lasalle type theorem and its applications to infinite horizon optimal control of discrete time nonlinear stochastic systems based on discrete martingale theory the lasalle type theorem for general discrete time stochastic systems is obtained and the almost sure stability is in turn discussed as applications infinite horizon nonlinear optimal regulator is investigated and a dynamical programming equation called the hamilton jacobi bellman equation is also derived for discrete time nonlinear stochastic optimal control

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

asynchronous l 2 l filtering for discrete time stochastic markov jump systems with randomly occurred sensor nonlinearities this paper is concerned with the problem of asynchronous l 2 l filtering for discrete time stochastic markov jump systems with sensor nonlinearity the sensor nonlinearity is assumed to occur randomly according to a stochastic variable satisfying the bernoulli distribution a sufficient condition is first given such that the resultant filtering error system which is a kind of nonhomogeneous markov jump system is stochastically stable with a guaranteed l 2 l performance index then the existence criterion of the desired asynchronous filter with piecewise homogeneous markov chain is proposed in terms of a set of linear matrix inequalities a numerical example is given to show the effectiveness and potential of the developed theoretical results

fuzzy real numbers abstract the paper gives a contribution to the study of fuzzy real numbers continuing that in eklund and gahler 1988 gahler and gahler 1992 and gahler 1992 we start with investigating algebraic properties using two types of decompositions of fuzzy numbers and the notions of fuzzy norm of symmetric difference and of supplements of fuzzy numbers by means of these notions two special fuzzy metrics on the fuzzy real line are given the fuzzy distance function d and the extended fuzzy distance function d the main interest concerns fuzzy topological properties of the fuzzy real line in particular there are studied the fuzzy topologies t t d and t d t is closely related to the fuzzy topology of hutton s fuzzy unit interval and t d and t d are canonically generated by the fuzzy metrics d and d respectively

estimation and synthesis of reachable set for switched linear systems this paper focuses on the problems of reachable set estimation and state feedback controller design for discrete time switched linear systems under bounded peak disturbances for the reachable set estimation problem a lyapunov based inequality is developed based on the multiple lyapunov strategy by choosing appropriate lyapunov functions the ellipsoidal reachable set estimation conditions of discrete time switched linear systems are obtained in order to make the estimated ellipsoids as small as possible three optimization approaches are proposed specifically the genetic algorithm is used to search for the optimal parameters satisfying the obtained reachable set estimation conditions in addition the state feedback controller design problem for discrete time switched linear systems is considered the function of the controller is to manipulate the reachable set of the closed loop system to lie within a given ellipsoid or make the reachable set small finally the effectiveness of the obtained results is verified through some numerical examples

stability of positive switched linear systems weak excitation and robustness to time varying delay this article investigates the stability of positive switched linear systems we start from motivating examples and focus on the case when each switched subsystem is marginally stable in the sense that all the eigenvalues of the subsystem matrix are in the closed left half plane with those on the imaginary axis simple instead of asymptotically stable a weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable an extension to the case without dwell time assumption is also presented then we study the influence of time varying delay on the stability of the considered positive switched linear system we show that the proposed weak excitation condition for the delay free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time varying delay in addition it is shown that the convergence rate is exponential if there exists an upper bound for the delay irrespective of the magnitude of this bound the motivating examples are revisited to illustrate the theoretical results

technical communique reachable set bounding for delayed systems with polytopic uncertainties the maximal lyapunov krasovskii functional approach this paper deals with the problem of reachable set estimation for time delay systems subject to both polytopic parameter uncertainties and bounded peak inputs the maximal lyapunov krasovskii functional is constructed as the pointwise maximum of a family of lyapunov krasovskii functionals each functional corresponds to a vertex of uncertain polytope some criteria bounding the reachable set are derived this approach shows great advantages over the traditional methods based on the common lyapunov functionals by introducing a modified integral inequality the limitation imposed on the derivative of time delay being less than one is relaxed furthermore the obtained results involve less computational burden when the number of vertices of the polytope is small two examples are given to illustrate the theoretical results

technical communique further result on reachable set bounding for linear uncertain polytopic systems with interval time varying delays the problem of reachable set estimation of linear uncertain polytopic time varying delay systems subject to bounded peak inputs is studied in this paper the delays considered in this paper are assumed to be non differentiable and vary within an interval where the lower and upper bounds are known based on the lyapunov krasovskii method and delay decomposition technique a sufficient condition for the existence of a ball that bounds the reachable set of the system is proposed in terms of matrix inequalities containing only one scalar which can be solved by using an one dimensional search method and matlab s lmi toolbox and allow us to find the smallest radius a numerical example is given to illustrate the effectiveness of the proposed result

