反阻尼一维波动方程的稳定性

Stabilization of a 1-D wave equation with anti-damping

文章利用边界控制法研究了一类反阻尼一维波动方程的稳定性.首先,通过边界控制的backstepping方法,引入包含有两个核函数的积分变换,将控制系统转化为稳定的目标系统.核函数个数的增加导致核函数方程更加复杂,文中运用了一系列的数学计算技巧求解出核函数,从而得到反馈控制器;其次,运用类似的方法找到变换的逆变换;最后,选择合适范数,利用变换及逆变换的有界性证明得到闭环系统的稳定性.

This paper is to stabilize a 1-D wave equation with an anti-damping by boundary control. To use the backstepping method of boundary control, a new transformation of two kernel functions is introduced. The equations of kernel functions are more complicated mathematically. By some mathematical skill, solutions of the kernel equations are constructed. Finally, the inverse transformation is attained. Through boundedness of the transformation and its inverse,stability of the closed-loop system is established.