radial basis function approximation from gridded centres to scattered centres abstract the paper studies l infinity ir d norm approximations from a space spanned by a discrete set of translates of a basis function theta attention here is restricted to functions theta whose fourier transform is smooth on ird 0 and has a singularity at the origin examples of such basis functions are the thin plate splines and the multiquadrics as well as other types of radial basis functions that are employed in approximation theory the above approximation problem is well understood in case the set of points used for translating theta forms a lattice in ir d and many optimal and quasi optimal approximation schemes can already be found in the literature in contrast only few mostly specific results are known for a set of scattered points the main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non uniform case we introduce a single relatively simple conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature more precisely to almost all stationary schemes in anticipation of future introduction of new schemes for uniform grids an effort is made to impose only a few mild conditions on the function theta which still allow for a unified error analysis to hold in the course of the discussion here the recent results of budl on scattered center approximation are reproduced and improved upon

control design of uncertain quantum systems with fuzzy estimators an approach of control design using fuzzy estimators fes is proposed for quantum systems with uncertainties two types of quantum control problems are considered 1 control of a pure state quantum system in the presence of uncertainties and 2 control design of quantum systems with initial mixed states and uncertainties for the first type of tasks a partial feedback control scheme with an fe is presented to design controllers in this scheme an fe is trained to estimate the quantum state for feedback control of a quantum system and controlled projective measurement is used to assist in controlling the system for the second type of quantum control tasks a probabilistic fuzzy estimator pfe is trained to estimate the quantum state for control design of a quantum system with an initial mixed state and a corresponding control algorithm is proposed to design a control law that drives the system from the mixed state to a target pure state two examples of two spin 1 2 systems are also presented and analyzed to demonstrate the process of control design and potential applications of the proposed approach

identification of unknown parameters for a class of two level quantum systems this technical note studies the identification of unknown decoherence rates for a class of two level quantum systems undergoing spontaneous emission our previous work shows that estimates of the unknown decoherence rates can be obtained from the ensemble averages by imposing constant control and monitoring a sequence of identical systems continuously inspired by the work this technical note further investigates the identification based on a single quantum system it is shown that a mean ergodic solution exists for a single system under constant control and convergence to an ergodic process can be guaranteed by introducing some noise into the control signals the ensemble average can thus be replaced by the time average of a single system and the parameter estimate can be obtained from a single implementation

optimal control of dynamic systems application to spline approximations generally classical polynomial splines tend to exhibit unwanted undulations in this work we discuss a technique based on control principles for eliminating these undulations and increasing the smoothness properties of the spline interpolants we give a generalization of the classical polynomial splines and show that this generalization is in fact a family of splines that covers the broad spectrum of polynomial trigonometric and exponential splines a particular element in this family is determined by the appropriate control data it is shown that this technique is easy to implement several numerical and curve fitting examples are given to illustrate the advantages of this technique over the classical approach finally we discuss the convergence properties of the interpolant

a quasi separation principle and newton like scheme for coherent quantum lqg control abstract this paper is concerned with the coherent quantum linear quadratic gaussian lqg control problem of constructing an optimal controller for linear quantum plants using a quadratic performance criterion coherent quantum feedback control does not employ classical measurements which inherently entail the loss of quantum information a coherent quantum controller is itself a quantum system and this imposes physical realizability pr constraints on the quantum stochastic differential equation which governs such a controller pr corresponds to the equivalence of the controller to an open quantum harmonic oscillator whereby its state space matrices are related to the hamiltonian and coupling operators of the oscillator the hamiltonian parameterization of the controller is combined with frechet differentiation of the lqg cost with respect to the state space matrices in order to obtain equations for the optimal controller a quasi separation property for the gain matrices of the quantum controller is established and a newton like iterative scheme for the numerical solution of the equations is outlined

reproduction of exponential polynomials by multivariate non stationary subdivision schemes with a general dilation matrix we study scalar multivariate non stationary subdivision schemes with a general integer dilation matrix we characterize the capability of such schemes to reproduce exponential polynomials in terms of simple algebraic conditions on their symbols these algebraic conditions provide a useful theoretical tool for checking the reproduction properties of existing schemes and for constructing new schemes with desired reproduction capabilities and other enhanced properties we illustrate our results with several examples

