Stationary Solutions for a boundary controlled Burgersʹ equation

This paper is concerned with the stationary solutions of a one-parameter family of boundary control problems for a forced viscous Burgersʹ equation. We assume that the forcing term possesses a special symmetry that greatly aids in our analysis. The parameter characterizing the family enters as a scalar gain in a proportional error boundary feedback control scheme. We show that as the gain varies from zero to infinity, the stationary solutions undergo an interesting bifurcation. Namely, when the gain is zero, there are infinitely many stationary solutions, the one-dimensional subspace of all constants. 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.