The extension of sequential game theory to the continuous-time case is
called *differential game theory* (or
*dynamic game theory* [59]), a subject introduced by
Isaacs [477]. All of the variants considered in Sections
9.3, 9.4, 10.5 are possible:

- There may be any number of players.
- The game may be zero-sum or nonzero-sum.
- The state may or may not be known. If the state is unknown, then interesting I-spaces arise, similar to those of Section 11.7.
- 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 [59].

in which , , and are constant, real-valued matrices of dimensions , , and , 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 Section 15.2.2 and [59].

The original work of Isaacs [477] contains many interesting
examples of *pursuit-evasion
differential games*. One of the most
famous is described next.

(13.205) | ||||

The state space is is , and the action spaces are and .

The task is to determine whether the pursuer can come within some prescribed distance of the evader:

(13.206) |

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 values of , , , and , the state space can be partitioned into two regions that correspond to whether the pursuer or evader wins [59,477]. 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 [694].

The pursuit-evasion game becomes very interesting if both players are restricted to be Dubins cars.

Steven M LaValle 2020-08-14