non markovian quantum feedback networks ii controlled flows the concept of a controlled flow of a dynamical system especially when the controlling process feeds information back about the system is of central importance in control engineering in this paper we build on the ideas presented by bouten and van handel quantum stochastics and information statistics filtering and control world scientific 2008 and develop a general theory of quantum feedback we elucidate the relationship between the controlling processes z and the measured processes y and to this end we make a distinction between what we call the input picture and the output picture we should note that the input output relations for the noise fields have additional terms not present in the standard theory but that the relationship between the control processes and measured processes themselves is internally consistent we do this for the two main cases of quadrature measurement and photon counting measurement the theory is general enough to include a modulating filter which post processes

on polynomial ideals of finite codimension with applications to box spline theory we investigate the relation between an ideal i of finite codimension in the space of multivariate polynomials and ideals which are generated by lower order perturbations of some generators for i of particular interest are the codimension of these ideals and the local approximation order of their kernels the discussion stimulated by recent results in approximation theory allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines we describe their structure dimension and local approximation order and an algorithm for their construction the resulting theory is extended to subspaces of the above exponential polynomial spaces

a frequency domain condition for the physical realizability of linear quantum systems a recently emerging approach to the feedback control of linear quantum systems involves the use of a controller which itself is a quantum linear system this approach to quantum feedback control referred to as coherent quantum feedback control has the advantage that it does not destroy quantum information is fast and has the potential for efficient implementation an important issue which arises both in the synthesis of linear coherent quantum controllers and in the modeling of linear quantum systems is the issue of physical realizability this issue relates to the property of whether a given set of linear quantum stochastic differential equations corresponds to a physical quantum system satisfying the laws of quantum mechanics under suitable assumptions the paper shows that the question of physical realizability is equivalent to a frequency domain j j unitary condition this is important in controller synthesis since it is the transfer function matrix of the controller which determines the closed loop system behavior

steady turbulent flow computations using a low mach fully compressible scheme a recently proposed modification to fully compressible schemes significantly improves the resolution of low mach number features tackling the problem of excessive numerical dissipation as the mach number reduces this paper explores the application of this modification to reynolds averaged navier stokes simulations using second and fifth order in space schemes following verification of the modeling approach four test cases are employed to highlight the scheme performance including a backward facing step a two dimensional lid driven cavity the naca 4412 airfoil incorporating a trailing edge separation and finally a mach 2 25 oblique shock wave boundary layer interaction it demonstrates first that the converged solution is grid and scheme independent as expected and that the low mach number correction provides a significant improvement in low mach number regions because this correction is easy to implement computationally inexpensive and stable the conclusion is that it should be recommend

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

transfer function approach to quantum control part i dynamics of quantum feedback systems this paper gives a unified approach to feedback control theory of quantum mechanical systems of bosonic modes described by noncommutative operators a quantum optical closed loop including a plant and controller is developed and its fundamental structural properties are analyzed extensively from a purely quantum mechanical point of view in order to facilitate the use of control theory in microscopic world described by quantum theory in particular an input output description of quantum mechanical systems which is essential in describing the behavior of the feedback systems is fully formulated and developed this would then provide a powerful tool for quantum control and pave an avenue that connects control theory to quantum dynamics this paper is divided into two parts the first part is devoted to the basic formulation of quantum feedback control via quantum communication and local operations on an optical device cavity that can be regarded as a unit of quantum dynamics of bosonic modes the formulation introduced in this paper presents the feature intrinsic in quantum feedback systems based on quantum stochastic differential equations the input output description provides a basis for developing quantum feedback control through the transfer function representation of quantum feedback systems in the follow up paper the quantum mechanical representation of feedback is further elaborated to yield the control theoretical representation of fundamental notions of quantum theory uncertainty principle e g and some applications are presented