robust exponential convergence of a class of linear delayed systems with bounded controllers and disturbances this brief paper deals with the problem of robust exponential stabilization of a class of dynamic systems subject to time varying delays external disturbances and control saturation a linear memoryless state feedback controller is synthesized to guarantee that there is a domain of admissible initial conditions from which all solutions of the class of systems considered converge exponentially to a ball with a prespecified convergence rate the problem is formulated as a two step optimization problem with matrix inequality constraints to enlarge the size of the ball of admissible initial conditions

technical communique improved ellipsoidal bound of reachable sets for time delayed linear systems with disturbances we consider the problem of finding an ellipsoidal reachable set that bounds the states of time delayed linear systems with bounded peak disturbances the considered time delay is time varying but it has an upper bound in magnitude and rate based on the modified lyapunov krasovskii l k type functional we derive a delay dependent result expressed in the form of matrix inequalities containing only one non convex scalar the result is extended to uncertain time delayed linear systems finally we show the usefulness of our result by a comparative numerical example

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

barrier lyapunov functions for the output tracking control of constrained nonlinear switched systems abstract in this paper we investigate the output tracking control problem of constrained nonlinear switched systems in lower triangular form first when all the states are subjected to constraints we employ a barrier lyapunov function blf which grows to infinity whenever its arguments approach some finite limits to prevent the states from violating the constraints based on the simultaneous domination assumption we design a continuous feedback controller for the switched system which guarantees that asymptotic output tracking is achieved without transgression of the constraints and all closed loop signals remain bounded provided that the initial states are feasible then we further consider the case of asymmetric time varying output constraints by constructing an appropriate blf finally the effectiveness of the proposed results is demonstrated with a numerical example

h and h 2 filtering for linear systems with uncertain markov transitions this paper is concerned with h and h 2 filtering for markovian jump linear systems with uncertain transition probabilities motivated by the fact that the existing results either impose severe restrictions on some key matrices or introduce some unnecessary matrix variables this paper is focused on developing a new approach to systematically relax these restrictions for filter design by applying a novel technique to eliminate the product terms between the lyapunov matrices and the filter parameters an improved condition is first obtained for analyzing the h performance of the filtering error system then sufficient conditions in terms of linear matrix inequalities are presented for designing filters with a guaranteed h filtering performance level the proposed method is further extended to h 2 filtering theoretical analyses followed by a few numerical examples show that the proposed filter design method outperforms some existing results with respect to reduction of conservatism or variables needed for computation the filter design problems for both continuous time and discrete time markovian jump linear systems are addressed in a unified framework

on l proximities of the internal type in this paper the initial and final l proximities for a notion of l proximity introduced by the author and others in 1998 are investigated this l proximity is called l proximity of the internal type this paper shows that all initial and final lifts in the category l pri of l proximity spaces of the internal type and hence all initial and final l proximities of the internal type exist the expected relation between the initial l proximities of the internal type and the initial l topologies the initial global l neighborhood structures the initial l uniform structures is verified that is the l topology global l neighborhood structure associated with the initial of a family of l proximities of the internal type coincides with the initial of the family of l topologies global l neighborhood structures associated with these l proximities of the internal type moreover the l proximity of the internal type associated with the initial of a family of l uniform structures coincides with the initial of the family of l proximities of the internal type associated with these l uniform structures

the theory of global fuzzy neighborhood structures part i the general case abstract this paper is devoted to global fuzzy neighborhood structures we introduce three types of these structures defined by means of fuzzy filters in some sense the first type is more general than the second one and the second type is more general than the third one in the second case only homogeneous fuzzy filters are used in the third case homogeneous fuzzy filters are used which are representable by a prefilter or equivalently prefilters are used which represent special homogeneous fuzzy filters all fuzzy topologies and stratified fuzzy topologies are global fuzzy neighborhood structures of the first and second type respectively they appear in a canonical way as interior operators fuzzy neighborhood structures introduced by lowen fuzzy sets and systems 7 1982 165 are defined by means of prefilters the definition of these structures is in some sense similar to a characterization of those global fuzzy neighborhood structures of the third type which can be identified with fuzzy topologies however the related fuzzy topological approach differs fuzzy neighborhood structures in sense of lowen are characterized canonically as fuzzy closure operators in this paper the relations between the three types of global fuzzy neighborhood structures and their associated fuzzy topologies and also some relations to the fuzzy neighborhood structures in the sense of lowen are investigated moreover this paper deals with initial and final structures of global fuzzy neighborhood structures in two subsequent papers part ii and part iii fuzzy topogenous orders and fuzzy uniform structures will be investigated respectively all regular fuzzy topogenous orders that is all fuzzy topogenous structures and in particular all fuzzy proximities are global fuzzy neighborhood structures as is shown by examples there exist global fuzzy neighborhood structures which are not fuzzy topogenous structures hence the notion of global fuzzy neighborhood structure is more rich fuzzy uniform structures defined analogously to a weil s definition of a uniform structure as fuzzy filters generate in a canonical way global fuzzy neighborhood structures some results on global fuzzy neighborhood structures will also be published in gahler et al submitted in particular those related to stratifications

