A null controllability problem with a finite number of constraints on the normal derivative for the semilinear heat equation

• Main Author: Carole Louis-Rose
• Format: Article
• Language: English
• Published: University of Szeged 2012-11-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: null controllability; semilinear heat equation; constraint on the normal derivative; carleman inequality; observability inequality
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=1813

We consider the semilinear heat equation in a bounded domain of $\mathbb{R}^m$. We prove the null controllability of the system with a finite number of constraints on the normal derivative, when the control acts on a bounded subset of the domain. First, we show that the problem can be transformed into a null controllability problem with constraint on the control, for a linear system. Then, we use an appropriate observability inequality to solve the linearized problem. Finally, we prove the main result by means of a fixed-point method.