global robust servomechanism problem of lower triangular systems in the general case the global robust servomechanism problem alternatively global robust output regulation problem for lower triangular systems has been studied for two special cases the first case assumes that the systems only contain polynomial nonlinearities and the second case limits the exogenous signals and the unknown parameters to be within a known bounded set this paper presents the solvability conditions of the global robust servomechanism problem for the lower triangular systems for the most general case where neither of the above two assumptions is needed our approach consists of two steps in the first step we convert the problem into a global adaptive regulation problem for lower triangular systems subject to both dynamic and static uncertainties in the second step we derive the solvability conditions of the problem by appealing to the recent result on the solvability of the global adaptive regulation problem for lower triangular systems with both dynamic and static uncertainties

sharp estimates of unimodular fourier multipliers on wiener amalgam spaces abstract we study the boundedness on the wiener amalgam spaces w s p q of fourier multipliers with symbols of the type e i for some real valued functions whose prototype is with 0 2 under some suitable assumptions on we give the characterization of w s p q w p q boundedness of e i d for arbitrary pairs of 0 p q our results are an essential improvement of the previous known results for both sides of sufficiency and necessity even for the special case with 1 2

limit sets zero dynamics and internal models in the problem of nonlinear output regulation in this paper we lay the foundations for a nonequilibrium theory of nonlinear output regulation giving a more general nonequilibrium definition of the problem deriving necessary conditions and using these necessary conditions we present a set of sufficient conditions and a design methodology for the solution of the problem in question our analysis leads to a nonequilibrium enhancement of the internal model principle

certainty equivalence in nonlinear output regulation with unmeasured error in this article we consider a nonlinear output regulation problem in which the controlled output and the measured output are not the same it is assumed that the controlled plant has a single control input and that it can be transformed into gauthier kupka s observability canonical form then it is shown that a design based on certainty equivalence is effective for determining a controller that solves the given problem

adaptive control variates for finite horizon simulation adaptive monte carlo methods are simulation efficiency improvement techniques designed to adaptively tune simulation estimators most of the work on adaptive monte carlo methods has been devoted to adaptively tuning importance sampling schemes we instead focus on adaptive methods based on control variate schemes we introduce two adaptive control variate methods and develop their asymptotic properties the first method uses stochastic approximation to adaptively tune control variate estimators it is easy to implement but it requires some nontrivial tuning of parameters the second method is based on sample average approximation tuning is no longer required but it can be computationally expensive numerical results for the pricing of barrier options are presented to demonstrate the methods

adaptive importance sampling monte carlo simulation for general multivariate probability laws we establish a parametric adaptive importance sampling variance reduction method for general multivariate probability laws employing the principle of bypass distributions makes it possible to develop adaptive algorithms without relying on particular properties of the target and proposal laws both of which in the proposed framework are as general as the uniform law on the unit hypercube without changing the sampling distribution at each iteration we establish the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity although implementation of the proposed methodology requires a small amount of initial work it has the potential to yield substantial improvements in estimator efficiency in various general problem settings to illustrate the applicability and effectiveness we provide numerical results throughout in which we apply exponential and normal bypass distributions as well as demonstrate that well known adaptive importance sampling formulations in the literature can be easily rewritten in the proposed framework

on the existence of a kazantzis kravaris luenberger observer we state sufficient conditions for the existence on a given open set of the extension to nonlinear systems of the luenberger observer as it has been proposed by kazantzis and kravaris we prove it is sufficient to choose the dimension of the system giving the observer less than or equal to 2 twice the dimension of the state to be observed we show that it is sufficient to know only an approximation of the solution of a pde needed for the implementation we establish a link with high gain observers finally we extend our results to systems satisfying an unboundedness observability property

brief mixed internal model based and feedforward control for robust tracking in nonlinear systems the purpose of this paper is to compare different features of two approaches to the problem of tracking prescribed trajectories in a nonlinear system and to propose a possible merger of these approaches in what is called here the problem of asymptotically robust perfect tracking the latter is the problem of designing a feedforward error feedback control law achieving perfect tracking in case the initial conditions of the system exosystem are known and compatible with perfect tracking and asymptotic tracking in all other cases

