Appl Math Optim 55:241–253 (2007)
2007 Springer Science+Business Media, Inc.
On the Non-Validity of the Order Reduction Method
for Singularly Perturbed Control Systems
Department of Mathematics, Technion,
Haifa 32000, Israel
Abstract. The order reduction method for singularly perturbed optimal control
systems consists of setting the small parameter equal to zero and employing the dif-
ferential system thus obtained. Although in many situations this provides the correct
variational limit problem, it is established in this paper that when considering sys-
tems with non-scalar fast variables, the set of systems for which the order reduction
method is invalid is dense in the class of systems under consideration. This extends
previous results, where only systems with linear fast variables were considered. The
present result complements a result established in a joint work with Artstein, where
it was established that the order reduction method is valid for singularly perturbed
optimal control systems with scalar fast variable.
Key Words. Optimal control, Singular perturbations, Order reduction, Variational
AMS Classiﬁcation. 34E15, 49J15.
Singularly perturbed (SP) control systems form a natural model for the control of coupled
slow and fast motions. Such is the following SP optimal control problem:
c(x(t), y(t), u(t)) dt (1.1)
˙x(t) = f (x, y, u), x(0) = x
ε˙y(t) = g(x, y, u), y(0) = y