Some Results and Open Problems on the Controllability of Linear and Semilinear Heat Equations

Abstract

In this paper we address the problem of null-controllability of heat equations in two different cases: (a) The semilinear heat equation in bounded domains and (b) The linear heat equation in the half line. Concerning the first problem (a) we show that a number of systems in which blow-up arises may be controlled by means of external forces which are localized in an arbitrarily small open set. In the frame of problem (b) we prove that compactly supported initial data may not be driven to zero if the control is supported in a bounded set. This shows that although the velocity of propagation in the heat equation is infinite, this is not sufficient to guarantee null-controllability properties.We also include a list of open problems.KeywordsHeat EquationMoment ProblemApproximate ControllabilityCarleman EstimateOpen Nonempty SubsetThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.