Limitations on the control of Schrödinger equations

Illner, Reinhard; Lange, Horst; Teismann, Holger

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 615-635.

We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control $\left(E\right(t)·x)u$ is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, and we give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.

Classification : 35Q40, 35Q55, 81Q99, 93B05
Mots clés : Schrödinger equations, exact and approximate control, quantum control

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