a new algorithm for testing diagnosability of fuzzy discrete event systems failure diagnosis and detection of fuzzy discrete event systems play a significant role in the study of complex systems in this paper we investigate the diagnosability of fuzzy discrete event systems by proposing a new algorithm based on the concept of undistinguishable strings moreover a necessary and sufficient condition for fuzzy diagnosability is obtained in terms of certain properties of the diagnoser which is constructed with respect to the minimal observable event the computing process to check the diagnosability of fuzzy dess and some examples serving to illuminate the applications are developed and described

on continuous dependence for an inverse initial boundary value problem for the wave equation abstract we show that the bounded coefficient q x of the wave equation u tt u qu x t 0 t u x 0 u t x 0 0 x u v x x x t 0 t can be continuously determined by its neumann to dirichlet map in certain function spaces

conditions for minimal fuzzy deterministic finite automata via brzozowski s procedure this paper deals with the application of brzozowski s minimization procedure to fuzzy finite automata with truth values in a complete residuated zero divisor free lattice for a given fuzzy finite automaton mathcal a the procedure computes the automaton d r d r mathcal a where d mathcal a is a fuzzy determinization of mathcal a and r mathcal a is the reverse automaton of mathcal a it is observed that the size of the resulting automaton is strongly dependent on the determinization method used in the procedure consequently this paper studies necessary and sufficient conditions for obtaining minimal fuzzy deterministic finite automata via brzozowski s procedure the study is accomplished for determinization methods based on a generalized notion of accessible fuzzy subset construction the obtained conditions determine that brzozowski s procedure returns a minimal fuzzy deterministic finite automaton when the determinization method does not produce proportional fuzzy states this is the reason to obtain minimal fuzzy deterministic finite automata using brzozowski s procedure for fuzzy finite automata with truth values in the product structure when the determinization method is the method based on factorization of fuzzy states

a categorical approach to minimal realization for a fuzzy language abstract in this study we provide a general theory of minimal realization for a given fuzzy language in a category theory setting we introduce categorical characterizations of a run map reachability map and observability map for a crisp deterministic fuzzy automaton finally we show that all minimal realizations for a given fuzzy language are isomorphic to the nerode realization for the same case

algorithmic and logical characterizations of bisimulations for non deterministic fuzzy transition systems abstract bisimulation is a well known behavioral equivalence for discrete event systems and has been developed in fuzzy systems quickly in this paper we adopt an approach of the relational lifting that is one of the most used methods in the field of bisimulation research to define it for a non deterministic fuzzy transition system an o s 4 2 algorithm is given for testing bisimulation where s is the number of states and the number of transitions in the underlying transition systems two different modal logics are presented one is two valued and indicates whether a state satisfies a formula which is an extension of hennessy milner logic the other is real valued and shows to what extent a state satisfies a formula they both characterize bisimilarity soundly and completely interestingly the second characterization holds under a class of fuzzy logics in addition this real valued logic allows us to conveniently define a logical metric that captures the similarity between states or systems that is the smaller distance the more states alike although the work is inspired by the corresponding work in probabilistic systems it is obviously different in particular the real valued logic in this paper remains unexplored in fuzzy systems even in probabilistic systems

characterizations of fuzzy finite automata in this paper we establish some important concepts in fuzzy finite automata ffas with bifuzzy property and clarify their essential relationships first we present a number of basic definitions and properties in ffas we then define a bifuzzy family of subautomata bifuzzy source and successor operators as well as demonstrate the bifuzzy topological characterization of ffas also we clarify the relationships between our results and previous ones and thus derive a relation figure visualizing their connections afterwards we investigate further the bifuzzy separability retrievability and homomorphism fuzzy continuous mapping and open mapping are also incorporated in particular we prove a number of equivalent characterizations for these concepts and expound the relationships amongst them this also concludes that both the bifuzzy source and successor operators are fuzzy closure operators we discover that our investigation generalizes to some extent the algebraic fuzzy automata elaborated by malik mordeson and others finally the main results obtained are summarized the potential applications are indicated and a number of related problems for further study are addressed

