on the limiting zeros of sampled multivariable systems for time invariant linear systems with single input single output it has been shown 3 that as the sampling period goes to 0 the zeros of the corresponding discrete time system obtained by sampling converge to some specific constant values determined only be the degrees of the finite and the infinite zeros of the original continuous time system it is natural to inquire the extent to which this result may be carried over to the case of multivariable systems in this note it is shown that the result is true also for multivariable systems if the difference among the degrees of infinite elementary divisors of the pencil corresponding to the original continuous system is less than two when the last condition is not satisfied the limiting values of the zeros of the discrete system depend in general not only on the integers cited above but also on system parameters

conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

stable zeros of a discrete system obtained by sampling a continuous time plant with a time delay criteria are presented that ensure that all zeros of a discrete system are inside the unit disc when a continuous time plant with a time delay is sampled by use of a zero order hold a necessary and sufficient condition developed in this paper is a natural extension of the results for the case with no time delays this paper also derives sufficient conditions for stable discrete zeros in terms of the coefficients of a continuous time transfer function they guarantee that all the sampled zeros stay inside the unit disc as long as the sampling period is selected to satisfy a simple relation

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

fredholm differential operators with unbounded coefficients abstract we prove that a first order linear differential operator g with unbounded operator coefficients is fredholm on spaces of functions on r with values in a reflexive banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both r and r and a pair of the ranges of the dichotomy projections is fredholm and that the fredholm index of g is equal to the fredholm index of the pair the operator g is the generator of the evolution semigroup associated with the evolution family in the case when the evolution family is the propagator of a well posed differential equation u t a t u t with generally unbounded operators a t t r the operator g is a closure of the operator d dt a t thus this paper provides a complete infinite dimensional generalization of well known finite dimensional results by palmer and by ben artzi and gohberg

perturbation and an abstract characterization of evolution semigroups abstract evans and howland used an abstract characterization of evolution semigroups and a perturbation result for c 0 semigroups to treat non autonomous schrodinger equations and problems in scattering theory here we proceed in a similar way at first we characterize evolution semigroups on spaces of vector valued functions e x which are induced by a strongly continuous evolution family u t s t s defined on the banach space x from this and a perturbation result of voigt we obtain a sufficient condition for an operator on l p r x to be the generator of an evolution semigroup this will be applied to non autonomous heat equations with absorption

modified euler frobenius polynomials with application to sampled data modelling the broad class of polynomials generally known as eulerian or euler frobenius polynomials has a rich history in mathematics and engineering they have been generalised in several directions and find application in diverse areas ranging from sampled data modelling and polynomial interpolation to wavelets here we describe a modified class of euler frobenius polynomials which are functions of two independent variables we show that the sampling zero dynamics of general linear and nonlinear systems can be described using this new class of polynomials when considering several non standard sampled data modelling problems including when generalised hold functions are used to generate the input and when the continuous system has a time delay

strong stability of bounded evolution families and semigroups abstract we prove several characterizations of strong stability of uniformly bounded evolution families u t s t s 0 of bounded operators on a banach space x i e we characterize the property lim t u t s x 0 for all s 0 and all x x these results are connected to the asymptotic stability of the well posed linear nonautonomous cauchy problem u t a t u t t s 0 u s x x x in the autonomous case i e when u t s t t s for some c 0 semigroup t t t 0 we present in addition a range condition on the generator a of t t t 0 which is sufficient for strong stability this condition is more general than the condition in the ablv theorem involving countability of the imaginary part of the spectrum of a

an extension of some theorems of l barreira and c valls for the nonuniform exponential dichotomous evolution operators abstract the aim of this paper is to study the connection between the non uniform exponential dichotomy of a non uniform exponentially bounded strongly continuous evolution family and the admissibility of some function spaces thus there are extended recent results established by l barreira and c valls 2010 2011 in 10 and 11

