periodic solutions of some infinite dimensional hamiltonian systems associated with non linear partial difference equations ii this is our second paper devoted to the study of some non linear schrodinger equations with random potential we study the non linear eigenvalue problems corresponding to these equations we exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails contrary to our results in the first paper the results established in the present paper hold for an arbitrary strength of the non linear cubic term in the non linear schrodinger equation

effective hamiltonians and phase diagrams for tight binding models we present rigorous results for several variants of the hubbard model in the strong coupling regime we establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are in fact leading order terms of well defined volume independent unitarily equivalent interactions in addition in the very asymmetric falicov kimball regime we are able to apply recently developed phase diagram technology quantum pirogov sinai theory to conclude that the zero temperature phase diagrams obtained for the leading classical part remain valid except for thin excluded regions and small deformations for the full fledged quantum interaction at zero or low temperature moreover the phase diagram is stable against addition of arbitrary but sufficiently small further quantum terms that do not break the ground state symmetries this generalizes and unifies a number of previous results on the subject in particular published results on the zero temperature phase diagram of the falikov kimball model with and without magnetic flux are extended to small temperatures and or small ionic hopping we give explicit expressions for the first few orders in the hopping amplitude of equivalent interactions and we describe the resulting phase diagram our approach yields algorithms to compute equivalent interactions to arbitrarily high order in the hopping amplitude

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

minimax linear observers and regulators for stochastic systems with uncertain second order statistics the problem of minimax design of linear observers and regulators for linear time varying multivariable stochastic systems with uncertain models of their second order statistics is treated in this paper general classes of allowable covariance matrices and means of the process and observation noises and of the random initial condition are considered a game formulation of the problem is adopted and it is shown that the optimal filter for the least favorable set of covariances is minimax robust for each of the filtering situations analyzed conditions satisfied by the saddle point solutions are given and their utility for finding the worst case covariances is illustrated by way of several examples of uncertainty classes of practical interest

characterization of robust controllers abstract this paper presents a procedure for designing a robust controller which is able to stabilize a multivariable system with interval uncertainties in the system parameters the uncertain parameters are assumed to lie within known intervals the fundamental idea behind the procedure is to characterize the set of robust controllers as a convex set by the new matrix inequalities result a suitable controller can then be chosen by means of an optimization algorithm the controller thus chosen will be able to ensure that the closed loop matrix lies within desired intervals for all the prescribed interval uncertainties in the system parameters

analysis of continuous time kalman filtering under incorrect noise covariances abstract the behavior of the continuous time kalman filter under incorrect noise covariances is analyzed the filter performance is quantified by the actual state error covariance through this quantity the characteristic of the filter is examined convergence and divergence analyses of the actual state error covariance are given the significance of the results presented in the paper is that they help one to understand and be able to predict certain behavior of the kalman filter when inexact values of noise covariances are used

a unique representation of polyhedral types centering via mobius transformations for n 3 distinct points in the d dimensional unit sphere there exists a mobius transformation such that the barycenter of the transformed points is the origin this mobius transformation is unique up to post composition by a rotation we prove this lemma and apply it to prove the uniqueness part of a representation theorem for 3 dimensional polytopes as claimed by ziegler 1995 for each polyhedral type there is a unique representative up to isometry with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere

stability and performance robustness for multivariable linear systems abstract this paper deals with both aspects of stability and performance robustness of linear systems in the time domain a linear time invariant feedback control law which yields the largest set of parameter variations is first determined it is then used as an initial guess for designing a robust control law which produces satisfactory performance for each of several operating points of the system given a priori the stability robustness properties of the resulting control law is also studied the proposed approach is applied to an example yielding larger stability bounds including satisfactory performance in comparison with recently reported results

stability radii of positive linear functional differential equations under multi perturbations we study stability radii of linear retarded systems described by general linear functional differential equations a lower and an upper bound for the complex stability radius with respect to multi perturbations are given furthermore in some special cases concerning the structure matrices the complex stability radius can precisely be computed via the associated transfer function then the class of positive linear retarded systems is studied in detail it is shown that for this class complex real and positive stability radius under multi perturbations or multi affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

