| In 1892, the Russian Mathematician Liapunov introduced the definition of motion and its general method in his paper "The general problems in the stability of motion", which founded the basis for the stability theory. With the development of science, the Liapunov stability theory has been developed and extended with its applications in Physics, Engineering, Ecological systems, Control theory, and Neural networks. Due to the limitation in technology, to study the stability of all the variables is not easy. On the other hand, due to the specific needs in practical problems, it is not necessary to study all the variables. It becomes important to discuss the partial variables.This paper uses the Liapunov direct method to discuss the stability of certain nonlinear operator-valued functional differential equations with infinite delay and the stability of its partial variable. The absolute stability of certain nonlinear control systems is analyzed using the Liapunov function.This paper includes four chapters.Chapter 1, "Basic Concepts", primarily gives the definition, categorization, and corollary for differential equations with delay, and the definition of the absolute stability for the nonlinear control systems.In Chapter 2, "The stability of Operator-valued Nonlinear Systems with Infinite delay", the theorem of the asymptotic stability of zero solution for operator-valued nonlinear systems with infinite delay is obtained using the Liapunov function satisfiying certain conditions, without the assumption of the boundedness of F.In Chapter 3, "The Stability of Nonlinear Neutral Functional Differential Equation with respect to the partial variable", by the Liapunov function, the asymptotic stability of certain operator-valued neutral system. with infinite delay is obtained.In Chapter 4, "The Absolute Stability of Nonlinear Control System", the absolute stablility of certain nonlinear control systems which concerns unsteady motion is studied and extends the currently relevant results.
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