Local exact bilinear control of the Schrödinger equation

Puel, Jean-Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1264-1281.

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 $N\le 3$. For a general domain we will require a non degeneracy condition of the normal derivative of the eigenfunction on a part $\Gamma {}_0$ 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 $\Gamma {}_0$ and the linear controllability results with real and sufficiently regular controls.

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