Stability and Boundedness for RFDE with Bounded Delay

Abstract

The extensions of Lyapunov’s second method to differential equations with delay have been carried out using both Lyapunov functionals and Lyapunov functions. The method of Lyapunov functionals demands the knowledge of solutions while the method of Lyapunov functions requires choosing minimal classes of functions to estimate the derivative of the Lyapunov function relative to the given delay system. If we examine the Lyapunov functionals constructed for all the examples that have been discussed 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 functionals respectively. As a result, the discussion of the examples is not really in the spirit of the method of Lyapunov functionals. This observation leads to the development of the method of Lyapunov functions on the product spaces for studying stability properties of equations with delay where knowledge of solutions is not demanded. We follow in this chapter, the development of ideas in a sequential order.