Based on the common disturbed motion equation of projectiles, Liapunov functionV(X)=XH eX in plural number mode is constituted with the help of Hermite matrix H e and Liapunov s direct method.

Furthermore,according to the method of Liapunov functions in the stability theory,both the globally asymptotic stability and the attractive regions of the nonnegative equilibrium in the model are discussed.

It is discussed that the methods of discrimination of the stability of the singular point of nonlinear autonomous system by using of eigenvalue,center manifold and Liapunov function.

Judgement on the existence of a common quadratic Lyapunov function for a kind of systems;

This paper uses a piecewise linear Lyapunov function and derives a new estimate for solutions of linear time-varying systems.

After Lyapunov function is de- rived,with theoretical analysis,energy-based control design method is discussed in order to solve the global stability problem.

Exponential stability for a class of stochastic differential equations with time-varying delay is analyzed by constructing suitable Lyapunov function and using the nonnegative semi-martingale convergence theorem.