分数阶脉冲微分方程研究

项目名称： 分数阶脉冲微分方程研究

项目编号： No.11201091

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 王锦荣

作者单位： 贵州大学

项目金额： 22万元

中文摘要： 分数阶导数为描述记忆性质和遗传过程提供了有力工具，分数阶脉冲微分方程适于描述脉冲现象与具有记忆性质过程和遗传过程相互交织影响的实际问题，广泛应用于粘弹性、电化学、控制、电磁学等领域。 我们将创新方法克服分数阶导数和脉冲扰动带来的本质困难，首先通过研究给出Caupto/Hadamard分数导数所决定的脉冲微分方程与脉冲发展方程片段连续解的合理定义，进而讨论分数阶脉冲Langevin方程边值问题和分数阶脉冲发展方程周期边值问题；之后引进局部稳定性、渐近周期解、Ulam稳定性和可稳化新概念，重点讨论解的存在性和局部稳定性、渐近周期解、Ulam稳定性和可稳化的充分条件。 本项目研究结果将为分析脉冲效应下双势阱和变化的磁场中带电粒子运动规律提供理论依据，也为讨论分数阶脉冲受控系统可控性和最优控制打下理论基础。

中文关键词： 分数阶导数；脉冲微分方程；存在性；稳定性；渐近周期解

英文摘要： Fractional derivatives provide an excellent tool for the description of memory properties and hereditary processes. As for problem of memory properties and hereditary processes mingles with impulse phenomena, it can be described by fractional order impulsive differential equations. Numerous applications can be found in viscoelasticity, electrochemistry, control, electromagnetic, and so forth. We will use novel methods to overcome some possible difficulties from fractional order derivative and impulsive perturbation. Firstly, we investigate and give a suitable defintion for piecewise continuous solutions of impulsive differential equations invovling Caupto/Hadamard fractional order derivative and we will discuss boundary value problems for fractional order impulsive Langevin equations and periodic boundary value problems for fractional evolution equations; Next, we will introduce local stability, asymptotically periodic solutions, Ulam stability and stabilizability and pay attention on sufficient conditions of local stability of the solutions, asymptotically periodic solutions, Ulam stability and stabilizability. The research results will not only provide a theoretical foundation for analysising on the movement law of charged particle in the double potential wells and fluctuating mangnetic field with impulsive

英文关键词： Fractional order derivative；Impulsive differential equations；Existence；Stability；Asymptotic periodic solution