Boundary Control Of Reaction-diffusion Equation

Stability of temperature is a requirement in the engineering problem such as chemical reaction or biological fermentation. Generally, the temperature is modeled by the reaction-diffusion equation. In this paper, we focus on boundary control problems of reaction-diffusion equation. We mainly study two types of reaction diffusion equation.Controllers is designed through the backstepping method and we obtain the Stabilization of the closed loop system.In this paper, based on the Volterra transformation a new backstepping trans-formation is introduced. The new transformation contains both Volterra integrals and Fredholm integrals, in which there are two kernels. Since the number of kernels is increased the kernel equations and computation are more complicated, especially, it is a challenge to solve the kernel equations. Fortunately, through a series of mathematical tricks, the exact solutions of kernels are obtained. A control law is obtained specifically via these kernels. To establish the stability of the closed loop, the inverse transformation is derived by the same procedure, and the stability of the closed loop system established through the boundedness of transformation and the inverse transformation. Simulation results show that the closed loop system is stable.