Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high‐level modelling and simulation package COMSOL Multiphysics. Using the first‐order optimality conditions, projection and semi‐smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space–time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole–Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.