The extension of sequential game theory to the continuous-time case is
called differential game theory (or
dynamic game theory ), a subject introduced by
Isaacs . All of the variants considered in Sections
9.3, 9.4, 10.5 are possible:
See  for a thorough overview of differential games.
, can be engaged in a differential
game in which each has a continuous set of actions. Let and
denote the action spaces of
, respectively. A state
transition equation can be defined as
- There may be any number of players.
- The game may be zero-sum or
- The state may or may not be known. If the state is unknown,
then interesting I-spaces arise, similar to those of Section
- Nature can interfere with the game.
- Different equilibrium concepts, such as saddle points and Nash
equilibria, can be defined.
in which is the state, , and .
Linear differential games are an
important family of games because many techniques from optimal control
theory can be extended to solve them .
(Linear Differential Games)
The linear system model (13.37
) can be extended to
incorporate two players. Let
be a phase space. Let
be an action spaces for
. A linear differential game
is expressed as
are constant, real-valued matrices of
respectively. The particular solution to such games depends on the
cost functional and desired equilibrium concept. For the case of
a quadratic cost, closed-form solutions exist. These extend techniques
that are developed for linear systems with one decision maker; see
The original work of Isaacs  contains many interesting
examples of pursuit-evasion
differential games. One of the most
famous is described next.
In the homicidal chauffeur
the pursuer is a Dubins car
and the evader is a point robot that
can translate in any direction. Both exist in the same world,
. The speeds of the car and robot are
respectively. It is assumed that
, which means that
the pursuer moves faster than the evader. The transition equation is
given by extending (13.15
) to include two state
variables that account for the robot position:
The state space is
, and the action spaces
The task is to determine whether the pursuer can come within some
prescribed distance of the evader:
If this occurs, then the pursuer wins; otherwise, the evader wins.
The solution depends on the
, and the
initial state. Even though the pursuer moves faster, the evader may
escape because it does not have a limited turning radius. For given
, the state space
be partitioned into two regions that correspond to whether the pursuer
or evader wins [59
]. To gain some intuition about
how this partition may appear, imagine the motions that a bullfighter
must make to avoid a fast, charging bull (yes, bulls behave very much
like a fast Dubins car when provoked).
Another interesting pursuit-evasion game arises in the case of one car
attempting to intercept another .
(A Game of Two Cars)
Imagine that there are two simple cars that move in the same world,
. Each has a transition equation given by
). The state transition equation for the game is
The pursuit-evasion game becomes very interesting if both players are
restricted to be Dubins cars.
Steven M LaValle