finite time stability of quantum systems governed by schrodinger s equation in this paper we investigate the finite time stability fts and finite time boundedness ftb of quantum systems which are governed by schrodinger s equation we first study the fts and fts at one target state and propose new results for the mentioned quantum systems in the form of sufficient conditions then we present sufficient conditions for the ftb and ftb at one target state in terms of matrix inequalities one numerical simulation is presented to illustrate the effectiveness of the obtained theoretical result

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

engineering stable discrete time quantum dynamics via a canonical qr decomposition we analyze the asymptotic behavior of discrete time markovian quantum systems with respect to a subspace of interest global asymptotic stability of subspaces is relevant to quantum information processing in particular for initializing the system in pure states or subspace codes we provide a linear algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace we then construct a design algorithm for discrete time feedback control that allows to stabilize a target subspace proving that if the control problem is feasible then the algorithm returns an effective control choice in order to prove this result a canonical qr matrix decomposition is derived and also used to establish the control scheme potential for the simulation of open system dynamics

practical stabilization of a quantum particle in a one dimensional infinite square potential well we consider a nonrelativistic charged particle in a one dimensional infinite square potential well this quantum system is subjected to a control which is a uniform in space time depending electric field it is represented by a complex probability amplitude solution of a schrodinger equation on a one dimensional bounded domain with dirichlet boundary conditions we prove the almost global practical stabilization of the eigenstates by explicit feedback laws

sliding mode control of two level quantum systems this paper proposes a robust control method based on sliding mode design for two level quantum systems with bounded uncertainties an eigenstate of the two level quantum system is identified as a sliding mode the objective is to design a control law to steer the system s state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system hamiltonian we propose a controller design method using the lyapunov methodology and periodic projective measurements in particular we give conditions for designing such a control law which can guarantee the desired robustness in the presence of the uncertainties the sliding mode control method has potential applications to quantum information processing with uncertainties

control of quantum states in decoherence free subspaces the problem of quantum state control in open quantum systems based on the lyapunov method is studied the control laws are designed to drive the system to a pure target state in decoherence free subspace dfs in the interaction picture where the average of an observable operator p is chosen as a lyapunov function by analyzing the positive limit set via barbalat s lemma we propose a sufficient condition on the observable operator such that the positive limit set only contains the target state under this condition it is proved that the control laws can drive the open system to any pure target state in dfs in the interaction picture also we provide a method to construct the observable operator via schmidt orthogonalization numerical simulations on a three level system validate the effectiveness of the proposed control strategy

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

bang bang control design for quantum state transfer based on hyperspherical coordinates and optimal time energy control we present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates unlike many control schemes based on factorization of unitary operators the scheme gives explicit expressions for all generalized euler angles in terms of the hyperspherical coordinates of the initial and final states the factorization when applicable has added benefits that phase rotations can be combined and performed concurrently the control procedure can be realized using simple bang bang or square wavefunction controls optimal time energy control is considered to find the optimal control amplitude the extension of the scheme to implement arbitrary unitary operators is also discussed

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

stability analysis of discontinuous quantum control systems with dipole and polarizability coupling closed quantum systems under the influence of a laser field whose interaction is modeled by a schrodinger equation with a coupling control operator containing both a linear dipole and a quadratic polarizability term are analyzed discontinuous feedbacks obtained by a lyapunov trajectory tracking procedure have recently been proposed to control these types of system the purpose of this paper is to study the asymptotic stability by considering the solutions in the filippov sense the analysis is developed by applying a variant of the lasalle invariance principle for differential inclusions numerical simulations are included to illustrate the efficiency of the discontinuous control

controllability of quantum systems with switching control 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

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

solutions in closed form and as power series to the real lorenz equations using the method of the lie theory of extended groups and for the parameter values b 1 and r 0 we construct explicitly the general exact solution to the real lorenz equations in terms of jacobian elliptic functions in the context of our approach further possible completely integrable cases of the lorenz system are discussed by considering the result of the painleve analysis for 1 b 2 and r 1 9 and negative values of r the latter case r 0 and 0 not following from the painleve test for other positive parameter values and in the form of appropriate power series we find some particular exact solutions which do not possess the painleve property

analysis of effectiveness of lyapunov control for non generic quantum states a lyapunov based control design for natural trajectory tracking problems is analyzed for quantum states where the analysis in the generic case is not applicable using dynamical systems tools we show almost global asymptotic stability for stationary target states subject to certain conditions on the hamiltonians and discuss effectiveness of the design when these conditions are not satisfied for pseudo pure target states the effectiveness of the design is studied further for both stationary and non stationary states using alternative tools