identification problem of two operators for nonlinear systems in banach spaces abstract we consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations for the identification problem we show the existence of optimal solutions and present necessary optimality conditions we illustrate the approach on two examples

distribution based behavioral distance for nondeterministic fuzzy transition systems modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems in this paper we first present a new notion of bisimulation for nondeterministic fuzzy transition systems which is distribution based and coarser than state based bisimulation appeared in the literature then we define a distribution based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations we also propose an on the fly algorithm for computing the bisimilarity metric moreover we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric interestingly this characterization holds for a class of fuzzy modal logics in addition we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric which makes compositional verification possible

on the stability in terms of two measures for perturbed impulsive integro differential equations this paper establishes several stability criteria for perturbed impulsive integro differential equations with fixed moments of impulsive effect by using a new comparison theorem which connects the solutions of perturbed system and the unperturbed one some sufficient conditions for the stability in terms of two measures are obtained for the perturbed system while unperturbed one dissatisfied which because of the effect of the perturbed terms

diagnosability of fuzzy discrete event systems a fuzzy approach in order to more effectively cope with the real world problems of vagueness fuzzy discrete event systems fdess were proposed by lin and ying recently then we and cao and ying investigated the supervisory control of fdess independently in this paper we are concerned with another important issue of fdess the failure diagnosis more specifically 1 we propose a ldquofuzzy diagnosabilityrdquo approach by introducing a fuzzy diagnosability function to characterize the diagnosability degree which takes values in the interval 0 1 rather than 0 1 2 based on the observability of events we formalize the construction of the diagnosers that are used to perform fuzzy diagnosis 3 a number of basic properties of the diagnosers are investigated in particular we present a necessary and sufficient condition for failure diagnosis of fdess our results generalize the important consequences of the diagnosability for crisp discrete event systems dess introduced by sampath et al the newly proposed approach allows us to deal with the problem of diagnosability for both crisp dess and fdess 4 in addition a method for checking the fuzzy diagnosability for fdess is proposed also some examples are provided to illustrate the application of the diagnosability of fdess

further improvements of determinization methods for fuzzy finite automata in this paper we provide further improvements of determinization methods for fuzzy finite automata these methods perform better than all previous determinization methods for fuzzy finite automata developed by b lohlavek 3 li and pedrycz 38 ignjatovic et al 25 and jancic et al 33 in the sense that they produce smaller automata while require the same computation time the only exception is the brzozowski type determinization algorithm developed recently by jancic and iric 34 which produces a minimal crisp deterministic fuzzy automaton but the algorithms created here can also be used within the brzozowski type algorithm and improve its performance

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

bisimilarity enforcement for discrete event systems using deterministic control for discrete event systems the control to enforce bisimilarity with respect to a given specification has been studied in in this note we consider the case when the control is required to be deterministic while a deterministic control is restrictive compared to a nondeterministic one the case of deterministic control has its own practical significance as it is easier to implement and computationally less expensive to verify we provide a necessary and sufficient condition for the existence of a bisimilarity enforcing deterministic control and discuss its computational complexity we also study properties related to the synthesis of subspecification and superspecification

polynomial time algorithms for computing distances of fuzzy transition systems behaviour distances to measure the resemblance of two states in a nondeterministic fuzzy transition system have been proposed recently in literature such a distance defined as a pseudo ultrametric over the state space of the model provides a quantitative analogue of bisimilarity in this paper we focus on the problem of computing these distances we first extend the definition of the pseudo ultrametric by introducing discount such that the discounting factor being equal to 1 captures the original definition we then provide polynomial time algorithms to calculate the behavioural distances in both the non discounted and the discounted setting the algorithm is strongly polynomial in the former case

approximate bisimulations and state reduction of fuzzy automata under fuzzy similarity measures abstract for the purpose of reducing the number of states of fuzzy automata we study approximate bisimulations of fuzzy automata under fuzzy similarity measures which are induced by residuums of left continuous t norms for a real number 0 1 and a fuzzy similarity measure s we define approximate bisimulation under s between two fuzzy automata and prove that the degree of similarity between them is more than or equal to if there exists an approximate bisimulation between them furthermore we put forward the notion of surjective functional approximate bisimulations between two fuzzy automata under s using the notion of surjective functional approximate bisimulations under s the approximate bisimulations for a fuzzy automaton under s is defined as an equivalence relation with two additional properties according to an approximate bisimulation for a fuzzy automaton an aggregated fuzzy automaton has been constructed and the degree of similarity between them is also more than or equal to it is significant for us to find that there might not possess the greatest approximate bisimulation for a fuzzy automaton and we provide a polynomial time algorithm for computing all maximal approximate bisimulations finally we provide the conditions of the existence of the greatest approximate bisimulation

