Full paper in PDF:
$%A. Yu. Khapalov, Global controllability properties for the semilinear heat equation with
superlinear
term, Rev. Mat. Complut. 12 (1999), no. 2, 511–535.%$
ABSTRACT
We discuss several global approximate controllability properties for the
semilinear heat equation with superlinear reaction-convection term, governed in
a bounded domain by locally distributed controls. First, based on the asymptotic
analysis in vanishing time, we study the steering of the projections of its solution
on any finite dimensional space spanned by the eigenfunctions for the truncated
linear part. We show that, if the control-supporting area is properly chosen,
then they can approximately be controlled globally at any time in the topology
induced by
. Then, based on the
-estimates as
for the
control functions solving the first problem, we prove that its global approximate
controllability from any
is also possible at any time in certain
topology, which is weaker than that of
. (It is known that this result does
not hold in
.) Finally, based on Altman’s fixed point theorem and some
of the above asymptotic-type results, we show that if the nonlinearity is purely
of reaction type and is locally Lipschitz, then the global exact controllability in
finite dimensions holds as well.
1991 Mathematics Subject Classification: 35, 93.