Feedback Control for a Two-Dimensional Burgers' Equation System Model

In this paper, we consider the problem of controlling a system governed by a two-dimensional nonlinear partial differential equation. Motivation for the problem is the development of control methodologies for fluid flow, where the dynamics of the system are governed by the nonlinear Navier-Stokes equations. An initial boundary value problem described by the twodimensional Burgers’ equation is formulated to model a right-travelling shock over an obstacle. We focus on implementing feedback control via Dirichlet boundary conditions on the obstacle. We formulate a control problem for the system model, and examine two different methods of finding the control. The first method involves obtaining the solution of an algebraic Riccati equation. The second method involves obtaining a steady-state solution of the Chandrasekhar equations. Numerical approximations are developed to numerically simulate solutions of the problem with and without control. Numerical examples are presented to illustrate the efficacy, as well as the shortcomings, of the control method. Additionally, the influence of boundary condition on the functional gains, and the resulting controls, is demonstrated through numerical examples. Avenues of future work are presented.