razumikhin type theorems on general decay stability of stochastic functional differential equations with infinite delay to the best of the authors knowledge there are no results based on the so called razumikhin technique via a general decay stability for any type of stochastic differential equations in the present paper the razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay the obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given an example is presented to illustrate the usefulness of the theory

couple group consensus for second order multi agent systems with fixed and stochastic switching topologies abstract this paper deals with the couple group consensus problem for second order discrete time multi agent systems both the fixed topology case and the stochastic switching topology case are considered the couple group consensus problem is converted into the stability problem of the error system by a linear transformation for the fixed topology case we obtain two different conditions of couple group consensus for the stochastic switching topology case we obtain a necessary and sufficient condition of mean square couple group consensus algorithms are provided to design the allowable control gains finally simulation examples are given to show the usefulness of the theoretical results

uniformly factoring weakly compact operators abstract let x and y be separable banach spaces suppose y either has a shrinking basis or y is isomorphic to c 2 n and a is a subset of weakly compact operators from x to y which is analytic in the strong operator topology we prove that there is a reflexive space with a basis z such that every t a factors through z likewise we prove that if a l x c 2 n is a set of operators whose adjoints have separable range and is analytic in the strong operator topology then there is a banach space z with separable dual such that every t a factors through z finally we prove a uniform version of this result in which we allow the domain and range spaces to vary

analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients this paper discusses exponential stability of solutions for highly nonlinear hybrid pantograph stochastic differential equations psdes two criteria are proposed to guarantee exponential stability of the solution the first criterion is a khasminskii type condition involving general lyapunov functions the second is developed on coefficients of the equation in virtue of m matrix techniques based on the second criterion robust stability of a perturbed hybrid psde is also investigated the theory shows how much an exponentially stable hybrid psde can tolerate to remain stable

delay dependent exponential stability of stochastic systems with time varying delay nonlinearity and markovian switching the problem of delay dependent stability in the mean square sense for stochastic systems with time varying delays markovian switching and nonlinearities is investigated both the slowly time varying delays and fast time varying delays are considered based on a linear matrix inequality approach delay dependent stability criteria are derived by introducing some relaxation matrices which can be chosen properly to lead to a less conservative result numerical examples are given to illustrate the effectiveness of the method and significant improvement of the estimate of stability limit over some existing results in the literature

delay dependent stochastic stability criteria for markovian jumping neural networks with mode dependent time varying delays and partially known transition rates abstract in this paper the problem of stochastic stability criterion of markovian jumping neural networks with mode dependent time varying delays and partially known transition rates is considered some new delay dependent stability criteria are derived by choosing a new class of lyapunov functional the obtained criteria are less conservative because free weighting matrices method and a convex optimization approach are considered finally a numerical example is given to illustrate the effectiveness of the proposed method

dissipativity analysis for discrete stochastic neural networks with markovian delays and partially known transition matrix the problem of dissipativity analysis for a class of discrete time stochastic neural networks with discrete and finite distributed delays is considered in this paper system parameters are described by a discrete time markov chain a discretized jensen inequality and lower bounds lemma are employed to reduce the number of decision variables and to deal with the involved finite sum quadratic terms in an efficient way a sufficient condition is derived to ensure that the neural networks under consideration is globally delay dependent asymptotically stable in the mean square and strictly z s g a dissipative next the case in which the transition probabilities of the markovian channels are partially known is discussed numerical examples are given to emphasize the merits of reduced conservatism of the developed results

sampled data state estimation for markovian jumping fuzzy cellular neural networks with mode dependent probabilistic time varying delays in this paper the problem of state estimation for markovian jumping fuzzy cellular neural networks fcnns using sampled data with mode dependent probabilistic time varying delays is investigated by developing a delay decomposition approach the information of the delayed states can be taken into full consideration by introducing a stochastic variable with a bernoulli distribution the information of probability distribution of the time varying delay is considered and transformed into one with deterministic time varying delay the main purpose of this paper is to estimate the neuron states through available output measurements such that the dynamics of the estimation error is globally asymptotically stable in the mean square based on the lyapunov krasovskii functional including triple integral terms and decomposed integral intervals delay distribution dependent stability criteria are obtained in terms of linear matrix inequalities lmis finally two numerical examples are given to illustrate the effectiveness of the proposed theoretical results