the global cauchy problem for the nls and nlkg with small rough data abstract by using the unit cube decomposition to the frequency spaces we study the cauchy problem for the nonlinear schrodinger equation and the nonlinear klein gordon equation some global well posedness results are obtained for the small cauchy data in some modulation spaces m p 1 s

a new approach to adaptive nonlinear regulation this paper shows how the theory of adaptive observers can be effectively used in the design of internal models for nonlinear output regulation the main result obtained in this way is a new method for the synthesis of adaptive internal models which substantially enhances the existing theory of adaptive output regulation by allowing nonlinear internal models and more general classes of controlled plants

adaptive importance sampling and control variates abstract we construct and investigate an adaptive variance reduction framework in which both importance sampling and control variates are employed the three lines monte carlo averaging and two variance reduction parameter search lines run in parallel on a common sequence of uniform random vectors on the unit hypercube given that these two variance reduction techniques are effective often in a complementary way their combined application is well expected to widen the applicability of adaptive variance reduction we derive convergence rates of the theoretical estimator variance towards its minimum as a fixed computing budget increases when stochastic approximation runs with optimal constant learning rates we derive sufficient conditions for the proposed algorithm to attain the minimal estimator variance in the limit by stochastic approximation with decreasing learning rates or by sample average approximation when computing budget is unlimitedly available numerical results support our theoretical findings and illustrate the effectiveness of the proposed framework

independent parameterization of two degree of freedom compensators in general robust tracking systems consideration is given to a general robust tracking problem in which the controlled object is a linear multivariable system having measured variables that are not necessary coincident with the output variables to be controlled a necessary and sufficient condition for the two degree of freedom compensator to achieve a robust tracking system with internal stability is given an existence condition is derived for such a compensator and the class of all such compensators is parameterized so that two specifications with respect to responses for reference commands and feedback properties can be satisfied independently by the two free parameters

phase space analysis of semilinear parabolic equations abstract we present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients as a consequence we prove local wellposedness of the corresponding cauchy problem in spaces of low regularity namely the modulation spaces assuming a nonlinearity of analytic type as another application we deduce that the corresponding phase space flow decreases the global wave front set we also consider the action on spaces of analytic functions provided the coefficients are analytic themselves

non resonance conditions for uniform observability in the problem of nonlinear output regulation abstract this paper deals with the design of an internal model based semiglobal output feedback regulator for possibly non minimum phase nonlinear systems taking advantage of the design tool proposed in isidori ieee trans automat control 45 10 2000 we show how the problem of output regulation can be reformulated into an output feedback stabilization problem of a suitably defined extended auxiliary system the output feedback stabilization of the extended auxiliary system is addressed in the second part of the paper where an observer based stabilizer is proposed the existence of the latter is characterized in terms of necessary and sufficient conditions which can be interpreted as nonlinear non resonance conditions between the modes of the exosystem and the zero dynamics of the controlled plant

the cauchy problem for the hartree type equation in modulation spaces we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces mp q rd we establish global well posedness results in m1 1 rd when k x x r 0 min 2 d 2 in mp q rd 1 q min p p where p is the holder conjugate of p 1 2 when k is in fourier algebra fl1 rd and local well posedness result in mp 1 rd 1 p when k m1 rd

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

canonical forms of unconditionally convergent multipliers multipliers are operators that combine frame like analysis a multiplication with a fixed sequence called the symbol and synthesis they are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing it is known that bounded symbols and bessel sequences guarantee unconditional convergence in this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers in particular we show that under mild conditions unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence into multipliers with symbol 1 and bessel sequences called multipliers in canonical form

remarks on fourier multipliers and applications to the wave equation exploiting continuity properties of fourier multipliers on modulation spaces and wiener amalgam spaces we study the cauchy problem for the nlw equation local wellposedness for rough data in modulation spaces and wiener amalgam spaces is shown the results formulated in the framework of modulation spaces refine those in a benyi k a okoudjou local well posedness of nonlinear dispersive equations on modulation spaces preprint april 2007 available at arxiv 0704 0833v1 the same arguments may apply to obtain local wellposedness for the nlkg equation

