complexity of stability and controllability of elementary hybrid systems in this paper we consider simple classes of nonlinear systems and prove that basic questions related to their stability and controllability are either undecidable or computationally intractable np hard as a special case we consider a class of hybrid systems in which the state space is partitioned into two halfspaces and the dynamics in each halfspace correspond to a different linear system

on the limit products of a family of matrices in this paper we direct attention at bounded families of complex n n matrices in order to study their asymptotic behaviour we recall from linear algebra appl 322 2001 162 the concept of limit spectrum maximizing product and show that nondefective families always admit such limit products then we consider defective families in loc cite we proved that for finite families of 2 2 matrices defectivity is equivalent to the existence of defective such limit products this result led us to conjecture the validity of this property also for higher dimensions n 3 here instead by making use of the results obtained by bousch and mairesse j am math soc 15 2002 77 that disproved the well known finiteness conjecture we find some counterexamples to our conjecture in loc cite for all n 3 2002 elsevier science inc all rights reserved

absolute convergence of rational series is semi decidable this paper deals with absolute convergence of real valued rational series i e mappings r r computed by weighted automata an algorithm is provided that takes a weighted automaton a as input and halts if and only if the corresponding series ra is absolutely convergent hence absolute convergence of rational series is semi decidable a spectral radius like parameter r is introduced which satisfies the following property a rational series r is absolutely convergent iff r 1 we show that if r is rational then r can be approximated by convergent upper estimates then it is shown that the sum w r w can be estimated to any accuracy rate this result can be extended to any sum of the form w r w p for any integer p

the mathematics of microstructure and the design of new materials the pathological energy function e u u 2 for u 0 e 0 1 has no minimizer as u decreases to 0 the energy also decreases but there is no way to achieve the value 0 although examples like this might seem to be unimaginably far from scientific thought they are at the heart of a new approach 1 to understand the complex microstructure and macroscopic response of materials that undergo phase transformations the free energy of such materials typically has no minimizer and the observed microstructures complex fine scale patterns of domains of different atomic lattice structure as shown below in a micrograph of cualni by c chu and r d j fig 1 have their origin in the material s ultimately futile attempt to find the minimum energy state 2 the lack of a ground state prohibits prediction of the macroscopic response from microscopic data via the standard procedure determine the free energy find the minimizing state and evaluate its macroscopic properties emerging mathematical methods linked to profound work in the 1940s by l c young and recently surveyed in 3 nevertheless deliver well defined macroscopic quantities obtained via averaging over all low energy states one area where predictions obtained in this new way have played a role is the recent synthetization of a new magnetostrictive material 4 5 whose magnetostrictive strain is 50 times larger than that of giant magnetostrictive materials formerly those with the largest strain

konig chains for submultiplicative functions and infinite products of operators we generalize the so called weighted konig lemma due to mate for a submultiplicative function on a subset of the union n n where is a set and n is the cartesian product of n copies of instead of a combinatorial argument as done by mate our proof uses tychonoff s compactness theorem to show the existence of a konig chain for a submultiplicative function as a consequence we obtain an extension of the daubechies lagarias theorem concerning a finite set of matrices with right convergent products here we replace matrices by banach algebra elements and we substitute compactness for finiteness of the last result yields new generalizations of the kelisky rivlin theorem on iterates of the bernstein operators on the banach space c 0 1

convergence analysis of c2 hermite interpolatory subdivision schemes by explicit joint spectral radius formulas abstract subdivision schemes are popular iterative processes to build graphs of functions curves and surfaces we analyze the 2 point hermite c 2 subdivision scheme introduced by merrien in 26 for the analysis of its convergence and its smoothness properties we are concerned with the computation of the joint spectral radius of a family of 2 matrices associated with the scheme in this paper by an explicit computation of the joint spectral radius of such pairs of matrices we determine necessary and sufficient conditions for the scheme to be c 2 convergent whenever it reproduces cubic polynomials in addition we present two one parameter families of convergent subdivision schemes belonging to the class in 26 possessing interesting properties from the shape control point of view

on continuous weighted finite automata abstract we investigate the continuity of the functions and real functions defined by weighted finite automata wfa we concentrate on the case of average preserving wfa we show that every continuous function definable by some wfa can be defined by an average preserving wfa and then characterize minimal average preserving wfa whose function or function and real function are continuous we obtain several algorithmic reductions for wfa related decision problems in particular we show that deciding whether the function and real function of an average preserving wfa are both continuous is computationally equivalent to deciding stability of a set of matrices we also present a method for constructing wfa that compute continuous real functions

