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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Melikyan, A., Bernhard, P. Geometry of Optimal Paths around Focal Singular Surfaces in Differential Games. Appl Math Optim 52, 23–37 (2005). https://doi.org/10.1007/s00245-004-0816-8
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DOI: https://doi.org/10.1007/s00245-004-0816-8