control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

characterizations of compact sets in fuzzy sets spaces with l_p metric abstract compactness criteria in fuzzy set spaces endowed with the l p metric have been studied for several decades total boundedness is a key feature of compactness in metric spaces however comparing existing compactness criteria in fuzzy set spaces endowed with the l p metric with the arzela ascoli theorem the latter gives compactness criteria by characterizing totally bounded sets while the former does not characterize totally bounded sets currently compactness criteria are only presented for three particular fuzzy set spaces under assumptions of convexity or star shapedness general fuzzy sets have become more important in both theory and applications therefore this paper presents characterizations of totally bounded sets relatively compact sets and compact sets in general fuzzy set spaces equipped with the l p metric but which do not have any assumptions of convexity or star shapedness subsets of these general sets include common fuzzy sets such as fuzzy numbers fuzzy star shaped numbers with respect to the origin fuzzy star shaped numbers and general fuzzy star shaped numbers existing compactness criteria are stated for fuzzy numbers space the space of fuzzy star shaped numbers with respect to the origin and the space of fuzzy star shaped numbers endowed with the l p metric respectively constructing completions of fuzzy set spaces with respect to the l p metric is a problem closely dependent on characterizing totally bounded sets based on characterizations of total boundedness and relatively compactness and some discussion of the convexity and star shapedness of fuzzy sets we show that the completions of fuzzy set spaces studied here can be obtained using the l p extension we also clarify relationships among the ten fuzzy set spaces studied here the five pairs of original spaces and their corresponding completions we show that the subspaces have parallel characterizations of totally bounded sets relatively compact sets and compact sets finally we discuss properties of the l p metric on fuzzy set space as an application of our results and review compactness criteria proposed in previous work

exponential bounds for discrete time singularly perturbed markov chains abstract this paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed markov chains in discrete time by considering two time scale in the markov chains asymptotic analysis is carried out the cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first the upper bounds of a sequence of scaled occupation measures are derived then extensions to markov chains involving transient states and or nonhomogeneous transition probabilities are dealt with the results enable us to further our understanding of the underlying markov chains and related dynamic systems which is essential for solving many control and optimization problems

approximation and decomposition of singularly perturbed stochastic hybrid systems this paper considers a singularly perturbed stochastic hybrid system whose state equations are governed by a stochastic switching process which is modelled as a nearly completely decomposable continuous time finite state markov chain approximate models for the slow mode subsystem are derived over the intervals of rapid switching and their accuracy the mean squared errors between the approximate models and the original systems is quantified aggregate models for the subsystem are obtained over a longer period the almost surely exponential stability of the subsystem is studied the behaviour of the fast subsystem which depends on the relative size of the perturbation parameters is analysed the decomposition of the overall system and the switching process together into slow and fast subsystems is investigated the results derived in the paper are shown to hold when the switching process is stationary and irreducible and each group of strongly interacting states is irreducible and time reversible fina

general fuzzy least squares abstract a model for least squares fitting of fuzzy valued data is described a previous process obtained for triangular fuzzy numbers is generalized and improved to include all fuzzy numbers represented by single maxima piecewise continuous functions with compact support the new model is compared with the previous procedure and its applicability is demonstrated

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

aggregation of the policy iteration method for nearly completely decomposable markov chains a steady state optimal control problem is considered for nearly completely decomposable markov chains in order to apply the policy iteration method of r a howard dynamic programming and markov processes cambridge ma mit press 1960 a high dimensional ill conditioned system of algebraic equations must be solved in the value determination step although algorithms exist for aggregation of the steady state probability distribution problem they only provide methods for computing the cost not the dual variables using a singular perturbation approach an aggregation method for the value determination equation is developed the aggregation method is developed in three steps first a class of similarity transformations that transform the system into a singularly perturbed form is developed second an aggregation method to compute the steady state probability distribution is derived third this aggregation method is applied to the value determination step of howard s method 1960

entropy stable and entropy dissipative approximations of a fourth order quantum diffusion equation structure preserving numerical schemes for a nonlinear parabolic fourth order equation modeling the electron transport in quantum semiconductors with periodic boundary conditions are analyzed first a two step backward differentiation formula bdf semi discretization in time is investigated the scheme preserves the nonnegativity of the solution is entropy stable and dissipates a modified entropy functional the existence of a weak semi discrete solution and in a particular case its temporal second order convergence to the continuous solution is proved the proofs employ an algebraic relation which implies the g stability of the two step bdf second an implicit euler and q q step bdf discrete variational derivative method are considered this scheme which exploits the variational structure of the equation dissipates the discrete fisher information or energy numerical experiments show that the discrete relative entropies and fisher information decay even exponentially fast to zero