functions operating on modulation spaces and nonlinear dispersive equations abstract the aim of this paper is two fold we show that if a complex function f on c operates in the modulation spaces m p 1 r n by composition then f is real analytic on r 2 c this answers negatively the open question posed in ruzhansky et al 2012 21 regarding the general power type nonlinearity of the form u u we also characterise the functions that operate in the modulation space m 1 1 r n the local well posedness of the nls nlw and nlkg equations for the real entire nonlinearities are also studied in some weighted modulation spaces m s p q r n

time frequency analysis of the dirac equation abstract the purpose of this paper is to investigate several issues concerning the dirac equation from a time frequency analysis perspective more precisely we provide estimates in weighted modulation and wiener amalgam spaces for the solutions of the dirac equation with rough potentials we focus in particular on bounded perturbations arising as the weyl quantization of suitable time dependent symbols as well as on quadratic and sub quadratic non smooth functions hence generalizing the results in 40 we then prove local well posedness on the same function spaces for the nonlinear dirac equation with a general nonlinearity including power type terms and the thirring model for this study we adopt the unifying framework of vector valued time frequency analysis 56 most of the preliminary results are stated under general assumptions and hence they may be of independent interest

a remark on the problem of semiglobal nonlinear output regulation this paper shows how the method for semiglobal stabilization by output feedback based on high gain observer and saturation and the method for structurally stable regulation based on system immersion can be effectively coupled in order to solve problems of robust output regulation in the presence of parameter uncertainties ranging over compact sets

multipliers for von neumann schatten bessel sequences in separable banach spaces abstract in this paper we introduce the concept of von neumann schatten bessel multipliers in separable banach spaces and obtain some of their properties we study their behavior when the symbol belongs to l p also we investigate the continuous dependency of hilbert schmidt bessel multipliers on their parameters special attention is devoted to the study of invertible hilbert schmidt frame multipliers which is an extensive class of ordinary frame multipliers and includes g frame multipliers and fusion frame multipliers as elementary examples among other things the invertibility of hilbert schmidt riesz multipliers is completely characterized in particular we show that all the existing frames in hilbert spaces is uniquely determined by the set of its dual frames

acceleration on adaptive importance sampling with sample average approximation we construct and analyze acceleration techniques for adaptive monte carlo simulations for general multivariate probability laws when the sample average approximation is employed for optimal parameter search our goal is to accelerate the adaptive monte carlo estimation by leading the parameter search line based on the sample average approximation to reach a nearly optimal realm at its small sample size stage first we introduce an auxiliary parameter into the parameter search line and update it wisely to aim for small sample convergence all three lines of the algorithm the monte carlo averaging the importance sampling parameter search and the auxiliary parameter updating run concurrently in a fully automated manner with a common set of random vectors without ad hoc tuning of the simulation system we next propose and examine various criteria for the auxiliary parameter updating all under which the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter hol

nonlinear observer design in the siegel domain we extend the method of kazantzis and kravaris systems control lett 34 1998 pp 241 247 for the design of an observer to a larger class of nonlinear systems the extended method is applicable to any real analytic observable nonlinear system it is based on the solution of a first order singular nonlinear pde this solution yields a change of state coordinates which linearizes the error dynamics under very general conditions the existence and uniqueness of the solution is proved lyapunov s auxiliary theorem and siegel s theorem are obtained as corollaries the technique is constructive and yields a method for constructing approximate solutions

global adaptive observers for nonlinear systems via filtered transformations the problem of determining global adaptive observers for a class of single output nonlinear systems which are linear with respect to an unknown constant parameter vector is addressed sufficient conditions are given to observe asymptotically an equivalent state without persistency of excitation under additional geometric conditions the original state can be observed as well the results obtained are based on a nonlinear change of coordinates driven by auxiliary filters filtered transformations

