AbstractWhen economical, biological or chemical systems are modelled by descriptor systems, in which the state x describes concentrations, populations of species, or numbers of cells, then the solution is a nonnegative vector function. Hence, the numerical methods for the control or simulation should respect this special structure. We define and characterise positivity for descriptor systems. Furthermore, for a special case, we show a Schur complement decoupling technique that reduces a descriptor system to the ODE case. We use this technique to present a characterisation of the existence of positive solutions to generalised Lyapunov equations. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)