classifications of baire 1 functions and spreading models we prove that if for a bounded function f defined on a compact space k there exists a bounded sequence fn of continuous functions with spreading model of order 1 co the index rnd as defined by a kechris and a louveau as a corollary of this result we have that the banach spaces ve k 1 4 defining the notion of nullcoefficient sequences of order 1 4 coi we prove that every bounded sequence fn of continuous functions converging pointwise to a function f with rnd f co is a null coefficient sequence of order as a corollary to this we have the following co spreading model theorem every nontrivial weak cauchy sequence in a banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of co or is a nullcoefficient sequence of order 1

genericity and amalgamation of classes of banach spaces abstract we study universality problems in banach space theory we show that if a is an analytic class in the effros borel structure of subspaces of c 0 1 of non universal separable banach spaces then there exists a non universal separable banach space y with a schauder basis that contains isomorphs of each member of a with the bounded approximation property the proof is based on the amalgamation technique of a class c of separable banach spaces introduced in the paper we show among others that there exists a separable banach space r not containing l 1 0 1 such that the indices and r n d are unbounded on the set of baire 1 elements of the ball of the double dual r of r this answers two questions of h p rosenthal we also introduce the concept of a strongly bounded class of separable banach spaces a class c of separable banach spaces is strongly bounded if for every analytic subset a of c there exists y c that contains all members of a up to isomorphism we show that several natural classes of separable banach spaces are strongly bounded among them the class of non universal spaces with a schauder basis the class of reflexive spaces with a schauder basis the class of spaces with a shrinking schauder basis and the class of spaces with schauder basis not containing a minimal banach space x

stability analysis for stochastic hybrid systems this survey addresses stability analysis for stochastic hybrid systems shs which are dynamical systems that combine continuous change and instantaneous change and that also include random effects we re emphasize the common features found in most of the models that have appeared in the literature which include stochastic switched systems markov jump systems impulsive stochastic systems switching diffusions stochastic impulsive systems driven by renewal processes diffusions driven by levy processes piecewise deterministic markov processes general stochastic hybrid systems and stochastic hybrid inclusions then we review many of the stability concepts that have been studied including lyapunov stability lagrange stability asymptotic stability and recurrence next we detail lyapunov based sufficient conditions for these properties and additional relaxations of lyapunov conditions many other aspects of stability theory for shs like converse lyapunov theorems and robustness theory are not fully developed hence we also formulate some open problems to serve as a partial roadmap for the development of the underdeveloped pieces

a classification of baire class 1 functions we study in this paper various ordinal ranks of bounded baire class 1 functions and we show their essential equivalence this leads to a natural classification of the class of bounded baire class 1 functions b 1 in a transfinite hierarchy b 1 1 of small baire classes for which for example an analysis similar to the hausdorff kuratowski analysis of 0 2 sets via transfinite differences of closed sets can be carried out the notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a baire class 1 function are introduced and used in this study

razumikhin method and exponential stability of hybrid stochastic delay interval systems this paper deals with the exponential stability of hybrid stochastic delay interval systems also known as stochastic delay interval systems with markovian switching the known results in this area see e g x mao exponential stability of stochastic delay interval systems with markovian switching ieee trans automat control 47 10 2002 1604 1612 require the time delay to be a constant or a differentiable function and the main reason for such a restriction is due to the analysis of mathematics the main aim of this paper is to remove this restriction to allow the time delay to be a bounded variable only the razumikhin method is developed to cope with the difficulty arisen from the nondifferentiability of the time delay

exponential stability of stochastic delay interval systems with markovian switching in the past few years a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with markovian switching however little research has been on the stability of interval systems with markovian switching which is the topic of this paper the system discussed is the stochastic delay interval system with markovian switching it is a very advanced system and takes all the features of interval systems ito equations and markovian switching as well as time lag into account the theory developed is applicable in many different and complicated situations so the importance of the paper is clear