stability radius and internal versus external stability in banach spaces an evolution semigroup approach in this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems these include autonomous and nonautonomous systems modeled with unbounded state space operators acting on banach spaces this approach allows one to apply the classical theory of strongly continuous semigroups to time varying systems in particular the complex stability radius may be expressed explicitly in terms of the generator of an evolution semigroup examples are given to show that classical formulas for the stability radius of an autonomous hilbert space system fail in more general settings upper and lower bounds on the stability radius are proven for banach space systems in addition it is shown that the theory of evolution semigroups allows for a straightforward operator theoretic analysis of internal stability as determined by classical frequency domain and input output operators even for nonautonomous banach space systems in particular for the nonautonomous setting internal stability is shown to be equivalent to input output stability for stabilizable and detectable systems for the autonomous setting an explicit formula for the norm of input output operator is given

robust stability of delay difference systems under fractional perturbations in infinite dimensional spaces in this article we study the robust stability of difference systems with delays under fractional perturbations in infinite dimensional spaces first the estimates of the complex stability radius are addressed second it is shown that for positive linear systems the complex real and positive stability radii coincide and can be computed by a simple formula finally a simple example is given to illustrate the obtained results

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

how to get a conservative well posed linear system out of thin air part ii controllability and stability let a0 be a possibly unbounded positive operator on the hilbert space h which is boundedly invertible let c0 be a bounded operator from cal d a 0 1 2 with the norm z 1 2 2 langle

stability theory for semigroups using l p l q fourier multipliers abstract we study polynomial and exponential stability for c 0 semigroups using the recently developed theory of operator valued l p l q fourier multipliers we characterize polynomial decay of orbits of a c 0 semigroup in terms of the l p l q fourier multiplier properties of its resolvent using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space we do not assume that the semigroup is uniformly bounded our results depend only on spectral properties of the generator as a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability and we also obtain a new theorem on exponential stability for positive semigroups

evolution semigroups and sums of commuting operators a new approach to the admissibility theory of function spaces this paper is concerned with conditions for the admissibility of a translation invariant function space m with respect to a well posed linear evolution equation du dt au f t t r we propose a new approach to this problem by considering the sum of two commuting operators d dt dm and the operator of multiplication by a on m on the one hand the closure of this operator is the infinitesimal generator of the so called evolution semigroup associated with on the other hand the generator g of this semigroup relates a mild solution u of to the forcing term f by the rule gu f consequently various spectral criteria of the type dm a for the admissibility of the function space m with respect to can be proved in an elegant manner moreover they can be naturally extended to general classes of differential equations including higher order and abstract functional differential equations applications and examples are provided to illustrate the obtained results

on solutions of differential inclusions in homogeneous spaces of functions in this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions these spaces include a large number of functional spaces like periodic functions and bohr and stepanov almost periodic functions as an application we consider some existence results for feedback control systems governed by degenerate differential equations of sobolev type in a banach space

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

analytic study on the intrinsic zeros of sampled data systems this paper investigates the properties of the mapping from the simple zero spl gamma of a scalar continuous time system to the corresponding zero spl gamma t of the sampled data system that results by its discretization using a zero order hold where t is the sampling period it is shown that spl gamma t admits a taylor expansion with respect to t and that it coincides with that of exp spl gamma t at least up to the second order term in general and at least up to the third order term if the relative degree of the continuous time system is greater than or equal to two the result is applied to derive a new stability condition of spl gamma t for sufficiently small sampling periods

properties of zeros in digital control systems with computational time delay abstract a digital control system with computational time delay is considered in which the controlled plant is a continuous time linear system and the control input is calculated by a digital computer based on the data of the output measured before a certain time not greater than the sampling period some qualitative properties of the zeros such as the characteristics of the limiting zeros the parity relation between real poles and zeros and the behaviour of the zeros with respect to the computational time delay are investigated and the parity interlacing property and its dual are derived a method of stabilization by a stable digital compensator using the inter sample data is also proposed finally the first order hold case is discussed

stability radii of delay difference systems under affine parameter perturbations in infinite dimensional spaces abstract in this paper we study the stability radii of positive difference systems with delays under arbitrary affine parameter perturbations in infinite dimensional spaces the obtained results are extensions of the recent results in p h a ngoc n k son stability radii of positive linear difference equations under affine parameter perturbation appl math comput 134 2003 577 594 a fischer stability radii of infinite dimensional positive systems math control signals syst 10 1997 223 236

