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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.

MSC:

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

Citations:

Zbl 0838.35013
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Full Text: DOI

References:

[1] K. Beauchard, Local controllability of a 1D Schrödinger equation. J. Math. Pures Appl.84 (2005) 851-956. · Zbl 1124.93009
[2] K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations. J. Math. Pures Appl.94 (2010) 520-554. · Zbl 1202.35332
[3] K. Beauchard and C. Laurent, Local exact controllability of the 2D Schrödinger-Poisson system. Preprint hal-01333627 (2016). · Zbl 1375.35419
[4] J. Ball, J. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Cont. Optim.20 (1982) 575-597. · Zbl 0485.93015
[5] S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B14 (2010) 1375-1401. · Zbl 1219.93011
[6] S. Jaffard, Contrôle interne exact des vibrations d’une plaque rectangulaire. Port. Math.47 (1990) 423-429. · Zbl 0718.49026
[7] G. Lebeau, Contrôle de l’equation de Schrödinger. J. Math. Pures Appl.71 (1992) 267-291. · Zbl 0838.35013
[8] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilization des systèmes distribués. Tome 1, Contrôlabilité exacte. Collection R.M.A 8, Masson (1988).
[9] E. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Control Optim.32 (1994) 24-34. · Zbl 0795.93018
[10] J.-P. Puel, A regularity property for Schrödinger equations on bounded domains. Rev. Mat. Complut.26 (2013) 183-192. · Zbl 1276.35049
[11] G. Tenenbaum, M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation. Trans. Amer. Math. Soc.361 (2009) 951-977. · Zbl 1158.93013
[12] H. Weyl, Das asymptotisch Verteilungsgezetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann.71 (1912) 441-479. · JFM 43.0436.01
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