a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

stability analysis for stochastic differential equations with infinite markovian switchings abstract this paper is concerned with stability analysis of linear ito stochastic differential equations with countably infinite markovian switchings a spectral criterion is proposed for exponential stability of the considered models by means of the established spectral criterion the relationship between exponential stability and stochastic l 2 stability is clarified moreover under the disturbance of random signals with finite energy a sufficient condition is presented for l 2 input output stability of the perturbed stochastic differential equations

comment on subgraphs in random networks we point out biases in the algorithms used by itzkovitz et a phys rev e 68 026127 2003 to assess their approximate formulas for the average number of occurrences of certain subgraphs in random graphs with prescribed degree sequences

the switch markov chain for sampling irregular graphs and digraphs abstract the problem of efficiently sampling from a set of undirected or directed graphs with a given degree sequence has many applications one approach to this problem uses a simple markov chain which we call the switch chain to perform the sampling the switch chain is known to be rapidly mixing for regular degree sequences both in the undirected and directed setting we prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree d max which satisfies 3 d max 1 3 m where m is the sum of the degrees the mixing time bound obtained is only a factor n larger than that established in the regular case where n is the number of vertices our result covers a wide range of degree sequences including power law density bounded graphs with parameter 5 2 and sufficiently many edges for directed degree sequences such that the switch chain is irreducible we prove that the switch chain is rapidly mixing when all in degrees and out degrees are positive and bounded above by 1 4 m where m is the number of arcs and not all in degrees and out degrees equal 1 the mixing time bound obtained in the directed case is an order of m 2 larger than that established in the regular case

asymptotic stability in the pth moment for stochastic differential equations with levy noise abstract this paper is devoted to study a class of stochastic differential equations with levy noise in comparison to the standard gaussian noise levy noise is more versatile and interesting with a wider range of applications however levy noise makes the analysis more difficult owing to the discontinuity of its sample paths in this paper we attempt to overcome this difficulty we propose several sufficient conditions under which we investigate the long time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability also we discuss two types of continuity of the solution continuous in probability and continuous in the pth moment finally we provide two examples to illustrate the effectiveness of the theoretical results

on the number of matrices and a random matrix with prescribed row and column sums and 0 1 entries abstract we consider the set r c of all m n matrices having 0 1 entries and prescribed row sums r r 1 r m and column sums c c 1 c n we prove an asymptotic estimate for the cardinality r c via the solution to a convex optimization problem we show that if r c is sufficiently large then a random matrix d r c sampled from the uniform probability measure in r c with high probability is close to a particular matrix z z r c that maximizes the sum of entropies of entries among all matrices with row sums r column sums c and entries between 0 and 1 similar results are obtained for 0 1 matrices with prescribed row and column sums and assigned zeros in some positions

optimal control for infinite dimensional stochastic differential equations with infinite markov jumps and multiplicative noise abstract in this paper we solve an infinite horizon linear quadratic control problem for a class of differential equations with countably infinite markov jumps and multiplicative noise the global solvability of the associated differential riccati type equations is studied under detectability hypotheses a nonstochastic operatorial approach is used some properties of the linear stochastic systems such as stability stabilizability and detectability are also discussed on the basis of a new solution representation result a generalized ito s formula which applies to infinite dimensional stochastic differential equations with countably infinite markov jumps is also provided

approximate counting uniform generation and rapidly mixing markov chains the paper studies effective approximate solutions to combinatorial counting and uniform generation problems using a technique based on the simulation of ergodic markov chains it is shown that for self reducible structures almost uniform generation is possible in polynomial time provided only that randomised approximate counting to within some arbitrary polynomial factor is possible in polynomial time it follows that for self reducible structures polynomial time randomised algorithms for counting to within factors of the form 1 n are available either for all fl e r or for no fi e r a substantial part of the paper is devoted to investigating the rate of convergence of finite ergodic markov chains and a simple but powerful characterisation of rapid convergence for a broad class of chains based on a structural property of the underlying graph is established finally the general techniques of the paper are used to derive an almost uniform generation procedure for labelled graphs with a given degree sequence which is valid over a much wider range of degrees than previous methods this in turn leads to randomised approximate counting algorithms for these graphs with very good

controllability of semilinear stochastic systems in hilbert spaces we study the weak approximate and complete controllability properties of semilinear stochastic systems assuming controllability of the associated linear systems the results are obtained by using the banach fixed point theorem applications to stochastic heat equation are given