analysis and numerical solution of coupled volume surface reaction diffusion systems with application to cell biology abstract we consider the numerical solution of coupled volume surface reaction diffusion systems having a detailed balance equilibrium based on the conservation of mass an appropriate quadratic entropy functional is identified and an entropy entropy dissipation inequality is proven this allows us to show exponential convergence to equilibrium by the entropy method we then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error that hold uniformly in time some numerical tests are presented to illustrate these theoretical results the analysis and the numerical approximation are discussed in detail for a simple model problem the basic arguments however apply also in a more general context this is demonstrated by investigation of a particular volume surface reaction diffusion system arising as a mathematical model for asymmetric stem cell division

model order reduction via real schur form decomposition abstract a new algorithm is presented for computing moore s reduced order transfer function matrix without calculating the balancing transformation which tends to be ill conditioned especially when the original system is non minimal or when it has very nearly uncontrollable or unobservable modes the algorithm is based on finding the eigenspaces associated with large eigenvalues of the cross gramian matrix wco using the real schur form decomposition the algorithm does not require a minimal model to start with the state space realization obtained by this method is related to the balanced realization by a non singular matrix an example is presented to illustrate the proposed algorithm

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

a two factor stochastic production model with two time scales a two factor production model where one factor is characterized by a continuous state variable and corresponds to the fast mode via a controlled diffusion process whereas the second factor is characterized by a discrete variable and corresponds to the slow mode via a controlled jump process

singular perturbation modelling of continuous and discrete physical systems the major shortcoming of the singular perturbation theory preventing its application to practical problems is that the system described by state equations is seldom given in the singularly perturbed form this paper presents a systematic procedure which converts the state equations of a large scale time invariant system to singularly perturbed form assuming that such a classification is possible this is accomplished by appropriate scaling and grouping of the state variables into slow and fast groups while retaining as many physical state variables as possible in the new state description an improved slow fast grouping of the states can result by allowing linear combination of certain available physical variables as new state variables several examples are given to illustrate the method

singular perturbation modelling of large scale systems with multi time scale property the problem is investigated of singular perturbation modelling which plays an important role in cases of the application of the singular perturbation method to the design and analysis of large scale systems in both continuous and discrete time versions modelling concepts based on structural properties are studied and computer oriented algorithms are developed the results of two time scale systems are extended to multi time scale cases furthermore in order to verify the validity of these studies some numerical examples are given

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

energy estimates for continuous and discretized electro reaction diffusion systems abstract we consider electro reaction diffusion systems consisting of continuity equations for a finite number of species coupled with a poisson equation we take into account heterostructures anisotropic materials and rather general statistic relations we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly the same properties are shown for an implicit time discretized version of the problem moreover we provide a space discretized scheme for the electro reaction diffusion system which is dissipative the free energy decays monotonously on a fixed grid we use for each species different voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species

a characterization of relatively compact sets of fuzzy sets abstract a purely topological characterization of relatively compact sets is given for the metric space k y d of upper semicontinuous compact supported normal fuzzy subsets of a metric space y the considered metric d is that of the distance between fuzzy subsets which is the supremum of the hausdorff distances of the corresponding level sets in the given proof the compactness of a variational convergence which was introduced by de giorgi and franzoni is fundamental

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

paper on coherency based decomposition algorithms a new algorithm of coherency based decomposition is presented and applied contrary to avramovic and coworkers algorithm it does not necessarily imply in practice the knowledge of the eigenvalues and eigenvectors of the system or an a priori knowledge of the number of coherent areas as long as two classes of dynamics slow and fast exist its applicability to the case of a 48 machine system is demonstrated

a real schur form method for modeling singularly perturbed systems a method is presented for modeling a two time scale system in the singularly perturbed form the method uses an ordered real schur decomposition which can be efficiently computed using standard subroutines from eispack three results are given first it is shown that any two time scale system can be modeled in the singularly perturbed form by a transformation into an ordered real schur form followed by balancing second under some conditions on the ordered real schur decomposition a procedure is given to achieve the modeling task with all fast variables chosen from the original state variables third necessary and sufficient conditions are given to achieve modeling by permutation of the original state variables

