stability and stabilization of semi markov jump linear systems with exponentially modulated periodic distributions of sojourn time this paper is concerned with a class of discrete time semi markov jump linear systems s mjlss subject to exponentially modulated periodic emp probability density function pdf of sojourn time and the problems of stability and stabilization are addressed setting a relatively large period the considered systems are capable of approximating the general s mjlss without any requirements on sojourn time pdfs for which numerically testable stability and stabilization conditions are rather difficult to obtain necessary and sufficient criterion for mean square stability of the general s mjlss is first derived which involves an infinite number of conditions and as such not checkable however the developments lay a foundation to further establish the numerically testable conditions for the systems when the pdf of sojourn time is emp albeit the sojourn time can tend to infinity the derivations explicitly depend on the pdf of sojourn time which circumvents the difficulty in obtaining the memory transition probabilities of s mjlss the adopted lyapunov function is sojourn time dependent std by which the existence conditions of std controller are developed as well using certain techniques that can eliminate the terms of power of matrices in the stability conditions two illustrative examples including a class of population ecological systems are presented to show the validity and applicability of the developed theoretical results

sign changing multi bump solutions for nonlinear schrodinger equations with steep potential wells we study the nonlinear schrodinger equations p u 2 a x 1 u u p 1 u u h 1 r n where p 1 is a subcritical exponent a x is a continuous function satisfying a x 0 0 lim inf x a x lim sup x a x and a 1 0 consists of 2 connected bounded smooth components 1 and 2 we study the existence of solutions u of p which converge to 0 in r n 1 u 2 and to a prescribed pair v 1 x v 2 x of solutions of the limit problem v i v i v i p 1 v i in i i 1 2 as oo

uniform bounds for strongly competing systems the optimal lipschitz case for a class of systems of semi linear elliptic equations including delta u i f i x u i beta u i sum j neq i a ij u j p quad i 1 dots k for p 2 variational type interaction or p 1 symmetric type interaction we prove that uniform l infty boundedness of the solutions implies uniform boundedness of their lipschitz norm as beta to infty that is in the limit of strong competition this extends known quasi optimal regularity results and covers the optimal case for this class of problems the proofs rest on monotonicity formulae of alt caffarelli friedman and almgren type in the variational setting and on the caffarelli jerison kenig almost monotonicity formula in the symmetric one

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

brief paper on almost sure stability of continuous time markov jump linear systems in this paper we study the almost sure stability of continuous time jump linear systems with a finite state markov form process a sufficient condition for almost sure stability is derived that refers to the statistics of the transition matrix over m switches it is shown that if the system is exponentially almost sure stable there exists a finite m such that the criterion is satisfied in order to evaluate the expected value appearing in the condition an efficient monte carlo algorithm is worked out

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

on a critical schrodinger system in r4 with steep potential wells abstract in this paper we study the following two component elliptic system u a x a 0 u u 3 v 2 u 0 in r 4 v b x b 0 v v 3 u 2 v 0 in r 4 u v h 1 r 4 h 1 r 4 where a 0 b 0 r are constants 0 and r are parameters a x b x 0 are potentials by using variational methods we prove the existence and nonexistence of general ground state solutions maybe semi trivial and ground state solutions of the above system under some further conditions on the potentials a x b x and the parameters it is worth pointing out that the cubic nonlinearities and the coupled terms of the above system are all of critical growth owing to the sobolev embedding theorem furthermore by introducing some ideas which are different from that in the literature the phenomenon of phase separation of ground state solutions of the above system is also obtained without any symmetry conditions

stochastic stability of systems with semi markovian switching this paper examines stochastic stability of switched dynamics in continuous time the time evolution of the so called continuous state is at all times determined by the dynamics indexed by the switching process or discrete state the main contribution of this paper appears as stochastic stability results for switched dynamics with semi markovian switching the notion of moment stability in the wide sense msws is applied as a generalization of moment stability a sufficient criterion for msws is presented for the above class of systems where each subsystem is assumed to be characterized by a lyapunov function candidate together with an associated growth rate equation for the set of lyapunov functions a compatibility criterion is assumed to be fulfilled bounding the ratio between pairs of lyapunov functions

asymptotic behavior of nonhomogeneous semi markov systems abstract we study the asymptotic behavior of a nonhomogeneous semi markov system population in discrete time after a series of definitions lemmas and theorems we firstly establish the conditions under which the ergodic behavior of a nonhomogeneous semi markov chain exists and then find the limit of the basic matrix of the chain q n s in closed form finally the existence of the asymptotic population structure of the nonhomogeneous semi markov system is studied and the limit is provided in closed analytic form

