带有局部干扰的Euler-Bernoulli梁方程的稳定性分析

摘要

In order to enrich the system stability theory of the control theories,taking Euler-Bernoulli beam equation as the research subject,the stability of Euler-Bernoulli beam equation with locally distributed disturbance is studied.A feedback con-troller based on output is designed to reduce the effects of the disturbances.The well-posedness of the nonlinear closed-loop system is investigated by the theory of maximal monotone operator,namely the existence and uniqueness of solutions for the closed-loop system.An appropriate state space is established,an appropriate inner product is defined,and a non-linear operator satisfying this state space is defined.Then,the system is transformed into the form of evolution equation.Based on this,the existence and uniqueness of solutions for the closed-loop system are proved.The asymptotic stability of the system is studied by constructing an appropriate Lyapunov function,which proves the asymptotic stability of the closed-loop system.The result shows that designing proper anti-interference controller is the foundation of investigating the system stability,and the research of the stability of Euler-bernoulli beam equation with locally distributed disturbance can prove the asymptotic stability of the system.This method can be extended to study the other equations such as wave equation,Timoshenko beam equation,Schrod-inger equation,etc.%为了丰富控制理论中关于系统稳定性问题的理论,以Euler-Bernoulli梁方程为研究对象,研究了带有局部干扰的Euler-Bernoulli梁方程的稳定性问题.设计了一个基于输出的反馈控制器用于抑制干扰产生的影响,采用极大单调算子理论证明非线性闭环系统的适定性,即证明闭环系统的解的存在性与唯一性.设立适当的状态空间,定义适当的内积,进一步定义了符合此状态空间的非线性算子,将系统转化为抽象发展方程的形式,在此基础上,证明了闭环系统的解的存在性与唯一性.通过构造合适的Lyapunov函数,对闭环系统的稳定性问题进行研究,证明了闭环系统的渐近稳定性.结果表明,设计出合适的抗干扰控制器是研究系统稳定性的基础,研究带有局部干扰的Euler-Bernoulli梁方程的稳定性能够证明系统是具有渐进稳定性的,此方法可以推广到对诸如波方程、Timoshenko梁方程、薛定谔方程等系统的研究.