stabilization of a class of stochastic differential equations with markovian switching stability of stochastic differential equations with markovian switching has been studied quite extensively for a number of years for example by basak et al j math anal appl 202 1996 604 622 ji and chizeck ieee trans automat control 35 1990 777 788 mariton jump linear systems in automatic control marcel dekker new york 1990 mao et al stochastic process appl 79 1999 45 67 bernoulli 6 2000 73 90 and yuan and lygeros in r alur g pappas eds hybrid systems computation and control seventh international workshop hscc 2004 lecture notes in computer science vol 2993 springer berlin 2004 pp 646 659 by contrast the problem of designing controllers to stabilize systems of this type has received relatively little attention in this paper we study the problem of mean square exponential stabilization for a class of stochastic differential equations with markovian switching

on stabilization of bilinear uncertain time delay stochastic systems with markovian jumping parameters in this paper we investigate the stochastic stabilization problem for a class of bilinear continuous time delay uncertain systems with markovian jumping parameters specifically the stochastic bilinear jump system under study involves unknown state time delay parameter uncertainties and unknown nonlinear deterministic disturbances the jumping parameters considered here form a continuous time discrete state homogeneous markov process the whole system may be regarded as a stochastic bilinear hybrid system that includes both time evolving and event driven mechanisms our attention is focused on the design of a robust state feedback controller such that for all admissible uncertainties as well as nonlinear disturbances the closed loop system is stochastically exponentially stable in the mean square independent of the time delay sufficient conditions are established to guarantee the existence of desired robust controllers which are given in terms of the solutions to a set of either linear matrix inequalities lmis or coupled quadratic matrix inequalities the developed theory is illustrated by numerical simulation

stabilizability of a class of stochastic bilinear hybrid systems abstract the problem of the stabilizability of stochastic nonlinear hybrid systems with a markovian or any switching rule is considered using the lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found in particular the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a markovian or any switching rule is discussed and a closed loop controller is found also the sufficient conditions for the exponential mean square stabilizability for bilinear hybrid systems with any switching based on the lie algebra approach are formulated and an open loop controller is designed the obtained results are illustrated by examples and simulations

on a common quadratic lyapunov function for widely distant systems a sufficient condition concerning the construction of a common quadratic lyapunov function for widely distant linear systems is presented the condition is set on a mutual relation matrix whose size is just equal to the number of the coefficient matrices

geometry of the limit sets of linear switched systems this paper is concerned with asymptotic stability properties of linear switched systems under the hypothesis that all the subsystems share a nonstrict quadratic lyapunov function we provide a large class of switching signals for which a large class of switched systems are asymptotically stable for this purpose we define what we call nonchaotic inputs which generalize the different notions of inputs with dwell time next we turn our attention to the behavior for possibly chaotic inputs finally we give a sufficient condition for a system composed of a pair of hurwitz matrices to be asymptotically stable for all inputs

on decomposition of bott samelson sheaves abstract we give an exact algorithm to calculate under some gkm restriction the matrix describing the embedding b s x b s x where the first module is the costalk and the second one is the stalk at x of a bott samelson module sheaf b s this allows us to calculate the first few terms of the decomposition of b s into a sum of indecomposable modules sheaves and to calculate the characters of braden macpherson sheaves in some previously unknown cases

local hodge theory of soergel bimodules we prove the local hard lefschetz theorem and local hodge riemann bilinear relations for soergel bimodules using results of soergel and kubel one may deduce an algebraic proof of the jantzen conjectures we observe that the jantzen filtration may depend on the choice of non dominant regular deformation direction

lie algebraic stability conditions for nonlinear switched systems and differential inclusions abstract we present a stability criterion for switched nonlinear systems which involves lie brackets of the individual vector fields but does not require that these vector fields commute a special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third order lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching this generalizes a known fact for switched linear systems and provides a partial solution to the 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 university press nj 2004 pp 203 207 to prove the result we consider an optimal control problem which consists in finding the most unstable trajectory for an associated control system and show that there exists an optimal solution which is bang bang with a bound on the total number of switches this property is obtained as a special case of a reachability result by bang bang controls which is of independent interest by construction our criterion also automatically applies to the corresponding relaxed differential inclusion

