All-at-once method for the optimal control of Burgers equation

Optimal control problems (OCP) for Burgers equation are important for the development of numerical methods for optimal control of more complicated models in fluid dynamics such as Navier-Stokes equations. In this work, we analyze variant preconditioners for the saddle point problem arising from boundary control of unsteady Burgers equation. As for a solution approach for the discretization of the OCPs, we use discretize-then-optimize. In the discretize-then-optimize approach the state equation is discretized and then the optimality system for the finite dimensional optimization problem is derived. This approach is also referred to as the black-box approach. In other words, an existing algorithm for the solution of the state equation is embedded into an optimization loop. The black-box approach is easy to use because it requires no modification to an existing partial differential equation (PDE) integrator. As in the case of Burgers equation, the repeated costly solution of the state equation is needed. Treating the control and state as independent of optimization variables the discrete optimality conditions yield a system