how to cage an egg this paper proves that given a convex polyhedron p subset mathbb r 3 and a smooth strictly convex body k subset mathbb r 3 there is some convex polyhedron q combinatorically equivalent to p which midscribes k that is all the edges of q are tangent to k furthermore with some stronger smoothness conditions on k the space of all such q is a six dimensional differentiable manifold

improved measures of stability robustness for linear state space models in this paper the aspect of stability robustness of linear systems is analyzed in the time domain a bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a lyapunov matrix equation solution the resulting bound is shown to be an improved bound over the ones recently reported in the literature also special cases of the nominal system matrix are considered for which the bound is given in terms of the nominal matrix thereby avoiding the solution of the lyapunov matrix equation examples given include comparison of the proposed approach with the recently reported results

uniqueness of the ground state in weak perturbations of non interacting gapped quantum lattice systems we consider a general weak perturbation of a non interacting quantum lattice system with a non degenerate gapped ground state we prove that in a finite volume the dependence of the ground state on the boundary condition exponentially decays with the distance to the boundary which implies in particular that the infinite volume ground state is unique also equivalent forms of boundary conditions for ground states of general finite quantum systems are discussed

robust properties of periodic discrete and multirate systems for discrete periodic and multirate systems new bounds are derived for three different robustness problems disturbance attenuation and stability with time varying structured and unstructured parameter variations these results generalize the single rate results to allow multirate capability

comments on delay independent robust stability of uncertain linear systems abstract the purpose of this note is to give a counterexample to a possible implication of the main result of the above paper 1

stability robustness of the continuous time lqg system under plant perturbation and noise uncertainty abstract in this paper the robustness of the continuous time lqg problem is studied the system to be controlled is described by the usual state space formulation that includes the structure of plant parameter perturbations and noise uncertainty a simple method is given to prove the saddle point inequality of the closed loop system performance to cope with noise uncertainty one sufficient condition which is applicable to general linear systems for system stability robustness under plant perturbation is presented

perturbation bounds for robust stability in linear state space models abstract this paper addresses the aspect of stability robustness of linear systems in the time domain upper bounds on the linear perturbation of an asymptotically stable linear system are obtained to maintain stability both for structured as well as unstructured perturbations using the lyapunov approach for structured perturbation the resulting bound is such that it garners the structural information about the nominal as well as the perturbation matrix into a single unified expression in the case of unstructured perturbations special features of the nominally stable matrix are exploited resulting in simpler expressions for the bound without the need to solve the lyapunov equation improvement of the proposed measures is illustrated with the help of examples

characterization of allowable perturbations for robust stability the problem of determining stability for a closed loop linear multivariable process in the presence of ill defined process perturbations is considered it is shown that m matrices may be used to characterize a class of allowable perturbations

characterization of allowable perturbations of a closed loop system with pole placement controller the problem of determining stability of a closed loop transfer function in the presence of coefficient perturbations is considered the results of interval polynomial theory are used to characterize the allowable interval of variation

a first order lyapunov robustness method for linear systems with uncertain parameters a method for stability robustness analysis based on a quadratic lyapunov function that varies linearly with uncertainty parameters is derived linear time invariant systems with structured uncertainties are discussed the lyapunov function is optimized numerically to maximize the robustness region in parameter space numerical results are given for four examples in which the first order method is compared to previous lyapunov methods for robustness analysis while the zero order method is slightly better than the first order method for one example the first order method is clearly superior in the other three more realistic examples the first order method is especially superior for applications to active control of flexible structures where robustness with respect to unmodeled coupling between modeled modes are important issues for such applications the first order method is much better at detecting the increased robustness associated with increased separation between frequencies

lqg optimal control system design under plant perturbation and noise uncertainty a state space approach abstract in this paper the stochastic linear dynamic system with uncertain parametric perturbations and uncertain noise covariances is studied based on minimax theory and the bellman gronwall inequality a sufficient condition for robust stability is derived finally a robust lqg optimal control design technique is employed to treat this problem

minimax control of linear stochastic systems with noise uncertainty the problem of linear quadratic gaussian control of multivariable linear stochastic systems with uncertain second order statistical properties is considered uncertainty is modeled by allowing process and observation noise spectral density matrices to vary arbitrarily within given classes and a minimax control formulation is applied to the quadratic objective functional general theorems proving the existence and characterization of saddle point solutions to this problem are presented and the relationship of these results to earlier results on minimax state estimation are discussed to illustrate the analytical results the specific example of regulating a double integrator plant is treated in detail

