modeling and analysis of timed petri nets using heaps of pieces the authors show that safe timed petri nets can be represented by special automata over the max semiring which compute the height of heaps of pieces this extends to the timed case the classical representation a la mazurkiewicz of the behavior of safe petri nets by trace monoids and trace languages for a subclass including all safe free choice petri nets we obtain reduced heap realizations using structural properties of the net covering by safe state machine components the authors illustrate the heap based modeling by the typical case of safe jobshops for a periodic schedule the authors obtain a heap based throughput formula which is simpler to compute than its traditional timed event graph version particularly if one is interested in the successive evaluation of a large number of possible schedules

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

stability of frobenius direct images over surfaces let x be a smooth projective surface over an algebraically closed field k of characteristic p 0 with omega x 1 semistable and mu omega x 1 0 for any semistable resp stable bundle w of rank r we prove that f w is semistable resp stable when p ge r r 1 2 1

direct images of bundles under frobenius morphism let x be a smooth projective variety of dimension n over an algebraically closed field k with char k p 0 and f x x1 be the relative frobenius morphism for any vector bundle w on x we prove that instability of f w is bounded by instability of w tl 1x 0 l n p 1 corollary 4 9 when x is a smooth projective curve of genus g 2 it implies f w being stable whenever w is stable

analysis of discrete time linear switched systems a variational approach a powerful approach for analyzing the stability of continuous time switched systems is based on using tools from optimal control theory to characterize the most unstable switching law this reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific most unstable switching law more generally this so called variational approach was successfully applied to derive nice reachability type results for both linear and nonlinear continuous time switched systems motivated by this we develop in this paper an analogous approach for discrete time linear switched systems we derive and prove a necessary condition for optimality of the most unstable switching law this yields a type of discrete time maximum principle mp we demonstrate by an example that this mp is in fact weaker than its continuous time counterpart to overcome this we introduce the auxiliary system of a discrete time linear switched system and show that regularity properties of time optimal controls for the auxiliary system imply nice reachability results for the original discrete time linear switched system using this approach we derive several new lie algebraic conditions guaranteeing nice reachability results these results and their proofs turn out to be quite different from their continuous time counterparts

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

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 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

on the regularity analysis of interpolatory hermite subdivision schemes abstract it is well known that the critical holder regularity of a subdivision schemes can typically be expressed in terms of the joint spectral radius jsr of two operators restricted to a common finite dimensional invariant subspace in this article we investigate interpolatory hermite subdivision schemes in dimension one and specifically those with optimal accuracy orders the latter include as special cases the well known lagrange interpolatory subdivision schemes by deslauriers and dubuc we first show how to express the critical holder regularity of such a scheme in terms of the joint spectral radius of a matrix pair f 0 f 1 given in a very explicit form while the so called finiteness conjecture for jsr is known to be not true in general we conjecture that for such matrix pairs arising from hermite interpolatory schemes of optimal accuracy orders a strong finiteness conjecture holds f 0 f 1 f 0 f 1 we prove that this conjecture is a consequence of another conjectured property of hermite interpolatory schemes which in turn is connected to a kind of positivity property of matrix polynomials we also prove these conjectures in certain new cases using both time and frequency domain arguments our study here strongly suggests the existence of a notion of positive definiteness for non hermitian matrices

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

fitting longitudinal reduced rank regression models by alternating least squares an alternating least squares method for iteratively fitting the longitudinal reduced rank regression model is proposed the method uses ordinary least squares and majorization substeps to estimate the unknown parameters in the system and measurement equations of the model in an example with cross sectional data it is shown how the results conform closely to results from eigenanalysis optimal scaling of nominal and ordinal variables is added in a third substep and illustrated with two examples involving cross sectional and longitudinal data

