Appl Math Optim 44:163–175 (2001)
2001 Springer-Verlag New York Inc.
Dynamics for Linear Feedback Controlled Two-Dimensional
B´enard Equations with Distributed Controls
Hyung-Chun Lee and Byeong Chun Shin
Department of Mathematics, Ajou University,
Suwon 442-749, Korea
Communicated by R. Temam
Abstract. The long-time behavior of solutions for some feedback distributed con-
trol problems associated with the B´enard equations is studied. Some linear feedback
solutions for the B´enard equations are constructed. Then we prove that these feed-
back solutions possess the decay (in time) properties.
Key Words. Optimal control, Navier–Stokes equation, B´enard equations.
AMS Classiﬁcation. 35B40, 35B37, 35Q30, 65M60.
The study of optimal ﬂow control problems in an inﬁnite time interval is of great im-
portance in many physical applications such as turbulence and drag minimization in
the entire life span of a ﬂow. In this article we consider the long-time behavior of the
solutions for feedback control problems associated with the B´enard equations on the
inﬁnite time interval. Dynamics and approximations for controlled Navier–Stokes equa-
tions were considered in  and . This work is motivated by the desire to steer, over
time, a candidate velocity ﬁeld u and ﬂuid temperature θ to target velocity ﬁeld U and
target ﬂuid temperature , respectively, by appropriately controlling the body force f and
the density of external heat source τ. We consider the following minimization problem:
The ﬁrst author was supported by Grant No. 2000-1-10300-001-5 from the Basic Research Program
of the KOSEF. The second author was supported by KOSEF.