On the stability of some systems with aftereffect (Q1342580)

A method of constructing Lyapunov functionals for some systems with an arbitrary aftereffect is suggested. Such an aftereffect may be observed in models for biotechnical systems -- an example treated in the paper --, where the actual growth rate of microorganisms may be not only dependent on the actually available substrate concentration but also on the recent history of this state variable. The method is developed for linear systems and then extended to certain nonlinear systems. An interesting result lies in the remark that for some systems, the aftereffect seems to be necessary for their stabilization.