Some Extensions of Liapunov’s Second Method*

IAPUNOV’S second method has long been recognized in the Soviet Union as the most general method for the study of the stability of equlibirum positions of systems described by differential or difference equations. The method was first presented by Liapunov in his now classical memoir,’ which appeared in Russian in 1892 and was translated into French in 1907. Good sources for the statements and proofs of the mathematical theorems underlying the method can be found in works by Hahn,’ Antosiewicz,3 and Cesari.4 These references also contain extensive bibliographies. By way of introduction, let us consider first Liapunov’s asymptotic stability theorem for autonomous systems. Let the systems of differential equations be (li: = dxldt)