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Applied Mathematics and Optimization

, Volume 43, Issue 2, pp 169–186 | Cite as

An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

  • Y. Fujita
  • 22 Downloads

Abstract.

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.}

\amsclass{49L20, 35G20, 93E20.}

Key words. Bellman equation, Auxiliary equation, Ergodic control. AMS Classification. 49L20, 35G20, 93E20. 

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Copyright information

© Springer-Verlag New York Inc. 2001

Authors and Affiliations

  • Y. Fujita
    • 1
  1. 1.Department of Mathematics, Faculty of Science, Toyama University, Toyama 930-8555, Japan JP

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