zonotopal algebra and forward exchange matroids abstract zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors x it connects objects from combinatorics geometry and approximation theory the origin of zonotopal algebra is the pair d x p x where d x denotes the dahmen micchelli space that is spanned by the local pieces of the box spline and p x is a space spanned by products of linear forms the first main result of this paper is the construction of a canonical basis for d x we show that it is dual to the canonical basis for p x that is already known the second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by ardila postnikov holtz ron holtz ron xu li ron and others we call the underlying combinatorial structure of those spaces forward exchange matroid a forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom

quantum feedback networks hamiltonian formulation a quantum network is an open system consisting of several component markovian input output subsystems interconnected by boson field channels carrying quantum stochastic signals generalizing the work of chebotarev and gregoratti we formulate the model description by prescribing a candidate hamiltonian for the network including details of the component systems the field channels their interconnections interactions and any time delays arising from the geometry of the network we show that the candidate is a symmetric operator and proceed modulo the proof of self adjointness the model is non markovian for finite time delays but in the limit where these delays vanish we recover a markov model and thereby deduce the rules for introducing feedback into arbitrary quantum networks the type of feedback considered includes that mediated by the use of beam splitters we are therefore able to give a system theoretic approach to introducing connections between quantum mechanical state based input output systems and give a unifying treatment using non commutative fractional linear or mobius transformations

on physical realizability of nonlinear quantum stochastic differential equations in this article we study physical realizability for a class of nonlinear quantum stochastic differential equations qsdes physical realizability is a property in which a qsde corresponds to the dynamics of an open quantum system we derive a sufficient and necessary condition for a nonlinear qsde to be physically realizable

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

non markovian quantum feedback networks i quantum transmission lines lossless bounded real property and limit markovian channels the purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems is via a non markovian channel this is a general feature of quantum transmission lines our ultimate objective is to extend the network rules that have been developed for markovian models to this end we recall the hamiltonian description of such non markov models of transmission lines and their quantization these have occurred in the context of non equilibrium thermodynamics but our interest is in the transmission lines as carriers of information rather than heat baths we show that there is an analytic scattering matrix associated with these models and that stability may be formulated in terms of the lossless bounded real property noting that the input and output fields do not separately satisfy a non self demolition principle we discuss the rigorous limit in which such models appear markov and so amenable to standard approaches of quantum filtering an

a discrete invitation to quantum filtering and feedback control the engineering and control of devices at the quantum mechanical level such as those consisting of small numbers of atoms and photons is a delicate business the fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise making this a novel field of application for stochastic estimation and control theory in this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance the model is extremely rich and allows a full development of the theory while remaining completely within the setting of finite dimensional hilbert spaces thus avoiding the technical complications of the continuous theory we introduce discretized models of an atom in interaction with the electromagnetic field obtain filtering equations for photon counting and homodyne detection and solve a stochastic control problem using dynamic programming and lyapunov function methods

small time controllability of systems on compact lie groups and spin angular momentum in this article we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact lie groups the main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group this under suitable assumptions is proved to be a lie subgroup of the underlying lie group we apply these results to the analysis of the controllability of particles with spin the results are motivated by and generalize the results in another work d d alessandro sys control lett 41 213 221 2000 where the specific model of a spin 12 particle system in an electro magnetic field was considered

on lagrange and hermite interpolation in rk a method for the construction of a set of data of interpolation in several variables is given the resulting data which are either function values or directional derivatives values give rise to a space of polynomials in such a way that unisolvence is guaranteed the interpolating polynomial is calculated using a procedure which generalizes the newton divided differences formula for a single variable

coherent h infty control for a class of annihilation operator linear quantum systems this paper considers a coherent h control problem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only for this class of quantum systems a solution to the h control problem can be obtained in terms of a pair of complex algebraic riccati equations in addition the question of physical realizability of the resulting quantum controllers is related to a bounded real property

