Local exact bilinear control of the Schrödinger equation. (English) Zbl 1354.35126

Summary: 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 \(N \leq 3\). 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. (9) 71, No. 3, 267–291 (1992; Zbl 0838.35013)]) 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.


35Q41 Time-dependent Schrödinger equations and Dirac equations
35B65 Smoothness and regularity of solutions to PDEs
93C20 Control/observation systems governed by partial differential equations
35P99 Spectral theory and eigenvalue problems for partial differential equations


Zbl 0838.35013
Full Text: DOI


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