带干扰项的ARZ交通流模型的时滞控制与ISS稳定性  

The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances

作　　者：高彩霞[1] 赵东霞 Gao Caixia;Zhao Dongxia(School of Mathematics,North University of China,Taiyuan 030051)

机构地区：[1]中北大学数学学院,太原030051

出　　处：《数学物理学报（A辑）》2024年第4期960-977,共18页Acta Mathematica Scientia

基　　金：山西省基础研究计划(20210302123046);国家自然科学基金青年基金(12001343)。

摘　　要：针对线性化之后的ARZ交通流模型,已有文献通常基于如下假设:一是系统的平衡态恰好等于自由流的速度,二是进入上游路段的交通量恰好等于交通需求的数学期望,三是边界反馈不考虑时滞因素的影响.该文抛开各类限制条件,结合时滞边界控制策略,建立了具有模型漂移项和边界扰动项的PDE-PDE无穷维耦合闭环系统.具体地,采用算子半群理论,将闭环系统改写为抽象发展方程的形式;结合线性系统解与控制算子的允许性理论,证明了闭环系统的适定性;构造加权ISS-Lyapunov函数,证明了闭环系统的输入-状态稳定性(ISS),得到了反馈参数的耗散性条件.通过数值仿真实验,进一步验证本文设计的时滞控制器的有效性与参数条件的可行性.For the linearized ARZ traffic flow model,the existing literatures are usually based on the following assumptions:First,the equilibrium state of the system is exactly equal to the speed of free flow;Second,the trafic flow entering the upstream section is exactly equal to the mathematical expectation of traffic demand;Third,the boundary feedback doesn't consider the impact of time delay factors.In this paper,a PDE-PDE infinite-dimensional coupled closed-loop system with model drift term and boundary disturbance term is established without these constraints by combining the time-delay boundary control strategy.Specifically,the closed-loop system is transformed into an abstract evolution equation by using operator semigroup theory.The well-posedness of the closed-loop system is proved by combining the admissible theory of linear system solutions and control operators.The weighting ISS-Lyapunov function is constructed,and the input-to-state stability(ISS)of the closed-loop system is proved.The dissipative conditions of the feedback parameters are obtained.The effectiveness of the proposed controller and the feasibility of the parameter conditions are further verified by numerical simulation experiments.

关 键 词：ARZ交通流模型 时滞边界控制 ISS-Lyapunov函数 ISS稳定性 

分 类 号：O231.4[理学—运筹学与控制论]