基于积分型Lyapunov泛函的积分时滞系统的稳定性研究

Integral delay systems (IDSs) are dynamics systems described by a class of functional (algebraic) equations and are closely related with the model transformation approaches and (pseudo) predictor feedback approaches for normal time-delay systems, and the stability of neutral time-delay systems. Hence IDSs have very important value in theory. Since the state equations of IDSs do not contain the term of the time-derivative of the state vector, the corresponding Lyapunov functional must contain only integral functions of the state. By taking this essential property of IDSs into consideration, this project will build a general stability theory for IDSs by utilizing the so-called integral Lyapunov functionals. After building a general Lyapunov stability theory for IDSs with general nonlinear integral kernels，by using different integral inequalities, matrix inequalities, the delay partition technique, different linearization techniques, and the positive operator theory based analysis approach developed by the investigator recently, this project will study extensively the stability analysis and stabilization of general linear IDSs with both time-varying and time-invariant integral kernels. Finally, the developed stability analysis and stabilization approaches for IDSs will be utilized to solve several problems for normal time-delay systems that are closely related with IDSs. The work to be done can not only complete the stability theory of time-delay systems by providing new ideas for studying other theoretic problems, but can also offer certain technical supports and theoretical guarantee for solving some other practical engineering problems.

积分时滞系统（IDS）是一类用泛函（代数）方程描述的动态系统，它与常规时滞系统的模型转换方法、（伪）预估器反馈方法以及中立型时滞系统的稳定性等问题密切相关，具有重要的理论价值。由于IDS的状态方程不含状态的导数，相应的Lyapunov泛函只能是状态的积分函数，本项目通过充分考虑IDS的这一根本特征，建立基于积分型Lyapunov泛函的稳定性理论。在建立具有一般非线性积分核的IDS的Lyapunov稳定性理论的基础上，综合运用各种积分不等式、矩阵不等式、时滞分割方法、线性化方法以及负责人在前期研究工作中建立的基于正算子理论的分析方法，研究具有时变和定常积分核的线性IDS的稳定性理论和镇定方法，最后将所建立的稳定性分析和镇定方法用于研究与IDS相关的几类常规时滞系统的控制问题。这些即将展开的研究工作不但能够丰富时滞系统的稳定性理论，而且也为解决一些实际的工程问题提供一定的技术支持和理论保证。

积分时滞系统由于与常规时滞系统的模型转换方法、（伪）预估器反馈方法以及中立型时滞系统的稳定性等问题密切相关，具有重要的理论与应用价值。本项目主要研究了一般形式的积分时滞系统的稳定性问题。首先，针对一类具有多重时滞的积分时滞系统的稳定性问题，通过构造新颖的Lyapunov泛函提出了基于线性矩阵不等式的充分条件，进而利用正算子理论，建立了基于谱半径的充分条件，该条件在后续的研究工作中发挥了重要的作用。其次，针对一类具有多重指数积分核的积分时滞系统的稳定性问题，建立了基于耦合线性矩阵不等式的充分条件；此外，对该类系统的几种特殊情形，分别建立了基于特征方程的充分必要条件，并研究了参数存在不确定性时的鲁棒稳定性问题。第三，针对带有多重时滞的线性连续时间时滞差分系统的稳定性及鲁棒稳定性问题，通过分析多重点时滞之间的关系，构造了完整Lyapunov泛函，建立了时滞独立的稳定性条件。最后，利用所建立的积分时滞系统的稳定性分析方法，研究了几类随机中立型时滞系统的稳定性问题，通过构造遍历全部可能且包含项数最少的二次积分型泛函，建立了保证该类系统均方稳定的充分条件。本项目的研究成果一方面丰富了微分方程，特别是积分时滞系统稳定性的理论成果，另一方面为一些相关的实际工程问题提供了一定的理论保证和技术支持。