a belief evolution based approach for online control of fuzzy discrete event systems under partial observation in this paper we investigate the partially observed supervisor synthesis problem in the framework of fuzzy discrete event systems dess the goal is to synthesize a fuzzy supervisor such that the behavior of the closed loop system is within a given fuzzy language a new approach for solving this problem is proposed based on the idea of belief evolution specifically we propose an algorithm that can be implemented in an online manner we show that the proposed algorithm is both sound and complete i e it effectively solves the supervisor synthesis problem to the best of our knowledge this is the first algorithm with such a property for fuzzy dess as previous works on this topic mostly focus on the supervisor existence condition rather than the supervisor synthesis problem

on the generating sequences of regular languages on k symbols the main result is a characterization of the generating sequences of the length of words in a regular language on k symbols we say that a sequence s of integers is regular if there is a finite graph g with two vertices i t such that sn is the number of paths of length n from i to t in g thus the generating sequence of a regular language is regular we prove that a sequence s is the generating sequence of a regular language on k symbols if and only if both sequences s sn n 0 and t kn sn n 0 are regular

theory of extended fuzzy discrete event systems for handling ranges of knowledge uncertainties and subjectivity in 2001 we originated a theory of fuzzy discrete event systems fdess that generalized the conventional crisp discrete event systems dess vagueness and imprecision concerning states and event transitions of dess were represented by membership grades and computed via fuzzy logic our application of the fdes theory to computerized human immunodeficiency virus acquired immune deficiency syndrome treatment regimen selection although preliminarily successful suggests that a more comprehensive fdes theory is needed to address two general issues critically important not only to biomedical applications but also to real world problems in other industries first domain experts should have means other than point estimates and type 1 fuzzy sets mandated in the current framework to describe uncertainties subjectivity and imprecision in their complex knowledge and experience second when a group of experts with distinct opinions is involved they should not be forced to reach consensus for the sake of system development this is because collective consensus may not be achievable which is often the case in medicine where individual expertspsila opinions should be equally respected since the underlying ground truth is unknown most of the time the theory of extended fdes presented in this paper addresses both the problems and contains the fdes theory as a special case experts are now allowed to use interval numbers and type 1 and type 2 fuzzy sets to intuitively and quantitatively express their diverse knowledge and experience which will then be processed by the new theory to form fuzzy state vectors and fuzzy event transition matrices accordingly we have established mathematical operations that cover the computations of fuzzy states fuzzy event transitions and parallel composition numerical examples are provided

finite l fuzzy machines abstract for finite l fuzzy machines we develop behaviour reduction and minimization theory here l stands for l 0 1 where l is a totally ordered set with universal bounds 0 and 1 and the operations are join and meet the most essential results include investigating behaviour equivalence reduction and minimization problems and their algorithmical decidability

epsilon bisimulation relations for fuzzy automata this paper investigates the approximate bisimulation relations for fuzzy automata in order to study the approximate minimization problem of fuzzy automata for a small positive real number epsilon we introduce the notion of epsilon bisimulation relations between two fuzzy automata and prove that the behavior of a fuzzy automaton a differs by epsilon from the behavior of a fuzzy automaton b under a epsilon bisimulation relation between them also the notion of surjective functional epsilon bisimulation relations between two fuzzy automata is defined according to surjective functional epsilon bisimulation relations we discuss epsilon bisimulation relations for a fuzzy automaton a construction of aggregated fuzzy automaton by the given epsilon bisimulation for a fuzzy automaton is given furthermore we find that there might not exist the greatest epsilon bisimulation relation for a fuzzy automaton and we novelly give an effective algorithm to construct all maximal epsilon bisimulation relations for the given fuzzy automaton finally we point out that bisimulation relations for a fuzzy automaton are also epsilon bisimulation relations the conditions for the real number epsilon to ensure the existence of the greatest epsilon bisimulation are also discussed