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

robust matrix commutator conditions for stability of switched linear systems under restricted switching this article treats global uniform exponential stability gues of discrete time switched linear systems under restricted switching given admissible minimum and maximum dwell times we provide sufficient conditions on the subsystems under which they admit a set of switching signals that obeys the given restrictions on dwell times and preserves stability of the resulting switched system our analysis relies on combinatorial arguments applied to matrix commutators the proposed set of stabilizing switching signals is characterized in terms of duration of activation of schur stable subsystems and non consecutive activation of distinct unstable subsystems

on the m dimensional cayley hamilton theorem and its application to an algebraic decision problem inferred from the h2 norm analysis of delay systems abstract we consider a recursion formula for multi dimensional powers of a finite set of matrices which can be interpreted as a natural generalization of the celebrated cayley hamilton theorem and we show how it allows to solve an algebraic decision problem on a semigroup of matrices which bears similarities to the observability problem of a switched linear system this problem appears in the computation of the h 2 norm of a stable system described by a class of linear time invariant delay differential equations ddaes with multiple delays the h 2 norm of a ddae may not be finite even if there are seemingly no direct feedthrough terms we show that necessary and sufficient conditions for a finite h 2 norm consist of an infinite number of linear equations to be satisfied inducing the algebraic decision problem and that using the generalized cayley hamilton theorem checking these conditions can be turned into a check of a finite number of equations we conclude with some comments on the computation of the h 2 norm whenever it is finite and by stating an open problem

on the stabilization of bilinear systems via constant feedback we study the problem of stabilization of a bilinear system via a constant feedback the question reduces to an eigenvalue problem on the pencil a 0b of two matrices using the idea of simultaneous triangularization of the matrices involved some easily checkable conditions for the solvability of this question are obtained algorithms for checking these conditions are given and illustrated by a few examples

natural boundaries and spectral theory we present and exploit an analogy between lack of absolutely continuous spectrum for schrodinger operators and nat ural boundaries for power series among our new results are gen eralizations of hecke s example and natural boundary examples for random power series where independence is not assumed

a stochastic jurdjevic quinn theorem the purpose of this paper is to state sufficient conditions for the stabilizability of stochastic differential systems when both the drift and diffusion terms are affine in the control this result extends to stochastic differential systems the well known theorem of jurdjevic quinn j differential equations 28 1978 pp 381 389 and incorporates earlier stabilization results proved in p florchinger stochastic anal appl 12 1994 pp 473 480 and r chabour and m oumoun stochastic anal appl 16 1998 pp 43 50

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

the steinberg quotient of a tilting character let g be a simple algebraic group over an algebraically closed field of prime characteristic if m is a finite dimensional g module that is projective over the frobenius kernel of g then its character is divisible by the character of the steinberg module in this paper we study such quotients showing that if m is an indecomposable tilting module then the multiplicities of the orbit sums appearing in its steinberg quotient are well behaved

stabilization of switched systems with nonlinear impulse effects and disturbances in this note we investigate the stabilization of switched systems with nonlinear impulse effects and disturbances based on the estimate on transition matrices and gronwall inequality feedback laws are designed to achieve the exponential stability of the closed loop system with arbitrary switching frequency

stabilization of switched linear systems in this note we study the stabilization problem of systems that switch among a finite set of controllable linear systems with arbitrary switching frequency for both cases of known and unknown switching functions feedback laws are designed to achieve exponential stability for the later case a method combining on line adaptive estimation and feedback stabilization is developed in the controller design

sufficient conditions for generic feedback stabilizability of switching systems via lie algebraic solvability we address the stabilization of switching linear systems slss with control inputs under arbitrary switching a sufficient condition for the stability of autonomous without control inputs slss is that the individual subsystems are stable and the lie algebra generated by their evolution matrices is solvable this sufficient condition for stability is known to be extremely restrictive and therefore of very limited applicability our main contribution is to show that in contrast to the autonomous case when control inputs are present the existence of feedback matrices for each subsystem so that the corresponding closed loop matrices satisfy the aforementioned lie algebraic stability condition can become a generic property hence substantially improving the applicability of such lie algebraic techniques in some cases since the validity of this lie algebraic stability condition implies the existence of a common quadratic lyapunov function cqlf for the sls our results yield an analytic sufficient condition for the generic existence of a control cqlf for the sls

exponential stabilization of switched time varying systems with delays and disturbances abstract this paper deals with exponential stabilization for a class of switched time varying systems by taking time varying delays and nonlinear disturbances into consideration time dependent switching signals have been characterized in terms of metzler matrices such that the resulting system is globally exponentially stable compared with preceding works we introduce a model transformation and an approach without involving the lyapunov krasovskii functional to derive new exponential stability criteria for switched time varying systems under the average dwell time switching numerical examples show that the obtained theoretical results can be applied to some cases not covered by some existing results

a sufficient condition for the stability of discrete time systems with state dependent coefficient matrices there is no systematic method of checking the stability of general nonlinear systems in this study we focus on discrete time autonomous systems with state dependent coefficient matrices sufficient conditions for local and global asymptotic stabilities are given for upper triangularizable systems the transformation matrix for triangularization is not required explicitly for checking such conditions moreover the system may have nondifferentiability at the origin which is an advantage of our approach over lyapunov s indirect method