singularly perturbed markov chains with two small parameters a matched asymptotic expansion this work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed markov chains with two small parameters it is motivated by the model of a cost minimizing firm involving production planning and capacity expansion and a two level hierarchical decomposition our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation different from previous work on singularly perturbed markov chains the inner expansion terms are constructed by solving certain partial differential equations the methods of undetermined coefficients are used the error bound is obtained

diffusion on a curved surface coupled to diffusion in the volume an algorithm is presented for solving a diffusion equation on a curved surface coupled to diffusion in the volume a problem often arising in cell biology it applies to pixilated surfaces obtained from experimental images and performs at low computational cost in the method the laplace beltrami operator is approximated locally by the laplacian on the tangential plane and then a finite volume discretization scheme based on a voronoi decomposition is applied convergence studies show that mass conservation built in the discretization scheme and cancellation of sampling error ensure convergence of the solution in space with an order between 1 and 2 the method is applied to a cell biological problem where a signaling molecule g protein rac cycles between the cytoplasm and cell membrane thus coupling its diffusion in the membrane to that in the cell interior simulations on realistic cell geometry are performed to validate and determine the accuracy of a recently proposed simplified quantitative analysis of fluorescence loss in photobleaching the method is implemented within the virtual cell computational framework freely accessible at www vcell org

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

stability bounds of singularly perturbed discrete systems the authors propose a systematic approach to determine the exact stability bound of singularly perturbed discrete time systems by combining three conditions of the critical stability criterion with a bialternate product for discrete time systems the stability problems of two types of singularly perturbed discrete time systems can be solved via the derived scheme several examples are given to illustrate the proposed results

robust stability of singularly perturbed discrete delay systems this paper presents new sufficient conditions for the robust stability of singularly perturbed discrete systems subjected to multiple delayed perturbations for a given spl epsi spl isin 0 spl epsi sub 0 where spl epsi sub 0

a method to calculate steady state distributions of large markov chains by aggregating states this paper develops an efficient iterative algorithm to calculate the steady state distribution of nearly all irreducible discrete time markov chains computational experiences suggest that for large markovian systems more than 130 states the proposed algorithm can be ten times faster than standard gaussian elimination in finding solutions to an accuracy of 0 1 the proposed algorithm is developed in three stages first we develop a very efficient algorithm for determining steady state distributions of a restricted class of markovian systems a second result establishes a relationship between a general irreducible markovian system and a system in the restricted class of markovian systems finally we combine the two results to produce an efficient iterative algorithm to solve markov systems the paper concludes with a discussion of the observed performance of the algorithm

stabilization bound of discrete two time scale systems abstract in this paper the exact bound problem in various types of discrete two time scale tts system is considered a set of stability criteria based on the frequency domain representation is derived two interesting examples are used to demonstrate the proposed scheme

optimal control of a particular class of singularly perturbed nonlinear discrete time systems this note studies a class of discrete time nonlinear systems which depend on a small parameter using the singular perturbation theory in a systematic way we give a trajectory approximation result based on the decomposition of the model into reduced and boundary layer models this decomposition is used to analyze optimal control via maximum principle of such systems as a result significant order reduction of optimal control problems is achieved

time scale decomposition of linear implicit singularly perturbed systems an implicit singularly perturbed system dot x a epsilon x where a 0 is singular is considered this note presents an algorithm for decomposing the system into a set of isolated subsystems at different time scales the decomposed form of the system is used to construct a uniform asymptotic approximation to exp a epsilon t

characterizations of endograph metric and γ convergence on fuzzy sets abstract this paper is devoted to the relationships and properties of the endograph metric and the convergence the main contents can be divided into three closely related parts firstly on the class of upper semi continuous fuzzy sets with bounded cuts we find that an endograph metric convergent sequence is exactly a convergent sequence satisfying the condition that the union of cuts of all its elements is a bounded set in r m for each 0 secondly based on investigations of level characterizations of fuzzy sets themselves we present level characterizations level decomposition properties of the endograph metric and the convergence it is worth mentioning that using the condition and the level characterizations given above we discover the fact the endograph metric and the convergence are compatible on a large class of general fuzzy sets which do not have any assumptions of normality convexity or star shapedness its subsets include common particular fuzzy sets such as fuzzy numbers compact and noncompact fuzzy star shaped numbers compact and noncompact and general fuzzy star shaped numbers compact and noncompact thirdly on the basis of the conclusions presented above we study various subspaces of the space of upper semi continuous fuzzy sets with bounded cuts equipped with the endograph metric we present characterizations of total boundedness relative compactness and compactness in these fuzzy set spaces and clarify relationships among these fuzzy set spaces it is pointed out that the fuzzy set spaces of noncompact type are exactly the completions of their compact counterparts under the endograph metric

