Optimal Control and Vanishing Viscosity for the Burgers Equation

We revisit an optimization strategy recently introduced by the authors to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the one-dimensional (1-D) Burgers equation. This new descent strategy, called the alternating descent method, in the inviscid case, distinguishes and alternates descent directions that move the shock and those that perturb the profile of the solution away from it. In this chapter we analyze the optimization problem for the viscous version of the Burgers equation. We show that optimal controls of the viscous equation converge to those of the inviscid one as the viscosity parameter tends to zero and discuss how the alternating descent method can be adapted to this viscous frame. Optimal control for hyperbolic conservation laws is a difficult topic which requires a considerable analytical effort and is computationally expensive in practice. In the last years a number of methods have been proposed to reduce the computational cost and to render this type of problem affordable. In particular, recently, the authors have introduced the alternating descent method, which takes into account possible shock discontinuities. This chapter is devoted to revisit this method in the context of the viscous Burgers equation. We focus on the 1-D Burgers equation although most of our results extend to more general equations with convex fluxes. Most of the ideas we develop here, although they need further developments at a technical level, apply to multi-dimensional scenarios, too. To be more precise, given a finite time horizon T > 0, we consider the following inviscid Burgers equation: { ∂tu+ ∂x( 2 2 ) = 0, in R× (0, T ), u(x, 0) = u0(x), x ∈ R. (7.1)