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

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 convergence of linear switched systems this paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems we provide a collection of results that use weak dwell time dwell time strong dwell time permanent and persistent activation hypothesis the obtained results are shown to be tight by counterexample finally we apply our result to the three cell converter

on the asymptotic regularity of a family of matrices abstract in this paper we consider bounded families f of complex n n matrices we give sufficient conditions under which the sequence k f 1 k k 1 where k f is the supremum of the spectral radii of all possible products of k matrices chosen in f is convergent to its supremum f the so called generalized spectral radius of f we also illustrate a possible practical application

asymptotic stability and stabilizability of special classes of discrete time positive switched systems abstract in this paper we consider discrete time positive switched systems switching among autonomous subsystems characterized either by monomial matrices or by circulant matrices necessary and sufficient conditions are provided guaranteeing either global uniform asymptotic stability or stabilizability i e the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence such conditions lead to simple algorithms that allow to easily detect under suitable conditions whether a given positive switched system is not stabilizable

subadditivity generalized products of random matrices and operations research an elementary theorem on subadditive sequences provides the key to a far reaching theory of subadditive processes one important instance of this theory is the limit theory for products of stationary random matrices this paper shows that the subadditive inequality that governs the log of the norm of ordinary matrix products also governs other functions of several generalized matrix products these generalized matrix products are used to calculate minimal cost transportation routes schedules in manufacturing and minimal and maximal probabilities of multistage processes the application of subadditive ergodic theory to generalized products of stationary random matrices yields new information about the limiting behavior of generalized products exact calculations of the asymptotic behavior are possible in some examples

a gap result for the norms of semigroups of matrices let be a matrix norm on md c and let a be a finite set of matrices in md c we define mn a to be the maximum norm of a product of n elements of a we show that there is a gap in the possible growth of mn a showing that mn a grows either at least exponentially or is bounded by a polynomial in n of degree at most d 1 moreover we show that the growth is bounded by a polynomial if and only if every element of the semigroup generated by a has all of its eigenvalues on or inside the unit circle

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

on the complexity of computing the capacity of codes that avoid forbidden difference patterns some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed the maximal number of n bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n the coefficient of the exponent is the capacity of the forbidden patterns in this paper new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy the computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given subsequently a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be np hard when the sets of forbidden patterns are defined over an extended set of symbols finally the existence of extremal norms is proved for any set of matrices arising in the capacity computation based on this result a second capacity approximating algorithm is proposed the usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately

a tree based approach to joint spectral radius determination abstract we suggest a novel method to determine the joint spectral radius of finite sets of matrices by validating the finiteness property it is based on finding a certain finite tree with nodes representing sets of matrix products our approach accounts for cases where one or several matrix products satisfy the finiteness property moreover is potentially functional even for reducible sets of matrices

stability of positive linear switched systems on ordered banach spaces abstract we provide sufficient criteria for the stability of positive linear switched systems on ordered banach spaces the switched systems can be generated by finitely many bounded operators in infinite dimensional spaces with a general class of order inducing cones in the discrete time case we assume an appropriate interior point of the cone whereas in the continuous time case an appropriate interior point of the dual cone is sufficient for stability this is an extension of the concept of linear lyapunov functions for positive systems to the setting of infinite dimensional partially ordered spaces we illustrate our results with examples

the characterization of continuous four coefficient scaling functions and wavelets four coefficient dilation equations are examined and results converse to a theorem of daubechies lagarias are given these results complete the characterization of those four coefficient dilation equations having a continuous solution

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

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

konig chain for compact matrix sets abstract a topological proof via the generalized gelfand spectral radius formula is given of the fact that every compact set of complex n n matrices admits a konig chain

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

an inequality for the spectral radius of an interval matrix abstract for an n n interval matrix a aij we say that a is majorized by the point matrix a a ij if aij aij when the jth column of a has the property that there exists a power a m containing in the same jth column at least one interval not degenerated to a point interval and aij aij otherwise denoting the generalized spectral radius in the sense of daubechies and lagarias of a by a and the usual spectral radius of a by a it is proved that if a is majorized by a then a a this inequality sheds light on the asymptotic stability theory of discrete time linear interval systems

the optimal coefficients in daubechies wavelets abstract the construction of wavelets begins with a dilation equation x ck 2x k the choice of the coefficients ck controls the accuracy and orthogonality of wavelet expansions we show how the daubechies coefficients which give pth order accuracy with 2p coefficients the minimum number follow from orthogonality and accuracy conditions applied to the polynomial ckeik

the normed finiteness property of compact contraction operators we study the joint spectral radius given by a finite set of compact operators on a hilbert space it is shown that the normed finiteness property holds in this case that is if all the compact operators are contractions and the joint spectral radius is equal to 1 then there exists a finite product that has a spectral radius equal to 1 we prove an additional statement in that the requirement that the joint spectral radius be equal to 1 can be relaxed to the asking that the maximum norm of finite products of a length norm is equal to 1 the length of this product is related to the dimension of the subspace on which the set of operators is norm preserving

