Appl Math Optim 47:143–149 (2003)
2003 Springer-Verlag New York Inc.
A Linear PDE Approach to the Bellman Equation of
Ergodic Control with Periodic Structure
Department of Mathematics, Faculty of Science,
Toyama University, Toyama 930-8555, Japan
Communicated by A. Bensoussan
Dedicated to Professor Makiko Nisio on her seventieth birthday
Abstract. In this paper we give a new proof of the existence result of Bensoussan
[1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic struc-
ture. This Bellman equation is a nonlinear PDE, and he constructed its solution by
using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two
linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.
Key Words. A linear PDE approach, Bellman equation of ergodic control, Peri-
odic structure, The Schauder ﬁxed point theorem.
AMS Classiﬁcation. 49L20, 35G60, 93E20.
For n ∈ N and q ≥ 2, let W
() denote the Sobolev space with the norm
, u ∈ W
where = ]0, 1[
for d ≥ 2. In this paper we are concerned with the following Bellman
αϕ(x) + min
[γ · Dϕ(x) +|γ |
] + h(x) = λ,
ϕ ∈ W
() is periodic and λ is scalar,