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

hierarchical zonotopal spaces zonotopal algebra interweaves algebraic geometric and combinatorial properties of a given linear map x of basic significance in this theory is the fact that the algebraic structures are derived from the geometry via a non linear procedure known as the least map and that the statistics of the algebraic structures e g the hilbert series of various polynomial ideals are combinatorial i e computable using a simple discrete algorithm known as the valuation function on the other hand the theory is somewhat rigid since it deals for the given x with exactly two pairs each of which is made of a nested sequence of three ideals an external ideal the smallest a central ideal the middle and an internal ideal the largest in this paper we show that the fundamental principles of zonotopal algebra as described in the previous paragraph extend far beyond the setup of external central and internal ideals by building a whole hierarchy of new combinatorially defined zonotopal spaces ams subject classification 13f20 13a02 16w50 16w60 47f05 47l20 05b20 05b35 05b45 05c50 52b05 52b12 52b20 52c07 52c35 41a15 41a63

decoherence free linear quantum subsystems this paper provides a general theory for characterizing and constructing a decoherence free df subsystem for an infinite dimensional linear open quantum system the main idea is that based on the heisenberg picture of the dynamics rather than the commonly taken schrodinger picture the notions of controllability and observability in control theory are employed to characterize a df subsystem a particularly useful result is a general if and only if condition for a linear system to have a df component this condition is used to demonstrate how to actually construct a df dynamics in some specific examples it is also shown that as in the finite dimensional case we are able to do coherent manipulation and preservation of a state of a df subsystem

on multivariate e splines abstract in a recent interesting paper a ron 14 initiated the study of cardinal exponential splines in several variables beyond settling some open problems which were suggested by his results this paper addresses a variety of additional questions of an algebraic nature thus some of the results we obtained elsewhere on cube spline surfaces appear now as a special case of the present general setting our approach is critically based on some new bounds and formulas for the dimension of certain intersections of null spaces of commuting linear operators where the particular way of forming these intersections is determined by a matroid structure specifically this allows us to characterize the local linear independence of translates of the exponential cube spline as well as to study local algebraic properties of functions spanned by them only special cases were covered by ron s investigation furthermore we consider in detail the construction of dual bases linear projectors linear dependence relations among these exponential splines and subdivision algorithms for their approximate evaluation the analysis of these latter algorithms is based on the concept of discrete exponential splines we investigate to what extent these discrete objects possess analogous properties to their continuous counterparts finally the combinatorial interpretation of the discrete splines their relationship to their continuous counterparts and the machinery we develop leads to interesting combinatorial applications for enumerating the number of solutions to linear diophantine systems and corresponding reciprocity relations an announcement of these results appeared in 10

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

coherent robust h control of linear quantum systems with uncertainties in the hamiltonian and coupling operators abstract this paper considers a class of uncertain linear quantum systems subject to uncertain perturbations both in the system hamiltonian and in the coupling operators a sufficient condition is provided such that these uncertain quantum systems can be guaranteed to be robustly strict bounded real with a given disturbance attenuation parameter a scaled h problem is shown to have a connection with the stability of the given uncertain quantum system and can be used as an auxiliary system to design a robust h quantum controller a pair of riccati equations is used to give explicit formulas for the parameters of a desired controller illustrative examples show that for the given uncertain quantum system the method presented in this paper has improved performance over the existing quantum h control results without considering uncertainties

translates of exponential box splines and their related spaces exponential box splines eb splines are multivariate compactly supported functions on a regular mesh which are piecewise in a space z spanned by exponential polynomials this space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients the main part of this paper is devoted to algebraic analysis of the space h of all entire functions spanned by the integer translates of an eb spline this investigation relies on a detailed description of z and its discrete analog s the approach taken here is based on the observation that the structure of z is relatively simple when z is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process also we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of fourier analysis thus while generalizing the known theory of polynomial box splines the results here offer a simpler approach and a new insight towards this important special case we also identify and study in detail several types of singularities which occur only for complex eb splines the first is when the fourier transform of the eb spline vanishes at some critical points the second is when z cannot be embedded in y and the third is when h is a proper subspace of z we show among others that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh

the series product and its application to quantum feedforward and feedback networks the purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks these methods build on and generalize existing methods for series or cascade connections by allowing for more general interfaces and by introducing an efficient algebraic tool the series product we also introduce another product which we call the concatenation product that is useful for assembling and representing systems without necessarily having connections we show how the concatenation and series products can be used to describe feedforward and feedback networks a selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology

illustrating the geometry of coherently controlled unital open quantum systems we extend standard markovian open quantum systems quantum channels by allowing for hamiltonian controls and elucidate their geometry in terms of lie semigroups for standard dissipative interactions with the environment and different coherent controls we particularly specify the tangent cones lie wedges of the respective lie semigroups of quantum channels these cones are the counterpart of the infinitesimal generator of a single one parameter semigroup they comprise all directions the underlying open quantum system can be steered to and thus give insight into the geometry of controlled open quantum dynamics such a differential characterization is highly valuable for approximating reachable sets of given initial quantum states in a plethora of experimental implementations