asymptotic properties of feedback solutions for a class of quantum control problems control of quantum systems described by the schrodinger equation is considered feedback control laws are developed for orbit tracking via controlled hamiltonians and their asymptotic properties are analyzed numerical integration via time splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

observer based hamiltonian identification for quantum systems a symmetry preserving observer based parameter identification algorithm for quantum systems is proposed starting with a 2 level quantum system qubit where the unknown parameters consist of the atom laser frequency detuning and coupling constant we prove an exponential convergence result the analysis is inspired by lyapunov and adaptive control techniques and is based on averaging theory the observer is then extended to the multi level case where all the atom laser coupling constants are unknown the extension of the convergence analysis is discussed through some heuristic arguments the relevance and the robustness with respect to various noises are tested through numerical simulations

stabilization for an ensemble of half spin systems feedback stabilization of an ensemble of non interacting half spins described by the bloch equations is considered this system may be seen as an interesting example for infinite dimensional systems with continuous spectra we propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin 1 2 or 1 2 the proof of the convergence is done locally around the equilibrium in the h 1 topology this local convergence is shown to be a weak asymptotic convergence for the h 1 topology and thus a strong convergence for the c 0 topology the proof relies on an adaptation of the lasalle invariance principle to infinite dimensional systems numerical simulations illustrate the efficiency of these feedback laws even for initial conditions far from the equilibrium

a unification in the theory of linearization of second order nonlinear ordinary differential equations in this letter we introduce a new generalized linearizing transformation glt for second order nonlinear ordinary differential equations snodes the well known invertible point ipt and non point transformations npt can be derived as sub cases of the glt a wider class of nonlinear odes that cannot be linearized through npt and ipt can be linearized by this glt we also illustrate how to construct glts and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this glt for the snodes we demonstrate the theory with two examples which are of contemporary interest

switching control of closed quantum systems via the lyapunov method in this paper we consider the state transfer problem for closed quantum systems under a degenerate case where the linearized system around the target state is not controllable it is known that the traditional lyapunov control methods may fail to guarantee the convergence to the target state under the degenerate case hence we propose to use multiple lyapunov functions and design a switching control strategy to achieve more accurate state transfer it is shown that the system can converge to the intersection of invariant sets including the target state the explicit analysis of the convergence is provided to design the switching law moreover the effectiveness of open loop lyapunov control is discussed simulation studies are presented to show the improved control performance

on the determination of non local symmetries the importance of non local symmetries of differential equations lies in their manifestation as lie point symmetries of the equations resulting from reduction of order the reason for the determination of these symmetries in second order equations with only one lie point symmetry is self evident however the disadvantage of non local symmetries is that no systematic approach to their determination exists we present such an approach applicable to differential equations of any order and apply it to some second order ordinary differential equations and show that they have a rich occurrence we also look at possible generalizations of the concept of non local symmetries

feedback stabilization of isospectral control systems on complex flag manifolds application to quantum ensembles 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

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

the gradient flow for control of closed quantum systems the gradient flow is commonly used to numerically solve the problem of maximizing quantum observables for finite dimensional closed quantum systems in this note we analyze the asymptotic behavior of the gradient flow we show that under a regularity assumption on the controls the flow almost always converges to a solution of the maximization problem

a class of second order differential equations and related first order systems a class of second order nonlinear differential equations which arises in several branches of mathematical physics is considered it is shown that equations of this class may be factorised into first order equations of riccati type conditions are obtained on the coefficient functions of the second order equations for the first order equations to be of matrix riccati form whose solutions have a finite superposition property the factorisation into first order equations is then not unique and there is an alternative first order set of equations whose solutions do not have this superposition property a second order equation arising in the theory of pellet fusion processes is investigated in detail solutions are obtained when the corresponding first order equations are of matrix riccati from and shown to be equivalent to solutions derived by alternative methods lagrangian systems giving rise to equations of the class are also considered

implicit lyapunov control of finite dimensional schrodinger 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

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

analysis of lyapunov method for control of quantum states 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

stabilization of stochastic quantum dynamics via open and closed loop control in this paper we investigate parametrization free solutions of the problem of quantum pure state preparation and subspace stabilization by means of hamiltonian control continuous measurement and quantum feedback in the presence of a markovian environment in particular we show that whenever suitable dissipative effects are induced either by the unmonitored environment or by non hermitian measurements there is no need for feedback as open loop time invariant control is sufficient to achieve stabilization of the target set in probability constructive necessary and sufficient conditions on the form of the control hamiltonian can be provided in this case when time invariant control is not sufficient state stabilization can be attained by the addition of filtering based feedback control

