Necessity of the Terminal Condition in the Infinite Horizon Dynamic Optimization Problems with Unbounded Payoff

Agnieszka Wiszniewska-Matyszkiel*, Rajani Singh

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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In this paper, we prove the necessity of a terminal condition for a solution of the Bellman Equation to be the value function in dynamic optimization problems with unbounded payoffs. We also state the weakest sufficient condition, which can be applied in a large class of problems, including economic growth, resource extraction, or human behaviour during an epidemic. We illustrate the results by examples, including simple linear–quadratic problems and problems of resource extraction, with multiple solutions to the Bellman Equation or the maximizer of the right hand side of the Bellman Equation with the actual value function being the worst control instead of being optimal.
Original languageEnglish
Article number109332
Number of pages10
Publication statusPublished - Jan 2021

Bibliographical note

Published online: 13 November 2020.


  • Dynamic optimization
  • Infinite horizon
  • Unbounded payoffs
  • Feedback controls
  • Bellman Equation
  • Terminal condition
  • Necessity
  • Sufficiency
  • Resource extraction
  • Fish Wars

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