Optimal Control for Burgers Equation using Particle Methods

TL;DR

This paper demonstrates the convergence of particle method discretizations for optimal control problems constrained by the viscous Burgers equation, providing theoretical analysis and numerical verification of convergence rates.

Contribution

It introduces a particle-based discretization approach for the Burgers equation in optimal control, establishing convergence results and rates with rigorous proofs.

Findings

Convergence of particle discretization for the control problem

Derived convergence rates for the discretized solutions

Numerical verification confirms theoretical results

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.