Controls Insensitizing the Norm of the Solution of a Semilinear Heat-Equation

• Published in: Journal of mathematical analysis and applications Vol. 195; no. 3; pp. 658 - 683
• Main Authors: Bodart, O., Fabre, C.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.11.1995; Elsevier
• Subjects: Calculus of variations and optimal control; Exact sciences and technology; Mathematical analysis; Mathematics; Partial differential equations; Sciences and techniques of general use; Controllability; Semi linear equation; Linear system; Fixed point theorem; Heat equation; Linearization
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1006/jmaa.1995.1382

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 functional 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 nonlinearity and the observation subset. It is shown that the insensitivity conditions are equivalent to a particular nonlinear exact controllability problem for parabolic equations. Due to the smoothing effects of this type of equation, 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 nonlinear approximate controllability problem of a special type. That is done using a linearization and fixed point method. Solving the linear problem leads to the proof of a nontrivial uniqueness property which is abo used to characterize a particular subset of the admissible controls. This characterization is made thanks to a convex duality theorem and allows us to solve a fixed-point problem and get the result for the nonlinear case. Various comments and conclusions are eventually given, with other (approximately) insensitizing problems that can be solved by our methods.