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

optimal control of two level quantum systems we study the manipulation of two level quantum systems this research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two level quantum system a quantum bit we consider the problem of driving the evolution operator to a desired state while minimizing an energy type cost mathematically this problem translates into an optimal control problem for systems varying on the lie group of special unitary matrices of dimension two with cost that is quadratic in the control we develop a comprehensive theory of optimal control for two level quantum systems this includes in particular a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions the impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed

on realization theory of quantum linear systems the purpose of this paper is to study the realization theory of quantum linear systems it is shown that for a general quantum linear system its controllability and observability are equivalent and they can be checked by means of a simple matrix rank condition based on controllability and observability a specific realization is proposed for general quantum linear systems in which an uncontrollable and unobservable subspace is identified when restricted to the passive case it is found that a realization is minimal if and only if it is hurwitz stable computational methods are proposed to find the cardinality of minimal realizations of a quantum linear passive system it is found that the transfer function g s of a quantum linear passive system can be written as a fractional form in terms of a matrix function s moreover g s is lossless bounded real if and only if s is lossless positive real a type of realization for multi input multi output quantum linear passive systems is derived which is related to its controllability and observability decomposition two realizations namely the independent oscillator realization and the chain mode realization are proposed for single input single output quantum linear passive systems and it is shown that under the assumption of minimal realization the independent oscillator realization is unique and these two realizations are related to the lossless positive real matrix function s

quantum linear coherent controller synthesis a linear fractional representation approach abstract this paper is concerned with a linear fractional representation approach to the synthesis of linear coherent quantum controllers for a given linear quantum plant the plant and controller represent open quantum harmonic oscillators and are modelled by linear quantum stochastic differential equations the feedback interconnections between the plant and the controller are assumed to be established through quantum bosonic fields in this framework conditions for the stabilization of a given linear quantum plant via linear coherent quantum feedback are addressed using a stable factorization approach the class of all stabilizing quantum controllers is parameterized in the frequency domain coherent quantum weighted 2 and control problems for linear quantum systems are formulated in the frequency domain finally a projected gradient descent scheme is outlined for the coherent quantum weighted 2 control problem

power spectrum identification for quantum linear systems abstract we investigate system identification for general quantum linear systems in the situation where the input field is prepared as stationary squeezed quantum noise in this regime the output field is characterised by the power spectrum which encodes covariance of the output state we address which parameters can be identified from the power spectrum and how to construct a system realisation from the power spectrum the power spectrum depends on the system parameters via the transfer function we show that the transfer function can be uniquely recovered from the power spectrum so that equivalent systems are related by a symplectic transformation

local approximation by certain spaces of exponential polynomials approximation order of exponential box splines and related interpolation problems local approximation order to smooth complex valued functions by a finite dimensional space 7b h spanned by certain products of exponentials by polynomials is investigated the results obtained together with a suitable quasi interpolation scheme are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline the analysis of a typical 7b h is based here on the identification of its dual with a certain space 7b p of multivariate polynomials this point of view allows us to solve a class of multivariate interpolation problems by the polynomials from 7b p

quantum dissipative systems and feedback control design by interconnection the purpose of this paper is to extend j c willems theory of dissipative systems to open quantum systems described by quantum noise models this theory which combines ideas from quantum physics and control theory provides useful methods for analysis and design of dissipative quantum systems we describe the interaction of the plant and a class of external systems called exosystems in terms of feedback networks of interconnected open quantum systems our results include an infinitesimal characterization of the dissipation property which generalizes the well known positive real and bounded real lemmas and is used to study some properties of quantum dissipative systems we also show how to formulate control design problems using network models for open quantum systems which implements willems control by interconnection for open quantum systems this control design formulation includes for example standard problems of stabilization regulation and robust control