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

robust filtering and prediction for linear systems with uncertain dynamics a game theoretic approach we examine the existence and behavior of game theoretic solutions for robust linear filters and predictors our basic uncertainty class includes m th order time varying discrete time systems with uncertain dynamics uncertain initial state covariance and uncertain nonstationary input and observation noise covariance our results include recursive kalman filter predictor realizations for the resulting robust procedures our approach is based on saddle point theory we emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique in this paper we highlight the role such distributions with finite support play in these decision models in particular we demonstrate that in these decision models the least favorable prior distribution is always discrete

unitary dressing transformations and exponential decay below threshold for quantum spin systems parts i and ii this is the continuation of a series of articles concerning a class of quantum spin systems with hamiltonian operators of the form h lambda sum limits x in lambda s x sum limits yo subset lambda lambda left yo right c 1 t yo where is a graph is a small parameter andsx has a gap 1 for all x e i in the singular seti the gap ofsx can be arbitrarily small part iii is devoted to the proof of a preliminary result while in part iv we consider the case in whichi is a subset of finite density of this completes the first iteration step of the deterministic part of the proof of localization in the ground state of the random field quantumxy model

robust stability of uncertain time delay systems and its stabilization by variable structure control in this paper two main results for uncertain time delay systems are derived the first result is the presentation of a new robust stability criterion for uncertain time delay systems the second result is the successful application of the concept of variable structure control to the stabilization problem of uncertain time delay systems the robustness and stability degree of perturbed systems in the sliding mode are also discussed last some examples are included to illustrate our results

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

on stability independent of delay for linear systems a new criterion for asymptotic stability of solutions of certain linear differential difference equations independent of the delay duration is given in terms of solutions of a complex lyapunov matrix equation

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

automatic alignment of genus zero surfaces a new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero the algorithm takes as input a pair of surfaces that are topological 2 spheres each surface given by a distinct triangulation the algorithm then constructs a map f between the two surfaces first each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm the two mappings are then composed with a mobius transformation to generate the function f the mobius transformation is chosen by minimizing an energy that measures the distance of f from an isometry we illustrate our approach using several real life data sets we show first that the algorithm allows for accurate automatic and landmark free nonrigid registration of brain surfaces we then validate our approach by comparing shapes of proteins we provide numerical experiments to demonstrate that the distances computed with our algorithm between low resolution surface based representations of proteins are highly correlated with the corresponding distances computed between high resolution atomistic models for the same proteins

mott transition and sign problem for a model of lattice fermions we consider a model of spinless fermions on the square latticez2 with an interaction potential of strengthu 0 at distance one and strengthj at distance two in the largeu limit t j u wheret is the hopping amplitude as the chemical potential is varied ift t 0 we find three different phases corresponding to full half and zero filling fractions we study the system at low temperaturet 0 by a method involving a canonical transformation and a functional integral representation ift 0 we locate the phase boundaries of the mott metal insulator transition for all j u with upper and lower bounds show that mean field theory is valid ifj 0 we have only one sided bounds for the phase boundaries and we can t validate mean field theory in casej 0 we introduce a new resummation scheme for low temperature expansions which yields finite and convergent perturbation series and permits us to study issues like the sign problem our algorithm gives an optimal canonical transformation for the functional integral such that the expectation of the sign observables is exp c cdot t cdot ve frac beta 2 wherev is the volume and t 1

stabilization of linear time invariant interval systems via constant state feedback control this paper investigates the stabilization problem of linear uncertain systems via constant state feedback control the systems under consideration contain time invariant uncertain parameters whose values are unknown but bounded in given compact sets and are thus called interval systems the criterion for the asymptotic stability of the closed loop system obtained when a state feedback control is applied is that all the eigenvalues of the resulting system matrix are in the strict left half of the complex plane first the author shows that to insure an interval system stabilizable some entries of the system matrices must be sign invariant more precisely the number of the least required sign invariant entries in system matrices is equal to the system order then the author studies the stabilizability of a set of interval systems called standard systems which contain sufficient numbers of sign invariant entries in proper locations after dividing all standard systems into some subsets by the uncertainty locations the author then derives necessary and sufficient conditions under which every system in a subset is stabilizable regardless of its parameter varying bounds the conditions show that all uncertain entries in system matrices should form a particular geometrical pattern called a generalized antisymmetric stepwise configuration for an interval system satisfying the stabilizability conditions a computational control design procedure is also provided and illustrated via an example the result is further generalized for nonstandard systems via linear transformation

