Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games |
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Authors: | Arik Melikyan Pierre Bernhard |
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Affiliation: | (1) Institute for Problems in Mechanics, Russian Academy of Sciences, 101-1 Vernadsky Ave., 119526 Moscow, Russia;(2) University of Nice-Sophia Antipolis/CNRS I3S, ESSI, 930 Route des Colles, BP 145, 06903 Sophia Antipolis Cedex, France |
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Abstract: | We investigate a special type of singularity in non-smooth solutions of first-order partial differential equations, with emphasis on Isaacs’ equation. This type, called focal manifold, is characterized by the incoming trajectory fields on the two sides and a discontinuous gradient. We provide a complete set of constructive equations under various hypotheses on the singularity, culminating with the case where no a priori hypothesis on its geometry is known, and where the extremal trajectory fields need not be collinear. We show two examples of differential games exhibiting non-collinear fields of extremal trajectories on the focal manifold, one with a transversal approach and one with a tangential approach. |
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Keywords: | Differential games Isaacs’ equation Singular surfaces Singular characteristics |
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