Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> A new approach to the steering problem for n-level quantum systems is described. Feedback control strategies for the Schrodinger equation of a multilevel quantum system are considered that eventually produce the decrease of the distance between the state and the target. Once these controls have been constructed via numerical integration of the nonlinear evolution, they can be implemented in open-loop. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> We propose feedback control laws for quantum systems, based on Lyapunov design. These feedback laws locally asymptotically stabilize any desired eigenstate of the system Hamiltonian. The target eigenstate and the size of its region of attraction can be tailored by the choice of design parameters. These feedback control laws are used to design open loop controls for state transfer problems. We illustrate the method with a transfer problem between bound states of a molecular Morse oscillator. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> A Lyapunov-based approach for trajectory tracking of the Schrodinger equation is proposed. In the finite dimensional case, convergence is precisely analyzed. Connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied. When the linearized system is controllable, such a feedback ensures almost global asymptotic convergence. When the linearized system is not controllable, the stability of the closed-loop system is not asymptotic. To overcome such lack of convergence, we propose, when the reference trajectory is an eigenstate, a modification based on adiabatic invariance. Simulations illustrate the simplicity and also the interest for trajectory generation. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> Local time control methods are used in the simulation of quantum control phenomena because they conveniently ensure an increase of a predefined performance index and also avoid singularities associated with tracking procedures. However, the drawback of the existing implementations is that they only take into account one-photon, direct transitions and may stop at nonoptimal values of the index. We propose in this paper a modification of the currently used algorithms that addresses this issue and explain how the convergence is improved. Furthermore, when iterations are required, we show that this approach can be inserted into a monotonically convergent algorithm. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> The feedback stabilization problem for ensembles of coupled spin 1/2 systems is discussed from a control theoretic perspective. The noninvasive nature of the bulk measurement allows for a fully unitary and deterministic closed loop. The Lyapunov-based feedback design presented does not require spins that are selectively addressable. With this method, it is possible to obtain control inputs also for difficult tasks, like suppressing undesired couplings in identical spin systems. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> The convex set of density operators of an N-level quantum mechanical system foliated as a complex flag manifold, where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix. For an isospectral bilinear control system evolving on such an orbit, the state feedback stabilization problem admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ldquoroot-spacerdquo-like structure of the cone of density operators. The converging conditions are time independent but depend on the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing topological obstructions to global stabilizability. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> 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. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Lyapunov-based control design <s> The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective. <s> BIB008
Lyapunov-based control methods are powerful tools for feedback controller design in classical control theory. In quantum control, the acquisition of feedback information through measurements usually destroys the state being measured, which makes it difficult to directly apply Lyapunov approaches to quantum feedback controller design. However, we may first complete the feedback control design by simulation on a computer, which will give a sequence of controls. Then we can apply the control sequence to the quantum system to be controlled in an open-loop form BIB003 , BIB007 , BIB001 - BIB006 . This is a "feedback design and open-loop control" strategy. This strategy is especially useful for some difficult quantum control tasks BIB005 . The most important aspects in Lyapunov-based control design include the construction of the Lyapunov function, the determination of the control law and the analysis of asymptotic convergence. To design an open-loop control law for quantum systems, several types of Lyapunov functions have been considered. For example, if the target state is |ψ f , we may select one of the following Lyapunov functions: BIB003 , or V 3 (t) = ψ(t)|P |ψ(t) , where P is a positive semidefinite Hermitian operator BIB002 . It is clear that V j (t) ≥ 0 (j = 1, 2, 3). We can now select the control function to guarantee that the first-order time derivative of the Lyapunov function is negative semidefinite. That is, we may determine the control law using the conditionV (t) ≤ 0 on the Lyapunov function V (t). For the case that the target state is an eigenstate of the free Hamiltonian H 0 , several algorithms have been developed to find corresponding control laws and their convergence has been analyzed BIB007 ,  , BIB002 . For the case that target states are not eigenstates of H 0 , the control problem can be formulated in terms of reference trajectory tracking BIB003 , BIB004 , BIB008 . Lyapunov-based techniques have also been formulated using density operators for the control of a spin ensemble BIB005 , BIB006 . LaSalle's invariance principle is also useful to characterize the asymptotic behavior of the system dynamics  .