Studies Of Boundary Control Of A Kind Nonlinear Pde

Boundary control is one kind of distributed parameter controls, which has been emphasized in the Control Theory and has been extensively studied and developed. Recently people more and more take notice on the boundary control of Burgers equation, KdV equation, KdVB equation, and K-S equation. In this paper we mainly study the boundary control of the sufficiently nonlinear Korteweg-de Vries-Burgers equation and Kuramoto-Sivashinsky equation with an external excitation. In chapter 3 we consider the problem of stabilization by boundary feedback conditions for the sufficiently nonlinear Korteweg-de Vries-Burgers equation on the domain [0,1] .We use a control law of the formu(o, t) = u_x(1, t)=0,u_xx(1, t)=k₁u(1, t)^2m+1 + k₂u(1, t)to analysis the problem of global boundary stabilization .we mainly prove it exists an unique solution, and show it guarantees L² -global exponentialstability, H³-global asymptotic stability, and H³-semi global exponentialstability. In chapter 4 we mainly study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on afinite domain, firstly shows that under the given boundary feedback conditions it admits an unique solution and the solution is stable. Secondly it proves that if the external excitation f is time periodic function, then the system under theboundary conditions admits a unique time periodic solution and its period is the same as the f 's, and show the time periodic solution is the global attractor ofthe space L² .