admissibility of observation functionals the concept of infinite time admissibility of unbounded observation functionals is introduced under the assumption of exponential stability of the semigroup it is equivalent to finite time admissibility recently investigated in weiss 1988 b 1989 necessary and sufficient criteria for admissibility are given in particular it is shown that the ho russell weiss 1988 b test for admissibility of observation functionals control vectors can be derived in an elementary way without invoking the geometric interpretation of the carleson measure while the criterion in weiss 1991 can easily be deduced from the carleson embedding theorem some practically applicable sufficient conditions guaranteeing admissibility are discussed in 3 the results are illustrated by a feedback system containing rlcg transmission line

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

robust stability ofc0 semigroups and an application to stability of delay equations abstract let a be a closed linear operator on a complex banach space x and let a be a fixed element of the resolvent set of a let u and y be banach spaces and let d l u x and e l x y be bounded linear operators we define r a d e by formula and prove that formula we give two applications of this result the first is an exact formula for the so called stability radius of the generator of a c 0 semigroup of linear operators on a hilbert space it is derived from a precise result about robustness under perturbations of uniform boundedness in the right half plane of the resolvent of an arbitrary semigroup generator the second application gives sufficient conditions on the norm of the operators b j l x such that the classical solutions of the delay equation formula are exponentially stable in l p h 0 x

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

geometric aspects of the ode im correspondence this review describes a link between lax operators embedded surfaces and thermodynamic bethe ansatz equations for integrable quantum field theories this surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off critical generalisation of the ode im correspondence which initially involved only conformal invariant quantum field theories we will mainly focus of the kdv and sinh gordon models however various aspects of other interesting systems such as affine toda field theories and non linear sigma models will be mentioned we also discuss the implications of these ideas in the ads cft context involving minimal surfaces and wilson loops this work is a follow up of the ode im review published more than ten years ago by jpa before the discovery of its off critical generalisation and the corresponding geometrical interpretation partially based on lectures given at the young researchers integrability school 2017 in dublin

the operator equationax xb c admissibility and asymptotic behavior of differential equations it is shown that there exists a natural relationship between the regular admissibility of a translation invariant subspace m ofbuc r e the space of uniformly continuous bounded functions onrwith values in a banach spacee w r t the differential equationu t au t f t and the unique solvability of a special operator equation of lyapunov s typeax xdm 0 where dmis the restriction of the operator d d dtonto m and 0 f f 0 this leads to some spectral conditions for the regular admissibility applications to questions of exponential dichotomy exponential stability the existence of periodic and almost periodic solutions of equation as well as analogous questions for some nonlinear equations and functional differential equations are presented

optimal control on the doubly infinite continuous time axis and coprime factorizations we study the problem of existence of weak right or left or strong coprime factorizations in h infinity over the right half plane of an analytic function defined in some subset of the right half plane we give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous time axis in particular we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic riccati equation of an arbitrary realization that need not be well posed have a solution in general unbounded and not even densely defined and that a coupling condition involving these two solutions is satisfied the proofs that we give are partly based on corresponding discrete time results which we have recently obtained

bethe ansatz and the spectral theory of affine lie algebra valued connections ii the non simply laced case we study the ode im correspondence for ode associated to widehat mathfrak g valued connections for a simply laced lie algebra mathfrak g we prove that subdominant solutions to the ode defined in different fundamental representations satisfy a set of quadratic equations called psi system this allows us to show that the generalized spectral determinants satisfy the bethe ansatz equations

robustness of stability of time varying linear systems abstract this paper introduces the concept of stability radius for time varying linear systems invariance properties of the stability radius are analysed for the group of bohl transformations we also explore the relationship between the stability radius the norm of a certain perturbation operator and the solvability of a nonstandard differential riccati equation as an application we construct robust lyapunov functions and show how they can be used to analyze robustness with respect to nonlinear perturbations

characterization of exponential stability of a semigroup of operators in terms of its action by convolution on vector valued function spaces overr abstract it is proved that a c 0 semigroup t t t t 0 of linear operators on a banach space x is uniformly exponentially stable if and only if it acts boundedly on one of the spaces l p r x or c 0 r x by convolution as an application it is shown that t is uniformly exponentially stable if and only if formula where ap r x is the space of x valued almost periodic functions on r

