Optimal and robust control of Burgers equation and heat equation with random forcing

Optimal and robust control of Burgers equation with disturbance are presented in this paper. The quadratic performance index is employed to find the optimal controller for Burgers equation. It is shown that the governing optimal controller for the system is determined by a Hamilton-Jacobi-Bellman (HJB) equation. This equation can be reduced to a partial differential Riccati equation by assuming that the solution of the closed loop Lyapunov function in the HJB equation is a non-negative integral function of time and the state variable with a costate vector. Due to this arrangement, the optimal controller can be found using numerical approach, so that the Burgers equation is controlled very well. The Isaac inequality is used for robust control of Burgers equation such that the penalty performance from the output to the disturbance is satisfied. By taking time derivative of the Isaac inequality, this inequality can be reduced to the HJB equation. After properly selecting the closed loop Lyapunov function, the robust controller is required to satisfy partial differential integral form of the Riccati equation instead of the HJB equation. This allows Burgers equation with random forcing to be stabilized by the robust controller. Finally a special case, feedback control of a heat transfer problem, is given to illustrate and support the theory for both optimal control and robust control.