# Level Sets of the Value Function in Differential Games with the Homicidal Chauffeur Dynamics

### V.S. Patsko, V.L. Turova

Three differential games with the dynamics of the homicidal chauffeur
are considered.
The first problem is the Isaacs' homicidal chauffeur
differential game.
In this game, a pursuer
*P* minimizes the capture time of an evader *E*.
The objective of the evader is to prevent the capture or to maximize
the capture time.
The magnitude of the velocity is constant for the pursuer, and
his maneuverability is bounded through a minimal turn radius.
The maneuverability of the evader is not bounded.
The pursuer's control is the rate of turn; the evader steers by
choosing directions of his velocity. The main difference of the
second problem is that the size of the constraint on the control parameter
of the evader depends on the position of the game.
The idea of such a modification
was suggested by Bernhard. The third problem is
a conic surveillance-evasion game studied by Lewin and Olsder.
In this game, the dynamics
is the same as in the Isaacs' problem, but the goals of the players differ
from the classic formulation: an evader *E*
minimizes the time of escaping from a detection set that is a
two-dimensional semi-infinite cone.
The detection set is attached to the velocity vector of a pursuer *P*
whose objective is to keep the evader within the detection set
for maximal time.
The paper describes the computation of
level sets of the value functions for these games. The algorithm proposed
by the authors is used.
An analysis of families of semipermeable
curves is carried out.
The results of this analysis are used to check
the correctness of the computation of level sets and
to explain the appearance of holes in victory domains of the pursuer
in the second problem.

## Electronic version of the article

IGTR2001_Patsko_Turova.pdf (5521 KB)

*Patsko V.S., Turova V.L.*
**Level Sets of the Value Function in Differential Games with the Homicidal Chauffeur Dynamics**
// International Game Theory Review, Vol. 3, No. 1, March 2001. pp. 67–112.

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