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 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.
Keywords: Schrödinger equations, exact and approximate control, quantum control
@article{COCV_2006__12_4_615_0, author = {Illner, Reinhard and Lange, Horst and Teismann, Holger}, title = {Limitations on the control of {Schr\"odinger} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {615--635}, publisher = {EDP-Sciences}, volume = {12}, number = {4}, year = {2006}, doi = {10.1051/cocv:2006014}, mrnumber = {2266811}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2006014/} }
TY - JOUR AU - Illner, Reinhard AU - Lange, Horst AU - Teismann, Holger TI - Limitations on the control of Schrödinger equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 615 EP - 635 VL - 12 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2006014/ DO - 10.1051/cocv:2006014 LA - en ID - COCV_2006__12_4_615_0 ER -
%0 Journal Article %A Illner, Reinhard %A Lange, Horst %A Teismann, Holger %T Limitations on the control of Schrödinger equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2006 %P 615-635 %V 12 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2006014/ %R 10.1051/cocv:2006014 %G en %F COCV_2006__12_4_615_0
Illner, Reinhard; Lange, Horst; Teismann, Holger. Limitations on the control of Schrödinger equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 615-635. doi : 10.1051/cocv:2006014. http://www.numdam.org/articles/10.1051/cocv:2006014/
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