ON LIAPUNOV FUNCTIONAL IN THEORY OF STABILITY OF SYSTEMS WITH TIME-LAG

It is well known,that in the theory of stability in differential equations,Liapunov’s second method may be the most important The center problem of Liapunov’s second method is construction of Liapunov function for concrete problems.Beyond any doubt,construction of Liapunov functions is an art.In the case of functional differential equations,there were also many attempts to establish various kinds of Liapunov type theorems.Recently Burton[2]presented an excellent theorem using the Liapunov functional to solve the asymptotic stability of functional differential equation with bounded delay. However,the construction of such a Liapunov functional is still very hard for concrete problems. In this paper, by utilizing this theorem due to Burton,we construct concrete Liapunov functional for certain and nonlinear delay differential equations and derive new sufficient conditions for asymptotic stability.Those criteria improve the result of literature[1]and they are with simple forms,easily checked and applicable.