  The standard Dynamic Programming (DP) formulation can be used to solve
Multi-Stage Optimization Problems (MSOP's) with additively separable objective
functions. In this paper we consider a larger class of MSOP's with
monotonically backward separable objective functions; additively separable
functions being a special case of monotonically backward separable functions.
We propose a necessary and sufficient condition, utilizing a generalization of
Bellman's equation, for a solution of a MSOP, with a monotonically backward
separable cost function, to be optimal. Moreover, we show that this proposed
condition can be used to efficiently compute optimal solutions for two
important MSOP's; the optimal path for Dubin's car with obstacle avoidance, and
the maximal invariant set for discrete time systems.
