Impulsive Relaxation of Continuity Equations and Modeling of Colliding Ensembles

  • Maxim Staritsyn
  • Nikolay Pogodaev
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)


The paper promotes a relatively novel class of multi-agent control systems named “impulsive” continuity equations. Systems of this sort, describing the dynamics of probabilistically distributed “crowd” of homotypic individuals, are intensively studied in the case when the driving vector field is bounded and sufficiently regular. We, instead, consider the case when the vector field is unbounded, namely, affine in a control parameter, which is only integrally constrained. This means that the “crowd” can be influenced by “shock” impacts, i.e., actions of small duration but very high intensity. For such control continuity equations, we design an impulsive relaxation by closing the set of solutions in a suitable coarse topology. The main result presents a constructive form of the relaxed system. A connection of the obtained results to problems of contact dynamics is also discussed along with applications to optimal ensemble control and other promising issues.


Multi-agent systems Ensemble control Mean-field type control Continuity equation Impulsive control 



The authors are grateful to Krzysztof Konewski for inspiration and careful attention to this work.


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Authors and Affiliations

  1. 1.Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of SciencesIrkutskRussia

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