Optimal Control of the Keilson-Storer Master Equation in a Monte Carlo Framework

Abstract

This paper is devoted to the formulation and numerical solution by Monte Carlo (MC) methods of an optimal control problem governed by the linear space-homogeneous Keilson-Storer (KS) master equation. The KS master equation is a representative model of the class of linear Boltzmann equations with many applications ranging from spectroscopy to transport theory. The purpose of the optimal control in the collision kernel of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. For this purpose, a KS optimality system characterizing the solution of the proposed optimal control problem is derived and used to construct a gradient-based optimization strategy in the framework of MC methods. This task requires to accommodate the resulting adjoint KS model in a form that is consistent with the kinetic formulation. Results of numerical experiments successfully validate the proposed control framework.

Keywords:

• Keilson-Storer master equation
• linear Boltzmann equation
• ensemble optimal control
• Monte Carlo methods

2010 Mathematics Subject Classification:

• 49M05
• 49N05
• 62P35
• 65C05
• 90C15

Acknowledgments

The authors are very grateful to Vittorio Romano for continued support. We thank very much Dejan Petkow and Maxim Gelin for helpful discussions, and the two anonymous Referees for suggestions for improvement.

Additional information

Funding

The author J.B. was partially financially supported by SPARC Industries S.A.R.L. Luxembourg. The author G.N. acknowledges the support from INDAM (GNFM) and Universita degli Studi di Catania, Piano della Ricerca 2016/2018 Linea di intervento 2.