The next task is to extend the definition of the cost-to-go under a fixed plan, which was given in Section 10.1.3, to the case of imperfect state information. Consider evaluating the quality of a plan, so that the ``best'' one might be selected. Suppose that the nondeterministic uncertainty is used to model nature and that a nondeterministic initial condition is given. If a plan is fixed, some state and action trajectories are possible, and others are not. It is impossible to know in general what histories will occur; however, the plan constrains the choices substantially. Let denote the set of state-action histories that could arise from applied to the initial condition .

The cost of a plan from an initial condition is
measured using *worst-case analysis* as

Note that includes , which is usually not known. It may be known only to lie in , as specified by . Let denote the set of all possible plans. An optimal plan using worst-case analysis is any plan for which (11.22) is minimized over all and . In the case of feasible planning, there are usually numerous equivalent alternatives.

Under probabilistic uncertainty, the cost of a plan can be measured
using *expected-case analysis* as

in which denotes the mathematical expectation of the cost, with the probability distribution taken over . The task is to find a plan that minimizes (11.23).

Steven M LaValle 2020-08-14