static output feedback control of nonhomogeneous markovian jump systems with asynchronous time delays this paper investigates the problem of static output feedback control for a class of nonhomogeneous markovian jump system nmjs with asynchronous time delays atds since the atds subject to uncertain transition probabilities tps are taking into account in a practical phenomenon new approaches are introduced to deal with the atds characterized by nonhomogeneous markov processes it is assumed that the communication links are not perfect due to its detrimental effect on the performance of systems stochastic variables are presented to characterize the data transmission which are depending on operation modes and satisfying the bernoulli distribution sets of slack variables are adopted to decouple the product terms between system matrices and lyapunov matrices based on an extended lyapunov function combined with finsler inequality approach the robust static output feedback controller is designed for the closed loop nmjs finally a numerical example is provided to verify the design method

mean square stochastic stability of linear time delay system with markovian jumping parameters this paper deals with the mean square stochastic stability of the class of linear time delay systems with markovian jumping parameters a delay independent sufficient condition for checking the stochastic stability of this class of systems is established a numerical example is given to show the usefulness of the proposed theoretical results

brief paper h estimation for discrete time piecewise homogeneous markov jump linear systems this paper concerns the problem of h estimation for a class of markov jump linear systems mjls with time varying transition probabilities tps in discrete time domain the time varying character of tps is considered to be finite piecewise homogeneous and the variations in the finite set are considered to be of two types arbitrary variation and stochastic variation respectively the latter means that the variation is subject to a higher level transition probability matrix the mode dependent and variation dependent h filter is designed such that the resulting closed loop systems are stochastically stable and have a guaranteed h filtering error performance index using the idea in the recent studies of partially unknown tps for the traditional mjls with homogeneous tps a generalized framework covering the two kinds of variations is proposed a numerical example is presented to illustrate the effectiveness and potential of the developed theoretical results

brief on robust stabilization of markovian jump systems with uncertain switching probabilities this brief paper is concerned with the robust stabilization problem for a class of markovian jump linear systems with uncertain switching probabilities the uncertain markovian jump system under consideration involves parameter uncertainties both in the system matrices and in the mode transition rate matrix first a new criterion for testing the robust stability of such systems is established in terms of linear matrix inequalities then a sufficient condition is proposed for the design of robust state feedback controllers a globally convergent algorithm involving convex optimization is also presented to help construct such controllers effectively finally a numerical simulation is used to illustrate the developed theory

brief paper markov jump linear systems with switching transition rates mean square stability with dwell time the stability of a class of markov jump linear systems characterized by piecewise constant transition rates and system dynamics is investigated for these switching markov jump linear systems mean square stability is analyzed through the time evolution of the second order moment of the state the main result is a sufficient condition that guarantees mean square stability under constraints on the dwell time between switching instants an alternative condition based on kronecker calculus is worked out it is shown that both the stability criteria admit an lmi implementation

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

homotopies exploiting newton polytopes for solving sparse polynomial systems this paper is concerned with the problem of finding all isolated solutions of a polynomial system the bkk bound defined as the mixed volume of the newton polytopes of the polynomials in the system is a sharp upper bound for the number of isolated solutions in mathbb c 0 n mathbb c 0 mathbb c backslash 0 of a polynomial system with a sparse monomial structure first an algorithm is described for computing the bkk bound following the lines of bernshtei n s proof the algorithmic construction of the cheater s homotopy or the coefficient homotopy is obtained the mixed homotopy methods can be combined with the random product start systems based on a generalized bezout number applications illustrate the effectiveness of the new approach

existence and symmetry of positive ground states for a doubly critical schrodinger system n n 2 and 1 1 satisfying 2 this problem is related to coupled nonlinear schrodinger equations with critical exponent for bose einstein condensate for different ranges of n and 0 we obtain positive ground state solutions via some quite different methods which are all radially symmetric it turns out that the least energy level depends heavily on the relations among and 2 besides for sufficiently small 0 positive solutions are also obtained via a variational perturbation approach note that the palais smale condition can not hold for any positive energy level which makes the study via variational methods rather complicated

stability and stabilization of discrete time semi markov jump linear systems via semi markov kernel approach this technical note is concerned with the problems of stability and stabilization for a class of discrete time semi markov jump linear systems s mjlss the discrete time semi markov kernel smk is introduced where the probability density function of sojourn time is dependent on both current and next system mode as a consequence different types of distributions and or different parameters in a same type of distribution of sojourn time depending on the target mode towards which the system jumps can coexist in each mode of a smk the underlying s mjlss are therefore more general than those considered in existing studies a new stability concept generalizing the traditional mean square stability is proposed such that numerically testable criteria on the basis of smk are obtained numerical examples are presented to illustrate the validity and advantage of the developed theoretical results