delay dependent robust stability of uncertain neutral type it stochastic systems with markovian jumping parameters this paper investigates the problem of mean square exponential stability for uncertain neutral stochastic systems with time delays and markovian jumping parameters based on the new results on expectations of stochastic cross terms containing the it o integral by song et al 2013 a new lyapunov krasovskii function is established and then an improved mean square exponential stability criterion is derived the derived results extend the conclusions recently presented in song et al 2013 in fact the system discussed in song et al 2013 is a special case of ours finally two examples are provided to demonstrate the effectiveness of the proposed results

almost sure exponential stability of recurrent neural networks with markovian switching this paper presents new stability results for recurrent neural networks with markovian switching first algebraic criteria for the almost sure exponential stability of recurrent neural networks with markovian switching and without time delays are derived the results show that the almost sure exponential stability of such a neural network does not require the stability of the neural network at every individual parametric configuration next both delay dependent and delay independent criteria for the almost sure exponential stability of recurrent neural networks with time varying delays and markovian switching parameters are derived by means of a generalized stochastic halanay inequality the results herein include existing ones for recurrent neural networks without markovian switching as special cases finally simulation results in three numerical examples are discussed to illustrate the theoretical results

some new criteria on p th moment stability of stochastic functional differential equations with markovian switching this note gives some new razumikhin type theorems on th moment stability of stochastic functional differential equations with markovian switching sfdewms by using auxiliary ordinary differential equation the main results of this note allow the diffusion operator associated with the underlying sfdewms of the lyapunov function along a solution of the system to be not always negative an example is provided to illustrate the effectiveness of the proposed results

a note on almost sure asymptotic stability of neutral stochastic delay differential equations with markovian switching in this paper we consider neutral stochastic delay differential equations with markovian switching our key aim is to establish lasalle type stability theorems for the underlying equations the key techniques used in this paper are the method of lyapunov functions and the convergence theorem of nonnegative semi martingales the key advantage of our new results lies in the fact that our results can be applied to more general non autonomous equations

stability of a random diffusion with linear drift we consider a linear system with markovian switching which is perturbed by gaussian type noise if the linear system is mean square stable then we show that under certain conditions the perturbed system is also stable we also show that under certain conditions the linear system with markovian switching can be stabilized by such noisy perturbation

a note on stability of hybrid stochastic differential equations the method is different from those usual methods used in previous literature some harsh restrictions imposed on diffusion operators have been removed the differentiability of delays has been removed we studied the robustness of exponential stability respect to delays in this paper the pth moment stability of hybrid stochastic differential equations is investigated several new sufficient conditions are derived by constructing an auxiliary delayed differential equation and using the comparison principle the proposed criteria remove some harsh restrictions imposed on the diffusion operators and improve some previous related works numerical examples and simulations are given to illustrate the effectiveness of theoretical results

interpolating hereditarily indecomposable banach spaces a banach space x is said to be hereditarily indecomposable h i if for any pair of closed subspaces y z of x with y z 0 y z is not a closed subspace throughout this section by the term subspace we mean a closed infinite dimensional subspace of x the h i spaces form a new and as we believe fundamental class of banach spaces the celebrated example of a banach space with no unconditional basic sequence due to w gowers and b maurey gm is the first construction of a h i space it is easily seen that every h i space does not contain any unconditional basic sequence actually the concept of h i spaces came after w johnson s observation that this was a property of the gowers maurey example to describe even further the peculiar structure of a h i space we recall an alternative definition of such a space a banach space x is a h i space if and only if for every pair of subspaces y z and e 0 there exist y y z z with y z 1 and y z e thus h i spaces are structurally irrelevant to classical banach spaces in particular to hilbert spaces other constructions of h i spaces already exist we mention argyros and deliyanni s construction of h i spaces which are asymptotic spaces ad2 v ferenczi s example of a uniformly convex h i space f2 and h i modified asymptotic spaces contained in adkm other examples of banach spaces which are h i or which have a h i subspace are given in g1 h os1 the construction of such a space requires several steps and it uses two fundamental ideas the first is tsirelson s recursive definition of saturated norms ts and the second is maurey rosenthal s construction of weakly null sequences without unconditional basic subsequences mr an important ingredient in the gowersmaurey construction is the schlumprecht space this is a tsirelson type banach space which is arbitrarily distortable and has been used in the solution of important problems thus beyond its use in the constructions of h i spaces it plays a central role in the solution of the distortion problem for hilbert spaces os the essential difference between schlumprecht and tsirelson spaces became more transparent in ad2 where the mixed tsirelson spaces were introduced it is natural to expect that h i spaces share special and interesting properties not found in the previously known banach spaces indeed the following theorem is proven in

