Appl Math Optim 41:425–445 (2000)
2000 Springer-Verlag New York Inc.
The Value Function of Singularly Perturbed Control Systems
and V. Gaitsgory
Department of Mathematics,
The Weizmann Institute of Science,
Rehovot 76100, Israel
School of Mathematics,
University of South Australia,
The Levels, Pooraka,
South Australia 5095, Australia
Abstract. The limit as ε → 0 of the value function of a singularly perturbed
optimal control problem is characterized. Under general conditions it is shown that
limit valuefunctions existand solve in a viscosity sense a Hamilton–Jacobi equation.
The Hamiltonian of this equation is generated by an inﬁnite horizon optimization
on the fast time scale. In particular, the limit Hamiltonian and the limit Hamilton–
Jacobi equation are applicable in cases where the reduction of order, namely setting
ε = 0, does not yield an optimal behavior.
Key Words. Singular perturbation, Optimal control, Hamilton–Jacobi equation,
AMS Classiﬁcation. 49L05, 34E15.
Z. Artstein is the Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics, and was
supported by a grant from the United States–Israel Binational Science Foundation (BSF) and by a grant
from the MINERVA Foundation, Germany. V. Gaitsgory was supported by the Australian Research Council