中立型分布延时微分方程可计算的稳定性判据

中立型分布延时微分方程广泛应用于各工程领域，对方程解的稳定性分析有重要意义。本项目将首先改进线性中立型分布延时微分方程相应的差分算子满足强稳定性的充分条件,扩展传统研究中的中立型方程的范围，使更广泛的一类中立型微分方程成为研究对象。在复变函数及矩阵理论的基础上，获得中立型方程理论解延时相关渐近稳定的可计算的充要条件，以降低以往结论中关于解渐近稳定性的充分条件的保守性。基于此稳定性判断，首次从几何角度确定出右半复平面内包含特征方程所有不稳定根的一个有界闭区域，并理论推导出特征方程不稳定根的个数，从而在包含所有不稳定特征根的有界闭区域内，设计有效的数值搜索算法求取满足精度要求的不稳定特征根。本项目的研究是对中立型分布延时微分方程基础理论研究的发展。

Neutral differential equations with distributed delays are widely applied in various engineering fields, so stability analysis for solutions of equations is of great importance. This project will first improve the sufficient condition under which the difference operator associated with the linear neutral differential equation with distributed delays is strongly stable, so as to expand the range of traditionally studied neutral equations. And thus a wider calss of neutral equations will be concerned. By employing complex variables and matrix theories, obtain computable sufficient and necessary conditions for asymptotically delay-dependent stability of analytical solutions of the neutral differential equations with distributed delays, and thus reduce the conservatism generated by earlier conclusions about sufficient conditions for stability of analytical solutions. Based on the judgement of stability, for the first time, accurately describe the bounded region that includes all unstable roots of the characteristic equation on the right-half complex plane from the viewpoint of geometry, and theoretically derive the number of all unstable characteristic roots. Furthermore, in the bounded region which contains all unstable characteristic roots, design efficient numerical algorithm to search for roots with required precision.This project is a development of the basic theoretical research for the neutral differential equations with distributed delays.