beyond the wkb approximation in pt symmetric quantum mechanics the mergings of energy levels associated with the breaking of symmetry in the model of bender and boettcher and in its generalization to incorporate a centrifugal term are analysed in detail even though conventional wkb techniques fail it is shown how the ode im correspondence can be used to obtain a systematic approximation scheme which captures all previously observed features nonperturbative effects turn out to play a crucial role governing the behaviour of almost all levels once the symmetry breaking transition has been passed in addition a novel treatment of the radial schrodinger equation is used to recover the values of local and non local conserved charges in the related integrable quantum field theories without any need for resummation even when the angular momentum is nonzero

y system for scattering amplitudes we compute super yang mills planar amplitudes at strong coupling by considering minimal surfaces in ads5 space the surfaces end on a null polygonal contour at the boundary of ads we show how to compute the area of the surfaces as a function of the conformal cross ratios characterizing the polygon at the boundary we reduce the problem to a simple set of functional equations for the cross ratios as functions of the spectral parameter these equations have the form of thermodynamic bethe ansatz tba equations the area is the free energy of the tba system we consider any number of gluons and in any kinematic configuration

from fuchsian differential equations to integrable qft we establish an intriguing correspondence between a special set of classical solutions of the modified sinh gordon equation i e hitchin s self duality equations on a punctured riemann sphere and a set of stationary states in the finite volume hilbert space of the integrable 2d quantum field theory introduced by va fateev an application of this correspondence to the problem of non perturbative quantization of classically integrable nonlinear sigma models is briefly discussed a detailed account of the results announced in this communication is contained in separate publications bazhanov and lukyanov 2014 arxiv 1310 4390 hep th and bazhanov et al 2014 j high energy phys jhep09 2014 147

stability of the limiting zeros of sampled data systems with zero and first order holds this paper is concerned with the zeros of sampled data systems resulting from continuous time systems preceded by a hold and followed by a sampler the holds we shall consider are a zero order hold and a first order hold for sufficiently small or large sampling periods such zeros are called limiting zeros for sufficiently small sampling periods they are known to consist of two different types of zeros the zeros of the first type correspond to the zeros of the original continuous time system while those of the second type have no continuous time counterparts we first show basic properties of the zeros of sample data systems for sufficiently small sampling periods next we clarify in more detail the correspondence between the former type zeros of the sampled data systems and the zeros of the original continuous time system including the stability property of these zeros we also study stability properties of the latter type zeros in addition we study limiting zeros for sufficiently large sampli

sampled data optimal control of continuous systems for quadratic criterion function taking account of delayed control action this paper is concerned with the sampled data control of a continuous system in the situation where there exists delay for the control action after the measurement of the output at each sampling instant the optimal control with the quadratic criterion function defined for the continuous system is adopted in this control problem this paper shows that the problem can be transformed into the optimal control of a discrete system for the quadratic criterion function and the variation of the control law to the sampling period and to the delay of the control action is clarified the deterioration of cost performance with respect to delay is derived a numerical example is also presented

evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations abstract we investigate spectral criteria for the existence of almost periodic solutions to linear 1 periodic evolution equations of the formdx dt a t x f t with in general unbounded a t and almost periodicf using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set e isp f then the above equation has an almost periodic mild solutionxfwhich is unique if one requiressp xf 2 k k z sp f we emphasize that our method allows us to treat the equations without assumption on the existence of floquet representation this improves recent results on the subject in addition we discuss some particular cases in which the spectrum of monodromy operator does not intersect the unit circle and apply the obtained results to study the asymptotic behavior of solutions finally an application to parabolic equations is considered

