Boundary Control Of Diffusion Reaction Equation In Different Spatial Domains

In physics,chemistry,biology and other engineering fields exist the diffusion reaction phenomena,diffusion reaction equation(also called heat equation)is a mathematical model which describes the diffusion reaction problems in these engineering fields.The boundary control problem of diffusion reaction equation has a wide application in physics,biology and engineering practice.such as the stability of temperature control,wet spinning carbon fiber forming process and chemical reaction process,Therefore has attracted the attention of researchers.The main goal about boundary control of diffusion reaction equation is to give the boundary control law,so that the system under different spatial domain can eventually be stabilized to the equilibrium state.In this paper,we mainly study the boundary control problem of diffusion equation in three different kinds of space,and establish the corresponding PDE model for the specific reaction diffusion system.Based on Backstepping control algorithm,the reversible transform into Volterra integral mapping,establish equivalence between the original system and a stable target system,the original system is transformed into an exponentially stable target system,using equivalence,nuclear function equation,using the link between boundary controller and kernel function,obtaining the accurate solution boundary controller.finally,based on the Lyapunov stability theory,the stability of system is proved.Firstly,studying the boundary control problem of reaction diffusion equations in n-dimensional spherically symmetric space domain,because many practical systems can use n-dimensional hyperspherical coordinates to describe and the system which is spherical symmetry,so the state changes through the research in the radius direction,get the global dynamic process of the system through the symmetric high dimensional system transformation the radial equation of one-dimensional equivalent,Backstepping method is used to design the boundary controller,obtaining the boundary state value that is easily measured,a state space observer was designed to estimate the system in all state space domain,so as to achieve the output feedback control.At the same time,using the Lyapunov method,it is proved that the closed-loop system with feedback control has exponential stability under the H1 norm.Secondly,this paper also studies the boundary control problem of two dimensional diffusion reaction equation in the cylindrical space.The bottom surface of the cylinder is a circular region,and the boundary of the cylinder is the surface area.The state variables on the center rotational symmetry,using the Fourier series expansion method,the two-dimensional system can be transformed into one-dimensional parabolic equation equivalent,which reduce the dimension of the system.The target system is stable,the introduction of Volterra integral reversible mapping,establish the equivalent links between the original system and the target system,using the equivalence of the original system and the target system,we can obtain the kernel functions,finally using the relationship between boundary controller and kernel function,obtaining the accurate solution boundary controller.In this paper,we design a boundary observer to estimate all the states of the whole system in the space of the cylinder by measuring boundary differential values.Finally,researching the boundary control problem of boundary control input time-delay diffusion reaction equation system,especially for time-delay system response coefficient depends on the space variable,by introducing the transport equation and diffusion reaction equation system that constitute the cascade system,the time delay is transferred to the transport equation of the system in the original system,Using the equivalence relation to obtain three kernel function equations,the solution of each kernel function is solved by using the separation variable method,and the solution of the boundary controller is obtained by the relation between the boundary controller and the kernel function.