带有黏性阻尼的波动方程的一致指数镇定和状态重构

Uniform exponential stabilization and the state reconstruction of the wave equation with viscosity

通过有限时间的输出观测信息,以向前-向后观测器为基础的时间反转法可以用于无穷维能量守恒系统的状态重构.本文利用Lyapunov函数法将上述结果推广到具有一般黏性和边界观测的一维波动方程,给出上述方法的数值逼近格式和收敛于初值的迭代序列.对于任何初始猜想值,迭代序列都强收敛到初值,进而实现状态重构的数值逼近.其中在状态重构的数值逼近的时间反转法中,离散系统和向前观测系统的误差系统的一致指数稳定性(关于离散步长)具有重要作用,因而先讨论相关系统的一致指数稳定性.本文通过将port-Hamilton理论和有限差分格式相结合,得到一种降阶型差分格式,利用与连续系统同样的验证方法得到离散系统的一致指数稳定性.

In this paper,through constructing a sequence of forward and backward observers,We reconstruct the state of a string system with viscosity from the measured output over a finite interval by means of the time reversal focusing approach.By the theory of the port-Hamiltonian system,we introduce a direct Lyapunov approach to the problem and extend the results of time reversal focusing to the wave equation with general viscosity and the boundary observation from a finite interval.Moreover,the approximating scheme of the method of time reversal focusing and the iterative sequence of the initial value are also presented.For any given guessed value,the iterative sequence converges strongly to the initial value and the numerical approximation of the state reconstruction is given.However,because the uniform exponential decay of the errors between the original continuous system and its forward observers plays an important role in applying the time reversal focusing approach to the discrete scheme,the uniform exponential stability of the involved system is firstly discussed.Combining the theory of port-Hamiltonian systems and the finite difference,we give the semi-discretization scheme of the type of finite difference and order reduction,and the uniform exponential stability of the semi-discretization systems is verified by the method parallel to that for the continuous system.