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Invariant Solutions of Differential Games with Measures: A Discontinuous Time Reparameterization Approach

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Abstract

The paper is devoted to bilevel qualitative games with impulsive dynamics described by a measure differential equation. We study games with fixed impulses, and games with impulsive control in the hands of the lower level player, provided that the upper level player aims to ensure the existence of invariant solutions to a measure differential equation under any admissible answer of the opponent. We give sufficient conditions for weak invariance of time-varying domains with respect to the impulsive dynamical games.

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Notes

  1. Here, \(y\) is an initial state, \(t\) is an initial time moment in the natural time scale, while \(\vartheta \) is introduced to characterize a time-like position within the interval \([0, T_{t}]\) (provided that \(\mu (\{t\})\ne 0\), and such an interval does exist).

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Acknowledgments

The work was partially supported by the Russian Foundation for Basic Research, Grants 13-08-00441, 14-08-00606, 15-31-20531.

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  1. Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russia

    Elena Goncharova & Maxim Staritsyn

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  1. Elena Goncharova
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  2. Maxim Staritsyn
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Correspondence to Elena Goncharova.

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Communicated by Josef Shinar.

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Goncharova, E., Staritsyn, M. Invariant Solutions of Differential Games with Measures: A Discontinuous Time Reparameterization Approach. J Optim Theory Appl 167, 382–400 (2015). https://doi.org/10.1007/s10957-014-0691-7

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  • DOI: https://doi.org/10.1007/s10957-014-0691-7

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