Boundary Control Of A Kind Nonlinear Evolution Equation

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, and K-S equation.Firstly, This paper Studies the problem of exponential stabilization by boundary control for the general K-S equation on the domain [0,1]. The global existence and uniqueness of the solutions with the help of the Banach fixed point theory and the theory of operator semi-group are verified. Using some usual inequalities and intergration by parts, we derive a control law of the and prove that it guarantees L² -global exponential stability.Secondly, we studies a type of important nonlinear evolution equation: KdV-MKdV-Burgers equation and its backstepping boundary control. With elaborate Lyapunov analysis; the closed-loop system, including the boundary dynamics is H³ global stable and L², H¹, H³ stability estimate of the system under boundary control are also proved; The equilibrium solution acquires global asymptotic stability and exponential stability on L₂[0,1] and Neumann boundary control condition. For control inputs under the boundary control bounded in L^∞, the equilibrium solution decay to zero by using nonlinear boundary control condition and input feedback control method.