具有动态边界阻尼的波方程的降阶型差分半离散化的一致指数稳定性

The uniform exponential stability of wave equation with dynamical boundary damping discretized by the order reduction finite difference

由于在最优控制的数值计算和可观性的反问题中,离散系统关于离散化参数的一致指数稳定性具有重要作用,因而一致指数稳定性得到广泛而深入的研究.众所周知,对指数稳定的波方程的空间变量用经典的有限差分或有限元离散化后,离散格式产生高频病态伪特征模,致使离散系统不是一致指数稳定的.为了恢复一致指数稳定性,学者们引入了添加数值粘性项法和滤波法等方法.然而对于具有动态边界条件的波方程的一致指数稳定性问题研究的较少.本文用降阶型差分方法对此问题进行研究,先对具有动态边界条件的波方程进行降阶处理,然后利用有限差分对其进行空间半离散化,不用再对其进行任何处理,引入合适的李雅普诺夫函数即可验证离散系统是一致指数稳定的.

Because the uniform exponential stabilities with respect to the discretized parameter play key roles in the computing of optimal control and the inverse problem of observability,they were broadly and intensively discussed.It is well known that,for the continuous wave equation,it is exponentially stable.If the continuous system is discretized in spacial variable by finite difference method,the numerical scheme yields spurious high frequency oscillations which induce the deficiency of the uniform exponential stability.To restore the uniform qualitative behaviors,researchers introduced the methods of vanishing viscosity terms and filtering.However,there are rare results on the uniform exponential stability of the wave equation with dynamical boundary condition.In this note,we shall apply finite difference approach of order reduction to study this question.That is to say,we reduce the order of the wave equation and then discretized spacial variable by finite difference method,the uniform exponential stability is tested by introducing suitable Lyapunov function and without any remedy.