To make a *differential game against nature* that extends Formulation 10.1 to
continuous time, suppose that nature actions are chosen
from . A differential model can be defined as

The state space and action space are used in the same way as throughout this chapter. The difference only comes in the state transition equation. State-dependent nature action spaces may also be used.

As observed repeatedly throughout Part III, nature can be
modeled nondeterministically or probabilistically. In the
nondeterministic case, (13.199) is equivalent to a
*differential inclusion* [53]:

Possible future values for can be computed using forward projections. Reachable sets, which will be introduced in Section 14.2.1, can be defined that characterize the evolution of future possible states over time. Plans constructed under this model usually use worst-case analysis.

At each time , nature chooses some . The state transition equation is

The cone shown in Figure 12.45 is just the nondeterministic forward projection under the application of a constant .

In the probabilistic case, restrictions must be carefully placed on
the nature action trajectory (e.g., a Weiner process
[910]). Under such conditions, (13.199)
becomes a *stochastic differential equation*. Planning in this
case becomes continuous-time stochastic control [567], and
the task is to optimize the expected cost.

in which the domain of must be extended to or other suitable restrictions must be imposed. At each time , a nature action

In a similar way, parameters that account for nature can be introduced virtually anywhere in the models of this chapter. Some errors may be systematic, which reflect mistakes or simplifications made in the modeling process. These correspond to a constant nature action applied at the outset. In this case, nature is not allowed to vary its action over time. Other errors could correspond to noise, which is expected to yield different nature actions over time.

Steven M LaValle 2012-04-20