state based decentralized diagnosis of bi fuzzy discrete event systems recently we have introduced bi fuzzy discrete event systems bfdess and established their supervisory control theory under full observations and partial observations respectively compared with the models of discrete event systems and fuzzy discrete event systems bfdess are more satisfactory models to characterize some high uncertainty systems in this paper we study the fault diagnosis problems of bfdess first the framework of state based decentralized diagnosis is introduced then the notion of state based codiagnosability is defined and a necessary and sufficient condition for state based codiagnosability is provided in particular a polynomial time algorithm for state based codiagnosability verification and an online decentralized diagnosis method are presented in addition a compact state based decentralized diagnosis framework is introduced and the notion of state based jointly diagnosability and its verification algorithm are proposed finally the implication relationships between the two diagnosabilities are investigated

logical characterizations of simulation and bisimulation for fuzzy transition systems simulations and bisimulations are known to be useful for abstracting and comparing formal systems and they have recently been introduced into fuzzy systems in this study we provide sound and complete logical characterizations for simulation and bisimulation which are defined over fuzzy labeled transition systems via two variants of the hennessy milner logic the logic for characterizing fuzzy simulation has neither negation nor disjunction which is very different from the well known logical characterizations of probabilistic simulations although the completeness proofs of our characterization results are inspired by relevant research in probabilistic concurrency theory the logic for characterizing fuzzy bisimulation also deviates from that for probabilistic bisimulations

from classic observability to a simple fuzzy observability for fuzzy discrete event systems in order to determine uncertainties from restricted available information fuzzy discrete event systems fdess or fuzzy discrete event dynamic systems fdedss were recently proposed these frameworks include fuzzy states and events occurring simultaneously with different membership degrees fuzzy states and events have been used to describe uncertainties that occur often in practical problems such as treatment planning for hiv aids patients sensory information processing for robotic control and fault diagnosis problems in order to measure information associated with fdess or fdedss the classical discrete event system des observability has been turned into fuzzy observability for fdess or fdedss the newly proposed method allows ease of defining degrees of observability so that uncertainties in fdess or fdedss can be dealt with effectively this gives an opportunity to design better decision making systems to calculate the observability degree a simple fuzzy observability checking method is introduced and two examples are elaborated upon to illustrate the presented method finally the newly proposed method is tested on a heating ventilating and air conditioning hvac system

on the stability in terms of two measures of perturbed neutral functional differential equations different concepts of stability are presented in the literature to recall a few we name the usual stability partial stability and the conditional stability the overall concepts of stability can be put forward in the notion of stability in terms of two different measures following movchan 12 many authors have successfully developed the theory of stability in terms of two measures which became important in the investigation of converse theorems see 1 3 8 9 10 13

reliable decentralized supervisory control of fuzzy discrete event systems decentralized supervisory control has been introduced to deal with a class of fuzzy discrete event systems des in which state transitions are uncertain in this paper we develop the work on fuzzy des to consider the reliable decentralized supervisor problem with possible failures of some local supervisors by presenting a new definition of co observable language the reliable decentralized supervisory control problem is solved since the co observable languages are not closed under union a new normal language that is co observable is defined to solve the synthesis problem finally a numerical example is given

model checking of linear time properties based on possibility measure using possibility measure we study model checking of linear time properties in possibilistic kripke structures first the notion of possibilistic kripke structures and the related possibility measure are introduced and then model checking of reachability and repeated reachability linear time properties in finite possibilistic kripke structures are studied standard safety properties and regular properties in possibilistic kripke structures are introduced the verification of regular safety properties and regular properties using finite automata are thoroughly studied it has been shown that the verification of regular safety properties and regular properties in a finite possibilistic kripke structure can be transformed into the verification of reachability properties and repeated reachability properties in the product possibilistic kripke structure that is introduced in this paper several examples are given to illustrate the methods that are presented in this paper

