Boundary Control Of Reaction-diffusion Equations On Symmetric Domains

The boundary control problem of reaction-diffusion equations in high-dimensional space has been a difficult problem in the field of distributed parameter system con-trol.For the general high-dimensional space region,there is no result of the control law at present.In this paper,Boundary control design of reaction-diffusion equations on two classes of higher dimensional domains,2-dimensional x-axis symmetric,and 3-dimensional z-axis symmetric domains are considered.The exponential stability of the closed-loop system with state feedback is provedIn this paper,the symmetric backstepping method for partial differential e-quations is introduced,and the original system is transformed into the selected exponentially stable target system by using the symmetric backstepping transfor-mation.The boundary control law is established.The kernel function is derived.and the solution of the kernel function is solved,and the analytic control law is ob-tained.Secondly,the target system is transformed into the original control system by inverse transformation.Through the exponential stability of the target system and the boundedness of the transformation and its inverse transformation,the ex-ponential stability of the closed-loop system controlled by boundary control law is established.In this paper,the boundary control input of the reaction-diffusion equation on two classes of higher dimensional domains,2-dimensional x-axis symmetric,and 3-dimensional z-axis symmetric domains are obtained,and the obtained results are applied to the boundary control of the reaction-diffusion equation in the circular domain and the spherical domain,respectively.The results obtained are different from those obtained through the establishment of ultra-spherical coordinates and spherical coordinate transformation,and have more application value in engineering technology.The results of numerical simulation verify the effectiveness of the designed con-troller.