Local exact bilinear control of the Schrödinger equation^∗

Laboratoire de Mathématiques de Versailles, Université de Versailles St Quentin, 78035 Versailles cedex, France.
jppuel@math.uvsq.fr

Received: 6 June 2016
Accepted: 7 June 2016

Abstract

We are going to prove the local exact bilinear controllability for a Schrödinger equation, set in a bounded regular domain, in a neighborhood of an eigenfunction corresponding to a simple eigenvalue in dimension N ≤ 3. For a general domain we will require a non degeneracy condition of the normal derivative of the eigenfunction on a part Γ₀ of the boundary satisfying the Geometric Control Condition (see [G. Lebeau. J. Math. Pures Appl. 71 (1992) 267–291]) and for a rectangle when N = 2 or an interval for N = 1 no further condition. In the general case we will use real potentials concentrated in the neighborhood of Γ₀ and the linear controllability results with real and sufficiently regular controls.