Construction Of Lyapunov Function And Stability Analysis Of A Class Of Fourth-order Nonlinear System

Construction of Lyapunov function and stability analysis of a class of fourth-order nonlinearsystemIn 1892, Lyapunov. a Russian mathematician,first gave the exact mathematic definition of movement stability and general method in his doctor's paper. It was founded that the second Lyapunov's method (Lyapunov's direct method for stability. The method effected the development of theories for differential equation greatly. Since 1950's, the theories of stability has been widely developed.The key to studying the stability of differential equation is to construct Lyapunov function. In 1972, S.Kasprzyk studied global stability of the following third-order nonlinear systems.(x|…)+a(x|¨)+b(x|·)+fx=0, f(0)=0(x|…)+a(x|¨)+f(x|·)+cx=0, f(0)=0 (x|…)+f(x|¨)+b(x|·)+cx=0, f(0)=0In1976, R.Reising improved S.Kasprzyk result. Hereafter, some scholar solved S.Kasprzyk'sproblem through an analogy. (some Lyapunov functions of thirdorder constant coefficient linear system are calculated with Barbashin's formula(x|…)+a(x|¨)+b(x|·)+cx=0Then Lyapunov function of third-order nonlinear system are analysed). This paper contains three chapters.In chapter 1, the basic concepts and the theorems of Lyapunov's direct method for stability are introduced. The theorem of Lyapunov stability is one of nucleus theorems.Theorem 1 For n-dimension nonautonomous systemdx/dt=f(t,x) (1)where x=(x₁,x₂,…,x_n)^T,f=(x₁,x₂,…,x_n)^T: C[I×Rⁿ,Rⁿ],and f(t,0)≡0,Let G_H={(t, x) :t≥t₀,‖x‖1=(?)+sum from i=1 to n(?)f_i(t,x)≤0then the zero solution of system (1)is stable.In chapter 2, we base on Barbashin's formula, and constant some Lyapunov functions of constant coeflicient linear equation.x⁽⁴⁾+a(x|…)+b(x|¨)+c(x|·)+dx=0There are primary conclusions of this paper in chapter 3.Stability condition of zero solution for three fourth-order nonlinear equations.x⁽⁴⁾+a(x|…)+b(x|¨)+c(x|·)+fx=0,f(0)=0 (2)x⁽⁴⁾+a(x|…)+b(x|¨)+f(x|·)+dx=0,f(0)=0 (3)x⁽⁴⁾+a(x|…)+f(x|¨)+c(x|·)+dx=0,f(0)=0 (4)Theorem 2 Assume a>0,b>0,c>0.and equation (2)satisfies three conditions:(i) f"(x)is continuos and bounded. (ii) xf(x)>0,when x≠0(iii) abc-c²-a²f'(x)≥ε>0 then the zero solution of equation (2)is stable.Theorem 3 Assume a>0,b>0,d>0, and equation (3)satisfies two conditions:(i) f"(x)is continuos and bounded,(ii) abf'(x)-(?)(x)-a²d≥ε>0 then the zero solution of equation (3)is stable.Theorem 4 Assume a>0,c>0,d>0.and equation (A)satisfies two conditions:(i) f"(x)is continuos and bounded,(ii) af'(x)c-c²-a²d≥ε> 0 then the zero solution of equation (4)is stable.