on factorized l fuzzy automaton and its l fuzzy topological characterization abstract this paper is towards the study of l fuzzy automata via theory of factorizations where l is a complete residuated lattice we study the concept of factorization of l fuzzy sets and factorization of collection of l fuzzy sets such factorizations are used i to study the concept of minimal l fuzzy automaton and ii to introduce l fuzzy topologies for the study of concepts associated with an l fuzzy automaton specifically we introduce the notion of factorized l fuzzy automaton corresponding to a given l fuzzy automaton and study the minimality of such automaton further we associate l fuzzy topologies for factorized l fuzzy automaton and use these to characterize some algebraic concepts

bisimulations for fuzzy transition systems there has been a long history of using fuzzy language equivalence to compare the behavior of fuzzy systems however the comparison at this level is too coarse recently a finer behavioral measure i e bisimulation has been introduced to fuzzy finite automata however the results obtained are applicable only to finite state systems in this paper we consider bisimulation for general fuzzy systems which may be infinite state or infinite event by modeling them as fuzzy transition systems ftss to help understand and check bisimulation we characterize it in three ways by enumerating whole transitions comparing individual transitions and using a monotonic function in addition we address composition operations subsystems quotients and homomorphisms of ftss and discuss their properties connected with bisimulation the results presented here are useful to compare the behavior of general fuzzy systems in particular this makes it possible to relate an infinite fuzzy system to a finite one which is easier to analyze with the same behavior

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

fuzzy ϵ approximate regular languages and minimal deterministic fuzzy automata ϵ accepting them abstract in this paper for a real number 0 1 we put forward the notion of fuzzy approximate regular languages and study their properties especially the pumping lemma in the context of this kind of languages by comparing sets of fuzzy approximate regular languages under the order of set inclusion we get an infinite hierarchy of fuzzy languages the smallest is the set of fuzzy regular languages and the biggest is the set of fuzzy languages and the sets of approximate regular languages are different for different 0 1 2 we also investigate whether operations closed in the set of fuzzy regular languages are still closed in the set of fuzzy approximate regular languages furthermore for a fuzzy approximate regular language f approximate equivalence relations for f are characterized in order to construct deterministic fuzzy automata accepting f if a fuzzy approximate regular language f is also a fuzzy regular language and accepted by a given accessible deterministic fuzzy automaton a then we give a polynomial time algorithm to construct at least one minimal deterministic fuzzy automaton accepting f by means of a finally we point out that the number of states of minimal deterministic fuzzy automata accepting a fuzzy regular language is smaller than or equal to that of minimal deterministic fuzzy automata accepting this language

weighted finite automata over strong bimonoids we investigate weighted finite automata over strings and strong bimonoids such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold we define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions moreover we show that in this case the given weighted automata can be determinized

computation of the greatest simulations and bisimulations between fuzzy automata recently two types of simulations forward and backward simulations and four types of bisimulations forward backward forward backward and backward forward bisimulations between fuzzy automata have been introduced if there is at least one simulation bisimulation of some of these types between the given fuzzy automata it has been proved that there is the greatest simulation bisimulation of this kind in the present paper for any of the above mentioned types of simulations bisimulations we provide an efficient algorithm for deciding whether there is a simulation bisimulation of this type between the given fuzzy automata and for computing the greatest one whenever it exists the algorithms are based on the method developed in ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 which comes down to the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations

fuzzy discrete event systems under fuzzy observability and a test algorithm in order to more effectively cope with the real world problems of vagueness impreciseness and subjectivity fuzzy discrete event systems fdess were proposed recently notably fdess have been applied to biomedical control for human immunodeficiency virus hiv acquired immune deficiency syndrome aids treatment planning and sensory information processing for robotic control qiu independently developed supervisory control theory of fdess we note that the controllability of events in qiu s work is fuzzy but the observability of events is crisp and the observability of events in cao and ying s work is also crisp although the controllability is not completely crisp since the controllable events can be disabled with any degrees motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees in this paper we establish the supervisory control theory of fdess with partial observations in which both the observability and controllability of events are fuzzy instead we formalize the notions of fuzzy controllability condition and fuzzy observability condition in addition controllability and observability theorem of fdess is set up in a more generic framework in particular we present a detailed computing flow to verify whether the controllability and observability conditions hold thus this result can decide the existence of supervisors also we use this computing method to check the existence of supervisors in the controllability and observability theorem of classical discrete event systems dess which is a new method and different from classical case a number of examples are elaborated on to illustrate the presented results