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

stability and almost periodicity of trajectories of periodic processes abstract we prove that if the monodromy operator v of a linear periodic process u t in a banach space e is power bounded has countable peripheral spectrum and if its peripheral point spectrum satisfies a certain natural and simple duality condition which always holds in reflexive spaces then every positive trajectory u u 0 x 0 x e is asymptotically almost periodic if in particular the peripheral point spectrum of v is empty then every positive trajectory is asymptotically stable we also obtain results on almost periodicity of complete bounded trajectories and consider conditions under which nontrivial complete trajectories exist

the arveson frequency response and systems theory a frequency domain theory which is applicable to all linear systems and a class of non linear systems is formulated in a hilbert resolution space setting our approach is predicated on an analytic characterization of certain non selfadjoint operator algebras due to arveson although originally used as an analytic tool arveson in fact formulated an operator valued frequency response function for the elements of these algebras which in particular include the causal operators defined on a hilbert resolution space the resultant arveson frequency response afr is well defined for all linear and non linear systems on a hilbert resolution space and in the linear case it is characterized by an analyticity theory which is identical to that of the classical time invariant frequency response concept moreover it yields computationally viable formulae for the most widely studied classes of linear systems including the time varying abcd systems finally the classical frequency response concepts and their ti

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

a note on stability and stabilizability of neutral systems this note presents frequency domain characterizations of exponential stability and stabilizability of neutral systems based on transfer function matrices and the existence of nice solutions of certain bezout equations it turns out that the existence of h sup spl infin solutions is not sufficient for exponential stabilizability but that they have to satisfy an additional growth assumption as well whilst the proofs of the authors results are based on an abstract infinite dimensional representation of the neutral system they emphasize that the results are expressed in terms of the original parameters of the neutral equation and do not require a reformulation of the system in an abstract state space form the sufficiency parts of the results hold even when the delay operator acting on the derivative contains a singular part

on the spectrum complete trajectories and asymptotic stability of linear semi dynamical systems abstract let t t t 0 be a c0 semigroup in a banach space x with generator a we prove that if t t t 0 is bounded and sun reflexive and the sun dual semigroup is not asymptotically stable then there exist bounded complete trajectories under t t t 0 provided i there is t0 0 such that ran t t0 is dense in x or ii a i r questions of almost periodicity of complete trajectories are also discussed and a new proof of our earlier theorem jointly with yu i lyubich on asymptotic stability is given

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

exponential dichotomy and dichotomy radius for difference equations abstract the aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces we classify the classes of input and output spaces respectively and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems by an example we show that the obtained results are the most general in this topic as an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input output operators acting on sequence spaces which are invariant under translations

schaffer spaces and exponential dichotomy for evolutionary processes a very general characterization of exponential dichotomy for evolutionary processes in terms of the admissibility of some pair of spaces which are translation invariant the so called schaffer spaces is given it includes as particular cases many interesting situations among which we note the results obtained by n van minh f rabiger and r schnaubelt and the authors concerning the connections between admissibility and dichotomy

spectral conditions for admissibility of evolution equations in hilbert space abstract we study properties of solutions of the evolution equation u t b u t f t where b is a closable operator on the space ap r h of almost periodic functions with values in a hilbert space h such that b commutes with translations the operator b generates a family b of closed operators on h such that b e i t x e i t b x whenever e i t x d b for a closed subset r we prove that the following properties i and ii are equivalent i for every function f ap r h such that f there exists a unique mild solution u ap r h of eq such that u ii b i is invertible for all and sup b i 1

asymptotically almost periodic solutions of evolution equations in banach spaces abstract tile linear abstract evolution equation u t au t t t r is considered where a d a e e is the generator of a strongly continuous semigroup of operators in the banach space e starting from analogs of kadets and loomis theorems for vector valued almost periodic functions we show that if a i r is countable and r e is asymptotically almost periodic then every bounded and uniformly continuous solution u to is asymptotically almost periodic provided e tu t has uniformly convergent means for all a i r related results on eberlein weakly asymptotically almost periodic periodic asymptotically periodic and c0 solutions of as well as on the discrete case of solutions of difference equations are included

