Appl Math Optim 46:313–330 (2002)
2002 Springer-Verlag New York Inc.
Averaging of Differential Equations Generating Oscillations and
an Application to Control
R. M. Temam
and D. Wirosoetisno
Laboratoire d’Analyse Num´erique et EDP,
Universit´e Paris-Sud, Orsay-91405, France
Institute for Scientiﬁc Computing and Applied Mathematics,
Indiana University, Bloomington, IN 47405, USA
Twente Institute of Mechanics, University of Twente,
7500AE Enschede, The Netherlands
Abstract. In this article we consider differential equations which generate oscil-
lating solutions. These oscillations are due to the presence of a small parameter
ε>0; however, they are not present in the coefﬁcients but instead they are caused
by a penalty term involving an antisymmetric operator. Our aims are twofold. In
the ﬁrst part we study asymptotics at all orders, for ε → 0, construct approximate
solutions, and derive estimates of the error between the exact solution and the ap-
proximate ones. One of the motivations of this part is the study to high orders of the
geostrophic asymptotics in atmospheric science, but there are many other possible
applications involving in particular the wave equation. The actual applications of
our results to atmospheric science will be discussed elsewhere [STW], as well as, on
the mathematical side, the application to partial differential equations [TW1]. In the
second part of this article we study a control problem involving such an equation and
study the behavior of the state equation, of the optimal control, and of the optimality
equation as ε → 0. For the control part we restrict ourselves to a linear equation and
to the ﬁrst order in the asymptotics ε → 0, leaving nonlinear problems and higher
orders to a future work.
Key Words. Higher-order averaging, Optimal control.
AMS Classiﬁcation. 34E05, 34H05, 49K15.
This work was supported in part by the National Science Foundation under Grant NSF-DMS-0074334
and by the Research Fund of Indiana University (RMT), and by a visitor’s stipend from the Twente Institute
of Mechanics (DW).