Appl Math Optim 49:113–121 (2004)
2004 Springer-Verlag New York Inc.
Stochastic Control for a Class of Random Evolution Models
Halil Mete Soner,
and Ludwig Streit
Institut de Production et Robotique / (LPM)
Ecole Polytechnique F´ed´erale de Lausanne,
CH-1015 Lausanne, Switzerland
Department of Mathematics, Koc University,
Fener Yolu Caddesi, Sariyer 80910, Istanbul, Turkey
CCM-Centro de Ciencias Matematicas, Universidade da Madeira,
P-9000 Funchal, Portugal
BiBoS, University of Bielefeld,
D-33615 Bielefeld, Germany
Abstract. We construct the explicit connection existing between a solvable model
of the discrete velocities non-linear Boltzmann equation and the Hamilton–Bellman–
Jacobi equation associated with a simple optimal control of a piecewise determin-
istic process. This study extends the known relation that exists between the Burgers
equation and a simple controlled diffusion problem. In both cases the resulting par-
tial differential equations can be linearized via a logarithmic transformation and
hence offer the possibility to solve physically relevant non-linear ﬁeld models in
Key Words. Piecewise deterministic evolutions, Stochastic optimal control, Log-
arithmic transformation, Nonlinear ﬁeld equations.
AMS Classiﬁcation. 60G40, 93C10.
This research was partially supported by the Funda¸cao para a Ciˆencia e a Tecnologica, Portugal, and
the ﬁrst author was partially supported by the Fonds National Suisse pour la Recherche Scientiﬁque.