A generalization of Bellman’s equation with application to path planning, obstacle avoidance and invariant set estimation

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 monotonically backward functions; functions being special case We propose necessary and sufficient condition, utilizing generalization Bellman’s equation, for solution MSOP, cost function, optimal. Moreover, show that proposed condition efficiently compute optimal solutions two important MSOP’s; the path Dubin’s car obstacle avoidance, maximal invariant set discrete time systems.