guaranteed performance in semi global trajectory tracking for matched disturbance nonlinear systems in this article we present a semi global trajectory tracking approach that guarantees a priori computable l 2 and l performance bounds for matched disturbance control affine systems the proposed controller is derived by combining a standard inverse control technique with an extended nonlinear robust state feedback the latter is based on a control lyapunov function used for stabilising one operating point inside the considered state space a difference gradient formulation of this lyapunov function is then applied to prove stabilisation along any trajectory in the considered state space results for l 2 and l bounded disturbances will be presented and further extended to the case of actuator uncertainties and disturbance offsets the theoretical contributions are verified applying them to a numerical example

nonlinear internal models for output regulation in this note we show how nonlinear internal models can be effectively used in the design of output regulators for nonlinear systems this result provides a significant enhancement of the nonequilibrium theory for output regulation which we have presented in a recent paper

a dissipativity based approach to output regulation of non minimum phase systems abstract the present paper presents a new contribution to the design of output regulators for a class of nonlinear systems characterized by a possibly unstable zero dynamics it is shown that the problem in question can be reduced to a stabilization problem with a supplementary gain constraint for a suitably defined reduced auxiliary plant

uniform practical nonlinear output regulation in this paper we present a solution to the problem of asymptotic and practical semiglobal regulation by output feedback for nonlinear systems a key feature of the proposed approach is that practical regulation is achieved uniformly with respect to the dimension of the internal model and to the gain of the stabilizer near the zero error manifold this property renders the approach interesting for a number of real cases by bridging the gap between output regulation theory and advanced engineering applications simulation results regarding meaningful control problems are also presented

erratum nonlinear observer design in the siegel domain there is an error in the proof of the main result of our paper siam j control optim 41 2002 pp 932 953 an additional assumption is needed for the main result to hold in this erratum we supply a corrected version of the main result

isometric decomposition operators function spaces ep qλ and applications to nonlinear evolution equations abstract by using the isometric decomposition to the frequency spaces we will introduce a new class of function spaces e p q which is a subspace of gevrey 1 class g 1 r n c r n for 0 and we will study the cauchy problem for the nonlinear schrodinger equation the complex ginzburg landau equation and the navier stokes equation some well posed results are obtained for the cauchy data in e 2 1 0 and the regularity behavior in e 2 1 ct g 1 r n for the complex ginzburg landau equation and the navier stokes equation is also obtained as time t 0

unimodular fourier multipliers on wiener amalgam spaces abstract we study the boundedness of unimodular fourier multipliers on wiener amalgam spaces for a real valued homogeneous function on r n of degree 2 we show the boundedness of the operator e i d between the weighted wiener amalgam space w s p q and w p q for all 1 p q and s n 2 1 p 1 2 n 1 p 1 q this threshold is shown to be optimal for regions max 1 q 1 2 1 p and min 1 q 1 2 1 p moreover we give sufficient conditions for the boundedness of e i d on w p q for 0 2

safe and effective importance sampling abstract we present two improvements on the technique of importance sampling first we show that importance sampling from a mixture of densities using those densities as control variates results in a useful upper bound on the asymptotic variance that bound is a small multiple of the asymptotic variance of importance sampling from the best single component density this allows one to benefit from the great variance reductions obtainable by importance sampling while protecting against the equally great variance increases that might take the practitioner by surprise the second improvement is to show how importance sampling from two or more densities can be used to approach a zero sampling variance even for integrands that take both positive and negative values

adaptive sinusoidal disturbance cancellation for unknown lti systems despite input delay solutions exist for the problems of canceling sinusoidal disturbances and compensating input delays in this paper two problems are considered simultaneously and an adaptive controller is designed to cancel unknown sinusoidal disturbances forcing an unknown linear time invariant system in controllable canonical form despite input delay the design is based on three steps 1 parametrization of the sinusoidal disturbance as the output of a known feedback system with an unknown output vector that depends on unknown disturbance parameters 2 representation of the delay as a transport pde 3 design of the adaptive controller by using the backstepping boundary control technique for pdes it is proven that the equilibrium of the closed loop system is stable and the state of the considered system converges to zero as t with perfect disturbance estimation the effectiveness of the controller is illustrated with a simulation example of a second order system

