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Differential Games in \(L^{\infty }\)

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Abstract

The Isaacs equations for differential games in \({L^{\infty }}\) first derived in Barron (Nonlinear Anal 14:971–989, 1990) are reformulated so that the Hamiltonians are continuous and result in a simpler problem to analyze numerically. Relaxed differential games in \({L^{\infty }}\) are considered. \(L^{\infty }\) differential games with time and state independent dynamics and convex or quasiconvex terminal data are solved explicitly using a type of Hopf–Lax formula. The stochastic differential game in \({L^{\infty }}\) connected to stochastic target problems is also discussed.

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Notes

  1. Precisely, G is quasiconvex if \(E_\alpha =\{x\;|\; G(x) \le \alpha \}\) is convex for all \(\alpha \). Equivalently, \(G(\lambda \,x+(1-\lambda )y) \le G(x) \vee G(y),\) for all \(x,y \in {{\mathbb {R}}}^n, 0 \le \lambda \le 1\).

  2. \(G^*(p)=\sup _x p\cdot x-G(x)\) is the Fenchel conjugate of G.

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Acknowledgments

I would like to thank the reviewers for a careful reading of the paper and for bringing several references to my attention.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA

    E. N. Barron

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  1. E. N. Barron
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Correspondence to E. N. Barron.

Additional information

The work was supported in part by a Grant NSF-DMS 1515871.

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Barron, E.N. Differential Games in \(L^{\infty }\) . Dyn Games Appl 7, 157–184 (2017). https://doi.org/10.1007/s13235-016-0183-5

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