Exact controllability for stochastic Schrödinger equations☆

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Abstract

This paper is addressed to studying the exact controllability of stochastic Schrödinger equations by two controls. One is a boundary control and the other is an internal control in the diffusion term. By means of the duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.

MSC

primary
93B05
secondary
93B07
93E20
60H15

Keywords

Stochastic Schrödinger equations
Exact controllability
Observability
Carleman estimate

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This work is partially supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7), the NSF of China under grant 11101070 and the Fundamental Research Funds for the Central Universities in China under grant ZYGX2012J115.

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