Quantum control theory and applications: A survey <s> Variable structure control <s> Abstract This paper formulates and solves a variable structure control problem for the state control of quantum systems. Following the concept of wavefunction controllability, we first present a specific quantum system which is wavefunction uncontrollable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. An extension and the potential applications of the proposed control strategy are briefly discussed and the general steps of variable structure control design for this class of quantum systems are also presented. The variable structure control scheme provides an alternative quantum engineering approach for the manipulation of quantum systems. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Variable structure control <s> This article focuses on wavefunction controllability of a class of quantum systems with switching control. First, we present a specific quantum system which is not wavefunction controllable under the action of any one of two controllers with different fixed structures. However, we prove that the system is wavefunction controllable if the control input is allowed to switch between the two controllers. Then we present a sufficient condition for wavefunction controllability of a class of quantum systems with switching control and briefly discuss the problem of control design for such quantum control systems. Finally, two illustrative examples are presented to demonstrate a possible control design process for the proposed method. <s> BIB002
For some quantum systems which are not controllable, we can introduce additional resources or special strategies to change the controllability of the quantum system or enhance our capability of controlling the quantum system. In this subsection, we introduce a variable structure control strategy to enhance our capability of controlling a class of quantum systems. In variable structure control, one can change the controller structure according to specified switching logic in order to obtain improved properties BIB001 . Consider an example corresponding to two special cases of (9): where A = diag{1.0, 1.2, 1.3, 2.0, 2.15}, and In BIB001 , it has been shown that the two models (28) and (29) are not individually pure state controllable but are controllable under a variable structure control strategy. Furthermore, consider a system (S m ): The corresponding connectivity graph of B u m is denoted as G m = (V m , E m ). It is clear that for arbitrary m, V m = V = {|φ 1 , . . . , |φ N }. Hence, we can write G m = (V, E m ) and define G = (V, E) as the combined graph of G m = (V, E m ) (m = 1, . . . , M ) where We then give the following theorem BIB002 .