessary and sufficient conditions for positivity of these equations. This characterization can be considered as a unification and generalization for some previous results. On the other hand, we study the exponential stability of positive implicit dynamic equations. Previously, this issue was not completely addressed. By using Krein–Rutman theorem, we show that a positive implicit dynamic equation on a time scale is uniformly exponentially stable if and only if the characteristic polynomial of the matrix pair defining the equation has all its coefficients of the same sign.