Appl Math Optim 43:169–186 (2001)
2001 Springer-Verlag New York Inc.
An Auxiliary Equation for the Bellman Equation
in a One-Dimensional Ergodic Control
Department of Mathematics, Faculty of Science,
Toyama University, Toyama 930-8555, Japan
Communicated by M. Nisio
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 solu-
tion under general assumptions. For this purpose we introduce an auxiliary equation
whose solution gives the invariant measure of the diffusion corresponding to an opti-
mal control. Using thissolution, we construct a solutionto 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 Markovcontrol explicitlyin manyexamples.
Key Words. Bellman equation, Auxiliary equation, Ergodic control.
AMS Classiﬁcation. 49L20, 35G20, 93E20.
Bellman equations in ergodic control have been considered by many authors , , ,
–, . In most of these papers, solutions were constructed by “the vanishing dis-
count rate method”. In this paper we consider the Bellman equation in a one-dimensional
ergodic control, but we do not use this method to construct its solution. We introduce
an auxiliary equation whose solution gives the invariant measure of the diffusion cor-
responding to an optimal control. Using this solution, we construct a solution to this
We are concerned with the Bellman equation
(x)|+ f (x), x ∈ R, (1.1)