the unconditional basic sequence problem we construct a banach space that does not contain any infinite un conditional basic sequence and investigate further properties of this space for example it has no subspace that can be written as a topological direct sum of two infinite dimensional spaces this property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity in par ticular it is either strictly singular or fredholm with index zero this implies that the space is not isomorphic to any proper subspace trinity college cambridge cb2 1tq england current address department of mathematics university college london gower street lon don wc1e 6bt england e mail address wtg l o phx cam ac uk universit paris vii u f r de mathtmatiques 2 place jussieu 75251 paris cedex 05 france current address equipe d analyse et mathematiques appliquees universite de mamne la vallee 93160 noisy le grand france e mail address maurey logiquejussieu fr this content downloaded from 157 55 39 105 on fri 07 oct 2016 04 30 48 utc all use subject to http about jstor org terms

stability of hybrid stochastic functional differential equations abstract in this paper a class of hybrid stochastic functional differential equations is investigated under some suitable assumptions the pth moment stability asymptotic stability and exponential stability are discussed by means of constructing an auxiliary functional differential equation and using the comparison principle the proposed criteria removes some conditions in some earlier publications moreover two examples and simulations are given to illustrate our results

on delay dependent stability for a class of nonlinear stochastic systems with multiple state delays abstract global asymptotic stability conditions for nonlinear stochastic systems with multiple state delays are obtained based on the convergence theorem for semimartingale inequalities without assuming the lipschitz conditions for nonlinear drift functions the lyapunov krasovskii and degenerate functionals techniques are used the derived stability conditions are directly expressed in terms of the system coefficients nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given

stabilization of stochastic functional differential systems with delayed impulses abstract this paper is concerned with the stabilization problem for a class of stochastic functional differential system with delayed impulses by applying a generalized comparison principle and the average dwell time method some sufficient criteria for input to state stable iss type and exponential stability properties are derived the coefficients of the upper bound for the diffusion operator of lyapunov functions are allowed to be time varying which improves the existing results greatly two numerical examples are presented to illustrate the effectiveness of the proposed results

razumikhin type exponential stability criteria of neutral stochastic functional differential equations the paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations by using the razumikhin type technique the main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with lyapunov functionals the analysis is based on paper x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 referring to mean square and almost sure exponential stability

quadratic congruential pseudorandom numbers distribution of lagged pairs abstract the present paper deals with the quadratic congruential method with power of two modulus for generating uniform pseudorandom numbers statistical independence properties of the generated sequences x n n 0 are studied based on the distribution of lagged pairs x n x n 2 upper and lower bounds for the discrepancy of the corresponding point sets in the unit square are established

banach spaces with separable duals it is proved that every banach space with a separable dual em beds into a space with a shrinking basis it follows that every separable re flexive space can be embedded in a reflexive space with a basis

finite time h static output control of markov jump systems with an auxiliary approach this paper considers the finite time h static output feedback control of markov jump systems with the help of introducing two new variables related to the original system state and output sufficient conditions for the closed loop system to be stochastic finite time boundedness with the prescribed h performance level are proposed in terms of linear matrix inequalities lmis meanwhile the static output control gains are solved explicitly by the proposed conditions it is shown that the proposed method is less or at least the same conservative than the existing result lastly a numerical example is given to demonstrate the effectiveness of the proposed method

robust stability and boundedness of nonlinear hybrid stochastic differential delay equations one of the important issues in the study of hybrid stochastic differential delay equations sddes is the automatic control with consequent emphasis being placed on the asymptotic analysis of stability and boundedness in the study of asymptotic properties the robust stability has received a great deal of attention the theory of robust stability shows how much perturbation a given stable hybrid sdde can tolerate so that its perturbed system remains stable almost all results so far on the robust stability require that the underlying sddes be either linear or nonlinear with linear growth condition however little is known on the robust stability of nonlinear hybrid sddes without the linear growth condition which is one of the key topics in this paper the other key topic is the robust boundedness the aim here is to answer the question how much perturbation can a given asymptotically bounded hybrid sdde tolerate so that its perturbed system remains asymptotically bounded

a note on banach spaces containing c0 or c it is shown that every separable banach space x containing a subspace isomorphic to c0 has a subspace y with basis such that xy c0c and the latter space has a shrinking basis and an unconditional fdd moreover it is shown that x c has a basis if x has the bounded approximation property

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

optimal sequences of continuous functions converging to a baire 1 function a proof of rosenthal s co index conjecture is given our approach uses optimal sequences of continuous functions converging to a baire 1 function their existence is obtained by the optimal sequences theorem stated and proved here for a sequence of functions and a countable ordinal the variation is also introduced if pointwise converges to f the relation between and is completely clarified finally optimal sequences associated to a baire 1 function are defined and their existence for every baire 1 function is provided

on the complexity of the uniform homeomorphism relation between separable banach spaces we investigate the uniform homeomorphism relation between separable banach spaces and the related relation of local equivalence we completely characterize the descriptive complexity of local equivalence in the borel reducibility hierarchy this also provides a lower bound for the complexity of the uniform homeomorphism

direct delay decomposition approach to synchronization of chaotic fuzzy cellular neural networks with discrete unbounded distributed delays and markovian jumping parameters in this paper the problem of synchronization of chaotic fuzzy cellular neural networks fcnns with discrete unbounded distributed delays and markovian jumping parameters mjps is investigated sufficient delay dependent stability criteria are obtained in terms of linear matrix inequalities lmis to ensure the chaotic delayed fcnns to be stochastic asymptotically synchronous with the help of free weighting matrix and some inequality techniques the information of the delayed plant states can be taken into full consideration here the delay interval is decomposed into two subintervals by using the tuning parameter such that 0 1 by developing a delay decomposition approach and constructing suitable lyapunov krasovskii functional lkf sufficient conditions for synchronization are established for each subinterval numerical example and its simulations are provided to demonstrate the effectiveness and less conservatism of the derived results

ξ completely continuous operators and ξ schur banach spaces abstract for each ordinal 0 1 we introduce the notion of a completely continuous operator and prove that for each ordinal 0 1 the class v of completely continuous operators is a closed injective operator ideal which is not surjective symmetric or idempotent we prove that for distinct 0 1 the classes of completely continuous operators and completely continuous operators are distinct we also introduce an ordinal rank v for operators such that v a 1 if and only if a is completely continuous and otherwise v a is the minimum countable ordinal such that a fails to be completely continuous we show that there exists an operator a such that v a if and only if 1 1 and there exists a banach space x such that v i x if and only if there exists an ordinal 1 such that finally prove that for every 0 1 the class a l v a is 1 1 complete in l the coding of all operators between separable banach spaces this is in contrast to the class v l which is 2 1 complete in l

a universal reflexive space for the class of uniformly convex banach spaces we show that there exists a separable reflexive banach space into which every separable uniformly convex banach space isomorphically embeds this solves problems raised by j bourgain b in 1980 and by w b johnson in 1977 jo we also give intrinsic characterizations of separable reflexive banach spaces which embed into a reflexive space with a block q hilbertian and or a block p besselian finite dimensional decomposition

on classes of banach spaces admitting small universal spaces we characterize those classes c of separable banach spaces admitting a separable universal space y that is a space y containing up to isomorphism all members of c which is not universal for all separable banach spaces the characterization is a byproduct of the fact proved in the paper that the class nu of non universal separable banach spaces is strongly bounded this settles in the affirmative the main conjecture from argyros and dodos 2007 our approach is based among others on a construction of l spaces due to j bourgain and g pisier as a consequence we show that there exists a family y 1 of separable non universal l spaces which uniformly exhausts all separable banach spaces a number of other natural classes of separable banach spaces are shown to be strongly bounded as well

the asymptotic stability and exponential stability of nonlinear stochastic differential systems with markovian switching and with polynomial growth abstract this paper discusses the asymptotic stability and exponential stability of nonlinear stochastic differential systems with markovian switching sdswmss the systems coefficients are assumed to satisfy local lipschitz condition and polynomial growth condition by applying some novel techniques we propose some conditions under which such sdswmss are asymptotically stable and exponentially stable nontrivial examples are provided to illustrate our results