stability analysis of markov switched stochastic differential equations with both stable and unstable subsystems abstract this paper discusses the asymptotic stability of markov switched stochastic differential equations by using the method of multiple lyapunov functions we provide sufficient conditions for stochastic asymptotic stability of markov switched stochastic differential equations with both stable and unstable subsystems via the inequality based on the multiple lyapunov functions and the stationary distribution of markovian switching process particularly our results include some existing results as special cases and improve some results in the literature two examples are given to illustrate the effectiveness of the obtained results

input output finite time mean square stabilization of nonlinear semi markovian jump systems this paper is concerned with the input output finite time mean square stabilization problem for a class of nonlinear semi markovian jump systems smjss with time varying delay both time varying factors of transition rates and states delay are considered to be bounded our main purpose is to derive some parameter dependent sufficient criteria by designing a state feedback controller to guarantee the input output finite time mean square stabilization for the underlying systems where the nonlinear term satisfies the conic sector constraint then the stabilization conditions are obtained by employing a novel lyapunov krasovskii functional differential form gronwall s inequality and an extra slack variable which is composed of the time varying transition rates finally a simulation example is provided to illustrate the effectiveness and merits of the proposed method

stability analysis and stabilization of discrete time non homogeneous semi markov jump linear systems a polytopic approach abstract this paper addresses the issues of stability analysis and stabilizing controller design for two classes of discrete time non homogeneous semi markov jump linear systems s mjlss the first class is concerned with the sojourn time probability mass functions that are independent of jump instants while the other considers the existence of probability mass functions of sojourn time depending on jump instants new techniques are developed respectively for both classes of systems by transforming their corresponding time varying semi markov kernels into more tractable forms based on the stochastic stability allowing for bounded sojourn time the derived stability and stabilization criteria containing the probability distribution information of random sojourn time can be numerically tested in order to reduce the possible conservatism of the obtained results a novel class of polytopic quadratic lyapunov function is constructed which depends on both the current mode and the elapsed time in the current mode finally the proposed control strategies are applied to an illustrative application of automotive electronic throttle valve to show the effectiveness of the theoretical results the results also reveal the significance of considering the inhomogeneity of s mjlss and the importance of our constructed lyapunov function

mixed h sub 2 h sub spl infin control of discrete time markovian jump linear systems in this paper we consider the mixed h sub 2 h sub spl infin control problem for the class of discrete time linear systems with parameters subject to markovian jump linear systems it is assumed that both the state variable and the jump variable are available to the controller the transition probability matrix may not be exactly known but belongs to an appropriate convex set for this controlled discrete time markovian jump linear system the problem of interest can be stated in the following way find a robust with respect to the uncertainty on the transition markov probability matrix mean square stabilizing state and jump feedback controller that minimizes an upper bound for the h sub spl infin norm under the restriction that the h sub spl infin norm is less than a prespecified value spl delta the problem of the determination of the smallest h sub spl infin norm is also addressed we present an approximate version of these problems via linear matrix inequality optimization

stochastic stabilization of a class of nonhomogeneous markovian jump linear systems this paper deals with the problem of stochastic stability analysis and control synthesis of a class of markovian jump linear systems mjls with time varying transition probabilities tps in the discrete time domain the time varying character is considered to be in a polytopic sense the approach followed in this note is based on the use of a parameter dependent stochastic lyapunov function we give a stability condition in terms of a linear matrix inequality lmi feasibility problem the obtained results are illustrated in a numerical example

positive least energy solutions and phase separation for coupled schrodinger equations with critical exponent in this paper we study the following coupled schrodinger system which can be seen as a critically coupled perturbed brezis nirenberg problem left begin array ll delta u lambda 1 u mu 1 u 3 beta uv 2 quad x in omega delta v lambda 2 v mu 2 v 3 beta vu 2 quad x in omega u geqq 0 v geqq 0 rm in omega quad u v 0 quad rm on partial omega end array right here omega subset mathbb r 4 is a smooth bounded domain lambda 1 omega 0 and beta neq 0 where lambda 1 omega is the first eigenvalue of with the dirichlet boundary condition note that the nonlinearity and the coupling terms are both critical in dimension 4 that is frac 2n n 2 4 when n 4 we show that this critical system has a positive least energy solution for negative positive small and positive large for the case in which lambda 1 lambda 2 we obtain the uniqueness of positive least energy solutions we also study the limit behavior of the least energy solutions in the repulsive case beta to infty and phase separation is expected these seem to be the first results for this schrodinger system in the critical case