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

robust minkowski lyapunov functions abstract a necessary and sufficient condition for solvability of the lyapunov inequality for polytopic linear inclusions over the space of minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is derived it is demonstrated that the introduced framework results in a nonconservative stability analysis of polytopic linear inclusions

contractibility of compact contractions in hilbert space for a finite set of compact contractions in a complex hilbert space h it is shown that r a 1 for all a in the multiplicative semigroup generated by if and only if there exists a positive integer n such that a 1 for all a in the multiplicative semigroup generated by with length greater than n here r a denotes the spectral radius of a as an application an answer is given to an infinite dimensional case of the finiteness conjecture for the generalized spectral radius attributed to j c lagarias and y wang linear algebra appl 214 1995 17

stability analysis of switched linear systems defined by regular languages in this work we study the stability of an autonomous discrete time linear switched system whose switching sequences are generated by a muller automaton this system arises in various engineering problems such as distributed communication and automotive engine control the asymptotic stability of this system referred to as regular asymptotic stability ras generalizes two well known definitions of stability of autonomous discrete time linear switched systems namely absolute asymptotic stability aas and shuffle asymptotic stability sas we also extend these stability definitions to robust versions we show that absolute asymptotic stability robust absolute asymptotic stability and robust shuffle asymptotic stability are equivalent to exponential stability in addition by using the kronecker product we prove that a robust regular asymptotic stability problem is equivalent to the conjunction of several robust absolute asymptotic stability problems

continuity of the generalized spectral radius in max algebra abstract let be an induced matrix norm associated with a monotone norm on r n and be the collection of all nonempty closed and bounded subsets of n n nonnegative matrices under this matrix norm for the hausdorff metric h between and is given by h max sup a inf b a b sup b inf a a b the max algebra system consists of the set of nonnegative numbers with sum a b max a b and the standard product ab for a b 0 for n n nonnegative matrices a b their product is denoted by a b where a b ij max 1 k n a ik b kj for each the max algebra version of the generalized spectral radius of is limsup m sup a m a 1 m where m a 1 a 2 a m a i here a is the maximum circuit geometric mean in this paper we prove that the max algebra version of the generalized spectral radius is continuous on the hausdorff metric space h the notion of the max algebra version of simultaneous nilpotence of matrices is also proposed necessary and sufficient conditions for the max algebra version of simultaneous nilpotence of matrices are presented as well

simultaneous schur stability let be a set of n n complex matrices denote by the multiplicative semi group generated by for an n n complex matrix a the spectrum of a and spectral radius of a are denoted by a and r a respectively motivated by a phenomenon in the closed unit disc of the complex plane we give a rigorous definition of simultaneous schur stability as follows is said to be simultaneously schur stable if r a 1 a and 1 a a is said to be asymptotically stable if there exists a norm on cn such that sup a a e 1 it is proved that for a bounded set of n n complex matrices is asymptotically stable if and only if it is simultaneously schur stable by way of simultaneous schur stability some applications are illustrated especially an analytic combinatorial proof of a recent result of considerable depth generalized gelfand spectral radius formula

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

compactification of a set of matrices with convergent infinite products abstract we generalize and unify some aspects of the work of i daubechies j c lagarias linear algebra appl 162 1992 227 263 on a set of matrices with right convergent products rcps we show that most properties of an rcp set pass on to its compactification i e its closure in the matrix space results on finite rcp sets generally hold for compact rcp sets among which is the existence of the konig chain an important tool for analyzing rcp sets

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

on the generalized spectral subradius in this note we study the concepts of generalized spectral subradius and joint spectral subradius of a family of matrices we show that both these numbers are equal and we present two different formulas for them we also explain the relation between them and maximal lyapunov exponent of discrete linear time varying system finally we show that the spectral subradius is less than one if and only if a discrete linear inclusion is stable in a certain sense

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non negative matrices abstract recently blondel nesterov and protasov proved 1 2 that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non negative matrices with independent row column uncertainty we show that this result can be obtained as a simple consequence of the so called hourglass alternative used in 3 by the author and his companions to analyze the minimax relations between the spectral radii of matrix products axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property which includes the sets of non negative matrices with independent row uncertainty this class of matrices supplemented by the zero and identity matrices forms a semiring with the minkowski operations of addition and multiplication of matrix sets which gives means to construct new sets of non negative matrices possessing the finiteness property for the generalized and the lower spectral radii

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

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

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

approximating the spectral radius of sets of matrices in the max algebra is np hard the lower and average spectral radii measure respectively the minimal and average growth rates of long products of matrices taken from a finite set the logarithm of the average spectral radius is traditionally called the lyapunov exponent when one performs these products in the max algebra we obtain quantities that measure the performance of discrete event systems we show that approximating the lower and average max algebraic spectral radii is np hard

convergence of inhomogenous products of matrices and coefficients of ergodicity abstract given a square matrix a and a norm the coefficient of ergodicity of a with respect to is defined as max x t all x r n x 1 xtf 0 with f as a matrix satisfying af 0 we demonstrate that for a bounded set of such matrices with all coefficients of ergodicity of the matrices in the set below 1 all sequences constructed through inhomogenous products of matrices from the set converge geometrically

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

tight bound for deciding convergence of consensus systems abstract we analyze the asymptotic convergence of all infinite products of matrices taken in a given finite set by looking only at finite or periodic products it is known that when the matrices of the set have a common nonincreasing polyhedral norm all infinite products converge to zero if and only if all infinite periodic products with period smaller than a certain value converge to zero moreover bounds on that value are available lagarias and wang 1995 we provide a stronger bound that holds for both polyhedral norms and polyhedral seminorms in the latter case the matrix products do not necessarily converge to zero but all trajectories of the associated system converge to a common invariant subspace we prove that our bound is tight for all seminorms our work is motivated by problems in consensus systems where the matrices are stochastic nonnegative with rows summing to one and hence always share a same common nonincreasing polyhedral seminorm in that case we also improve existing results

stability and stabilizability criteria for discrete time positive switched systems in this paper we consider the class of discrete time switched systems switching between p autonomous positive subsystems first sufficient conditions for testing stability based on the existence of special classes of common lyapunov functions are investigated and these conditions are mutually related thus proving that if a linear copositive common lyapunov function can be found then a quadratic positive definite common function can be found too and this latter in turn ensures the existence of a quadratic copositive common function secondly stabilizability is introduced and characterized it is shown that if these systems are stabilizable they can be stabilized by means of a periodic switching sequence which asymptotically drives to zero every positive initial state conditions for the existence of state dependent stabilizing switching laws based on the values of a copositive linear quadratic lyapunov function are investigated and mutually related too finally some properties of the patterns of the stabilizing switching sequences are investigated and the relationship between a sufficient condition for stabilizability the existence of a schur convex combination of the subsystem matrices and an equivalent condition for stabilizability the existence of a schur matrix product of the subsystem matrices is explored

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

a max version of the generalized spectral radius theorem abstract let be a bounded set of n n nonnegative matrices in max algebra in this paper we propose the notions of the max algebra version of the generalized spectral radius of and the max algebra version of the joint spectral radius of the max algebra version of the generalized spectral radius theorem is established we propose the relationship between the generalized spectral radius of in the sense of daubechies and lagarias and its max algebra version moreover a generalization of elsner and van den driessche s lemma is presented as well

limits of level and parameter dependent subdivision schemes in this paper we present a new matrix approach for the analysis of subdivision schemes whose non stationarity is due to linear dependency on parameters whose values vary in a compact set indeed we show how to check the convergence in c r s and determine the holder regularity of such level and parameter dependent schemes efficiently via the joint spectral radius approach the efficiency of this method and the important role of the parameter dependency are demonstrated on several examples of subdivision schemes whose properties improve the properties of the corresponding stationary schemes

stability and paracontractivity of discrete linear inclusions abstract we study stability properties of a finite set of n n matrices such as paracontractivity bv and left convergent products lcps stability and their relations to each other the conjecture on equivalence of paracontractivity and lcp stability is proved moreover we prove the equivalence of the uniform bv stability and the property of vanishing length of steps of any trajectory of

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

the eigenvalue problem for linear and affine iterated function systems abstract the eigenvalue problem for a linear function l centers on solving the eigen equation lx x this paper generalizes the eigenvalue problem from a single linear function to an iterated function system f consisting of possibly an infinite number of linear or affine functions the eigen equation becomes f x x where 0 is real x is a compact set and f x f f f x the main result is that an irreducible linear iterated function system f has a unique eigenvalue equal to the joint spectral radius of the functions in f and a corresponding eigenset s that is centrally symmetric star shaped and full dimensional results of barabanov and of dranisnikov konyagin protasov on the joint spectral radius follow as corollaries

on the accuracy of the ellipsoid norm approximation of the joint spectral radius the joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set this quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate we introduce in this paper an approximation hat rho that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval hat rho sqrt n hat rho where n is the dimension of the matrices we also provide a simple approximation for the special case where the entries of the matrices are non negative in this case the approximation is proved to be within a factor at most m m is the number of matrices of the exact value

