Implicit Lyapunov-based control strategy for closed quantum systems with dipole and polarizability coupling

In this paper, the state transfer of finite dimensional closed quantum systems with dipole and polarizability coupling in non-ideal cases is investigated. Two kinds of non-ideal systems are considered, where the internal Hamiltonian of the system is not strongly regular and not all the eigenvectors of the internal Hamiltonian are directly coupled with the target state. Such systems often exist in practical quantum systems such as the one-dimensional oscillator and coupled two-spin system. An implicit Lyapunov-based control strategy is proposed here with convergence analysis for quantum systems modeled by finite dimensional bilinear Schrodinger equations. Specifically, two kinds of Lyapunov functions are defined via implicit functions, and their existences are guaranteed with the help of a fixed point theorem. Then, the local convergence analysis is investigated with the explicit characterization of the largest invariant set by LaSalle invariance principle. Finally, the performance of the feedback design is illustrated by numerical simulations. Copyright (C) 2017 John Wiley & Sons, Ltd.