Optimal Control of the Viscous Burgers Equation by the Hopf-Cole Transformation and Its Properties

This paper discusses an optimal control problem for the viscous Burgers equation using the Hopf-Cole transformation. Using this transformation, an optimal control problem for the Burgers equation is transformed into that for a linear heat equation. Although this transformation makes the boundary condition complicated, we use a state feedback to overcome the difficulty. Using the state feedback, an exact solution of the heat equation is obtained by Laplace transformation which enables us to obtain the optimal control input for the heat equation exactly. In fact, it is obtained by a solution of a well-known Fredholm integral equation. Furthermore, we discuss how to choose an achievable desired terminal state from the controllability point of view. In addition, we exhibit the effectiveness of the proposed approach through a numerical simulation. It verifies that when the solution of the heat equation at terminal time converges to its desired value, that of the corresponding viscous Burgers equation also converges to the desired one.