some notes on the characterization of compact sets in e n d p in this paper our purpose is to modify the characterization of compact sets in en dp 1 p shown in p diamond and p kloeden metric spaces of fuzzy sets fuzzy sets and systems 35 1990 241 249

numerical solution of linear fredholm fuzzy integral equations of the second kind by adomian method using parametric form of fuzzy numbers we convert a linear fuzzy fredholm integral equation of the second kind to a linear system of integral equations of the second kind in crisp case we use adomian method and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the linear fuzzy fredholm integral equation of the second kind we apply the method to some examples

a unified view of aggregation and coherency in networks and markov chains abstract this paper presents a unified treatment of aggregability lumpability coherency reversibility partial balance and similar properties appearing in different fields including power systems and queueing theory a coherency condition well known in power systems implies the existence of a finite state filter for markov chains while aggregability and coherency yield a new condition for decentralized computation of the invariant measure

markov decision processes with slow scale periodic decisions we consider a class of discrete time dynamic decision making models which we refer to as periodically time inhomogeneous markov decision processes ptmdps in these models the decision making horizon can be partitioned into intervals calledslow scale cycles ofn 1 epochs the transition law and reward function are time homogeneous over the firstn epochs of each slow scale cycle but distinct at the final epoch the motivation for such models is in applications where decisions of different nature are taken at different time scales i e many low level decisions are made between less frequent high level ones for the ptmdp model we consider the problem of optimizing the expected discounted reward when rewards devalue by a discount factor at the beginning of each slow scale cycle whenn is large initially stationary policies i s p s are natural candidates for optimal policies similar to turnpike policies an initially stationary policy uses the same decision rule for some large number of epochs in each slow scale cycle followed by a relatively short planning horizon of time varying decision rules in this paper we characterize the form of the optimal value as a function ofn establish conditions ensuring the existence of near optimal i s p s and characterize their structure our analysis deals separately with the cases where the time homogeneous part of the system has state dependent and state independent optimal average reward as we illustrate the results in these two distinct cases are qualitatively different

asymptotic expansions of solutions of integro differential equations for transition densities of singularly perturbed switching diffusions rapid switchings this work develops asymptotic expansions for solutions of integro differential equations arising from transition densities of singularly perturbed switching diffusion processes the underlying processes have two components namely the switching component and the diffusion component processes with rapid switchings are examined here under suitable conditions it is proved that the asymptotic expansion is uniformly valid in the domain of interest

some notes on the characterization of compact sets of fuzzy sets with lp metric in 1990 diamond gave a characterization of compact sets in the space of fuzzy star shaped numbers with respect to the origin with l p metric for 1

numerical solutions of fuzzy differential and integral equations using the embedding method numerical procedures for solving fuzzy differential equations fdes and fuzzy integral equations fies with arbitrary kernels have been investigated sufficient conditions for convergence of the proposed algorithms are given and their applicability is illustrated with examples this work and its conclusions may narrow the gap between the theoretical research on fdes and fies and the practical applications already existing in the design of various fuzzy dynamical systems

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

stochastic dynamic job shops and hierarchical production planning this paper presents an asymptotic analysis of hierarchical production planning in a general manufacturing system consisting of a network of unreliable machines producing a variety of products the concept of a dynamic job shop is introduced by interpreting the system as a directed graph and the structure of the system dynamics is characterized for its use in the asymptotic analysis the optimal control problem for the system is a state constrained problem since the number of parts in any buffer between any two machines must remain nonnegative a limiting problem is introduced in which the stochastic machine capacities are replaced by corresponding equilibrium mean capacities as the rate of change in machine states approaches infinity the value function of the original problem is shown to converge to that of the limiting problem and the convergence rate is obtained furthermore near optimal controls for the original problem are constructed from near optimal controls of the limiting problem and an error estimate is obtained on the near optimality of the constructed controls

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space

decomposition of two time scale linear systems via the relevant eigenspace this paper presents a very reliable and computationally simple algorithm to transform a two time scale system in the singularly perturbed form by permutation followed by scaling assuming that certain conditions are satisfied the algorithm is based on computing orthonormal bases which span the left and the right invariant subspaces associated with the slow eigenvalues the algorithm does not require the system to be diagonalizable moreover the algorithm works even if the original system is badly scaled examples are given to illustrate the proposed algorithm

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods