一类空间非因果关系反应扩散方程的边界控制

Stabilization of a Spatially Non-Causal Reaction-Diffusion Equation by Boundary Control

运用Backstepping方法研究了一类空间非因果关系的反应扩散方程的稳定性问题,该系统的温度与整个空间有关系,而不是仅仅与空间中的某一点有关,因此,给控制器的设计带来了难度.为设计反馈控制器,首先,选用边界反馈控制的Backstepping方法进行设计;其次,在设计过程中增加核函数个数,引入改进后的Volterra型积分变换,再采用求解偏微分方程的变量分离法求解出核函数;最后,找到Volterra变换的逆变换,利用Volterra变换及其逆变换的有界性,可证明控制器的有效性.

Stabilization of a reaction-diffusion equation,in which the heat source depends on the temperature of the whole space,is considered by boundary control.The system is more complicated.For the new system,a new Volterra transformation is introduced,in which there are two kernels.Through a solution of separated variables,the exact solutions of kernels are obtained,and a control law is obtained specifically.The inverse transformation is derived,and stability of the closed loop system established via the boundedness of transformation and the inverse transformation.