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

feedback control of quantum state reduction feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement we formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter we explore the use of stochastic lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability

characteristic operator functions for quantum input plant output models and coherent control we introduce the characteristic operator as the generalization of the usual concept of a transfer function of linear input plant output systems to arbitrary quantum nonlinear markovian input output models this is intended as a tool in the characterization of quantum feedback control systems that fits in with the general theory of networks the definition exploits the linearity of noise differentials in both the plant heisenberg equations of motion and the differential form of the input output relations mathematically the characteristic operator is a matrix of dimension equal to the number of outputs times the number of inputs which must coincide but with entries that are operators of the plant system in this sense the characteristic operator retains details of the effective plant dynamical structure and is an essentially quantum object we illustrate the relevance to model reduction and simplification definition by showing that the convergence of the characteristic operator in adiabatic eliminatio

constructive control of quantum systems using factorization of unitary operators we demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or gaussian wavepackets to illustrate the technique it is applied to find control schemes to achieve population transfers for pure state systems complete inversions of the ensemble populations for mixed state systems create arbitrary superposition states and optimize the ensemble average of dynamic observables

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

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

an algorithm for les of premixed compressible flows using the conditional moment closure model a novel numerical method has been developed to couple a recent high order accurate fully compressible upwind method with the conditional moment closure combustion model the governing equations turbulence modelling and numerical methods are presented in full the new numerical method is validated against direct numerical simulation dns data for a lean premixed methane slot burner although the modelling approaches are based on non premixed flames and hence not expected to be valid for a wide range of premixed flames the predicted flame is just 10 longer than that in the dns and excellent agreement of mean mass fractions conditional mass fractions and temperature is demonstrated this new numerical method provides a very useful framework for future application of cmc to premixed as well as non premixed combustion

low frequency approximation for a class of linear quantum systems using cascade cavity realization abstract this paper presents a method for approximating a class of complex transfer function matrices corresponding to physically realizable complex linear quantum systems the class of linear quantum systems under consideration includes interconnections of passive optical components such as cavities beam splitters phase shifters and interferometers this approximation method builds on a previous result for cascade realization and gives good approximations at low frequencies

singular perturbation approximations for a class of linear quantum systems this technical note considers the use of singular perturbation approximations for a class of linear quantum systems arising in the area of linear quantum optics the technical note presents results on the physical realizability properties of the approximate system arising from singular perturbation model reduction

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

the transfer function of generic linear quantum stochastic systems has a pure cascade realization this paper establishes that generic linear quantum stochastic systems have a pure cascade realization of their transfer function generalizing an earlier result established only for the special class of completely passive linear quantum stochastic systems in particular a cascade realization therefore exists for generic active linear quantum stochastic systems that require an external source of quanta to operate the results facilitate a simplified realization of generic linear quantum stochastic systems for applications such as coherent feedback control and optical filtering the key tools that are developed are algorithms for symplectic qr and schur decompositions it is shown that generic real square matrices of even dimension can be transformed into a lower 2 2 block triangular form by a symplectic similarity transformation the linear algebraic results herein may be of independent interest for applications beyond the problem of transfer function realization for quantum systems numerical examples are included to illustrate the main results in particular one example describes an equivalent realization of the transfer function of a nondegenerate parametric amplifier as the cascade interconnection of two degenerate parametric amplifiers with an additional outcoupling mirror

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

h infty control of linear quantum stochastic systems the purpose of this paper is to formulate and solve a h infin controller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology the paper includes results on the class of such systems for which the quantum commutation relations are preserved such a requirement must be satisfied in a physical quantum system a quantum version of standard classical dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived this enables a quantum version of the two riccati solution to the h infin control problem to be presented this result leads to controllers which may be realized using purely quantum purely classical or a mixture of quantum and classical elements this issue of physical realizability of the controller is examined in detail and necessary and sufficient conditions are given our results are constructive in the sense that we provide explicit formulas for the hamiltonian function and coupling operator corresponding to the controller

non commutative markov chains and multi analytic operators journal of mathematical analysis and applications volume 364 issue 1 1 april 2010 pages 275 288