on initial and final fuzzy uniform structures this paper is devoted to study the initial and final fuzzy uniform structures of the fuzzy uniform structure defined by the author and others in 1998 in this paper we show that all initial and final lifts in the category fun of these fuzzy uniform spaces and hence the initial and final fuzzy uniform structures exist we introduce a characterization for the initial fuzzy uniform structures and as a special initial fuzzy uniform spaces the subspaces and product spaces of these fuzzy uniform spaces can be also characterized we also show that the fuzzy topology global homogeneous fuzzy neighborhood structure associated to the initial fuzzy uniform structure of a family of fuzzy uniform structures coincides with the initial fuzzy topology initial global homogeneous fuzzy neighborhood structure of the family of fuzzy topologies global homogeneous fuzzy neighborhood structures associated to these fuzzy uniform structures

on reachable set estimation of singular systems in this paper the problem of reachable set estimation of singular systems is investigated based on the lyapunov method a sufficient condition is established in terms of a linear matrix inequality lmi to guarantee that the reachable set of singular system is bounded by the intersection of ellipsoids then the result is extended to the problem for singular systems with time varying delay by utilizing the reciprocally convex approach the effectiveness of the obtained results in this paper is illustrated by numerical examples

the theory of global fuzzy neighborhood structures iii fuzzy uniform structures in this paper a new notion of fuzzy uniform structure is presented and investigated fuzzy uniform structures are given as special fuzzy filters of special interest are those fuzzy uniform structures which are sup fuzzy filters they can be characterized by a related notion of fuzzy syntopogenous structure as is shown fuzzy metrics gahler and gahler 1994 generate fuzzy uniform structures in our sense each fuzzy uniform structure has an associated global homogeneous fuzzy neighborhood structure and hence also an associated fuzzy topology in case of a 0 1 fuzzy uniform structure and even in a more general case a fuzzy uniform structure of the first type can be associated there are given relation to huttons s and katsaras notions of fuzzy uniform structure hutton 1977 katsaras 1979 and to the notion of fuzzy syntopogenous structure in sense of katsaras and petalas 1984

generalised nonlinear l 2 l filtering of discrete time markov jump descriptor systems this paper is devoted to the l2 l filter design problem for nonlinear discrete time markov jump descriptor systems subject to partially unknown transition probabilities the partially unknown transition probabilities are modelled via the polytopic uncertainties the objective is to propose a generalised nonlinear full order filter design method such that the resulting filtering error system is regular casual and stochastically stable and a prescribed l2 l attenuation level is satisfied for the autonomous discrete time descriptor system subject to lipschitz nonlinear condition by introducing some slack matrix variables a mode dependent stability criterion is established it cannot only ensure the regularity casuality and stochastic stability of system but also guarantee the considered system has a unique solution based on this obtained criterion a sufficient condition in terms of linear matrix inequalities lmis is derived such that the resulting filtering error system is regular casual s

the theory of global fuzzy neighborhood structures ii fuzzy topogenous orders abstract this paper deals with fuzzy topogenous orders in particular with fuzzy topogenous structures and with the more special fuzzy proximities these structures have been investigated ay katsaras and petalas 1983 1984 the notion of fuzzy proximity was introduced by katsaeas 1980 fuzzy topogenous structures and fuzzy proximities are represented in this paper as global fuzzy neighborhood structures fuzzy topogenous orders in general are characterized by the notion of global fuzzy neighborhood prestructure which is a weakening of that one of global fuzzy neighborhood structure in this paper moreover a modification of the notion of fuzzy proximity is considered called fuzzy proximity of the internal type whereas the notion of fuzzy proximity proposed by katsaras depends on a fixed order reversing involution of the related lattice l the notion of fuzzy proximity of the internal type is independent on such an involution the investigations intthis paper demonstrate that there are important global fuzzy neighborhood structures and prestructures different from fuzzy topologies and fuzzy pretopologies respectively

robust extended dissipative control for sampled data markov jump systems this paper investigates the problem of the sampled data extended dissipative control for uncertain markov jump systems the systems considered are transformed into markov jump systems with polytopic uncertainties and sawtooth delays by using an input delay approach the focus is on the design of a mode independent sampled data controller such that the resulting closed loop system is mean square exponentially stable with a given decay rate and extended dissipative a novel exponential stability criterion and an extended dissipativty condition are established by proposing a new integral inequality the reduced conservatism of the criteria is demonstrated by two numerical examples furthermore a sufficient condition for the existence of a desired mode independent sampled data controller is obtained by solving a convex optimisation problem finally a resistance inductance and capacitance rlc series circuit is employed to illustrate the effectiveness of the proposed approach