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

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

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

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

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

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

optimal periodic orbits of chaotic systems occur at low period invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis that indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor in particular our results indicate that if we consider that the function to be optimized depends on a parameter gamma then the lebesgue measure in gamma corresponding to optimal periodic orbits of period ital p or greater decreases exponentially with increasing ital p furthermore the set of parameter values for which optimal orbits are nonperiodic typically has zero lebesgue measure copyright ital 1996 the american physical society

brief paper a second order maximum principle for discrete time bilinear control systems with applications to discrete time linear switched systems a powerful approach for analyzing the stability of continuous time switched systems is based on using optimal control theory to characterize the most unstable switching law this reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific most unstable switching law for discrete time switched systems the variational approach received considerably less attention this approach is based on using a first order necessary optimality condition in the form of a maximum principle mp and typically this is not enough to completely characterize the most unstable switching law in this paper we provide a simple and self contained derivation of a second order necessary optimality condition for discrete time bilinear control systems this provides new information that cannot be derived using the first order mp we demonstrate several applications of this second order mp to the stability analysis of discrete time linear switched systems

stability analysis of second order switched homogeneous systems we study the stability of second order switched homogeneous systems using the concept of generalized first integrals we explicitly characterize the most destabilizing switching law and construct a lyapunov function that yields an easily verifiable necessary and sufficient condition for asymptotic stability using the duality between stability analysis and control synthesis this also leads to a novel algorithm for designing a stabilizing switching controller

stability of sheaves of locally closed and exact forms abstract for any smooth projective variety x of dimension n over an algebraically closed field k of characteristic p 0 with x 1 0 if the truncated symmetric powers t l x 1 0 l n p 1 of x 1 are semi stable then the sheaf b x 1 of exact 1 forms is stable when x is a surface with x 1 0 and x 1 is semi stable the sheaf b x 2 of exact 2 forms is also stable moreover under the same condition the sheaf z x 1 of closed 1 forms is stable when p 3 and z x 1 is semi stable when p 3

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

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

instability of truncated symmetric powers of sheaves abstract let x be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p 0 let f x x x be the absolute frobenius morphism and e a torsion free sheaf on x we give an upper bound of instability of truncated symmetric powers t l e 0 l rk e p 1 in terms of l max x 1 i x 1 and i e theorem 3 5 as an application we obtain an upper bound of the instability of frobenius direct image f x e and some sufficient conditions of f x e being slope semi stable in addition we study the slope semi stability of sheaves of locally exact resp closed forms b x i resp z x i

criteria for the stability of the finiteness property and for the uniqueness of barabanov norms a set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product the extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research in this article we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set but also for all sufficiently nearby sets of equal cardinality we also prove a theorem giving conditions under which the barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar and show that in certain cases these conditions are also persistent under small perturbations

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

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

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

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

finiteness property of pairs of 2 2 sign matrices via real extremal polytope norms this paper deals with the joint spectral radius of a finite set of matrices we s ay that a set of matrices has the finiteness property if the maximal rate of growth in the multiplicative semigroup it generates is given by the powers of a finite product here we address the problem of establishing the finiteness properties of p airs of 2 2 sign matrices such problem is related to the conjecture that pairs of sign ma trices fulfil the finiteness property for any dimension this would imply by a recent result by j ungers and blondel that finite sets of rational matrices fulfil the finiteness property wh ich would be very important property in terms of the computation of the joint spectral radius the finiteness property of n n sign matrices remains open but the knowledge of its validity in the 2 2 case gives the hope to solve it by using some induction argument on the dimension as a main tool of our proof we make use of a procedure to compute a so called real polytope extremal norm for the set in particular we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semig roup is spectrum maximizing for pairs of sign matrices we develop the computations exactly and hence are able to prove analytically the finiteness property on the other hand the algorithm c an be used in a floating point arithmetic and provide a general tool for approximating the join t spectral radius of a set of matrices

brief paper explicit construction of a barabanov norm for a class of positive planar discrete time linear switched systems we consider the stability under arbitrary switching of a discrete time linear switched system a powerful approach for addressing this problem is based on studying the most unstable switching law musl if the solution of the switched system corresponding to the musl converges to the origin then the switched system is stable for any switching law the musl can be characterized using optimal control techniques this variational approach leads to a hamilton jacobi bellman equation describing the behavior of the switched system under the musl the solution of this equation is sometimes referred to as a barabanov norm of the switched system although the barabanov norm was studied extensively it seems that there are few examples where it was actually computed in closed form in this paper we consider a special class of positive planar discrete time linear switched systems and provide a closed form expression for a corresponding barabanov norm and a musl the unit circle in this norm is a parallelogram

