无限时滞脉冲泛函微分方程及其在经济中的应用

项目名称： 无限时滞脉冲泛函微分方程及其在经济中的应用

项目编号： No.11201038

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： 宋学力

作者单位： 长安大学

项目金额： 22万元

中文摘要： 本项目致力于无限时滞和脉冲泛函微分方程的定性分析及其在索洛经济增长模型中的应用研究。首先，利用非线性测度，给出此类方程解的指数稳定性准则。特别地，对于稳定的时滞方程，给出脉冲扰动继承稳定性的条件；对于不稳定的时滞方程，给出脉冲扰动使之变成稳定的条件。其次，应用获得的成果研究无限时滞脉冲索洛经济增长模型，结合Hopf分歧理论，利用利润分配指标、资本贬值率和时滞参数给出指数稳定条件、不稳定条件和Hopf分歧条件及稳定化控制策略。和Lyapunov-Razumikhi方法相比，我们的方法不需要仅为使用Razumikhi技巧而做一些假设，应用中也不需要构造Lyapunov函数，只需验证方程系数算子的非线性测度的符号。基于此，获得的稳定性准则不但应用范围更广便于实现，而且还可以给出收敛速率。对经济增长模型的研究是本项目的另一个特色，将会为宏观经决策提供一定的理论参考，具有一定的现实意义。

中文关键词： 指数稳定性；相对非线性测度；非线性测度；脉冲泛函微分方程；时滞

英文摘要： The proposal is devoted to investigating qualitative analysis of impulsive functional differential equations with infinite delays and its applications to Solow growth model. Firstly, exponential stability criteria are given for this kind of equations by nonlinear measure method. Especially, we present the conditions where the exponential stability of the delayed equations will persist under impulsive perturbation and where impulse perturbation can make an unstable delayed equations become stable. Secondly, the dereived results are applied to Solow growth model with impulse and infinite delays. Based on Hopf bifurcation theory, the conditions of exponential stability, unstability and Hopf bifurcations and stabilization control strategies are presented for the model by means of profit share, the rate of capital depreciation and delayed index. Compared with Lyapunov-Razumikhi method, our method need not make some assumptions only for using Razumikhi techniques and need not construct Lyapunov function in applications, but only verify the sign of nonlinear measure of coefficient operators of the equations. Hence, our method is applied conveniently to more general problems and provide converence rate of solutions. Applicatons to Solow growth model is the other characteristic of the proposal, which will provide a ne

英文关键词： Exponential stability；Relative nonlinear measure；Nonlinear measure；Impulsive functional differential equation；Delay