cal h representation and applications to generalized lyapunov equations and linear stochastic systems this paper introduces an h representation method to express an n2 1 vector x as x h x tilde based on the introduced h representation approach several topics are extensively discussed including the generalized lyapunov equations gles arising from stochastic control stochastic observability generalized d stability and d stabilization weak stability and stabilization a necessary and sufficient condition for the existence and uniqueness of the symmetric and skew symmetric solutions of gles is presented respectively moreover the solution structure of gles is also clarified through the h representation method several necessary and sufficient conditions are also obtained for stochastic observability generalized d stability and d stabilization weak stability and stabilization

stochastic h sub 2 h sub spl infin control with state dependent noise this paper discusses the stochastic h sub 2 h sub spl infin control problem with state dependent noise by means of the stabilization exact observability and stochastic detectability of stochastic systems the infinite horizon stochastic h sub 2 h sub spl infin control design is developed for the finite horizon h sub 2 h sub spl infin control problem our results generalize the corresponding deterministic ones to the stochastic models finally the observer based suboptimal stochastic h sub 2 h sub spl infin control is discussed in which the state variables cannot be measured directly and a feasible design algorithm is proposed

stochastic analysis of a novel nonautonomous periodic siri epidemic system with random disturbances abstract in this paper a new stochastic nonautonomous siri epidemic model is formulated given that the incidence rates of diseases may change with the environment we propose a novel type of transmission function the main aim of this paper is to obtain the thresholds of the stochastic siri epidemic model to this end we investigate the dynamics of the stochastic system and establish the conditions for extinction and persistence in mean of the disease by constructing some suitable lyapunov functions and using stochastic analysis technique furthermore we show that the stochastic system has at least one nontrivial positive periodic solution finally numerical simulations are introduced to illustrate our results

robust stability and stabilization of linear stochastic systems with markovian switching and uncertain transition rates abstract this paper is concerned with the robust stabilization problem for a class of linear uncertain stochastic systems with markovian switching the uncertain stochastic system with markovian switching under consideration involves parameter uncertainties both in the system matrices and in the mode transition rates matrix new criteria for testing the robust stability of such systems are established in terms of bi linear matrix inequalities blmis and sufficient conditions are proposed for the design of robust state feedback controllers a numerical example is given to illustrate the effectiveness of our results

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

approximately counting hamilton paths and cycles in dense graphs we describe fully polynomial randomized approximation schemes for the problems of determining the number of hamilton paths and cycles in an n vertex graph with minimum degree frac 1 2 a n for any fixed a 0 we show that the exact counting problems are p complete we also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs

a unified approach for stochastic and mean square stability of continuous time linear systems with markovian jumping parameters and additive disturbances necessary and sufficient conditions for stochastic stability ss and mean square stability mss of continuous time linear systems subject to markovian jumps in the parameters and additive disturbances are established we consider two scenarios regarding the additive disturbances one in which the system is driven by a wiener process and one characterized by functions in l 2 m omega cal f mathbb p which is the usual scenario for the h infty approach the markov process is assumed to take values in an infinite countable set mathcal s it is shown that ss is equivalent to the spectrum of an augmented matrix lying in the open left half plane to the existence of a solution for a certain lyapunov equation and implies is equivalent for mathcal s finite asymptotic wide sense stationarity awss it is also shown that ss is equivalent to the state x t belonging to l 2 n omega cal f mathbb p whenever the disturbances are in l 2 m omega cal f mathbb p for the case in which mathcal s is finite ss and mss are equivalent and the lyapunov equation can be written down in two equivalent forms with each one providing an easier to check sufficient condition

on stabilizability and exact observability of stochastic systems with their applications this paper mainly studies the stabilizability and exact observability of stochastic linear controlled systems and their applications with the aid of the operator spectrum a necessary and sufficient condition is given for the stabilizability of stochastic systems some new concepts such as unremovable spectrum and strong solution are introduced an unremovable spectral theorem and a stochastic popov belevith hautus criterion for exact observability are presented as applications a comparison theorem for stochastic algebraic riccati equations and a result on lyapunov type equations are obtained

non classical derivation of the transport theorems for wave fronts abstract we establish a relationship between the derivatives of generalized functions and the transport theorems for moving surfaces of discontinuity we study the motion of a propagating and deforming wave front which itself can have a moving discontinuity on its surface with the help of the theory of generalized derivatives we present a general transport theorem which embodies the known results as special cases

observability and detectability of discrete time stochastic systems with markovian jump abstract this paper studies detectability and observability of markov jump discrete time linear stochastic systems with multiplicative noise mjdls for short the relations between some concepts of detectability and observability are established these relations together with equivalent expressions of observability gramian allow us to obtain some other sufficient and necessary conditions of detectability and observability as applications discrete time stochastic lyapunov equations and stochastic discrete algebraic riccati equations sdare are discussed