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

finite sections of random jacobi operators this article is about a problem in the numerical analysis of random operators we study a version of the finite section method for the approximate solution of equations ax b in infinitely many variables where a is a random jacobi i e tridiagonal operator in other words we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the large finite volume case via a particular truncation technique for most of the paper we consider non self adjoint operators a but we also comment on the self adjoint case when simplifications occur

on the 45 region and the uniform asymptotic stability of classes of second order parameter varying and switched systems in this paper we present sufficient conditions for the stability of a number of classes of time varying systems these results are of interest for two reasons firstly the conditions presented take the form of matrix pencil eigenvalue criteria and are therefore co ordinate independent and easily verifiable secondly we show a direct relationship between the form of our criteria and the choice of lyapunov function used to demonstrate stability quadratic or unic these results indicate the importance of the region of the complex plane known as the 45 region

on robust lie algebraic stability conditions for switched linear systems abstract this paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching which involve the commutators lie brackets among the given matrices generating the switched system the main novel feature of these stability criteria is that unlike their earlier counterparts they are robust with respect to small perturbations of the system parameters two distinct approaches are investigated for discrete time switched linear systems we formulate a stability condition in terms of an explicit upper bound on the norms of the lie brackets for continuous time switched linear systems we develop two stability criteria which capture proximity of the associated matrix lie algebra to a solvable or a solvable plus compact lie algebra respectively

sufficiency of favard s condition for a class of band dominated operators on the axis the purpose of this paper is to show that for a large class of band dominated operators on 1 z u with u being a complex banach space the injectivity of all limit operators of a already implies their invertibility and the uniform boundedness of their inverses the latter property is known to be equivalent to the invertibility at inflnity of a which on the other hand is often equivalent to the fredholmness of a as a consequence for operators a in the wiener algebra we can characterize the essential spectrum of a on p z u regardless of p 2 1 1 as the union of point spectra of its limit operators considered as acting on 1 z u

bounding ext for modules for algebraic groups finite groups and quantum groups given a finite root system we show that there is an integer c c such that dimextg1 l l c for any reductive algebraic group g with root system and any irreducible rational g modules l l there also is such a bound in the case of finite groups of lie type depending only on the root system and not on the underlying field for quantum groups a similar result holds for extn for any integer n 0 using a constant depending only on n and the root system when l is the trivial module the same result is proved in the algebraic group case thus giving similar bounded properties independent of characteristic for algebraic and generic cohomology a similar result holds for any choice of l l even allowing to vary provided the p adic expansion of lambda is limited to a fixed number of terms in particular because of the interpretation of generic cohomology as a limit for underlying families of finite groups the same boundedness properties hold asymptotically for finite groups of lie type the results both use and have consequences for kazhdan lusztig polynomials appendix a proves a stable version needed for small prime arguments of donkin s tilting module conjecture

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

technical communique on stability of a class of switched nonlinear systems this note is focused on the stability analysis for a class of switched nonlinear systems with disturbance input and delay sufficient conditions in terms of linear inequalities are presented such that the switched system is asymptotically stable for arbitrary switching any admissible sector nonlinearities and disturbances and any constant delay we not only drop a condition of one result given in aleksandrov chen platonov and zhang 2011 but also extend the main result to a more general switched nonlinear system with disturbance input and delay

improved overshoot estimation in pole placements and its application in observer based stabilization for switched systems in this note we first establish an improved and generalized overshoot estimation in pole placements then this estimation is applied to the observer based output feedback stabilization problem for switched linear systems with arbitrary switching frequency our results complement those results of cheng about the state feedback stabilization of switched linear systems

an affirmative answer to a core issue on limit operators abstract an operator on an l p space is called band dominated if it can be approximated in the operator norm by operators with a banded matrix representation it is known that a rich band dominated operator is p fredholm which is a generalization of the classical fredholm property if and only if all of its so called limit operators are invertible and their inverses are uniformly bounded we show that the condition on uniform boundedness is redundant in this statement

convergence domain for time varying switched systems abstract in this paper we investigate the convergence domain for time varying switched systems we construct multiple lyapunov functions and each lyapunov function is decreasing outside a ball whose radius is time varying we give a relatively accurate convergence domain for a general time varying switched system finally a numerical example a mass spring damper system is provided to show the effectiveness of the theoretical results

