中文摘要：

研究了二维圆盘上具有对称初始数据的反应扩散方程的边界控制.由于初始条件和边界条件关于圆心旋转对称,系统可以转化为等价的极坐标系下的一维抛物方程.此时,极点的奇异性成为了控制器设计中的难点.本文设计了一系列方程变换,消除了核函数方程中极点奇异性的影响,将其转化为修正的Bessel方程,求出了显式的核函数表达式和精确的边界反馈控制律,扩展了偏微分方程的backstepping方法.系统的收敛速度可通过改变控制器中的一个参数来调节.然后用Lyapunov函数法证明了闭环系统在H1范数下指数稳定,表明了系统对初值的连续依赖.最后用数值仿真验证了方法的有效性。

英文摘要：

This paper designs a boundary controller for a 2-D reaction-diffusion equation with symmetrical initial data. The system can be transformed to an equivalent 1-D parabolic equation in the polar coordinates due to the circle rotationally symmetric initial and boundary conditions. The singularity of the pole brings a great challenge to the design. This paper designs a series function transformations to destroy the impact of the singularity of the pole on the kernel function. Finally, the kernel function is transformed to a modified Bessel function, which produces an explicit kernel as well as an explicit boundary control law. The design process expands the existing backstepping methods for partial differential equations. Also, the convergence rate can be changed by adjusting a parameter in the controller.The exponential stability in H1 norm is proved, which implies that the system is continuous in terms of the initial conditions.The effectiveness of the method is illustrated with numerical simulations.