Stationary solutions for a boundary controlled Burgers' equation

This paper is concerned with the stationary solutions of a one-parameter fa mily of boundary control problems for a forced Viscous Burgers' equation. W e assume that the forcing term possesses a special symmetry that greatly ai ds in our analysis. The parameter characterizing the family enters as a sca lar gain in a proportional error boundary feedback control scheme. We show that as the gain varies from zero to infinity, the stationary solutions und ergo an interesting bifurcation. Namely, when the gain is zero, there are i nfinitely many stationary solutions, the one-dimensional subspace of all co nstants. When the gain is positive, the constants are no longer solutions. For small positive values of the gain, there are three distinct nonconstant stationary solutions, and for sufficiently large values of the gain there is a single, asymptotically stable equilibrium. (C) 2001 Elsevier Science L td. All rights reserved.