Stabilization Of An ODE-PDE Coupled System

This work studies the boundary control problem of a class of coupled systems of ordinary differential equations and partial differential equations.The system includes items such as reaction,convection,and diffusion.The boundary has heat exchange and conforms to Fourier's law.In industrial production,the coupled system represents issues such as biological fermentation and chemical reactions.Therefore,the research on the system has application value.For a coupled linear reaction diffusion system with a heat source inside the system,this paper introduces the Backstepping transformation to transform the original system into a selected exponentially stable target system.The vector-valued kernel function of the Backstepping transformation satisfies a coupled ODE-PDE equation set.In order to obtain To transform the kernel function,first convert the kernel equation into an integral equation,and then use some mathematical calculation methods to decouple this equation set,and then use mathematical induction and successive approximation methods to solve the vector-valued kernel function equation,and then find Get the specific analytical formula of the control law.By establishing the Backstepping inverse transformation and using the boundedness of the inverse transformation,the exponential stability of the closed-loop system under this control law is proved.Compared with the existing results,the system studied in this paper contains a heat source inside,and there is heat exchange at the system boundary,the system is more extensive,closer to the actual situation of the industry,and has certain practical significance.