a polyhedral method for solving sparse polynomial systems a continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations we introduce mixed subdivisions of newton polytopes and we apply them to give a new proof and algorithm for bernstein s theorem on the expected number of roots this results in a numerical homotopy with the optimal number of paths to be followed in this homotopy there is one starting system for each cell of the mixed subdivision and the roots of these starting systems are obtained by an easy combinatorial construction

uniform holder bounds for nonlinear schrodinger systems with strong competition for the positive solutions of the gross pitaevskii system u u 1 u 3 u v 2 v v 2 v 3 u 2 v we prove that l boundedness implies c 0 boundedness for every 0 1 uniformly as moreover we prove that the limiting profile as is lipschitz continuous the proof relies upon the blowup technique and the monotonicity formulae by almgren and alt caffarelli and friedman this system arises in the hartree fock approximation theory for binary mixtures of bose einstein condensates in different hyperfine states extensions to systems with k 2 densities are given

direct calculation of the information matrix via the em algorithm summary a simple explicit formula is provided for the matrix of second derivatives of the observed data log likelihood in terms of derivatives of the criterion function the conditional expectation of the complete data log likelihood given the observed data invoked by the em algorithm

razumikhin type theorem for stochastic functional differential equations with levy noise and markov switching abstractthis paper is devoted to study the well known razumikhin type theorem for a class of stochastic functional differential equations with levy noise and markov switching in comparison to the standard gaussian noise levy noise and markov switching make the analysis more difficult owing to the discontinuity of its sample paths in this paper we attempt to overcome this difficulty by using the razumikhin method and lyapunov functions we obtain several razumikhin type theorems to prove the pth moment exponential stability of the suggested system based on these results we further discuss the pth moment exponential stability of stochastic delay differential equations with levy noise and markov switching in particular the results obtained in this paper improve and generalise some previous works given in the literature finally an example is provided to illustrate the effectiveness of the theoretical results

state estimation and sliding mode control for semi markovian jump systems with mismatched uncertainties this paper is concerned with the state estimation and sliding mode control problems for phase type semi markovian jump systems using a supplementary variable technique and a plant transformation a finite phase type semi markov process has been transformed into a finite markov chain which is called its associated markov chain as a result phase type semi markovian jump systems can be equivalently expressed as its associated markovian jump systems a sliding surface is then constructed and a sliding mode controller is synthesized to ensure that the associated markovian jump systems satisfy the reaching condition moreover an observer based sliding mode control problem is investigated sufficient conditions are established for the solvability of the desired observer two numerical examples are presented to show the effectiveness of the proposed design techniques

the bkk root count in c n the root count developed by bernshtein kushnirenko and khovanskii only counts the number of isolated zeros of a polynomial system in the algebraic torus c n in this paper we modify this bound slightly so that it counts the number of isolated zeros in c n our bound is apparently significantly sharper than the recent root counts found by rojas and in many cases easier to compute as a consequence of our result the huber sturmfels homotopy for finding all the isolated zeros of a polynomial system in c n can be slightly modified to obtain all the isolated zeros in c n

a convex geometric approach to counting the roots of a polynomial system consider a polynomial system f f1 fn in n variables with complex coefficients a standard way to bound the number of isolated roots of f in cn is to multiply and add the degrees of the fi in a straightfoward computation described explicitly by some variant of bezout s theorem unfortunately even if one knows a priori which monomial terms will appear this upper bound can be far from exact in practice one frequently knows which monomial terms will appear in a polynomial system before one actually decides to count or approximate its roots so a fast accurate root count which takes this additional information into account is vital we propose a much tighter upper bound on the number of isolated roots of f in cn and give an explicit combinatorial geometric criterion for when it is exact as a corollary we obtain a geometric characterization of the average case computational complexity of root counting our root count is a formula derived from the theory of toric varieties and its computation involves finding the mixed volume of n shadowed polytopes in rn our formula generalizes the bkk bound bernshtein 1975 kushnirenko 1976 khovanskii 1978 canny and rojas 1991 in four ways 1 for any r 0 n our formula can be used to count the roots whose first r coordinates are nonzero the bkk bound only handles the r n case as a corollary we obtain that for fixed n all of these root counts for generic f can be done in time polynomial in the numbers of monomial terms of the fi 2 we refine bernshtein s 1975 algebraic criterion for exactness of the bkk bound into a complete classification of the subsets of the coefficients of f whose genericity guarantees exactness in our generalized bound 3 our results are stated in terms of arbitrary algebraically closed fields of any characteristic the bkk bound was originally stated only over the complex numbers 4 we generalize all of the above to results on the number of n k dimensional components of the zero set of f1 fk when k n

