Positivity for non-Metzler systems and its applications to stability of time-varying delay systems

In 1950, Wazewski obtained the following necessary and sufficient condition for positivity of linear systems of ordinary differential equations: the matrix of coefficients is Metzler (i.e. off-diagonal elements in the matrix of coefficients are nonnegative). No results on the positivity of solutions to delay systems in the case where the matrix is non-Metzler were expected to be obtained. It was demonstrated that for delay systems the Wazewski condition is not a necessary one. We prove results on positivity of solutions for non-Metzler systems of delay differential equations. New explicit tests for exponential stability are obtained as applications of results on positivity for non-Metzler systems. Examples demonstrate possible applications of our theorems to stabilization. For instance, in view of our results, the implicit requirement on dominance of the main diagonal can be skipped. Our approach is based on nonoscillation of solutions and positivity of the Cauchy functions of scalar diagonal delay differential equations.