Volterra型脉冲积分微分方程解的存在性和稳定性

Existence and Stability of Solution for a Class of Impulsive Volterra Integro-differential Equations

研究一类Volterra型脉冲积分微分方程的存在性、唯一性和稳定性问题。给出方程参数条件、相关定义和脉冲型Gronwall不等式的引理,利用函数序列的Weierstrass收敛定理,获得具有脉冲初始条件的Volterra型微分方程(不含脉冲项情形)解的存在性;在此基础上,利用迭代法和脉冲型Gronwall不等式,得到Volterra型脉冲积分微分方程解的存在性和唯一性;通过再次利用Gronwall不等式和分析技巧,获得该脉冲微分方程的指数稳定性的充分条件;举例说明即使连续的Volterra型微分方程不稳定,在一定脉冲扰动的情况下,系统可能变成指数稳定,这表明脉冲对微分方程稳定性的影响可能起决定作用。

This paper investigates the existence and uniqueness of solution to a class of impulsive Volterra integro-differential equations.Firstly,we give the parameters conditions,the related definition and impulsive-type Gronwall inequality impulsive Volterra integro-differential equations.Next,the existence of solution to the non-impulsive Volterra integro-differential equations with impulsive initial conditions is obtained by using Weierstrass criterion.Based on the result,we study the existence and uniqueness of the impulsive integro-dlfferential equations via iterative procedures and impulsive Grenwall inequality. Then, a sufficient condition ensuring to exponential stability is obtained in the impulsive Voherra integro-differential equations by utilizing the Gronwall inequality again and the analysis technique. Lastly, an example is given to illustrate that an impulsive systems may become exponentially stable even when the corresponding continuous systems is unstable, which shows the stability of the equation can be handled by impulsive effects.