direct calculation of the information matrix via the em a simple explicit formula is provided for the matrix of second derivatives of the observed data log likelihood in terms of derivatives of the criterion function conditional expectation of the complete data log likelihood given the observed data invoked by the em algorithm

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

transition probability bounds for the stochastic stability robustness of continuous and discrete time markovian jump linear systems this paper considers the robustness of stochastic stability of markovian jump linear systems in continuous and discrete time with respect to their transition rates and probabilities respectively the continuous time discrete time system is described via a continuous valued state vector and a discrete valued mode which varies according to a markov process chain by using stochastic lyapunov function approach and kronecker product transformation techniques sufficient conditions are obtained for the robust stochastic stability of the underlying systems which are in terms of upper bounds on the perturbed transition rates and probabilities analytical expressions are derived for scalar systems which are straightforward to use numerical examples are presented to show the potential of the proposed techniques

slow state variables feedback stabilization for semi markov jump systems with singular perturbations the slow state variables feedback stabilization problem for semi markov jump discrete time systems with slow sampling singular perturbations is discussed in this work a new fairly comprehensive system model semi markov jump system with singular perturbations which is more general than markov jump model is employed to describe the phenomena of random abrupt changes in structure and parameters of the systems based on a slow state variables feedback control scheme a novel technique to design the desired controller is presented and the allowed maximum of singular perturbation parameter can be calculated with the help of the discrete time semi markov kernel approach some sojourn time dependent and less conservative sufficient conditions are established via a novel matrix decoupling technique to ensure the solvability of the problem to be addressed finally an illustrative example is given to show the superiority and usefulness of the proposed method

computing isolated roots of sparse polynomial systems in affine space we present a symbolic probabilistic algorithm to compute the isolated roots in c n of sparse polynomial equation systems as some already known numerical algorithms solving this task our procedure is based on polyhedral deformations and homotopies but it amounts to solving a smaller number of square systems of equations and in fewer variables the output of the algorithm is a geometric resolution of a finite set of points including the isolated roots of the system the complexity is polynomial in the size of the combinatorial structure of the system supports up to a pre processing yielding the mixed cells in a subdivision of the family of these supports

brief continuous time state feedback h2 control of markovian jump linear systems via convex analysis continuous time h 2 control problem for the class of linear systems with markovian jumps mjls using convex analysis is considered in this paper the definition of the h 2 norm for continuous time mjls is presented and related to the appropriate observability and controllability gramians a convex programming formulation for the h 2 control problem of mjls is developed that enables us to tackle the optimization problem of mjls under the assumption that the transition rate matrix p p i j for the markov chain may not be exactly known but belongs to an appropriate convex set an equivalence between the convex formulation when p is exactly known and the usual dynamic programming approach of quadratic optimal control of mjls is established it is shown that there exists a solution for the convex programming problem if and only if there exists the mean square stabilizing solution for a set of coupled algebraic riccati equations these results are compared with other related works in the current literature

a hidden seasonal switching model for high resolution breakpoint rainfall data 1 a nonhomogeneous hidden semi markov model nhsmm for breakpoint rainfall data is proposed which extends the homogeneous hidden semi markov model hsmm of sansom and thomson 2001 to incorporate stochastic seasonality the nhsmm model is able to switch seasons at times that are earlier or later than expected and in this way is able to explain additional seasonal variability due to varying length seasons the model s hidden rainfall states align with precipitation mechanisms that are seasonally invariant but the state dynamics are assumed to vary with season recursions for constructing the likelihood are developed and the em algorithm used to fit the parameters of the model an application of the model to breakpoint rainfall measurements from invercargill new zealand is discussed and the results of fitting a number of different nhsmms are compared to those from fitting the non seasonal homogeneous hsmm

parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper we consider discrete time systems with polytopic time varying uncertainty we look for a class of parameter dependent lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter we show that extending the new discrete time stability condition proposed by de oliveira et al systems control lett 36 1999 135 to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a lyapunov function this allows to check asymptotic stability of the system under study the obtained linear matrix inequalities condition can also be used to cope with the control synthesis problem