Implicit Lyapunov control of finite dimensional Schrödinger equations

An implicit Lyapunov-based approach is proposed for generating trajectories of a finite dimensional controlled quantum system. The main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable. The controlled Lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem. The convergence analysis is done using LaSalle invariance principle. Closed-loop simulations illustrate the performance of such feedback laws for the open-loop control of a test case considered by chemists.     

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.     

Asymptotic stabilization with locally semiconcave control Lyapunov functions on general manifolds

Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configuration space is not a contractible manifold, we need to design a time-varying or discontinuous state feedback control for asymptotic stabilization at the desired equilibrium.     

Preservation of Lyapunov functions under bilinear mapping

Bilinear mapping preserves quadratic Lyapunov functions between continuous-time and discrete-time LTI systems. This note studies if this property holds for other types of Lyapunov functions specified by vector norms of the system state. Some implications of the obtained results in the common Lyapunov function problems are discussed.     

Controller design for nonlinear affine systems by control Lyapunov functions

This paper addresses the stabilization problems for nonlinear affine systems. First of all, the explicit feedback controller is developed for a nonlinear multiple-input affine system by assuming that there exists a control Lyapunov function. Next, based upon the homogeneous property, sufficient conditions for the continuity of the derived controller are developed. And then the developed control design methodology is applied to stabilize a class of nonlinear affine cascaded systems. It is shown that under some homogeneous assumptions on control Lyapunov functions and the interconnection term, the cascaded system can be globally stabilized. Finally, some interesting results of finite-time stabilization for nonlinear affine systems are also obtained.     

Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions

The constrained and unconstrained stabilisation problem of discrete-time bilinear systems is investigated. Using polyhedral Lyapunov functions, conditions for a polyhedral set to be both positively invariant and domain of attraction for systems with second-order polynomial nonlinearities are first established. Then, systematic methods for the determination of stabilising linear feedback for both constrained and unconstrained bilinear systems are presented. Attention is drawn to the case where no linear control law rendering the pre-specified desired domain of attraction positively invariant exists. For this case, an approach guaranteeing the existence of a possibly suboptimal solution is established.     

Lyapunov control methods of closed quantum systems

According to special geometric or physical meanings, the paper summarizes three Lyapunov functions in controlling closed quantum systems and their controller designing processes. Specially, for the average value-based method, the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity; for the error-based method, this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning. Also, we study the relations among the three Lyapunov functions and give a unified form of these Lyapunov functions. Finally, we compare the control effects of three Lyapunov methods by doing some simulation experiments.     

借助Lyapunov方法的量子系统平衡态的布居控制

This paper studies the population control problem associated with the equilibrium states of mixed-state quantum systems by using a Lyapunov function with degrees of freedom. The control laws are designed by ensuring the monotonicity of the Lyapunov function; main results on the largest invariant set in the sense of LaSalle are given; and the strict expression of any state in the largest invariant set is normally deduced in the framework of Bloch vectors. By analyzing the obtained largest invariant set and the Lyapunov function itself, this paper also discusses the determination problem of the degrees of freedom. Numerical simulation experiments on a three-level system show the validity of research results.     

Fourth-order algorithms for solving local Schrödinger equations in a strong magnetic field

We describe an efficient numerical method for solving eigenvalue problems associated with the one-body Schrödinger equation or the Kohn-Sham equations in an arbitrarily strong uniform external magnetic field. The eigenvalue problem is solved in real space by using a fourth order, forward factorization of the evolution operator e−εH, which is significantly more efficient than conventional second-order algorithms. In particular, the magnetic field is solved exactly by the decomposition process. The algorithm is applicable to any external potential, in addition to the magnetic field. We envision its primary application in the area of electronic structure calculations of quantum dots.     

理想条件下混合态量子系统的Lyapunov稳定化策略

在假定被控系统满足非退化、没有退化跃迁和完全连通的理想条件下,借助于带有附加自由度的Lyapunov函数研究了混合态量子系统的稳定化问题.基于LaSalle原理推导了闭环系统的最大不变集和任一初始态下的收敛状态集,给出了系统对最大不变集中任一平衡态渐近稳定化的自由度的构造原则.最后通过一个两能级系统的数值仿真,验证了所得理论结果的正确性.