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Structure of Viability Kernels for Some Linear Differential Games

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Abstract

A convenient form of necessary and sufficient conditions of viability for differential games with linear dynamics is proposed. These conditions are utilized to construct maximal viable subsets of state constraints, viability kernels, in two illustrative two-dimensional examples. These examples demonstrate the relative simplicity of the structure of the viability kernels. It was found that the boundaries of the viability kernels consist of segments of the boundary of the state constraint and of lines defined by the first integrals of the governing equations as the players use extremal constant controls. It is conjectured that such a structure holds in high dimensional cases too.

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Authors and Affiliations

  1. Technische Universität München, Boltzmannstrasse 3, 85748, Garching b. Munich, Germany

    N. D. Botkin

  2. Institute of Mathematics and Mechanics, Ural Department of Russian Academy of Sciences, S. Kovalevskaya str. 16, 620219, Ekaterinburg, Russia

    E. A. Ryazantseva

Authors
  1. N. D. Botkin
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  2. E. A. Ryazantseva
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Correspondence to N. D. Botkin.

Additional information

Communicated by F. Chernousko.

The authors remember with gratitude valuable ideas and remarks given by Dr. G. Sonnevend and Professor Dr. A.I. Subbotin during the work on the paper. The work was partly done when the first author was supported by the Alexander von Humboldt Foundation, Germany. The author is grateful to Prof. Dr. R. Bulirsch for the warm hospitality at the Technical University of Munich, Germany.

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Botkin, N.D., Ryazantseva, E.A. Structure of Viability Kernels for Some Linear Differential Games. J Optim Theory Appl 147, 42–57 (2010). https://doi.org/10.1007/s10957-010-9706-1

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