Some Aspects of the Asymptotic Behavior of the Solutions of the Semilinear Heat Equation in Vanishing Time and Approximate Controllability

This paper is concerned with sufficient conditions for approximate controllability in L2(Ω) of the semilinear heat equation with internal (in general, superlinear) nonlinearity in u and ∇ u. The approach employed focuses on the study of the asymptotic behavior in vanishing time of those solutions to the semilinear equation which are generated by control functions solving the corresponding linear controllability problem. A particular emphasis is given to the study of static distributed and time-dependent lumped controls.