ground states in relatively bounded quantum perturbations of classical lattice systems we consider ground states in relatively bounded quantum perturbations of classical lattice models we prove general results about such perturbations existence of the spectral gap exponential decay of truncated correlations analyticity of the ground state and also prove that in particular the aklt model belongs to this class if viewed on a large enough length scale this immediately implies a general perturbation theory about this model

the majorant lyapunov equation a nonnegative matrix equation for robust stability and performance of large scale systems a new robust stability and performance analysis technique is developed the approach involves replacing the state covariance by its block norm matrix i e the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics a bound i e majorant for the block norm matrix is given by the majorant lyapunov equation a lyapunov type nonnegative matrix equation existence uniqueness and computational tractability of solutions to the majorant lyapunov equation are shown to be completely characterized in terms of m matrices two examples are considered for a damped simple harmonic oscillator with uncertain but constant natural frequency the majorant lyapunov equation predicts unconditional stability and for a pair of nominally uncoupled oscillators with uncertain coupling the majorant lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators a result not predictable from quadratic or vector lyapunov functions

robust stabilizability for linear systems with both parameter variation and unstructured uncertainty the authors consider the problem of finding a single compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty an uncertain system model is proposed to characterize plants containing those two different uncertainties closed loop stability criteria are then proposed for the model robust stabilizability for single input single output siso systems is studied a number of assumptions describing the set of admissable siso plants are imposed the satisfaction of these assumptions guarantees the existence of a proper stable compensator c s achieving robust stabilization the algorithm used for controller design is recursive in nature and allows the designer to select one compensator coefficient at a time also the minimum order of the desired stabilizer c s depends only on the largest relative degree of the nominal plants which often leads to a lower order compensator

robust stability of delay dependence for linear uncertain systems a sufficient condition for the stability of the linear uncertain time delay systems is presented the result obtained can include information on the size of the delay and therefore can be a delay dependent stability condition the lyapunov method is employed to investigate the problem the result is demonstrated by an illustrative example

delay independent robust stability of uncertain linear systems in this paper we present some new sufficient conditions for robust stability of linear systems with uncertain time delay as well as structured parameter uncertainty the stability robustness bounds obtained from the new sufficient conditions are independent of the size of time delay some illustrative examples are given

stability of perturbed linear distributed parameter systems we present an estimate for a class of unbounded perturbations of the generator of an exponentially stable semigroup when this estimate is satisfied then the perturbed operator still generates an exponentially stable semigroup for generators of holomorhic semigroups the class of allowed perturbations contains unbounded operators of the form ha with the estimate is given in terms of the norm of the solution of the lyapunov equation which corresponds to the unperturbed system

independent of delay stability criteria for uncertain linear state space models new stability robustness criteria are presented for linear state space models with multiple uncertain time delays and structured parameter uncertainty using the lyapunov equation approach the proposed stability criteria are independent of the size of time delays some previous quadratic stability criteria for linear uncertain systems without time delays are included as special cases examples are used to illustrate the less conservative results of the proposed methods compared with existing ones in the literature

reduced conservatism in stability robustness bounds by state transformation this note addresses the issue of conservatism in the time domain stability robustness bounds obtained by the lyapunov approach a state transformation is employed to improve the upper bounds on the linear time varying perturbation of an asymptotically stable linear time invariant system for robust stability this improvement is due to the variance of the conservatism of the lyapunov approach with respect to the basis of the vector space in which the lyapunov function is constructed improved bounds are obtained using a transformation on elemental and vector norms of perturbations i e structured perturbations as well as on a matrix norm of perturbations i e unstructured perturbations for the case of a diagonal transformation an algorithm is proposed to find the optimal transformation several examples are presented to illustrate the proposed analysis