CONTROLS INSENSITIZING THE NORM OF THE SOLUTION
			 OF A SEMI-LINEAR HEAT EQUATION



		       Olivier BODART and Caroline FABRE



We consider here a semilinear heat equation with partially known initial and
boundary conditions.  The insensitizing problem consists in finding a control
function such that some functionnal of the state is locally insensitive to the
perturbations of these initial and boundary data.  In this paper the
insensitizing control of the norm of the observation of the solution in an open
subset of the domain is studied under appropriate assumptions on the non
linearity and the observation subset.  It is shown that the insensitivity
conditions are equivalent to a particular non linear exact controllability
problem for parabolic equations.  Due to the smoothing effects of this type of
equations, exact controllability is very hard to achieve and this is why it
seems natural to introduce the idea of approximately insensitizing control and
then to solve a non linear approximate controllability problem of a special
type.  That is done using a linearization and fixed point method.  Solving the
linear problem leads to prove a non trivial uniqueness property which is also
used to characterize a particular subset of the admissible controls.  This
characterization is made thanks to a convex duality theorem and allows to solve
a fixed point problem and get the result for the non linear case.  Various
comments and conclusions are eventually given, with other (approximately)
insensitizing problems that can be solved by our methods.

The whole paper is available as a compressed Postscript file by internet
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in the directory pub/RI/1994 under the name

        bodart_cfabre_291.jan.ps.gz