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

optimal bilinear control of stochastic nonlinear schrodinger equations mass sub critical case we study optimal control problems for stochastic nonlinear schrodinger equations in both the mass subcritical and critical case for general initial data of the minimal l 2 regularity we prove the existence and first order lagrange condition of an open loop control in particular these results apply to the stochastic nonlinear schrodinger equations with the critical quintic and cubic nonlinearities in dimensions one and two respectively furthermore we obtain uniform estimates of backward stochastic solutions in new spaces of type u 2 and v 2 adapted to evolution operators related to linear schrodinger equations with lower order perturbations these estimates yield a new temporal regularity of backward stochastic solutions which is crucial for the tightness of approximating controls induced by ekeland s variational principle

symmetry singularities and integrability in complex dynamics ii rescaling and time translation in two dimensional systems the explicit integrability of second order ordinary differential equations invariant under time translation and rescaling is investigated quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the painleve and singularity properties of the different systems the transformation contains a parameter and for critical values intimately related to the possession of the painleve property in the parent second order equation one finds a difference from the generic behaviour this study is a prelude to a full discussion of the class of transformations which preserve the painleve property in the construction of quadratic systems from scalar nth order ordinary differential equations invariant under time invariant under time translation and rescaling

notes on sliding mode control of two level quantum systems in dong d and petersen i r 2012 sliding mode control of two level quantum systems automatica 48 725 735 a sliding mode control approach has been proposed for two level quantum systems to deal with bounded uncertainties in the system hamiltonian this paper further extends these results in two directions one extension is to consider the effect of uncertainties during the process of driving the system s state back to the sliding mode domain from outside and we propose two approaches to accomplish this control task the other extension generalizes the previous design approach to consider uncertainties described as perturbations in the free hamiltonian

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

the probability density of spectral estimates based on modified periodogram averages welch s 1967 method for spectral estimation of averaging modified periodograms has been widely used for decades because such an estimate relies on random data the estimate is also a random variable with some probability density function here the pdf of a power estimate is derived for an estimate based on an arbitrary number of frequency bins overlapping data segments amount of overlap and type of data window given a correlated gaussian input sequence the pdfs of several cases are plotted and found to be distinctly non gaussian the asymptotic result of averaging frequency bins and or data segments using the kullback leibler distance as a measure for limited numbers of frequency bins or data segments the precise pdf is considerably skewed and will be important in applications such as maximum likelihood tests

wavefunction controllability for finite dimensional bilinear quantum systems we present controllability results for quantum systems interacting with lasers exact controllability for the wavefunction in these bilinear systems is proved in the finite dimensional case under very natural hypotheses

properties of second order ordinary differential equations invariant under time translation and self similar transformation the general form of a second order ordinary differential equation invariant under time translation and self similarity is obtained together with its solution in parametric form next two representatives of two different families of such equations are studied in terms of rescaled phase space variables the first example has a simple physical interpretation and its limit cycle properties and period are easily derived similar results are found for the second example but the physical interpretation is less obvious the invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self similar solution sss exists

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

lyapunov control of bilinear schrodinger equations 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

rapid lyapunov control of finite dimensional quantum systems abstract rapid state control of quantum systems is significant in reducing the influence of relaxation or decoherence caused by the environment and enhancing the capability in dealing with uncertainties in the model and control process bang bang lyapunov control can speed up the control process but cannot guarantee convergence to a target state this paper proposes two classes of new lyapunov control methods that can achieve rapidly convergent control for quantum states one class is switching lyapunov control where the control law is designed by switching between bang bang lyapunov control and standard lyapunov control the other class is approximate bang bang lyapunov control where we propose two special control functions which are continuously differentiable and yet have a bang bang type property related stability results are given and a construction method for the degrees of freedom in the lyapunov function is presented to guarantee rapid convergence to a target eigenstate being isolated in the invariant set several numerical examples demonstrate that the proposed methods can achieve improved performance for rapid state control of quantum systems

controllability of quantum mechanical systems by root space decomposition of su n the controllability property of the unitary propagator of an n level quantum mechanical system subject to a single control field is described using the structure theory of semisimple lie algebras sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve