robust stability and robust stabilizability for periodically switched linear systems abstract in this paper the problem of robust stability and robust stabilizability for periodically switched linear systems is studied the stability radius involving structured perturbations acting on both coefficient matrices and switching moments is introduced and investigated some lower bounds for stability radii of periodically switched linear systems are provided after that we derive the notion of fast and slow stabilizability for these systems some characterizations for robust stabilizability under structured perturbations are established several examples are given to illustrate the obtained results

stochastic stability of jump linear systems in this note some testable conditions for mean square i e second moment stability for discrete time jump linear systems with time homogenous and time inhomogenous finite state markov chain form processes are presented

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

on the relationship between the sample path and moment lyapunov exponents for jump linear systems in this paper we study the relationship between the sample and moment lyapunov exponents for jump linear systems using a large deviation theorem a modified version of arnold s formula for connecting sample path and moment lyapunov exponents for continuous time linear stochastic systems is extended to discrete time jump linear systems sample path stability properties of linear stochastic systems are determined by the top lyapunov exponent and relating sample and moment lyapunov exponents may be useful for developing computationally efficient methods for determining the almost sure sample path stability of linear stochastic systems

inequalities for numerical invariants of sets of matrices we prove three inequalities relating some invariants of sets of matrices such as the joint spectral radius one of the inequalities in which proof we use geometric invariant theory has the generalized spectral radius theorem of berger and wang as an immediate corollary

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

criteria of stabilizability for switching control systems with solvable linear approximations we study the stability and stabilizability of a continuous time switched control system that consists of the time invariant n dimensional subsystems dot x a ix b i x u x in mathbb r n t in mathbb r textrm and u in mathbb r m i where i in 1 dotsc n and a switching signal sigma boldsymbol cdot colon mathbb r rightarrow 1 dotsc n which orchestrates switching between these subsystems above where a i in mathbb r n times n n ge1 n ge2 m i ge1 and where b i boldsymbol cdot colon mathbb r n rightarrow mathbb r n times m i satisfies the condition b i x le boldsymbol beta x for all x in mathbb r n we show that if a 1 dotsc a n generates a solvable lie algebra over the field mathbb c of complex numbers and there exists an element boldsymbol a in the convex hull mathrm co a 1 dotsc a n in mathbb r n times n such that the affine system dot x boldsymbol a x is exponentially stable then there is a constan

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

parallel solution of the modified porous medium equation the present study deals with the numerical solution of the modified porous medium equation when the solution is subject to some constraints first of all we use a change of variables which leads to an evolutive problem where the nonlinear part is constituted by the derivative with respect to the time of a diagonal increasing operator then for the numerical solution we consider an implicit time marching scheme which leads to the solution of a sequence of stationary problems in fact each stationary problem is equivalent to a constrained minimization problem and for dirichlet and dirichlet neumann boundary conditions we show the existence and the uniqueness of the solution of our stationary constrained minimization problem moreover classically the solution of each stationary constrained problem can be characterized by the solution of a multivalued one the spatial discretization of the previous problem leads to the solution of large scale algebraic multivalued systems then we analyze in a unified approach the convergence of the sequential and parallel relaxation projected methods finally we present the results of numerical experiments

an explicit counterexample to the lagarias wang finiteness conjecture abstract the joint spectral radius of a finite set of real d d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set a set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth j c lagarias and y wang conjectured in 1995 that every finite set of real d d matrices satisfies the finiteness property however t bousch and j mairesse proved in 2002 that counterexamples to the finiteness conjecture exist showing in particular that there exists a family of pairs of 2 2 matrices which contains a counterexample similar results were subsequently given by v d blondel j theys and a a vladimirov and by v s kozyakin but no explicit counterexample to the finiteness conjecture has so far been given the purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the lagarias wang finiteness conjecture namely for the set a 1 1 0 1 1 0 1 1 we give an explicit value of 0 749326546330367557943961948091344672091327370236064317358024 such that a does not satisfy the finiteness property

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

extremal and barabanov semi norms of a semigroup generated by a bounded family of matrices abstract let s s i i i be an arbitrary family of complex n by n matrices where 1 n let s denote the joint spectral radius of s defined as s lim sup l sup i 1 i l i l s i 1 s i l 1 l which is independent of the norm used here a semi norm on c n is called extremal of s if it satisfies x 0 and x s i s x x x 1 x n c n and i i in this paper using an elementary analytical approach we show that if s is bounded in c n n then there always exists for s an extremal semi norm on c n if additionally s is compact in c n n this extremal semi norm has the barabanov type property i e to any x c n one can find an infinite sequence i n i with x s i 1 s i k s k x for each k 1 as a common starting point this directly implies the fundamental results barabanov s norm theorem berger wang s formula and elsner s reduction theorem

some criteria for spectral finiteness of a finite subset of the real matrix space rd d abstract in this paper we present some checkable criteria for the spectral finiteness of a finite subset of the real d d matrix space r d d where 2 d

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

a limit formula for joint spectral radius with p radius of probability distributions abstract in this paper we show a characterization of the joint spectral radius of a set of matrices as the limit of the p radius of an associated probability distribution when p tends to allowing the set to have infinitely many matrices the obtained formula extends the results in the literature based on the formula we then present a novel characterization of the stability of switched linear systems for an arbitrary switching signal via the existence of stochastic lyapunov functions of any higher degrees numerical examples are presented to illustrate the results

asynchronous iterations with flexible communication contracting operators the concept of flexible communication permits one to model efficient asynchronous iterations on parallel computers this concept is particularly useful in two practical situations firstly when communications are requested while a processor has completed the current update only partly and secondly in the context of inner outer iterations when processors are also allowed to make use of intermediate results obtained during the inner iteration in other processors in the general case of nonlinear or linear fixed point problems we give a global convergence results for asynchronous iterations with flexible communication whereby the iteration operators satisfy certain contraction hypotheses in this manner we extend to a contraction context previous results obtained for monotone operators with respect to a partial ordering

on asynchronous iterations asynchronous iterations arise naturally on parallel computers if one wants to minimize idle times this paper reviews certain models of asynchronous iterations using a common theoretical framework the corresponding convergence theory and various domains of applications are presented these include nonsingular linear systems nonlinear systems and initial value problems

a gel fand type spectral radius formula and stability of linear constrained switching systems abstract using ergodic theory in this paper we present a gel fand type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup s restricted to a subset that need not carry the algebraic structure of s this generalizes the berger wang formula using it as a tool we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one sided markov shift associated to s

rank one characterization of joint spectral radius of finite matrix family abstract in this paper we study the joint generalized spectral radius of a finite set of matrices in terms of its rank one approximation by singular value decomposition in the first part of the paper we show that any finite set of matrices with at most one element s rank being greater than one satisfies the finiteness property under the framework of invariant extremal norm formula for the computation of joint generalized spectral radius for this class of matrix family is derived based on that in the second part we further study the joint generalized spectral radius of finite sets of general matrices through constructing rank one approximations in terms of singular value decomposition and some new characterizations of joint generalized spectral radius are obtained several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach

uniformly stabilizing sets of switching sequences for switched linear systems a set of switching sequences of a discrete time switched linear system is said to be uniformly stabilizing if the system is uniformly exponentially stable over all switching sequences that belong to the set based on the recent result that a switched linear system is uniformly stable if and only if an increasing countable union of linear matrix inequality conditions is satisfied we characterize all uniformly stabilizing sets of switching sequences a discrete time hybrid automaton of the automobile engine is considered to illustrate the result

distributed asynchronous computation of fixed points we present an algorithmic model for distributed computation of fixed points whereby several processors participate simultaneously in the calculations while exchanging information via communication links we place essentially no assumptions on the ordering of computation and communication between processors thereby allowing for completely uncoordinated execution we provide a general convergence theorem for algorithms of this type and demonstrate its applicability to several classes of problems including the calculation of fixed points of contraction and monotone mappings arising in linear and nonlinear systems of equations optimization problems shortest path problems and dynamic programming

the finite step realizability of the joint spectral radius of a pair of d d matrices one of which being rank one abstract we study the finite step realizability of the joint generalized spectral radius of a pair of real square matrices s 1 and s 2 one of which has rank 1 where 2 d let a denote the spectral radius of a square matrix a then we prove that there always exists a finite length word i 1 i m 1 2 m for some finite m 1 such that s i 1 s i m m sup n 1 max i 1 i n 1 2 n s i 1 s i n n in other words there holds the spectral finiteness property for s 1 s 2 explicit formula for computation of the joint spectral radius is derived this implies that the stability of the switched system induced by s 1 s 2 is algorithmically decidable in this case

stability analysis for planar discrete time linear switching systems via bounding joint spectral radius abstract a pair of 2 by 2 matrices can be transformed into the joint forms which are bound together via an invariant that measures the deformation of their structures with respect to each other with this we obtain an approximation of the joint spectral radius of such a pair of matrices from above and then apply it to assessing stability for planar discrete time linear switching systems

stability and stabilization of discrete time switched systems this paper addresses two strategies for stabilization of discrete time linear switched systems the first one is of open loop nature trajectory independent and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition is expressed being non convex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

convergence of partially asynchronous block quasi newton methods for nonlinear systems of equations in this paper a partially asynchronous block broyden method is presented for solving nonlinear systems of equations of the form f x 0 sucient conditions that guarantee its local convergence are given in particular local convergence is shown when the jacobian f 0 x i s anh matrix where x is the zero point of f the results are extended to schubert s method a connection with discrete schwarz alternating procedure is also shown c 1999 elsevier science b v all rights reserved

common polynomial lyapunov functions for linear switched systems in this paper we consider linear switched systems dot x t a u t x t x in r n u in u a u u in u compact and the problem of asymptotic stability for arbitrary switching functions uniform with respect to switching uas given a uas system it is always possible to build a common polynomial lyapunov function our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems this result answers a question raised by dayawansa and martin a generalization to a class of piecewise polynomial lyapunov functions is given

on codes with local joint constraints we study the largest number of sequences with the property that any two sequences do not contain specified pairs of patterns we show that this number increases exponentially with the length of the sequences and that the exponent or capacity is the logarithm of the joint spectral radius of an appropriately defined set of matrices we illustrate a new heuristic for computing the joint spectral radius and use it to compute the capacity for several simple collections the problem of computing the achievable rate region of a collection of codes is introduced and it is shown that the region may be computed via a similar analysis

sets of matrices all infinite products of which converge an infinite product iit lmi of matrices converges on the right if limi m mi exists a set z ai i l of n x n matrices is called an rcp set rightconvergent product set if all infinite products with each element drawn from z converge such sets of matrices arise in constructing self similar objects like von koch s snowflake curve in various interpolation schemes in constructing wavelets of compact support and in studying nonhomogeneous markov chains this paper gives necessary conditions and also some sufficient conditions for a set x to be an rcp set these are conditions on the eigenvalues and left eigenspaces of matrices in 2 and finite products of these matrices necessary and sufficient conditions are given for a finite set z to be an rcp set having a limit function m d rit lad where d d d which is a continuous function on the space of all sequences d with the sequence topology finite rcp sets of column stochastic matrices are completely characterized some results are given on the problem of algorithmically deciding if a given set x is an rcp set

a rapidly converging lower bound for the joint spectral radius via multiplicative ergodic theory abstract we use ergodic theory to prove a quantitative version of a theorem of m a berger and y wang which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices the proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded

a converse lyapunov theorem for linear parameter varying and linear switching systems we study families of linear time varying systems where time variations have to satisfy restrictions on the dwell time that is on the minimum distance between discontinuities as well as on the derivative in between discontinuities such classes of systems may be formulated as linear flows on vector bundles the main objective of this paper is to construct parameter dependent lyapunov functions which characterize the exponential growth rate this is possible in the generic irreducible case as an application the gelfand formula is generalized to the class of systems studied here in other words the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time varying systems a perspective on the question of continuous dependence of the exponential growth rate on the data is given

technical communique a characterization of the generalized spectral radius with kronecker powers based on turan s power sum theory we extend a recent result obtained by blondel and nesterov blondel v d nesterov y 2005 computationally efficient approximations of the joint spectral radius siam journal on matrix analysis and applications 27 256 272 by deriving a new characterization of the generalized spectral radius in terms of kronecker powers

stabilization of planar switched systems this paper considers the problem of stabilization of single input planar switched systems we assume the switching law is observable a formula is presented which provides a necessary and sufficient condition for the system to be quadratically stabilizable a set of linear inequalities are given to describe the set of all quadratic lyapunov functions the solvability and the control design technique are clearly described in a straightforward computation algorithm

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

uniform stabilization of discrete time switched and markovian jump linear systems the uniform stability of discrete time switched linear systems possibly with a strongly connected switching path constraint and the existence of finite path dependent dynamic output feedback controllers uniformly stabilizing such a system are both shown to be characterized by the existence of a finite dimensional feasible system of linear matrix inequalities this characterization is based on the observation that a linear time varying system is uniformly stable only if there exists a finite path dependent quadratic lyapunov function the synthesis of a uniformly stabilizing controller is done without conservatism by solving any feasible system of linear matrix inequalities among an increasing family of systems of linear matrix inequalities the result carries over to the almost sure uniform stabilization of markovian jump linear systems

optimal disturbance attenuation for discrete time switched and markovian jump linear systems an exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete time domain via the union of an increasing family of linear matrix inequality conditions associated with each markovian jump linear system is a switched linear system so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for markovian jump linear systems as well the results lead to semidefinite programming based controller synthesis techniques from which optimal finite path dependent linear dynamic output feedback controllers arise naturally in particular under the notion of path by path optimal disturbance attenuation finite path dependent controllers can outperform the usual mode dependent ones

brief paper periodically switched stability induces exponential stability of discrete time linear switched systems in the sense of markovian probabilities the conjecture that periodically switched stability implies absolute asymptotic stability of random infinite products of a finite set of square matrices has recently been disproved under the guise of the finiteness conjecture in this paper we show that this conjecture holds in terms of markovian probabilities more specifically let s k c n x n 1 k k be arbitrarily given k matrices and s k k j j 1 1 k j k for each j 1 where n k 2 then we study the exponential stability of the following discrete time switched dynamics s x j s k j s k 1x 0 j 1 and x 0 c n where k j j 1 s k can be an arbitrary switching sequence for a probability row vector p p 1 p k r k and an irreducible markov transition matrix p r k x k with pp p we denote by m p p the markovian probability on s k corresponding to p p by using symbolic dynamics and ergodic theoretic approaches we show that if s possesses the periodically switched stability then i it is exponentially stable m p p almost surely ii the set of stable switching sequences k j j 1 s k has the same hausdorff dimension as s k thus the periodically switched stability of a discrete time linear switched dynamics implies that the system is exponentially stable for almost all switching sequences

when do several linear operators share an invariant cone abstract we establish a criterion for a finite family of matrices to possess a common invariant cone the criterion reduces the problem of existence of an invariant cone to equality of two special numbers that depend on the family in spite of theoretical simplicity the practical use of the criterion may be difficult we show that the problem of existence of a common invariant cone for four matrices with integral entries is algorithmically undecidable corollaries of the criterion which give sufficient and necessary conditions are derived finally we introduce a co directional number of several matrices we prove that this parameter is close to zero iff there is a small perturbation of matrices after which they get an invariant cone an algorithm for its computation is presented

characterization of joint spectral radius via trace the joint spectral radius for a bounded collection of the square matrices with complex entries and of the same size is characterized by the trace of matrices this characterization allows us to give some estimates concerning the computation of the joint spectral radius

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

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

on the universal family of hilbert schemes of points on a surface abstract for a smooth quasi projective surface x and an integer n 3 we show that the universal family z n over the hilbert scheme hilb n x of n points has non q gorenstein rational singularities and that the samuel multiplicity at a closed point on z n can be computed in terms of the dimension of the socle we also show that n

asymptotic stability and generalized gelfand spectral radius formula let be a set of n n complex matrices for m 1 2 let m be the set of all products of matrices in of length m denote by the multiplicative semigroup generated by is said to be asymptotically stable in the sense of dynamical systems if there is 0 1 such that there are bounded neighborhoods u v cn of the origin for which av mu for all a m m 1 2 for a bounded set of n n complex matrices it is shown that the following conditions are mutually equivalent i is asymptotically stable ii g9 lim supm supa m a 1 m 1 iii lim supm supa m a 1 m 1 where a stands for the spectral radius of a and iv there exists a positive number such that a 1 for all a this fact answers an open question raised by brayton and tong the generalized gelfand spectral radius formula that is g9 conjectured by daubechies and lagarias and proved by berger and wang using advanced tools from ring theory and then by elsner using analytic geometric tools follows immediately form the above asymptotic stability theorem

generalized joint spectral radius and stability of switching systems abstract this paper extends the notion of generalized joint spectral radius with exponents originally defined for a finite set of matrices to probability distributions we show that under a certain invariance condition the radius is calculated as the spectral radius of a matrix that can be easily computed extending the classical counterpart using this result we investigate the mean stability of switching systems in particular we establish the equivalence of mean square stability simultaneous contractibility in square mean and the existence of a quadratic lyapunov function also the stabilization of positive switching systems is studied numerical examples are given to illustrate the results

the berger wang formula for the markovian joint spectral radius abstract the berger wang formula establishes equality between the joint and generalized spectral radii of a set of matrices for matrix products whose multipliers are applied not arbitrarily but in accordance with some markovian law there are also known analogs of the joint and generalized spectral radii however the known proofs of the berger wang formula hardly can be directly applied in the case of markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products nevertheless as has been shown by x dai 1 the berger wang formula is valid for the case of markovian analogs of the joint and the generalized spectral radii too although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory in the paper we propose a matrix theory construction allowing to deduce the markovian analog of the berger wang formula from the classical berger wang formula

the finiteness conjecture for the generalized spectral radius of a set of matrices the generalized spectral radius g9 of a set of n n matrices is g9 lim supk g9k 1k where g9k sup a1a2 ak each ai the joint spectral radius g9 is g9 lim supk g9k 1k where g9k sup a1 ak each ai it is known that g9 g9 holds for any finite set of n n matrices the finiteness conjecture asserts that for any finite set of real n n matrices there exists a finite k such that g9 g9 g9k 1k the normed finiteness conjecture for a given operator norm asserts that for any finite set a1 am having all ai op 1 either g9 1 or g9 g9 g9k 1k 1 for some finite k it is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms the normed finiteness conjecture is proved for a large class of operator norms extending results of gurvits in particular for polytope norms and for the euclidean norm explicit upper bounds are given for the least k having g9 g9k 1k these results imply upper bounds for generalized critical exponents for these norms

m functions and parallel asynchronous algorithms the solution of nonlinear systems of equations fx z via parallel asynchronous algorithms is considered it is shown that when f is continuous off diagonally antitone and strictly diagonally isotone then point asynchronous iterations converge monotonically to a solution of the problem from supersolutions and subsolutions a global convergence result for asynchronous iterations when f is a continuous surjective m function is also presented

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

some convergence results for asynchronous algorithms nous presentons dans cet article des resultats de convergence des algorithmes asynchrones bases essentiellement sur la notion classique de contraction nous generalisons en particulier tous les resultats de convergence de ces algorithmes qui font l hypothese de contraction en norme vectorielle qui recemment a ete tres souvant utilisee par ailleurs l hypothese de contraction en norme vectorielle peut se trouver difficile voire impossible a verifier pour certains problemes qui peuvent etre cependant abordes dans le cadre de la contraction classique que nous adoptons in this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction we generalize in particular all convergence results for those algorithms which are based on the vectorial norm hypothesis in wide spread use recently certain problems for which the vectorial norm hypothesis can be difficult or even impossible to verify can nontheless be tackled within the scope of the classical contraction that we adopte

almost sure stability of discrete time switched linear systems a topological point of view in this paper we study the stability of discrete time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem a sufficient and necessary condition for mu a almost sure stability is derived where mu a is the parry measure of the topological markov chain with a prescribed transition 0 1 matrix a the obtained mu a almost sure stability is invariant under small perturbations of the system the topological description of stable processes of switched linear systems in terms of hausdorff dimension is given and it is shown that our approach captures the maximal set of stable processes for linear switched systems the obtained results cover the stochastic markov jump linear systems where the measure is the natural markov measure defined by the transition probability matrix two examples are provided to illustrate the theoretical outcomes of the paper

pointwise stability of discrete time stationary matrix valued markovian processes in this technical note we study the pointwise stability of a discrete time matrix valued and stationary markovian jump linear system when the system is restricted to a linear subspace we show that it is pointwise convergent if and only if it is pointwise exponentially convergent under the framework of probability and symbolic dynamics

the generalized spectral radius theorem an analytic geometric proof abstract let be a bounded set of complex matrices m a 1 a m a i the generalized spectral radius theorem states that where and are defined as follows lim sup m 1 m where m sup a a m with a the spectral radius lim sup m 1 m where m sup a a m with any matrix norm we give an elementary proof based on analytic and geometric tools which is in some ways simpler than the first proof by berger and wang

an intersection number for the punctual hilbert scheme of a surface we compute the intersection number between two cycles a and b of complementary dimensions in the hilbert scheme h parameterizingr subschemes of given finite length n of a smooth projective surface s the n 1 cycle a corresponds to the set of finite closed subschemes the support of which has cardinality 1 the n 1 cycle b consists of the closed subschemes the support of which is one given point of the surface since b is contained in a indirect methods are needed the intersection number is a b 1 n 1n answering a question by h nakajima

distributed asynchronous relaxation methods for convex network flow problems we consider the solution of the single commodity strictly convex network flow problem in a distributed asynchronous computation environment the dual of this problem is unconstrained differentiable and well suited for solution via gauss seidel relaxation we show that the structure of the dual allows the successful application of a distributed asynchronous method whereby relaxation iterations are carried out in parallel by several processors in arbitrary order and with arbitrarily large interprocessor communication delays

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

matrix products with constraints on the sliding block relative frequencies of different factors abstract one of fundamental results of the theory of joint generalized spectral radius the berger wang theorem establishes equal ity between the joint and generalized spectral radii of a set of matrices generalization of this theorem on products of matrices whose factors are applied not arbitrarily but are subjected to some constraints is connected with essential difficulties since known proofs of the berger wang theorem rely on the arbitrariness of appearance of different matrices in the related matrix products recently x dai 1 proved an analog of the berger wang theorem for the case when factors in matrix products are formed by some markov law we extend the concepts of joint and generalized spectral radii to products of matrices with constraints on the sliding block relative frequencies of occurrences of different factors and prove an analog of the berger wang theorem for this case

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

the set of stable switching sequences for discrete time linear switched systems in this paper we study the characterization of the asymptotical stability for discrete time switched linear systems we first translate the system dynamics into a symbolic setting under the framework of symbolic topology then by using the ergodic measure theory a lower bound estimate of hausdorff dimension of the set of asymptotically stable sequences is obtained we show that the hausdorff dimension of the set of asymptotically stable switching sequences is positive if and only if the corresponding switched linear system has at least one asymptotically stable switching sequence the obtained result reveals an underlying fundamental principle a switched linear system either possesses uncountable numbers of asymptotically stable switching sequences or has none of them provided that the switching is arbitrary we also develop frequency and density indexes to identify those asymptotically stable switching sequences of the system

weakly birkhoff recurrent switching signals almost sure and partial stability of linear switched dynamical systems let a1 ak cd d be arbitrary k matrices where k and d both 2 for any 0 we denote by l pc r k the set of all switching sequences u t n 1 k r satisfying tj tj 1 and 0 t0 t1 tj 1 tj with tj differently from the classical weak topology and l1 norm we equip l pc r k with the topology so that the one sided markov type shift l pc r k l pc r k defined by u j tj j 1 u j 1 tj 1 t1 j 1 is continuous which is different from and simpler than the classical continuous time translation we study the stability of the linear switched dynamics a x t au t x t x 0 cd and t 0 where u t j if tj 1 0 then for any ergodic probability p on l pc r k either limj 1jlog u tj 0for p a s u j tj j 1 or j u tj k tj 1 k j 0for p a s u j tj j 1 some applications are presented including i equivalence of various stabilities ii almost sure exponential stability of periodically switched stable systems iii partial stability and iv how to approach arbitrarily the stable manifold by that of periodically switched signals and how to select a stable switching signal for any initial data

bounded semigroups of matrices in this note are proved two conjectures of daubechies and lagarias the first asserts that if z is a bounded set of matrices such that all left infinite products converge then 8 generates a bounded semigroup the second asserts the equality of two differently defined joint spectral radii for a bounded set of matrices one definition involves the conventional spectral radius and one involves the operator norm

generalized spectral radius and its max algebra version abstract let c n n and r n n be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of respectively we apply a single matrix description of to give a new elementary and straightforward proof of the berger wang formula in max algebra and consequently a new short proof of the original berger wang formula in the case of bounded subsets of n n non negative matrices we also obtain a new description of in terms of the schur hadamard product and prove new trace and max trace descriptions of and in particular we show that limsup m sup a m tr a 1 m limsup m sup a m tr a 1 m and limsup m sup b m tr b 1 m limsup m sup b m tr b 1 m where tr a max i 1 n a ii and b b ij

asynchronous parallel methods for enclosing solutions of nonlinear equations we consider interval arithmetic based parallel methods for enclosing solutions of nonlinear systems of equations where processors are allowed to proceed asynchronously we present a general study on the convergence of asynchronous iterations in interval spaces and apply them to a variety of methods for the nonlinear equations case the convergence results turn out to be very similar to those known for synchronous methods several practical examples on shared memory architectures are included the asynchronous methods sometimes perform substantially better than their synchronous counterparts

algorithmes mixtes asynchrones etude de convergence monotone nous introduisons dans cet article deux classes d algorithmes iteratifs que nous appelons algorithmes mixtes asynchrones et nous en etudierons la convergence selon un ordre partiel ces algorithmes sont implementables aussi bien sur les monoprocesseurs que sur les multiprocesseurs et avec leur etude de convergence constituent une generalisation des algorithmes mixtes classiques newton relaxation in this paper we introduce two classes of iterative algorithms which we call asynchronous mixed algorithms and we study their convergence under partial ordering these algorithms can be implemented just as well on monoprocessors as on multiprocessors and along with their convergence study constitute a generalization of the mixed classical newton relaxation algorithms

a new class of asynchronous iterative algorithms with order intervals this paper deals with a new class of parallel asynchronous iterative algorithms for the solution of nonlinear systems of equations the main feature of the new class of methods preseted here is the possibility of flexible commmunication between processors in particular partial updates can be exchanged approximation of the asssociated fixed point mapping is also considered a detailed convergengence study is presented a connection with the schwarz alternating method is made for nonlinear boundary value problems computational results on a shared memory multiprocessotr ibm 3090 are presented