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

distributional analysis for discontinuous fields abstract the jump conditions that hold across singular surfaces for the fields having step function discontinuities do not in general apply if these surfaces themselves carry concentrated fields in this note the general situation when the surfaces of discontinuity carry multilayers and deform as they propagate is discussed formulas are presented for the first and second derivatives for these multilayers

generalized lyapunov equation approach to state dependent stochastic stabilization detectability criterion in this paper the generalized lyapunov equation approach is used to study stochastic stabilization detectability with state multiplicative noise some practical test criteria for stochastic stabilization and detectability such as stochastic popov belevitch hautus criterion for exact detectability are obtained moreover useful properties of the generalized lyapunov equation are derived based on critical stability and exact detectability introduced in this paper as applications first the stochastic linear quadratic regulator as well as the related generalized algebraic riccati equation are discussed extensively second the infinite horizon stochastic h 2 h infin control with state and control dependent noise is also investigated which extends and improves the recently published results

explicit solutions for an optimal stock selling problem under a markov chain model abstract this paper is concerned with explicit solutions for a classical optimal stock selling problem in contrast to almost all market models treated in the literature the underlying market is solely determined by a two state markov chain such markov chain model is strikingly simple and yet appears capable capturing various market movements ranging from close to brownian motion to no so brownian ones the purpose of this paper is to study the optimal selling rule under such a model and develop a set of analysis techniques useful for related optimal stopping problems in this paper the goal of the problem under consideration is to find an optimal stopping time to sell the stock so as to maximize an expected return explicit solutions to the associated variational inequalities are obtained these solutions are given in terms of a set of threshold levels verification theorems are provided to justify their optimality finally numerical examples are provided to illustrate the results

brief paper a delay decomposition approach to l2 l filter design for stochastic systems with time varying delay this paper investigates the problem of l 2 l filter design for a class of stochastic systems with time varying delay the addressed problem is the design of a full order linear filter such that the error system is asymptotically mean square stable and a prescribed l 2 l performance is satisfied in order to develop a less conservative filter design a new lyapunov krasovskii functional lkf is constructed by decomposing the delay interval into multiple equidistant subintervals and a new integral inequality is established in the stochastic setting then based on the lkf and integral inequality the delay dependent conditions for the existence of l 2 l filters are obtained in terms of linear matrix inequalities lmis the resulting filters can ensure that the error system is asymptotically mean square stable and the peak value of the estimation error is bounded by a prescribed level for all possible bounded energy disturbances finally two examples are given to illustrate the effectiveness of the proposed method

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

periodic solution and ergodic stationary distribution of two stochastic siqs epidemic systems abstract this paper proposes two stochastic siqs epidemic models with periodic parameters and markov switching we first prove that the stochastic non autonomous periodic system has a nontrivial positive periodic solution by using the khasminskii s theory then the sufficient conditions for extinction of the disease are obtained furthermore we construct suitable stochastic lyapunov functions with regime switching to prove the existence of ergodic stationary distribution of the stochastic siqs epidemic model at last some rigorous numerical simulations are presented to illustrate our theoretical results

on exponential almost sure stability of random jump systems this paper is concerned with a class of random jump systems represented by transition operators which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case a series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios the stability criteria obtained are further extended to markov jump linear systems with infinite states and hence a unified approach to describing the almost sure stability of mjlss is addressed under this context all the results in the work are developed for both the continuous and discrete time systems

generating random unlabelled graphs unlabelled graphs on n vertices can be generated uniformly at random without calculating the total numbers of such graphs but by using asymptotic enumeration results for large n the process is very efficient taking o n 2 steps on average in fact for n geqq 50 virtually all practical applications will require only the first step of the algorithm to be implemented this step is almost as simple as and has the appearance of the random generation of labelled graphs yet the net effect is the uniform generation of unlabelled graphs at random by similar methods one can generate random unlabelled graphs with n vertices and m edges in expected time o n 2 sqrt m for most values of m also random unlabelled r regular graphs can be generated in expected time o n when r geqq 3 is fixed but this is only so far practicable for a few small values of r

