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Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control. (English) Zbl 1202.35332
Summary: We consider a linear Schrödinger equation on a bounded interval with bilinear control that represents a quantum particle in an electric field (the control). We prove the exact controllability of this system in any positive time locally around the ground state.
Similar results were proved for particular models [K. Beauchard, J. Math. Pures Appl. (9) 84, No. 7, 851–956 (2005; Zbl 1124.93009); ESAIM, Control Optim. Calc. Var. 14, No. 1, 105–147 (2008; Zbl 1132.35446); K. Beauchard and J.-M. Coron, J. Funct. Anal. 232, No. 2, 328–389 (2006; Zbl 1188.93017)] in non-optimal spaces in long time and the proof relied on the Nash-Moser implicit function theorem in order to deal with an a priori loss of regularity.
In this article, the model is more general, the spaces are optimal, there is no restriction on the time and the proof relies on the classical inverse mapping theorem. A hidden regularizing effect is emphasized showing there is actually no loss of regularity.
Then, the same strategy is applied to nonlinear Schrödinger equations and nonlinear wave equations showing that the method works for a wide range of bilinear control systems.

MSC:
35Q93 PDEs in connection with control and optimization
35J10 Schrödinger operator, Schrödinger equation
35Q55 NLS equations (nonlinear Schrödinger equations)
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
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