supervisory control of fuzzy discrete event systems for simulation equivalence the supervisory control theory of fuzzy discrete event systems fdess for fuzzy language equivalence has been developed however in a way language equivalence has limited expressiveness therefore if the given specification can not be expressed by language equivalence then the control for language equivalence does not work in this paper we further establish the supervisory control theory of fdess for fuzzy simulation equivalence whose expressiveness is stronger than that of fuzzy language equivalence first we formalize the notions of fuzzy simulation and fuzzy simulation equivalence between two fdess then we present a method for deciding whether there is a fuzzy simulation or not in addition we also show several basic properties of fuzzy simulation relations afterward we put forward the notion of fuzzy simulation based controllability and in particular show that it serves as a necessary and sufficient condition for the existence of the fuzzy supervisors of fdess moreover we study the range control problem of fdess some examples are given to illustrate the main results obtained

fuzzy discrete event systems for multiobjective control framework and application to mobile robot navigation fuzzy discrete event systems fdess have been introduced in recent years to model systems whose discrete states or discrete state transitions can be uncertain and are hence determined by a possibility degree this paper develops an fdes framework for the control of sampled data systems that have to fulfill multiple objectives the choice of a fuzzy system representation is justified by the assumption of a controller realization that depends on various potentially imprecise sensor measurements the proposed framework consists of three basic steps that are performed at each sampling instant first the current fuzzy state of the system is determined by a sensor evaluation second the fuzzy state in the future sampling instant is predicted for all possible control actions of the system finally an original multiobjective weighting strategy is proposed to determine the control action to be applied in the current sampling instant the features of the proposed approach are demonstrated by a detailed mobile robot example which includes a simulation study

analysis and control of fuzzy discrete event systems using bisimulation equivalence in order to more effectively cope with the real world problems of vagueness impreciseness and subjectivity fuzzy discrete event systems fdess were proposed recently in this paper we deal with the analysis and control of fuzzy discrete event systems by using bisimulation equivalence in practice we design a supervisor such that the controlled system is bisimilar to the specification also a model theorem showing that a supervisor exists over a certain finite fuzzy state space is obtained furthermore the notion of fuzzy state controllability is introduced as part of a necessary and sufficient condition for the existence of a supervisor

observability and decentralized control of fuzzy discrete event systems fuzzy discrete event systems as a generalization of crisp discrete event systems have been introduced in order that it is possible to effectively represent uncertainty imprecision and vagueness arising from the dynamic of systems a fuzzy discrete event system has been modeled by a fuzzy automaton its behavior is described in terms of the fuzzy language generated by the automaton in this paper we are concerned with the supervisory control problem for fuzzy discrete event systems with partial observation observability normality and co observability of crisp languages are extended to fuzzy languages it is shown that the observability together with controllability of the desired fuzzy language is a necessary and sufficient condition for the existence of a partially observable fuzzy supervisor when a decentralized solution is desired it is proved that there exist local fuzzy supervisors if and only if the fuzzy language to be synthesized is controllable and co observable moreover the infimal controllable and observable fuzzy superlanguage and the supremal controllable and normal fuzzy sublanguage are also discussed simple examples are provided to illustrate the theoretical development

identification problem for the wave equation with neumann data input and dirichlet data observations we seek to identify the dispersive coefficient in a wave equation with neumann boundary conditions in a bounded space time domain from imprecise observations of the solution on the boundary of the spatial domain dirichlet data the problem is regularized and solved by casting it into an optimal control setting by letting the cost of the control tend to zero we obtain the limit of the corresponding control sequence which we identify with the sought dispersive coefficient the corresponding solution of the wave equation is interpreted as the possibly nonunique projection of the observation vector onto the range of the neumann to dirichlet maps corresponding to a single input neumann data as the dispersive coefficient is varied several numerical examples illustrate the merits and limitations of the procedure