边界条件含时滞的连续系统的稳定性问题

There has been an increasing interest in delayed dynamic systems due to the ubiquitous existence of delay effects in nature and engineering systems. For some specific engineering applications, partial delay differential equations with delayed boundary conditions are requied for accurate analysis of corresponding dynamic behaviors of the systems. However, not much efforts were devoted in this respect as a result of the complexity of these systems. Therefore, the project aims at presenting a systematic exact study about the stability of partial delay differential equations with delayed boundary conditions. By introducing the travelling wave solution, the continuous systems are transformed in the form of neutral delay differential equations. The exact stability analysis will be carried out for neutral delay differential equations with multiple constant delays and also for the case when state-dependent delays are involved. The theoretical stability results will then be applied for some specific engineering problems (an electroacoustic system, a real time substructuring test system, flight process of a rocket and a drilling system). The results will be compared and validated with the experiments (an electroacoustic experiment, a real time substructuring test and an elastic beam setup with follower force) and numerical computations. The exact stability results can served as a guide for the development of related control theory and as a test for the limits of some current numerical dynamic softwares.

时滞现象广泛存在于自然界和工程系统，考虑时滞因素的动力学研究受到广泛关注。在时滞动力系统中，有一类系统需用边界条件含时滞的偏微分方程描述。这类系统有广泛的工程背景，可较准确地描述一些工程技术领域。然而，由于该类系统的复杂性，目前相关研究甚少。本项目旨在对这类边界条件含时滞的连续系统的稳定性展开准确系统性分析。通过行波解将系统转化为中立型时滞微分方程，着重对含多个常时滞以及状态时滞的中立型时滞微分方程的稳定性进行分析，将准确理论研究结果应用于具体工程问题（电声学系统、实时子结构动态测试系统、火箭飞行过程、钻井系统），并通过数值计算和实验（电声学系统、实时子结构动态测试系统以及随从力作用下的弹性梁系统）对比相关结果。该准确稳定性分析结果将有助于指导相应控制理论的发展，并可应用于验证数值分析结果，特别是一些通过数值商业软件所得结果的正确性。

时滞现象广泛存在于自然界和工程系统，考虑时滞因素的动力学研究受到广泛关注。在时滞动力系统中，有一类系统需用边界条件含时滞的偏微分方程描述。本项目针对这类系统（电声学系统、车削加工系统），通过行波解将系统转化为多时滞中立型时滞微分方程，着重针对中立型时滞微分方程的稳定性及非线性动力学开展分析，并通过数值计算对比相关结果。准确稳定性分析结果将有助于指导相应控制理论的发展，并可应用于验证数值分析结果，特别是一些通过数值商业软件所得结果正确性。针对研究过程中发现的一类隐式中立型时滞微分方程，开展了稳定性及非线性动力学分析，加深了对该类系统的理解。