on the reachable set bounding of uncertain dynamic systems with time varying delays and disturbances in this paper improved conditions are proposed for finding an ellipsoidal bound of reachable sets which bound the state trajectories of uncertain dynamic systems with time varying delays and bounded peak disturbances a more generalized time derivative condition is considered in comparison with previous work by the use of convex hull properties and application of the lyapunov method sufficient conditions for finding an ellipsoidal bound of reachable sets of the concerned systems are established in terms of linear matrix inequalities lmis two numerical examples are included to show the superiority and effectiveness of our results

infinite horizon h2 h control for discrete time time varying markov jump systems with multiplicative noise in this paper we consider the infinite horizon h 2 h control problem for discrete time time varying linear systems subject to markov jump parameters and state multiplicative noise a stochastic version of a bounded real lemma is firstly developed for a general class of discrete time time varying markov jump systems with state and disturbance multiplicative noise by which we present a necessary and sufficient condition for the solvability of the h 2 h control problem in terms of four coupled discrete time riccati equations moreover the obtained design is applied to a macroeconomic problem to verify its effectiveness

a new approach to state bounding for linear time varying systems with delay and bounded disturbances abstract in this note the problem of state bounding for linear time varying systems with delay and bounded disturbances input is considered for the first time by using a novel approach which does not involve the lyapunov krasovskii functional method new explicit delay independent conditions are derived for the existence of a ball such that all the state trajectories of the system converge exponentially within it a numerical example is given to illustrate the effectiveness of the obtained result

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

an improved method of ultimate bound computation for linear switched systems with bounded disturbances in this note an improved method of ultimate bound computation for a linear switched system under arbitrary switching is presented an ultimate bound for a linear switched system can be computed by solving a class of linear matrix inequalities the effectiveness of the obtained results is illustrated by numerical examples

the general fuzzy filter approach to fuzzy topology ii in this and a subsequent paper by means of an extension of the notion of fuzzy filter a new approach to fuzzy topology is presented whereas this paper is devoted to the general theory of fuzzy filters in part ii this theory is applied on fuzzy topology

almost sure estimation and synthesis of ultimate bounds for discrete time markov jump linear systems this paper is concerned with the problems of almost sure ultimate bound estimation and controller design for markov jump linear systems with bounded stochastic disturbances by utilising a lyapunov based scheme proposed in this paper an almost sure estimation of ellipsoidal ultimate bound eub of the system is obtained through tractable matrix inequalities on the basis of the estimation results the problem of designing mode dependent state feedback controllers that make the closed loop system admit a prescribed ellipsoid as an eub is considered the obtained results on estimation and synthesis are then extended to the case of systems with deficient mode information finally a practical example in dc motor devices is presented to demonstrate the applicability of the obtained results

the a levels of a fuzzy uniform structure and of a fuzzy proximity there are different notions of fuzzy uniform structures and of fuzzy proximities that have been introduced in the literature in this paper we are interested in the fuzzy uniform structure u in the sense of gahler et al 1998 which is defined as some fuzzy filter and we are also interested in the fuzzy proximity n in the sense of gahler et al 1998 called the fuzzy proximity of the internal type that is defined by means of another notion of symmetry not depending on an order reversing involution here we introduce the level uniform structure u and the level proximity n u and n respectively weshow that there is one to one correspondence between a fuzzyuniform structure u and the family u l0 of uniform structuresthat fulfills certain conditions is given by u u and u u a u a u we also show that the topologies tu and tn associated with u and n coincides with the level topologies of the fuzzy topologies uand n associated to u and n respectively that is tu u and tn n moreover we assign for each fuzzyuniform structure u an associated fuzzy proximity of theinternal type nu and hence we get the relationbetween the levels of u and of nu which is given by nu nu

h2 filtering for discrete time hidden markov jump systems abstractin this paper we consider the h2 filtering problem for a discrete time markov jump linear systemin which the markov parameter is not available in the spirit of active fault tolerant control systems it is supposed that there is a discrete time hidden markov model k k in which the observable part k represents the information coming from a detector and available to the filter while the hidden part k of the process represents the dynamics of the real system several models found in the literature are encompassed by this framework like the complete observation case the clustering information case the mode independent case and markov models in fault tolerant control we start by analysing an auxiliary filtering problem in which it is assumed that both k k are available to the filter so that necessary and sufficient conditions in terms of lmi linear matrix inequalities for the existence of the optimal filter can be obtained in the sequel we consider the realistic case