Optimal Control for Burgers Equation using Particle Methods

Abstract: This papers shows the convergence of optimal control problems where the constraint function is discretised by a particle method. In particular, we investigate the viscous Burgers equation in the whole space $\mathbb R$ by using distributional particle approximations. The continuous optimisation problem is derived and investigated. Then, the discretisation of the state constraint and the resulting adjoint equation is performed and convergence rates are derived. Moreover, the existence of a converging subsequence of control functions, obtained by the discrete control problem, is shown. Finally, the derived rates are verified numerically.