High Gain Limits of Trajectories and Attractors for a Boundary Controlled Viscous Burgers' Equation

In this paper we consider a boundary control problem for a forced Burgers' equation on a nite interval. The controls enter as gain parameters in the boundary conditions as in [7, 6] and the forcing term is allowed to be time dependent and square integrable in the spatial variable for all time. The uncontrolled problem is obtained by equating the control parameters to zero while the zero dynamics is obtained by constraining the output to be zero. The main result of the paper is that for H-smooth initial data the trajectories of the closed loop system (positive gains) converge uniformly in space and time, to the trajectories of the zero dynamics system as the feedback gains are increased to in nity. This result is similar to the property of asymptotic phase for lumped nonlinear systems. For forcing terms which are independent of time, we also establish the existence of a compact local attractor for the nonlinear semigroup. Moreover, as a consequence of the uniform convergence of the trajectories, we show that the attracting sets converge to the attractor for the forced zero dynamics, which in this case always consists of a single point.