Niveau: Supérieur, Doctorat, Bac+8
New blow-up rates for fast controls of Schrodinger and heat equations G. Tenenbaum & M. Tucsnak Institut Elie Cartan Universite Henri Poincare Nancy 1, BP 239 54506 Vandœuvre-les-Nancy, France (version 16/4/2007, 12h27) Abstract: We consider the null-controllability problem for the Schrodinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see Miller [14], [15],[16] [17]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrodinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in Seidman, Avdonin and Ivanov [22]. These results are used, following [17], to deal with the case of several space dimensions. Keywords: null-controllability, Schrodinger equation, heat equation, series of expo- nentials. AMS subject classifications : 93C25, 93B07, 93C20, 11N36. 1 Introduction In this work we consider the boundary control of systems governed by the Schrodinger or by the heat equation. These systems can be written as an abstract infinite-dimensional linear control system described by the equations (1.1) w˙ = Aw +Bu, w(0) = ?, where w denotes the state.
- ?w?t??2 c2t ∫
- sional input
- ran ?t
- suitable constant
- let ? ?
- ?t ?
- control operator