Journal of Scientific Computing

, Volume 73, Issue 1, pp 178–202 | Cite as

Numerical Investigation of a Class of Liouville Control Problems



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.


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

Mathematics Subject Classification

35Q93 49K20 49J20 65M08 


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany

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