stability and disturbance attenuation for markov jump linear systems with time varying transition probabilities we address stability and disturbance attenuation for markov jump linear systems where transition probabilities vary in a finite set the time variation of the transition probabilities may be a priori known or unknown necessary and sufficient conditions for uniform stochastic stability and uniform stochastic disturbance attenuation are reported in both cases conditions are expressed as a set of finite dimensional linear matrix inequalities that can be solved efficiently

convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled data systems under dwell time constraints stability analysis and control of linear impulsive systems is addressed in a hybrid framework through the use of continuous time time varying discontinuous lyapunov functions necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas extensions to the stability of aperiodic systems under minimum maximum and ranged dwell times are then derived by exploiting further the particular structure of the stability conditions the results are non conservatively extended to quadratic stability analysis of linear uncertain impulsive systems these stability criteria are in turn losslessly extended to stabilization using a particular yet broad enough class of state feedback controllers providing then a convex solution to the open problem of robust dwell time stabilization of impulsive systems using hybrid stability criteria relying finally on the representability of sampled data systems as impulsive systems the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled data systems are straightforwardly solved using the same ideas several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach

uniform stability of switched linear systems extensions of lasalle s invariance principle this paper addresses the uniform stability of switched linear systems where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set we provide a collection of results that can be viewed as extensions of lasalle s invariance principle to certain classes of switched linear systems using these results one can deduce asymptotic stability using multiple lyapunov functions whose lie derivatives are only negative semidefinite depending on the regularity assumptions placed on the switching signals one may be able to conclude just asymptotic stability or uniform exponential stability we show by counter example that the results obtained are tight

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

control of linear switched systems with receding horizon modal information we provide an exact solution to two performance problems one of disturbance attenuation and one of windowed variance minimization subject to exponential stability considered are switched systems whose parameters come from a finite set and switch according to a language such as that specified by an automaton the controllers are path dependent having finite memory of past plant parameters and finite foreknowledge of future parameters exact convex synthesis conditions for each performance problem are expressed in terms of nested linear matrix inequalities the resulting semidefinite programming problem may be solved offline to arrive at a suitable controller a notion of path by path performance is introduced for each performance problem leading to improved system performance non regular switching languages are considered and the results are extended to these languages two simple physically motivated examples are given to demonstrate the application of these results

almost sure stability of markov jump linear systems with deterministic switching the technical note studies a class of linear systems whose piecewise constant dynamic matrix is subject to both stochastic jumps governed by a markov chain and deterministic switches these systems will be dubbed switching dynamics markov jump linear systems sd mjls sufficient conditions for exponential almost sure stability eas stability are established under either hard or average constraints on the dwell time between switching instants the proof relies on easy to check norm contractivity conditions and the ergodic law of large numbers

nonquadratic lyapunov functions for robust stability analysis of linear uncertain systems in this note we derive conditions of existence of a homogeneous polynomial lyapunov function of an arbitrary even degree establishing global asymptotic stability of linear system with box bounded uncertainty verification of these conditions is reduced to solving a convex minimization problem we produce numerical examples that demonstrate significant improvement in estimates of admissible uncertainty bounds compared with estimates obtained via the most commonly used quadratic lyapunov functions

on input to state stability of discrete time switched nonlinear time varying systems in this paper input to state stability iss for discrete time switched nonlinear time varying sntv systems is investigated starting with discrete time nonlinear time varying ntv systems some improved sufficient conditions are proposed to verify the iss of systems by using the weak implication form iss wi iss lyapunov function weak dissipative form iss wd iss lyapunov function and interval descent technique then the results obtained are extended to study the iss of discrete time sntv systems several relaxed conditions are given by using piecewise wi iss and wd iss lyapunov functions minimum dwell time and infinite switching methods respectively comparing with the existing results the obtained conditions release the requirement on negative definiteness of the differences of piecewise lyapunov functions moreover all subsystems are allowed to be unstable in the case of infinite switching finally a numerical example is given to illustrate the theoretical results

nonconservative lifted convex conditions for stability of discrete time switched systems under minimum dwell time constraint in this note a novel conception called virtual clock which is defined by an artificial timer over a finite cycle is introduced for stability analysis of discrete time switched linear systems under minimum dwell time constraint two necessary and sufficient conditions associated with a virtual clock with a sufficient length are proposed to ensure the global uniform asymptotic stability of discrete time switched linear systems for the two nonconservative stability criteria the lifted version maintains the convexity in system matrices based on the lifted convex conditions the extensions to ell 2 gain computation and mathcal h infty control problems are presented in the sequel in particular a novel virtual clock dependent controller is designed which outperforms the traditional mode dependent and common gain controllers several numerical examples are provided to illustrate our theoretic results

efficient evaluation of the fourier transform over products of slater type orbitals on different centers using the shift operator technique a compact formula for the fourier transform of a product of two slater type orbitals located on different atomic centers is derived the result is valid for arbitrary quantum numbers and was found to be numerically stable for a wide range of geometrical parameters and momenta details of the implementation are presented together with benchmark data for representative integrals we also discuss the assets and drawbacks of alternative algorithms available and analyze the numerical efficiency of the new scheme

