Appl Math Optim 52:23–37 (2005)
2005 Springer Science+Business Media, Inc.
Geometry of Optimal Paths around Focal
Singular Surfaces in Differential Games
and Pierre Bernhard
Institute for Problems in Mechanics, Russian Academy of Sciences,
101-1 Vernadsky Ave., 119526 Moscow, Russia
University of Nice-Sophia Antipolis/CNRS I3S, ESSI,
930 Route des Colles, BP 145, 06903 Sophia Antipolis Cedex, France
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.
Key Words. Differential games, Isaacs’ equation, Singular surfaces, Singular
AMS Classiﬁcation. 35F99, 49L99, 91A23.
Non-smooth solutions to HJBI equations (and generally, to non-linear ﬁrst-order PDEs)
have several interesting singularities representing singular trajectories in differential
games. One such singularity is the so-called focal surface, which is approached by regular
characteristics (trajectories) from both sides of the surface. The incoming ﬁelds may be
Both authors acknowledge the help of Institut Lyapunov (INRIA and Lomonosov Moscow State Uni-
versity) thanks to which this joint work was made possible.