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

Morgan Jones, Matthew M. Peet

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Article number109510
JournalAutomatica
Volume127
DOIs
StatePublished - May 2021

Keywords

  • Dynamic programming
  • GPU-accelerated computing
  • Maximal invariant sets
  • Path planning

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Electrical and Electronic Engineering

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