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

unified stability criteria for slowly time varying and switched linear systems this paper presents a unified approach to formulating stability conditions for slowly time varying linear systems and switched linear systems the concept of total variation is generalized to the case of matrix valued functions using this generalized concept a result extending existing stability conditions for slowly time varying linear systems is derived as special cases of this result two sets of stability conditions are derived for switched linear systems which match known results in the literature a numerical example is included to further illustrate the application of the main result

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

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

stability of discrete time switching systems with constrained switching sequences we introduce a novel framework for the stability analysis of discrete time linear switching systems with switching sequences constrained by an automaton the key element of the framework is the algebraic concept of multinorm which associates a different norm per node of the automaton and allows to exactly characterize stability building upon this tool we develop the first arbitrarily accurate approximation schemes for estimating the constrained joint spectral radius that is the exponential growth rate of a switching system with constrained switching sequences more precisely given a relative accuracy r 0 the algorithms compute an estimate of within the range 1 r these algorithms amount to solve a well defined convex optimization program with known time complexity and whose size depends on the desired relative accuracy r 0

approximation of the constrained joint spectral radius via algebraic lifting this paper studies the linear constrained switching system which is a discrete time switched linear system whose switching sequences are constrained by a deterministic finite automaton the stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate using the semi tensor product of matrices the matrix form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system then the constrained joint generalized spectral radius of a constrained switching system is proved to be equal to the joint generalized spectral radius of its lifted arbitrary switching system which can be approximated by off the shelf algorithms

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

stabilizing switching signals for switched systems this article deals with stability of continuous time switched systems under constrained switching given a family of systems possibly containing unstable dynamics we characterize a new class of switching signals under which the switched system generated by it and the family of systems is globally asymptotically stable our characterization of such stabilizing switching signals involves the asymptotic frequency of switching the asymptotic fraction of activation of the constituent systems and the asymptotic densities of admissible transitions among them our techniques employ multiple lyapunov like functions and extend preceding results both in scope and applicability

addition theorem of slater type orbitals a numerical evaluation of barnett coulson lowdin functions when using the one centre two range expansion method to evaluate multicentre integrals over slater type orbitals stos it may become necessary to compute numerical values of the corresponding fourier coefficients also known as barnett coulson lowdin functions bclfs bouferguene and jones 1998 j chem phys 109 5718 to carry out this task it is crucial to not only have a stable numerical procedure but also a fast algorithm in previous work bouferguene and rinaldi 1994 int j quantum chem 50 21 bclfs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming the present work aims at exploring another path in which bclfs are represented either by an infinite series involving modified bessel functions or by an integral whose integrand is a smooth function both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r as a consequence it is no longer necessary to split the integrals over r 0 into several components with a different analytical form in each of these a numerical study is also carried out to help select the most appropriate method to be used in practice

graph lyapunov function for switching stabilization and distributed computation abstract this paper studies stabilization of discrete time switched linear systems slss using the notion of graph control lyapunov functions gclfs a gclf is a set of lyapunov functions defined on a weighted digraph where each lyapunov function is represented by a node in the digraph and there is a lyapunov inequality associated with each subgraph consisting of a node and its out neighbors the weight of a directed edge indicates the decay or growth rate of the lyapunov functions it is proved that an sls is switching stabilizable if and only if there exists a gclf the main benefits of gclfs are reduced computational cost and conservatism for stabilizability tests besides we show that the proposed gclf framework unifies several control lyapunov functions and the related stabilization theorems moreover we propose a distributed algorithm to evaluate the stabilizability with reduced computational costs by taking benefits of the graph structure of gclfs several examples are given to demonstrate the efficiency of the algorithm

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

necessary and sufficient stability condition for second order switched systems a phase function approach abstractto find a unified approach for the stability analysis of second order switched system the concept of phase function is proposed in this paper first the basic properties of phase function are explored following this concept and its properties the phase based stability criterion is investigated based on the lyapunov theory and a necessary and sufficient stability condition is obtained in the phase function approach moreover the connection between phase based stability conditions and algebraic condition of system matrices is also discussed finally numerical examples are provided to exemplify the main result and make necessary comparisons with the existing methods