luenberger observers for nonautonomous nonlinear systems we show how the initial luenberger methodology presented by luenberger in observing the state of a linear system ieee transactions on military electronics vol 8 pp 74 80 1964 for linear systems can be used to design causal observers for controlled nonlinear systems their implementation relies on the resolution of a time varying pde the solutions of which transform the dynamics into linear asymptotically stable ones we prove the existence and injectivity after a certain time of such transformations under standard observability assumptions such as differential observability or backward distinguishability we show on examples how this pde can be solved and how the observability assumptions can be checked also we show that similarly to the high gain framework it is possible to use a time independent transformation when the system is observable for any input and strongly differentially observable of order the dimension of the system

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

brief paper geometric characterization on the solvability of regulator equations the solvability of the regulator equation for a general nonlinear system is discussed in this paper by using geometric method the feedback part of the regulator equation that is the feasible controllers for the regulator equation is studied thoroughly the concepts of minimal output zeroing control invariant submanifold and left invertibility are introduced to find all the possible controllers for the regulator equation under the condition of left invertibility useful results such as a necessary condition for the output regulation problem and some properties of friend sets of controlled invariant manifolds are also obtained

output stabilization via nonlinear luenberger observers the present paper addresses the problem of the existence of an output feedback law that asymptotically steers to zero prescribed outputs while keeping all state variables bounded for any initial conditions in a given compact set the problem can be viewed as an extension of the classical problem of semiglobally stabilizing the trajectories of a controlled system to a compact set the problem also encompasses a version of the classical problem of output regulation under only a weak minimum phase assumption it is shown that there exists a controller solving the problem at hand the paper is deliberately focused on theoretical results regarding the existence of such a controller practical aspects involving the design and the implementation of the controller are left to a forthcoming work

control functionals for monte carlo integration summary a non parametric extension of control variates is presented these leverage gradient information on the sampling density to achieve substantial variance reduction it is not required that the sampling density be normalized the novel contribution of this work is based on two important insights a trade off between random sampling and deterministic approximation and a new gradient based function space derived from stein s identity unlike classical control variates our estimators improve rates of convergence often requiring orders of magnitude fewer simulations to achieve a fixed level of precision theoretical and empirical results are presented the latter focusing on integration problems arising in hierarchical models and models based on non linear ordinary differential equations

modulation spaces and non linear hartree type equations abstract we study the cauchy problem for hartree equation with cubic convolution nonlinearity f u k u 2 u under a specified condition on potential k with cauchy data in modulation spaces m p q r n we establish global well posedness results in m p p r n with 1 p 2 n n when k x x r 0 m i n 2 n 2 in m p q r n with 1 q p 2 when k m 1 r n

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

nonlinear observers for autonomous lipschitz continuous systems this paper considers the state observation problem for autonomous nonlinear systems an observation mapping is introduced which is defined by applying a linear integral operator rather than a differential operator to the output of the system it is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems and to construct observers of a remarkably simple structure a linear state variable filter followed by a nonlinearity the observer is established by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist and by giving an explicit expression for its nonlinearity it is demonstrated that the existence conditions are satisfied and hence our results include a new observer which is not high gain for the wide class of smooth systems it is shown that the observer can as well be designed to realize an arbitrary finite accuracy rather than ultimate exactness on a compact region of the state space this requires only observability of the system a corresponding numerical design procedure is described which is easy to implement and computationally feasible for low order systems

nonlinear observer design using lyapunov s auxiliary theorem abstract the present work proposes a new approach to the nonlinear observer design problem based on the early ideas that influenced the development of the linear luenberger observer theory the proposed approach develops a nonlinear analogue the formulation of the observer design problem is realized via a system of singular first order linear pdes and a rather general set of necessary and sufficient conditions for solvability is derived by using lyapunov s auxiliary theorem the solution to the above system of pdes is locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

a note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback it is known that if a system can be robustly globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable uco then this system can be practically semiglobally stabilized by means of possibly dynamic output feedback this papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by uco functions is guaranteed the class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback in particular the hypothesis does not require the system to be minimum phase