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

right half plane poles and zeros and design tradeoffs in feedback systems this paper expresses limitations imposed by right half plane poles and zeros of the open loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed loop system the limitations are determined by integral relationships which must be satisfied by these functions the integral relationships are interpreted in the context of feedback design

a discrete perron ta li type theorem for the dichotomy of evolution operators following the perron ta li line of results we give a characterization of the uniform exponential dichotomy property for the abstract continuous time evolution families using a discrete time approach

robustness of linear systems abstract in this paper we consider linear systems which are stable and examine the robustness of this property the perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability we begin with an abstract analysis of the problem but in later sections specialize to three types of systems those whose mild solutions are given in terms of 1 a strongly continuous semigroups 2 b resolvent operators integrodifferential systems 3 c evolution operators time varying systems for b a banach space l2 t0 t b denotes the space of all square integrable functions defined on t0 t with values in b for b1 b2 banach spaces t0 t denotes the space of strongly measurable functions f t0 t l b1 b2 with ess sup f b 1 t0 l b1 b2 denotes the space of strongly measurable functions f t0 l b1 b2 with f t 0 and k l1 t0

meagre functions and asymptotic behaviour of dynamical systems a measurable function x j r x x a metric space is said to be c meagre if c x is non empty and for every closed set k x with k c x 1 k has 5nite lebesgue measure this concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems by way of motivation consider the initial value problem f x x 0 rn with f rn rn locally lipschitz and denote its unique solution by x t t de5ned on its maximal interval of existence i 0 let g rn r be continuous assume that x is a bounded solution of the initial value problem in which case i 0 then g x is uniformly continuous if in addition g x l1 then we may conclude by an elementary observation frequently referred to as barb alat s lemma 5 see also 18 section 21 lemma 1 that g x t 0 as t 0 since by boundedness x t approaches as t its non empty compact invariant limit set b we also conclude that b g 1 0 it follows that x t tends as t to the largest invariant subset of g 1 0 this is a short proof of theorem 1 2 of 6 the proof in 6 is based on arguments involving properties of the dow generated by the dieerential

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

linear state space systems in infinite dimensional space the role and characterization of joint stabilizability detectability several fundamental results from the theory of linear state space systems in finite dimensional space are extended to encompass a class of linear state space systems in infinite dimensional space the results treated are those pertaining to the relationship between input output and internal stability the problem of dynamic output feedback stabilization and the concept of joint stabilizability detectability a complete structural characterization of jointly stabilizable detectable systems is obtained the generalized theory applies to a large class of linear state space systems assuming only that i the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators ii the state space is hilbert space iii the input and output spaces are finite dimensional and iv the sensing and control operators are bounded general conclusions regarding the fundamental structure of control theoretic problems in infinite dimensional space can be drawn from these results

boundary stabilization for a hybrid system of viscoelasticity in this paper we study the boundary feedback stabilization of a hybrid system that is composed of a viscoelastic thin plate with one part of its edge clamped and the remaining free part attached to a viscoelastic rigid body by adopting the frequency domain method we prove that the boundary feedback controls together with the dissipation induced by the memory effect are strong enough to secure the result of the exponential stability of energy we also reach the result of the exponential stability of the hybrid system on the domain with corners without any extra hypotheses on the regularity of solutions

the spectrum of the kinematic dynamo operator for an ideally conducting fluid the spectrum of the kinematic dynamo operator for an ideally conducting fluid and the spectrum of the corresponding group acting in the space of continuous divergence free vector fields on a compact riemannian manifold are described we prove that the spectrum of the kinematic dynamo operator is exactly one vertical strip whose boundaries can be determined in terms of the lyapunov oseledets exponents with respect to all ergodic measures for the eulerian flow also we prove that the spectrum of the corresponding group is obtained from the spectrum of its generator by exponentiation in particular the growth bound for the group coincides with the spectral bound for the generator

weak lp stability of a linear semigroup on a hilbert space implies exponential stability george weiss department of theoretical mathematics the weizmann institute rehovot 76100 israel received september 24 1987 we prove that a strongly continuous semigroup of linear operators on a hilbert space is weakly p stable for some pe cl co if and only if the semigroup is exponentially stable as an application we prove that the cauchy problems associated with the semigroup are well posed on the infinite time interval co 0 if and only if the semigroup is exponentially stable

