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

A Generalization of Bellman's Equation with Application to Path Planning, Obstacle Avoidance and Invariant Set Estimation