stability of planar switched systems the linear single input case we study the stability of the origin for the dynamical system dot x t u t ax t 1 u t bx t where a and b are two 2 2 real matrices with eigenvalues having strictly negative real part x in mbox bf r 2 and u 0 infty to 0 1 is a completely random measurable function more precisely we find a coordinates invariant necessary and sufficient condition on a and b for the origin to be asymptotically stable for each function u the result is obtained without looking for a common lyapunov function but studying the locus in which the two vector fields ax and bx are collinear there are only three relevant parameters the first depends only on the eigenvalues of a the second depends only on the eigenvalues of b and the third contains the interrelation among the two systems and it is the cross ratio of the four eigenvectors of a and b in the projective line cp1 in the space of these parameters the shape and the convexity of the region in which there is stability are studied this bidimensional problem assumes particular interest since linear systems of higher dimensions can be reduced to our situation

corrigendum addendum to sets of matrices all infinite products of which converge abstract this corrigendum addendum supplies corrected statements and proofs of some results in our paper appearing in linear algebra appl 161 1992 227 263 these results concern special kinds of bounded semigroups of matrices it also reports on progress on the topics of this paper made in the last eight years

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

dynamic factor analysis of nonstationary multivariate time series a dynamic factor model is proposed for the analysis of multivariate nonstationary time series in the time domain the nonstationarity in the series is represented by a linear time dependent mean function this mild form of nonstationarity is often relevant in analyzing socio economic time series met in practice through the use of an extended version of molenaar s stationary dynamic factor analysis method the effect of nonstationarity on the latent factor series is incorporated in the dynamic nonstationary factor model dnfm it is shown that the estimation of the unknown parameters in this model can be easily carried out by reformulating the dnfm as a covariance structure model and adopting the ml algorithm proposed by joreskog furthermore an empirical example is given to demonstrate the usefulness of the proposed dnfm and the analysis

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

extremal norms for positive linear inclusions abstract for finite dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm in case the cone is simplicial a similar statement applies to absolute norms the semigroups under consideration may be generated by discrete time systems continuous time systems or continuous time systems with jumps the existence of extremal norms is used to extend results on the lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word

nice reachability for planar bilinear control systems with applications to planar linear switched systems corrected we consider planar bilinear control systems with measurable controls we show that any point in the reachable set can be reached by a ldquonicerdquo control specifically a control that is a concatenation of a bang arc with either 1 a bang bang control that is periodic after the third switch or 2 a piecewise constant control with no more than two discontinuities under the additional assumption that the bilinear system is positive or invariant for any proper cone we show that the reachable set is spanned by a concatenation of a bang arc with either 1 a bang bang control with no more than two discontinuities or 2 a piecewise constant control with no more than two discontinuities in particular any point in the reachable set can be reached using a piecewise constant control with no more than three discontinuities several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result we demonstrate this with an example

an algorithm for finding extremal polytope norms of matrix families in this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered we present an algorithm which under some suitable assumptions is able to check if a certain product in the multiplicative semigroup is spectrum maximizing the algorithm proceeds by attempting to construct a suitable extremal norm for the family namely a complex polytope norm as examples for testing our technique we first consider the set of two 2 dimensional matrices recently analyzed by blondel nesterov and theys to disprove the finiteness conjecture and then a set of 3 dimensional matrices arising in the zero stability analysis of the 4 step bdf formula for ordinary differential equations

alternative algorithms for the estimation of dynamic factor mimic and varying coefficient regression models this paper provides a general approach to the formulation and estimation of dynamic unobserved component models after introducing the general model two methods for estimating the unknown parameters are presented both are algorithms for maximizing the likelihood function the first is based on the method of scoring the second is the em algorithm a derivative free method each iteration of em requires a kalman filter and smoother followed by straightforward regression calculations the paper suggests using the em methods to quickly locate a neighborhood of the maximum scoring can then be used to pinpoint the maximum and calculate the information matrix