evolutionary semigroups and dichotomy of linear skew product flows on locally compact spaces with banach fibers abstract we study evolutionary semigroups generated by a strongly continuous semi cocycle over a locally compact metric space acting on banach fibers this setting simultaneously covers evolutionary semigroups arising from non autonomous abstract cauchy problems and c 0 semigroups and linear skew product flows the spectral mapping theorem for these semigroups is proved the hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew product flow to this end a banach algebra of weighted composition operators is studied the results are applied in the study of roughness of the dichotomy dichotomy and solutions of nonhomogeneous equations green s function for a linear skew product flow pointwise dichotomy versus global dichotomy and evolutionary semigroups along trajectories of the flow

a simple almost periodicity criterion and applications we introduce a new almost periodicity criterion for functions r x with x a complete metric space this result is used to establish asymptotic almost periodicity of precompact positive trajectories to some differential equations of the form dudt a t u t 0 where a t is periodic with respect to t r

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

equivalence of input output stability and exponential stability abstract for a large class of well posed infinite dimensional systems we show that if their state space realisation is exponentially stabilizable and detectable then exponential stability of the system operator is equivalent to input output stability

internal and external stability of linear time varying systems linear finite dimensional time varying systems are studied state variable representations of systems with a bounded input bounded output stability property which are uniformly stabilizable and detectable are shown to have their associated homogeneous state variable systems exponentially stable

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

the resolvent growth assumption for semigroups on hilbert spaces let x be a hilbert space and let t t1 be a c semigroup of linear operators on x with generator a the aim of this paper is to show how information about the behavior of t for large t can be obtained from information about the behavior of the resolvent sla for s with large imaginary part and vice versa in particular the type of u can easily be determined if we know on which vertical lines in the complex plane the resolvent is bounded we start by defining a sequence of numbers ou which give information about the behavior of the semigroup for large t

almost periodic solutions of first and second order cauchy problems abstract almost periodicity of solutions of first and second order cauchy problems on the real line is proved under the assumption that the imaginary resp real spectrum of the underlying operator is countable related results have been obtained by ruess v and basit our proof uses a new idea it is based on a factorisation method which also gives a short proof of the vector valued version of loomis classical theorem saying that a bounded uniformly continuous function from r into a banach spacexwith countable spectrum is almost periodic ifc0 x our method can also be used for solutions on the half line this is done in a separate paper

evolutionary semigroups and lyapunov theorems in banach spaces we present a spectral mapping theorem for continuous semigroups of operators on any banach space e the condition for the hyperbolicity of a semigroup on e is given in terms of the generator of an evolutionary semigroup acting in the space of e valued functions the evolutionary semigroup generated by the propagator of a nonautonomous differential equation in e is also studied a discrete technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity exponential dichotomy of the nonautonomous equation

robust stability of linear evolution operators on banach spaces this paper introduces a stability radius for a wide class of linear infinite dimensional time varying systems under structured time varying perturbations a framework is presented that allows the same degree of unboundedness in the perturbations as in the generator of the nominal model

bethe ansatz equations for the classical a 1 _ n affine toda field theories we establish a correspondence between classical affine toda field theories and an bethe ansatz systems we show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model

weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems the first ergodic theorems for semigroups of nonlinear operators date back to work done by j b baillon cl 23 in 1975 1976 where he proved in 2 that i i lim l t j t s i x dt exists for an odd nonlinear contraction semigroup s t a 0 in real hilbert space in subsequent years various authors have extended his results to motions and almost orbits of nonlinear strongly continuous semigroups in uniformly convex banach spaces e g see 4 10 22 24 26 27 30 31 33 351 as well as 4 10 241 for yet further references these results apply to strong and integral solutions of the quasi autonomous nonlinear cauchy problem

hyers ulam stability and discrete dichotomy abstract let m be a given positive integer and let a be an m m complex matrix we prove that the discrete system x n 1 a x n n z is hyers ulam stable if and only if the matrix a possesses a discrete dichotomy also we prove that the scalar difference equation of order m x n m a 1 x n m 1 a 2 x n m 2 a m x n n z is hyers ulam stable if and only if the algebraic equation z m a 1 z m 1 a m 1 z a m z c has no roots on the unit circle this latter result is essentially known for further details see for example 24 and 2 however our proofs are completely different and moreover it seems that our approach opens the way to obtain many other results in this topic