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

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

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

mode identifying time estimation and switching delay tolerant control for switched systems in this paper an elementary time unit etu method is proposed to study the problem of estimating the admissible delay in the identification of the active mode in the analysis and design of switched systems the activation interval of a subsystem is considered to consist of a finite time number of segments called etus by which a novel class of time scheduled lyapunov function is used to estimate the admissible delay in mode identification for switched systems further the etu method is applied for switching delay tolerant control problem and a class of time scheduled state feedback controllers are designed to achieve the exponentially stability several numerical examples are presented to validate the theoretic findings

on the stabilizability of discrete time switched linear systems novel conditions and comparisons in this paper we deal with the stabilizability property for discrete time switched linear systems a recent necessary and sufficient characterization of stabilizability based on set theory is considered as the reference for comparing the computation oriented sufficient conditions the classical bmi conditions based on lyapunov metzler inequalities are considered and extended novel lmi conditions for stabilizability derived from the geometric ones are presented that permit to combine generality with computational affordability for the different conditions the geometrical interpretations are provided and the induced stabilizing switching laws are given the relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity

modular intersection cohomology complexes on flag varieties we present a combinatorial procedure based on the w graph of the coxeter group which shows that the characters of many intersection cohomology complexes on low rank complex flag varieties with coefficients in an arbitrary field are given by kazhdan lusztig basis elements our procedure exploits the existence and uniqueness of parity sheaves in particular we are able to show that the characters of all intersection cohomology complexes with coefficients in a field on the flag variety of type an for n 7 are given by kazhdan lusztig basis elements by results of soergel this implies a part of lusztig s conjecture for sl n with n 7 we also give examples where our techniques fail in the appendix by tom braden examples are given of intersection cohomology complexes on the flag varities for sl 8 and so 8 which have torsion in their stalks or costalks

observability and controllability analysis of linear systems subject to data losses we provide algorithmically verifiable necessary and sufficient conditions for fundamental system theoretic properties of discrete time linear systems subject to data losses more precisely the systems in our modeling framework are subject to disruptions data losses in the feedback loop where the set of possible data loss sequences is captured by an automaton as such the results are applicable in the context of shared wireless communication networks and or embedded architectures where some information on the data loss behavior is available a priori we propose an algorithm for deciding observability or the absence of it for such systems and show how this algorithm can be used also to decide other properties including constructibility controllability reachability null controllability detectability and stabilizability by means of relations that we establish among these properties the main apparatus for our analysis is the celebrated skolem s theorem from linear algebra moreover we study the relation between the model adopted in this paper and a previously introduced model where instead of allowing dropouts in the feedback loop one allows for time varying delays

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

stability and stabilizability of discrete time dual switching systems with application to sampled data systems in this paper stability and stabilizability of discrete time dual switching linear systems is investigated the switched systems under consideration have two switching variables one of them is stochastic described by an underlying markov chain the other one can be regarded either as a deterministic disturbance or as a control input leading to stability or stabilizability problems respectively for the considered class of systems sufficient conditions for mean square stability with or without control gain synthesis and mean square stabilizability are provided in terms of matrix inequalities when the stochastic switching is driven by an independent identically distributed sequence we establish simpler conditions without additional conservatism then it is shown how the proposed framework can be used to study aperiodic sampled data systems with stochastic computation times the results are illustrated on examples borrowed from the literature

stability analysis and control synthesis for switched systems a switched lyapunov function approach this paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete time domain the approach followed in this paper looks at the existence of a switched quadratic lyapunov function to check asymptotic stability of the switched system under consideration two different linear matrix inequality based conditions allow to check the existence of such a lyapunov function the first one is classical while the second is new and uses a slack variable which makes it useful for design problems these two conditions are proved to be equivalent for stability analysis investigating the static output feedback control problem we show that the second condition is in this case less conservative the reduction of the conservatism is illustrated by a numerical evaluation

geometric langlands duality and representations of algebraic groups over commutative rings as such it can be viewed as a first step in the geometric langlands program the connected complex reductive groups have a combinatorial classification by their root data in the root datum the roots and the co roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs if g is a reductive group we write g for its companion and call it the dual group g the notion of the dual group itself does not appear in satake s paper but was introduced by langlands together with its various elaborations in li l2 and is a cornerstone of the langlands program it also appeared later in physics mo gno in this paper we discuss the basic relationship between g and g we begin with a reductive g and consider the affine grassmannian qx the grassmannian for the loop group of g for technical reasons we work with formal algebraic loops the affine grassmannian is an infinite dimen sional complex space we consider a certain category of sheaves the spherical perverse sheaves on r these sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a hopf algebra by what has come to be known as tannakian formalism the resulting hopf algebra turns out to be the ring of functions on g in this interpretation the spherical perverse sheaves on the affine grassman nian correspond to finite dimensional complex representations of g thus instead of defining g in terms of the classification of reductive groups we pro vide a canonical construction of g starting from g we can carry out our construction over the integers the spherical perverse sheaves are then those with integral coefficients but the grassmannian remains a complex algebraic object