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

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 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

third order nilpotency nice reachability and asymptotic stability we consider an affine control system whose vector fields span a third order nilpotent lie algebra we show that the reachable set at time t using measurable controls is equivalent to the reachable set at time t using piecewise constant controls with no more than four switches the bound on the number of switches is uniform over any final time t as a corollary we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching this provides a partial solution to an open problem posed in d liberzon lie algebras and stability of switched nonlinear systems in v blondel a megretski eds unsolved problems in mathematical systems and control theory princeton univ press 2004 pp 203 207

the output regulation problem with stability for linear switching systems a geometric approach abstract this work presents a solution for the output regulation problem with quadratic stability under arbitrary switching in linear switching systems the extension to other stability requirements like asymptotic stability in particular is also considered and it is shown to affect only few specific features of the proposed solution the main reason is that the geometric approach which is at the basis of the developed methodology establishes a neat separation between the structural aspects and the stability aspects of the problem for the same reason continuous time systems and discrete time systems are given a unified treatment as far as the structural issues are concerned while different technicalities characterize the discussion of the stability issues

a note on stability conditions for planar switched systems this article is concerned with the stability problem for the planar linear switched system where the real matrices a 1 a 2 2 2 are hurwitz and u 0 0 1 is a measurable function we give coordinate invariant necessary and sufficient conditions on a 1 and a 2 under which the system is asymptotically stable for arbitrary switching functions u the new conditions unify those given in previous papers and are simpler to be verified since we reduced to study 4 cases instead of 20 most of the cases are analysed in terms of the function

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

on the zero pattern properties and asymptotic behavior of continuous time positive system trajectories abstract in this paper the zero pattern properties and the asymptotic evolution of the trajectories of a continuous time positive system are investigated to this end we need to introduce some new tools and to derive some new results within the broad research area of nonnegative matrix theory which enable use to explore the zero pattern and the elementary modes of the exponential of a metzler matrix specifically a normal form for the exponential of a metzler matrix is provided and the concept of echelon basis consisting of eigenvectors and generalized eigenvectors of the metzler matrix is introduced by making use of these two ingredients a detailed result about the dominant mode of each single block appearing in the normal form of the exponential matrix is provided this allows to obtain a modal decomposition of the exponential matrix emphasizing the column dominant modes as a result the zero pattern as well as the asymptotic behavior of every free state evolution depending on the zero pattern of the initial state can be easily determined

generating functions of switched linear systems analysis computation and stability applications in this paper a unified framework is proposed to study the exponential stability of discrete time switched linear systems and more generally the exponential growth rates of their trajectories under three types of switching rules arbitrary switching optimal switching and random switching it is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong the weak and the mean generating functions respectively in particular necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence various properties of the generating functions are established and their relations are discussed algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples

on the stability of positive linear switched systems under arbitrary switching laws we consider n dimensional positive linear switched systems a necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is hurwitz several researchers conjectured that for positive linear switched systems this condition is also sufficient recently gurvits shorten and mason showed that this conjecture is true for the case n 2 but is not true in general their results imply that there exists some minimal integer np such that the conjecture is true for all n np but is not true for n np we show that np 3

dynamic structural systems under indirect observation identifiability and estimation aspects from a system theoretic perspective in this partly expository paper the parameter identifiability and estimation of a general dynamic structural model under indirect observation will be considered from a system theoretic perspective the general dynamic model covers dynamic factor analytic models dynamic mimic models and joreskog s linear structural model as special cases its reduced form is under a slightly different specification known in system theory and econometrics as the stochastic stationary version of the state space model by using concepts and methods from system theory such as the observability and controllability concept the steady state kalman filter and a general nonlinear ml estimation procedure known as prediction error estimation the general dynamic model will be identified it will be shown that joreskog s lisrel procedure is a special case of the prediction error estimation procedure

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

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

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