Numerical Investigation of a Class of Liouville Control Problems

Abstract

An accurate and efficient numerical scheme for solving a Liouville optimal control problem in the framework of the Pontryagin’s maximum principle (PMP) is presented. The Liouville equation models the time-evolution of a density function that may represent a distribution of non-interacting particles or a probability density. In this work, the purpose of the control is to maximize the measure of a target set at a given final time. In order to solve this problem, a high-order accurate conservative and positive preserving discretization scheme is investigated and a novel iterative optimization method is formulated that solves the PMP optimality condition without requiring differentiability with repsect to the control variable. Results of numerical experiments are presented that demonstrate the effectiveness of the proposed solution procedure.

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Acknowledgements

We would like to thank M. Annunziato, Andrei V. Dmitruk, Andrei V. Fursikov and Fredi Tröltzsch for helpful discussion. This project was supported in part by the BMBF Verbundprojekt 05M2013 ‘ROENOBIO: Robust energy optimization of fermentation processes for the production of biogas and wine’.

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Correspondence to Souvik Roy.

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Roy, S., Borzì, A. Numerical Investigation of a Class of Liouville Control Problems. J Sci Comput 73, 178–202 (2017). https://doi.org/10.1007/s10915-017-0410-2

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Keywords

  • Liouville equation
  • Optimal control theory
  • Pontryagin’s maximum principle
  • Sanders scheme

Mathematics Subject Classification

  • 35Q93
  • 49K20
  • 49J20
  • 65M08