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

on the stability of takagi sugeno fuzzy systems with time varying uncertainties this paper studies the problems of stability analysis of takagi sugeno free fuzzy systems with time varying uncertainties in our prior study we represented the time varying uncertainty incurred in characteristic interval matrices in terms of the stability of takagi sugeno free fuzzy systems with consequent parameter uncertainties based on mayer s convergent theorem for powers of single interval matrix and its generalization we further proposed some sufficient conditions for the takagi sugeno free fuzzy system with time varying uncertainties to be globally asymptotically stable in this paper we propose the notion of simultaneously nilpotent interval matrices to investigate the takagi sugeno free fuzzy system with time varying uncertainties to be strongly stable within steps where relates to the dimension of interval matrices moreover a unique situation for the deterministic takagi sugeno free fuzzy system to be strongly stable within steps is derived as well where relates to the dimension of characteristic matrices for the deterministic takagi sugeno free fuzzy system

hypocoercivity for a bgk model for gas mixtures abstract we consider a kinetic model for a two component gas mixture without chemical reactions our goal is to study hypocoercivity for the linearized bgk model for gas mixtures in continuous phase space by constructing an entropy functional we can prove exponential relaxation to equilibrium with explicit rates our strategy is based on the entropy and spectral methods adapting lyapunov s direct method as presented in 1 for the one species linearized bgk model in comparison to the one species case we start with two partial differential equations one for each species these equations are coupled due to interspecies interactions which requires additional estimates on these interspecies terms

computing the joint spectral radius this paper presents algorithms for finding an arbitrarily small interval that contains the joint spectral radius of a finite set of matrices it also presents a numerical criterion for verifying in certain cases that the joint spectral radius is the maximum of the spectral radii of the given matrices error bounds are derived for the case where calculations are done with finite precision and the matrices are not known exactly the algorithms are implemented and applied to estimate holder exponents of the orthonormal wavelets n constructed by daubechies for 3 n 8

regularity of non stationary subdivision a matrix approach in this paper we study scalar multivariate non stationary subdivision schemes with integer dilation matrix m and present a unifying general approach for checking their convergence and for determining their holder regularity latter in the case m mi m ge 2 m mi m 2 the combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius as an application we prove a recent conjecture by dyn et al on the holder regularity of the generalized daubechies wavelets we illustrate our results with several examples

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

a note on a gap result for norms of semigroups of matrices abstract let be a matrix norm and be a bounded set of n n complex matrices for each m 1 2 let m sup a a m where m a1a2 am ai bell proved that there is a gap in the possible growth of m if consists of finitely many n n complex matrices in this note based on a lemma by elsner we give an elementary proof of bell s theorem in bounded case

the generalised berger wang formula and the spectral radius of linear cocycles using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a banach space the first formula was previously obtained by v s shulman and yu v turovski using operator theoretic ideas the second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators this result may be seen as an extension of classical formulae for the essential spectral radius given by r nussbaum a lebow and m schechter both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system as a byproduct of our method we answer a question of j e cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle

bibo stability of linear switching systems in this paper we show that for linear switching systems of the form x sub n 1 a sub n x sub n u sub n 1 where the matrices a sub n are chosen arbitrarily from a given set of matrices bounded input bounded output stability implies uniform exponential stability

on the structure of finite dimensional paracontractions abstract we give a characterization of paracontractions with respect to a strictly convex norm on c n as an application we give a characterization of linear mappings on m n which preserve paracontractions with respect to p norms for 1 p in both directions

norm conditions for convergence of infinite products abstract we give a finite number of necessary and sufficient conditions on two matrices so that any infinite left product in them is convergent

optimal norms and the computation of joint spectral radius of matrices the notion of spectral radius of a set of matrices is a natural extension of spectral radius of a single matrix the finiteness conjecture fc claims that among the infinite products made from the elements of a given finite set of matrices there is a certain periodic product made from the repetition of the optimal product whose rate of growth is maximal fc has been disproved in this paper it is conjectured that fc is almost always true and an algorithm is presented to verify the optimality of a given product the algorithm uses optimal norms as a special subset of extremal norms several conjectures related to optimal norms and non decomposable sets of matrices are presented the algorithm has successfully calculated the spectral radius of several parametric families of pairs of matrices associated with compactly supported multi resolution analyses and wavelets the results of related numerical experiments are presented

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting

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

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