Appl Math Optim 41:155–170 (2000)
2000 Springer-Verlag New York Inc.
Exact Internal Controllability of Maxwell’s Equations
Institute of Mathematics, Fudan University,
Shanghai 200433, People’s Republic of China
Abstract. In this paper we obtain two exact internal controllability results of
Maxwell’s equations in a general region by using multiplier techniques. The ﬁrst
one is exact controllability in a short time, in which we obtain the “optimal” (observ-
ability) estimates when the location and the shape of the controller is ﬁxed. What
happens if we allow the controller to change? Under some conditions, we show that
by doing that the system can be exactly controllable within any given time duration,
which is our second exact controllability result.
Key Words. Maxwell’s equations, Exact internal controllability, Multiplier tech-
nique, Changing controller.
AMS Classiﬁcation. 93B05, 35L05.
In this paper we consider the following Maxwell’s equations with locally distributed
−∇×H = χ
+∇×E = 0in × R
∇·E =∇·H = 0in × R
ν × E = 0on × R
E(0) = E
, H(0) = H
Where is a bounded, open, and connected region in R
with a C
boundary ∂ = ,
and, for each t ∈ [0, ∞), G(t) is a subdomain of , χ
is the characteristic function
of the set G(t); ν is the unit normal vector to pointing into the exterior of ;
This work was partially supported by the Chinese State Education Commission Science Foundation.