Appl Math Optim (2009) 59: 37–73
Ergodic Type Bellman Equations of First Order
with Quadratic Hamiltonian
Hidehiro Kaise · Shuenn-Jyi Sheu
Published online: 13 May 2008
© Springer Science+Business Media, LLC 2008
Abstract We consider Bellman equations of ergodic type in ﬁrst order. The Hamil-
tonian is quadratic on the ﬁrst derivative of the solution. We study the structure of
viscosity solutions and show that there exists a critical value among the solutions. It
is proved that the critical value has the representation by the long time average of the
kernel of the max-plus Schrödinger type semigroup. We also characterize the critical
value in terms of an invariant density in max-plus sense, which can be understood as
a counterpart of the characterization of the principal eigenvalue of the Schrödinger
operator by an invariant measure.
Keywords Ergodic type Bellman equations · Structure of viscosity solutions ·
Critical value · Characterization of solutions
We consider the partial differential equation of ﬁrst order with the nonlinear quadratic
a∇W ·∇W + b ·∇W + V(x)= in R
H. Kaise supported by Grant-in-Aid for Young Scientists, No.17740052, JSPS.
S.-J. Sheu supported by National Science Council of Taiwan, NSC96-2119-M-001-002.
H. Kaise (
Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya 464-8601, Japan
Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan, ROC