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

on unified concepts of detectability and observability for continuous time stochastic systems this paper is concerned with detectability and observability of continuous time stochastic linear systems by adopting the idea used in defining these two concepts for time varying systems and markovian jump linear systems that have been studied in the literature corresponding definitions for continuous time stochastic linear systems are proposed these two definitions are not only able to unify some recent definitions on these two concept reported in the literature but also allow us to propose an efficient rank criterion to test observability of continuous time stochastic linear systems it seems that this rank criterion is quite analogous to the rank criterion for deterministic linear systems with the help of these two concepts and the new criteria the stochastic lyapunov equation is revisited and some recent published work on this equation are generalized numerical examples are given to illustrate the effectiveness of the proposed approach

on the observability and detectability of continuous time markov jump linear systems the paper introduces a new detectability concept for continuous time markov jump linear systems with finite markov space that generalizes previous concepts found in the literature the detectability in the weak sense is characterized as mean square detectability of a certain related stochastic system making both detectability senses directly comparable the concept can also ensure that the solution of the coupled algebraic riccati equation associated to the quadratic control problem is unique and stabilizing making other concepts redundant the paper also obtains a set of matrices that plays the role of the observability matrix for deterministic linear systems and it allows geometric and qualitative properties tests for weak observability and detectability of a system are provided the first consisting of a simple rank test similar to the usual observability test for deterministic linear systems

brief paper on detectability of stochastic systems we discuss notions of detectability for stochastic linear control systems of ito type a natural concept of detectability requires a non zero output if the state process is unstable we show that this property can equivalently be characterized by a generalized version of the hautus test the proof is based on spectral theory for positive operators

exponential stability and interval stability of a class of stochastic hybrid systems driven by both brownian motion and poisson jumps abstract a class of stochastic singular hybrid systems driven by both brownian motion and poisson jumps are studied this paper is devoted to discussing the exponential stability and interval stability of such stochastic singular hybrid systems the concept of interval admissibility is proposed sufficient conditions are given for exponential mean square admissibility and interval admissibility by using ito s formula h representation and spectrum technique finally two simulation cases are presented to demonstrate the theoretical results

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

on the random generation and counting of matchings in dense graphs in this work we present a fully randomized approximation scheme for counting the number of perfect matchings in a dense bipartite graphs that is equivalent to get a fully randomized approximation scheme to the permanent of a dense boolean matrix we achieve this known solution through novel extensions in the theory of suitable non reversible markov chains which mix rapidly and have a near uniform distribution

almost sure exponential stability of numerical solutions for stochastic pantograph differential equations abstract the sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered with a khasminskii type condition the almost sure exponential stability of the numerical solutions by the euler maruyama method and the backward euler maruyama method is also discussed based on the discrete semimartingale convergence theorem we present the sufficient conditions for the stability of the euler maruyama method with one extra condition when compared with the exact solution we show that the backward euler maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution

optimal harvesting strategy for a stochastic mutualism system in a polluted environment with regime switching abstract in this paper we stochastically perturb the harvesting mutualism system into stochastic system in a polluted environment by white noise telegraph noise and levy jump we first prove the existence and uniqueness of the global positive solution of the stochastic system and establish the sufficient condition for persistence in mean of the two cooperative species then using aggregation and ergodic methods the explicit forms of optimal harvesting strategy and the maximum of sustainable yield are obtained we also give some examples and numerical simulations to illustrate the theoretical results the impacts of toxicant in organism telegraph noise and levy jump on the persistence in mean and optimal harvesting strategy are revealed

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

comparison theorems for anticipated bsdes with non lipschitz coefficients abstract we study comparison theorems for one dimensional anticipated backward stochastic differential equations under one kind of non lipschitz assumption in the results the generator functions are allowed to contain the anticipated term of z neither generator function needs to be necessarily monotone in the anticipated term of y and the anticipated times of the anticipated terms of y z in one generator function can differ from those in the other

fast uniform generation of regular graphs abstract an algorithm is presented which randomly selects a labelled graph with specified vertex degrees from a distribution which is arbitrarily close to uniform the algorithm is based on simulation of a rapidly convergent stochastic process and runs in polynomial time for a wide class of degree sequences including all regular sequences and all n vertex sequences with no degree exceeding n 2 the algorithm can be extended to cover the selection of a graph with given degree sequence which avoids a specified set of edges one consequence of this extension is the existence of a polynomial time algorithm for selecting an f factor in a sufficiently dense graph a companion algorithm for counting degree constrained graphs is also presented this algorithm has exactly the same range of validity as the one for selection

