Study On φ0Practical Stability Of Some Nonlinear Systems

The nature of the stability problems is to study the influence of the interference to the motion state of the system. We discuss the stability and stable domains of the system by using Lyapunov. Lyapunov direct method is the main method of system stability research. While it does not solve the solution of the system, the system stability and its stability domain and its estimation are studied by constructing appropriate Lyapunov function. The practical stability is more realistic than Lyapunov stability to reflect the nature of the process of the system change. In recent years, many scholars began to study on the practical stability of the system, and made great achievements.In this paper, the φ0-stability of a class of nonlinear differential equations with initial time difference and the φ0-stability are studied by Lyapunov function and some the conditions of the φ0-stability of a class of nonlinear differential equations with initial time difference and the φ0-stability are given. Firstly, we study the solutions of the differential equation of φ0-stability and φ0-practical stability with variation in the initial time by comparison theorem, based on the existing φ0-stability theorems of the differential equations. Secondly, through the definition of Lyapunov function we gives the theorem about the φ0-stability of solutions of the differential equation. Thirdly, in this paper, the(h0,h) practical stability and (h0,h) practical instability of some nonlinear dynamical system are discussed by Lyapunov functions, mathematical theory and comparison theorem, some sufficient conditions of the (h0,h) practical stability and (h0,h) instability for dynamical system are given. The practical of the results are show by the examples.