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

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

the fourier transform of a two center product of exponential type functions and its efficient evaluation we analyze the analytical and numerical properties of the hitherto known formulas of the fourier transform of a two center product of exponentially declining functions exponential type functions etfs which are derived with the help of the fourier convolution theorem and feynman s identity in detail we consider b functions which are a special class of etfs having advantageous properties under fourier transformation other etfs orbitals can be expressed in terms of b functions by linear combinations in our derivations we use the properties of the differential operator ylm specifying a solid harmonic whose argument is the nabla operator r instead of the vector r in order to generate multicenter integrals over nonscalar functions from integrals over scalar functions applying the generating differential operator ylm we obtain a recently derived new formula for the fourier transform of a two center product of b functions in a much more straightforward manner furthermore we present an efficient procedure to compute this new formula which is valid for arbitrary quantum numbers and exponential parameters and report various numerical test values

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

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 converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

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

joint spectral radius and path complete graph lyapunov functions we introduce the framework of path complete graph lyapunov functions for approximation of the joint spectral radius the approach is based on the analysis of the underlying switched system via inequalities imposed among multiple lyapunov functions associated to a labeled directed graph inspired by concepts in automata theory and symbolic dynamics we define a class of graphs called path complete graphs and show that any such graph gives rise to a method for proving stability of the switched system this enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic common sum of squares path dependent quadratic and maximum minimum of quadratics lyapunov functions we compare the quality of approximation obtained by certain classes of path complete graphs including a family of dual graphs and all path complete graphs with two nodes on an alphabet of two matrices we derive approximation guarantees for several families of path complete graphs such as the de bruijn graphs this provides worst case performance bounds for path dependent quadratic lyapunov functions and a constructive converse lyapunov theorem for maximum minimum of quadratics lyapunov functions

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

dynamic output feedback control for positive roesser system under the switched and t s fuzzy rules abstract this paper is concerned with the dynamic output feedback controller design for the positive roesser type nonlinear system which is intrinsically characterized by the switched mechanism and subtly decomposed into the linear form under the takagi sugeno fuzzy rules firstly based on the co positive lyapunov function and the average dwell time method sufficient conditions are presented under which the resulting closed loop system is exponentially stable and has l 1 gain bound then explicit expressions are also given to derive the expected controller gain matrices with desired l 1 gain bound finally effectiveness of the proposed results is illustrated by two numerical examples

convex lifted conditions for robust l 2 stability analysis and l 2 stabilization of linear discrete time switched systems with minimum dwell time constraint stability analysis of discrete time switched systems under minimum dwell time is studied using a new type of lmi conditions these conditions are convex in the matrices of the system and shown to be equivalent to the nonconvex conditions proposed in geromel and colaneri 2006b the convexification of the conditions is performed by a lifting process which introduces a moderate number of additional decision variables the convexity of the conditions can be exploited to extend the results to uncertain systems control design and 2 gain computation without introducing additional conservatism several examples are presented to show the effectiveness of the approach

on torsion in the intersection cohomology of schubert varieties abstract we prove that the prime torsion in the local integral intersection cohomology of schubert varieties in the flag variety of the general linear group grows exponentially in the rank the idea of the proof is to find a highly singular point in a schubert variety and calculate the euler class of the normal bundle to the miraculously smooth fibre in a particular bott samelson resolution the result is a geometric version of an earlier result established using soergel bimodule techniques

stability analysis and l1 gain characterization for switched positive systems under dwell time constraint abstract this paper deals with the problems of stability analysis and l 1 gain characterization for continuous time switched systems consisting of positive subsystems with the aid of a discretized copositive lyapunov function a sufficient condition ensuring the asymptotic stability of continuous time switched positive systems under dwell time constraint is obtained which can be checked via linear programming furthermore the conservatism of the proposed approach is studied and the result with least conservatism in the framework of discretized copositive lyapunov function is obtained the result is then extended to l 1 gain characterization for switched positive systems with a prescribed dwell time an unweighted l 1 gain can be computed via solving a linear programming problem a numerical example and a practical traffic example are given to illustrate our results

on path complete lyapunov functions geometry and comparison we study optimization based criteria for the stability of switching systems known as path complete lyapunov functions and ask the question can we decide algorithmically when a criterion is less conservative than another our contribution is twofold first we show that a path complete lyapunov function which is a multiple lyapunov function by nature can always be expressed as a common lyapunov function taking the form of a combination of minima and maxima of the elementary functions that compose it geometrically our results provide for each path complete criterion an implied invariant set second we provide a linear programming criterion allowing to compare the conservativeness of two arbitrary given path complete lyapunov functions