基于一类积分不等式的非线性微分系统的一致Lipschitz稳定性

Uniform Lipschitz stability of nonlinear differential systems based on an integral inequality

为了判定非线性微分系统零解的Lipschitz稳定性,建立了一种指数均大于1的一类新型积分不等式。这类积分不等式与以往的积分不等式相比,形式上更广泛,可以处理更复杂的非线性微分系统。在新不等式的基础上,根据Lips-chitz稳定性理论,分析了一类非线性微分系统零解的一致Lipschitz稳定、一致Lipschitz渐近稳定,推广的指数渐近稳定等特性。通过上述积分不等式的运用,得到判定一类非线性微分系统Lipschitz稳定性的充分条件,且将相关性质推广到线性受扰系统。通过对一非线性微分系统零解的Lips-chitz稳定性的判别,说明了上述结论的实用性。

The integral inequality is an important method in Lipschitz stability theory of nonlinear differential systems. This paper presented a new integral inequality which effectively analyzes integral inequalities whose variable different moduli are greater than 1. Compared with previous integral inequalities, the new integral inequality can analyze more complicated differential equations. The new integral inequality was used to establish sufficient conditions for uniform Lipschitz stability, uniform Lipschitz asymptotic stability and generalized exponential stability of nonlinear differential systems. These conditions were generalized to linear perturbed differential systems. The uniform Lipschitz asymptotic stability of the origin of a system of nonlinear differential equations was illustrated using the results. The results demonstrate that the newly built integral inequality is more practical than previous inequalities.