further results on the robust tracking problem in two degree of freedom control systems the robust tracking problem in two degree of freedom control systems is formulated in a general setting which includes the case where the controlled output is not available directly a complete solution of the problem which is an extension of previous results is derived in a considerably simplified and transparent manner furthermore we show what kind of information about the output is needed to achieve robust tracking

rate of convergence for constrained stochastic approximation algorithms there is a large literature on the rate of convergence problem for general unconstrained stochastic approximations typically one centers the iterate theta n about the limit point bar theta and then normalizes by dividing by the square root of the step size epsilon n then some type of convergence in distribution or weak convergence of un the centered and normalized iterate is proved for example one proves that the interpolated process formed by the un converges weakly to a stationary gaussian diffusion and the variance of the stationary measure is taken to be a measure of the rate of convergence see the references in a benveniste m metivier and p priouret adaptive algorithms and stochastic approximation springer verlag berlin new york 1990 l gerencer siam j control optim 30 1992 pp 1200 1227 h j kushner and d s clark stochastic approximation for constrained and unconstrained systems springer verlag berlin new york 1978 h j kushner and g yin stochastic approximation algorithms and applications springer verlag berlin new york 1997 m t wasan stochastic approximation cambridge university press cambridge uk 1969 for algorithms where the step size either goes to zero or is small and constant large deviations provide an alternative approach to the rate of convergence problem p dupuis and h j kushner siam j control optim 23 1985 pp 675 696 p dupuis and h j kushner siam j control optim 27 1989 pp 1108 1135 p dupuis and h j kushner probab theory related fields 75 1987 pp 223 244 a p korostelev stochastic recurrent processes nauka moscow 1984 h j kushner and g yin stochastic approximation algorithms and applications springer verlag berlin new york 1997 when the iterates of the algorithm are constrained to lie in some bounded set the limit point is frequently on the boundary with the exception of the large deviations type p dupuis and h j kushner siam j control optim 23 1985 pp 675 696 p dupuis and h j kushner probab theory related fields 75 1987 pp 223 244 the rate of convergence literature is essentially confined to the case where the limit point is not on a constraint boundary when the limit point is on the boundary of the constraint set the usual steps are hard to carry out in particular the stability methods which are used to prove tightness of the normalized iterates cannot be carried over in general and there is the problem of proving tightness of the normalized process and characterizing the limit process this paper develops the necessary techniques and shows that the stationary gaussian diffusion is replaced by an appropriate stationary reflected linear diffusion whose variance plays the same role as a measure of the rate of convergence an application to constrained function minimization under inequality constraints q i x le 0 i le p is given where both the objective function and the constraints are observed in the presence of noise the iteration is on both the basic state variable and a lagrange multiplier which is constrained to be nonnegative if a limit multiplier value for an active constraint is zero then the classical method for computing the rate cannot be used but under appropriate conditions it is a special case of our results rate of convergence results are important because among other reasons they immediately yield the advantages of iterate averaging methods as noted in h j kushner and g yin stochastic approximation algorithms and applications springer verlag berlin new york 1997

gabor representations of evolution operators we perform a time frequency analysis of fourier multipliers and more generally pseudodierential operators with symbols of gevrey analytic and ultra analytic type as an application we show that gabor frames which provide optimally sparse decompositions for schrodinger type propagators 14 surprisingly reveal to be an equally ecient tool for representing solutions to hyperbolic and parabolic type dierential equations with constant coecients in fact the gabor matrix represen tation of the corresponding propagator displays super exponential decay away from the diagonal

high gain observers with limited gain power for systems with observability canonical form we consider the problem of state observation for systems having a well defined observability canonical form gauthier and kupka 2004 by means of high gain observers the main goal is to show that for this class of systems observers can be designed with the high gain parameter powered just up to the order 2 regardless the dimension of the state system in this way we substantially overtake the main limitations of standard design procedures in which the high gain parameter is powered up to the order of the system the observer structure which generalizes the ideas presented in astolfi and marconi 2015 can be used in all those contexts where fast state observation is required such as in the design of output feedback stabilizers by means of the nonlinear separation principle that is also specifically addressed in the paper