bounded real properties for a class of annihilation operator linear quantum systems this paper considers the bounded real properties for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only the paper considers complex quantum versions of the bounded real lemma the strict bounded real lemma and the lossless bounded real lemma for the class of quantum systems under consideration it is shown that the question of physical realizability is related to the bounded real and lossless bounded real properties

transfer function approach to quantum control part ii control concepts and applications based on the stochastic differential equation of quantum mechanical feedback obtained in the first part of this paper detailed control concepts and applications are discussed for quantum systems interacting with a noncommutative noise source a feedback system in our framework is purely nonclassical in the sense that feedback control is performed via local operation and quantum communication through a quantum channel the role of the controller is to alter the quantum dynamic characteristics of the plant through entanglement shared between the plant and controller by sending quantum states that is modulated by the hamiltonian on the controller the input output relation of quantum systems provides a natural extension of control theory to the quantum domain this enables one to present a control theoretical interpretation of some fundamental quantum mechanical notions such as the uncertainty principle but also to find applications of ideas and tools of control theory one of the most important applications is the production of squeezed states which has been an important issue of quantum theory in relation to quantum computation and quantum communication the method proposed here reduces the application to a conventional noise reduction problem with feedback the h sub spl infin control then leads to complete squeezing

coherent feedback control of linear quantum optical systems via squeezing and phase shift the purpose of this paper is to present a theoretic and numerical study of utiliz ing squeezing and phase shift in coherent feedback control of linear quantum optical systems a quadrature representation with built in phase shifters is proposed for such systems fundamen tal structural characterizations of linear quantum optical systems are derived in terms of the new quadrature representation these results reveal considerable insights into the issue of the physical realizability of such quantum systems the problem of coherent quantum linear quadratic gaussian lqg feedback control studied in h i nurdin m r james and i r petersen automatica ifac 45 2009 pp 1837 1846 g zhang and m r james ieee trans automat control 56 2011 pp 1535 1550 is reinvestigated in depth first the optimization methods in these papers are extended to a multistep optimization algorithm which utilizes ideal squeezers second a two stage optimization approach is proposed on the basis of controller parametrization numerical studies show that closed loop systems designed via the second approach may offer lqg control performance even better than that when the closed loop systems are in the vacuum state when ideal squeezers in a closed loop system are replaced by more realistic degenerate parametric amplifiers a sufficient condition is derived for the asymptotic stability of the resultant new closed loop system the issue of performance convergence is also discussed in the lqg control setting

brief paper cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control this paper presents a realization algorithm for a class of complex transfer function matrices corresponding to physically realizable linear quantum systems the aim of the realization algorithm is to enable a coherent quantum feedback controller which has been synthesized using methods such as quantum h control or quantum lqg control to be constructed using optical components such as cavities and phase shifters the class of linear quantum systems under consideration are passive linear quantum systems which can be described purely in terms of annihilation operators the proposed algorithm enables a complex transfer function matrix to be realized as a pure cascade connection involving only cavities and phase shifters

analysis of quantum linear systems response to multi photon states the purpose of this paper is to present a mathematical framework for analyzing the response of quantum linear systems driven by multi photon states both the factorizable namely no correlation among the photons in the channel and unfactorizable multi photon states are treated pulse information of a multi photon input state is represented in terms of tensor and the response of quantum linear systems to multi photon input states is characterized by tensor operations analytic forms of output correlation functions and output states are derived the proposed framework is applicable no matter whether the underlying quantum dynamic system is passive or active examples from the physics literature are used to illustrate the results presented

direct and indirect couplings in coherent feedback control of linear quantum systems the purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems a general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented fundamental properties of stability dissipation passivity and gain for this class of linear quantum models are presented and characterized using complex lyapunov equations and linear matrix inequalities lmis coherent h and lqg synthesis methods are extended to accommodate direct couplings using multistep optimization examples are given to illustrate the results

degrees of controllability for quantum systems and application to atomic systems precise definitions for different degrees of controllability for quantum systems are given and necessary and sufficient conditions for each type of controllability are discussed the results are applied to determine the degree of controllability for various atomic systems with degenerate energy levels and transition frequencies