一类内部点级联的PDE-ODE系统的边界控制

Boundary control for a partial differential equation-ordinary differential equation system cascaded at internal point

本文运用backstepping方法研究了一类偏微分方程与常微分方程(PDE-ODE)级联系统的能稳性.常见的级联系统在边界点x=0处级联,而本文所讨论的级联系统在内部点x0∈(0,1)处级联,级联点的改变使得新系统的控制问题更加复杂.针对新系统,首先,我们改进了backstepping方法中的常见变换,改进后的变换与常见变换相比,增加了变换中的核函数,且得到的是带有多个相容性条件的核方程组,给求解带来了困难.文中运用了一系列的技巧解出核函数,从而得到反馈控制器;其次,运用同样的方法找到改进变换的逆变换;最后,选择合适范数,利用变换的有界性证明得到闭环系统的稳定性.

Stabilization of a partial differential equation-ordinary differential equation （PDE-ODE） system cascaded at internal point is considered by boundary control. This system is more complicated than the conventional ones because the interconnection point is the internal point x0 ∈ （0, 1）. For this new system, a new backstepping transformation is introduced, which contains four kernel functions. Because the number of kernels is increased, the kernel equations with the compatibility conditions are more complicated. Fortunately, the kernel equations can be solved by a series of mathematical tricks to obtain their solutions. Then, the feedback controller is developed by using these kernel functions. The inverse transformation is derived by using the same procedure. Finally, we choose a proper norm and establish the stability of the closed-loop system by the boundedness of the transformation and the inverse transformation.