Lyapunov functions on product spaces and stability theory of delay differential equations

Abstract

If we examine the Lyapunov functionals constructed for all the-examples that have been discussed so far in the literature, we find that the investigators, inadvertently, employ a combination of a Lyapunov function and a functional in such a way that the corresponding derivative can be estimated suitably without demanding minimal classes of functions or the knowledge of solutions as in the case of Lyapunov functions or Lyapunov functionals, respectively [3-61. The method of Lyapunov functionals, however, demands the knowledge of solutions of delay differential equations considered and consequently the discussion of examples is not really in the spirit of the method. This observation leads us to develop the method of Lyapunov functions on product spaces for studying stability properties of equations with delay, where, except conceptually, the knowledge of solutions is not demanded. We shall also develop stability theory in terms of two measures which unifies several known stability concepts. Our presentation demonstrates the advantage of utilizing Lyapunov functions on product spaces. 391 0022-247X/91 $3.00