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

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.