CONTROLS INSENSITIZING THE NORM OF THE SOLUTION OF A SEMILINEAR HEAT-EQUATION

We consider here a semilinear heat equation with partially known initi al and boundary conditions. The insensitizing problem consists in find ing a control function such that some functional of the state is local ly insensitive to the perturbations of these initial and boundary data . In this paper the insensitizing control of the norm of the observati on of the solution in an open subset of the domain is studied under ap propriate assumptions on the nonlinearity and the observation subset. It is shown that the insensitivity conditions are equivalent to a part icular nonlinear exact controllability problem for parabolic equations . Due to the smoothing effects of this type of equation, exact control lability is very hard to achieve and this is why it seems natural to i ntroduce the idea of approximately insensitizing control and then to s olve a nonlinear approximate controllability problem of a special type . That is done using a linearization and fixed point method. Solving t he linear problem leads to the proof of a nontrivial uniqueness proper ty which is abo used to characterize a particular subset of the admiss ible controls. This characterization is made thanks to a convex dualit y theorem and allows us to solve a fixed-point problem and get the res ult for the nonlinear case, Various comments and conclusions are event ually given, with other (approximately) insensitizing problems that ca n be solved by our methods. (C) 1995 Academic Press, Inc.