一类2×2变系数双曲系统的PDP边界控制

PDP Boundary Control for a Class of 2×2 Hyperbolic Systems with Variable Coefficients

不同于比例反馈或PI反馈,文章提出将位置反馈和时滞位置反馈的线性组合设计控制器去镇定一类2×2变系数双曲守恒律系统.首先,考虑到时滞项可由一阶运输方程初值问题的解进行刻画,将系统转化为由四个偏微分方程构成的PDE-PDE无穷维串级耦合闭环系统的形式,并进一步改写为抽象发展方程的形式.其次,利用算子半群理论证明系统的适定性.第三,建立了系统算子的特征方程,这是一个包含三个指数项的超越方程,根据指数型多项式零点分布定理得到了与时滞无关的稳定性与不稳定性结论.第四,通过构造恰当的加权Lyapunov函数分析闭环系统的指数稳定性,建立了与时滞相关的参数耗散条件.最后,通过数值例子验证时滞控制器的有效性和所取参数的可行性.

evolution equation.Secondly,the well-posedness of the system is proven by using operator semigroup theory.Thirdly,the characteristic equation of the system operator,which is a transcendental equation with three exponential terms,is established,and the delay-independent stability and instability conclusions are obtained according to the theorem of the distribution of the zeros of exponential polynomials.Fourthly,the exponential stability of the closed-loop system is analyzed by constructing an appropriate weighted Lyapunov function,and the parameter dissipation condition related to the time delay value is established.Finally,the effectiveness of the time-delayed controller and the feasibility of the parameters are verified by numerical examples.