on the detectability and observability of continuous stochastic markov jump linear systems abstract this paper is concerned with the detectability and observability of continuous linear stochastic systems subject to markov jump by introducing some operators we investigate the relationship between some different concepts of detectability and observability and obtain their sufficient and necessary conditions continuous time stochastic lyapunov equations and stochastic continuous algebraic riccati equations are also discussed to serve as analysis tools

random k noncrossing rna structures in this paper we introduce a combinatorial framework that provides an interpretation of rna pseudoknot structures as sampling paths of a markov process our results facilitate a variety of applications ranging from the energy based sampling of pseudoknot structures as well as the ab initio folding via hidden markov models our main result is an algorithm that generates rna pseudoknot structures with uniform probability this algorithm serves as a steppingstone to sequence specific as well as energy based transition probabilities the approach employs a correspondence between pseudoknot structures parametrized in terms of the maximal number of mutually crossing arcs and certain tableau sequences the latter can be viewed as lattice paths the main idea of this paper is to view each such lattice path as a sampling path of a stochastic process and to make use of d finiteness for the efficient computation of the corresponding transition probabilities

stochastic stability of jump discrete time linear systems with markov chain in a general borel space necessary and sufficient conditions for stochastic stability ss of discrete time linear systems subject to markov jumps in the parameters are considered assuming that the markov chain takes values in a general borel space s it is shown that ss is equivalent to the spectrum radius of a bounded linear operator in a banach space being less than 1 or to the existence of a solution of a lyapunov type equation these results generalize several previous results in the literature which considered only the case of the markov chain taking values in a finite or infinite countable space

jump conditions for fields that have infinite integrable singularities at an interface jump conditions are derived for a basic set of first order partial differential equations whose fields have infinite integrable singularities as well as finite discontinuities at a moving and deforming interface these basic formulas are then applied to maxwell s electrodynamic equations and yield the jump conditions that hold when the electric and magnetic fields have such singularities these jump conditions are shown to be generalizations of formulas previously derived for double charge layers in electrostatics and for an interface with surface magnetization density in magnetostatics the basic formulas are also used to obtain jump conditions for the wave equation and other second order partial differential equations whose fields have finite discontinuities at an interface a similar application to second order vector identities yields a new set of jump identities these identities show that the normal and tangential components of the jump in a vector field are kinematically interdependent e g sh

on the detectability and observability of discrete time markov jump linear systems abstract this paper presents a new detectability concept for discrete time markov jump linear systems with finite markov state which generalizes the ms detectability concept found in the literature the new sense of detectability can similarly assure that the solution of the coupled algebraic riccati equation associated to the quadratic control problem is a stabilizing solution in addition the paper introduces a related observability concept that also generalizes previous concepts a test for detectability based on a coupled matrix equation is derived from the definition and a test for observability is presented which can be performed in a finite number of steps the results are illustrated by examples including one that shows that a system may be detectable in the new sense but not in the ms sense

robust h control with maximal decay rate for linear discrete time stochastic systems abstract the problems of robust stabilization and robust h control with maximal decay rate are investigated for discrete time stochastic systems with time varying norm bounded parameter uncertainties for the robust stabilization problem an optimal state feedback controller is designed which ensures that the closed loop system is robustly stochastically stable with maximal decay rate while for the robust h control problem the designed state feedback controller guarantees that the resulting closed loop system is with a specified h performance level besides robust stochastic stability with maximal decay rate the method based on the convex optimization and linear matrix inequality is developed to solve these problems two examples are provided to show the effectiveness of the proposed approach

brief paper detectability and observability of discrete time stochastic systems and their applications this paper studies detectability and observability of discrete time stochastic linear systems based on the standard notions of detectability and observability for time varying linear systems corresponding definitions for discrete time stochastic systems are proposed which unify some recently reported detectability and exact observability concepts for stochastic linear systems the notion of observability leads to the stochastic version of the well known rank criterion for observability of deterministic linear systems by using these two concepts the discrete time stochastic lyapunov equation and riccati equations are studied the results not only extend some of the existing results on these two types of equation but also indicate that the notions of detectability and observability studied in this paper take analogous functions as the usual concepts of detectability and observability in deterministic linear systems it is expected that the results presented may play important roles in many design problems in stochastic linear systems