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

singular value conditions for stability of dynamic switched systems abstract in this paper the stability of a certain class of time varying systems evolving over nonuniformly spaced discrete domains is analyzed switched systems used here in the context of dynamic equations over time scale domains arise naturally in applications when a continuous time system is discretized via a sample and hold method with multiple sample rates the stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems such as pairwise commutativity 24 simultaneous diagonalization 7 simultaneous triangularizability 8 or other lie algebraic conditions 1 which imply the existence of a common quadratic lyapunov function a novel approach is used here to determine the existence of quadratic lyapunov functions which does not rely on how the subsystems interact with each other this new method instead examines the role they each play in the aggregate system by way of singular value conditions results implying switched system stability and instability are developed under the two primary methods of examining such systems how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal several examples illuminating the results are provided throughout the text as well as important insight discussing certain bounds on these results as they apply to stability theory in general

on the spectrum of operator families on discrete groups over minimal dynamical systems it is well known that given an equivariant and continuous in a suitable sense family of selfadjoint operators in a hilbert space over a minimal dynamical system the spectrum of all operators from that family coincides as shown recently similar results also hold for suitable families of non selfadjoint operators in ell p mathbb z here we generalize this to a large class of bounded linear operator families on banach space valued ell p spaces over countable discrete groups we also provide equality of the pseudospectra for operators in such a family a main tool for our analysis are techniques from limit operator theory

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

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

stability of abstract linear switched impulsive differential equations linear impulsive switched differential equations in hilbert space are studied we propose a new method of stability investigation which is based in constructing an equivalent impulsive system without switching using operator valued and scalar lyapunov functions sufficient conditions for lyapunov stability of linear switched impulsive equations in a banach space are established the examples of studies of this class of equations with periodic switching are given

robust uniformly ultimate boundedness control for uncertain switched linear systems this paper presents uniformly ultimate boundedness uub control design for switched linear systems with parametric uncertainties only the possible bound of the uncertainty is needed under arbitrary switching laws a continuous state feedback control scheme is proposed in order to guarantee uniformly ultimate boundedness of every system response within an arbitrary small neighborhood of the zero state the design techniques are based on common lyapunov functions and lyapunov minimax approach

stability and stabilization of a class of switched nonlinear systems with time varying delay abstract this paper investigates the problems of exponential stability and robust stabilization for a class of switched nonlinear systems with time varying delay using the average dwell time method a new exponential stability criterion is derived in terms of linear matrix inequalities lmis it is shown that the obtained criteria can provide less conservative results than some existing ones then we propose a state feedback control law which ensures the exponential stability of the closed loop system finally numerical examples are given to illustrate the effectiveness of the proposed results

simultaneous triangularization of switching linear systems arbitrary eigenvalue assignment and genericity a sufficient condition for the stability of arbitrary switching linear systems slss without control inputs is that the individual subsystems are stable and their evolution matrices are simultaneously triangularizable st this sufficient condition for stability is known to be extremely restrictive and not robust and therefore of very limited applicability the situation can be radically different when control inputs are present indeed previous results have established that depending on the number of states inputs and subsystems the existence of feedback matrices for each subsystem so that the corresponding closed loop matrices are stable and st can become a generic property i e a property valid for almost every set of system parameters this note provides novel contributions along two lines first we give sufficient conditions for the genericity of the property of existence of feedback matrices so that the subsystem closed loop matrices are st not necessarily stable second we give conditions for the genericity of the property of existence of feedback matrices that in addition to achieving st enable arbitrary eigenvalue selection for each subsystem s closed loop matrix the latter conditions are less stringent than existing ones and the approach employed in their derivation can be interpreted as an extension to slss of specific aspects of the notion of eigenvalue controllability for non switching linear systems

koszul duality for kac moody groups and characters of tilting modules we establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p kazhdan lusztig polynomials for p h the coxeter number using results of andersen one may deduce a character formula for simple modules if p 2h 2 our results are a consequence of an extension to modular coefficients of a monoidal koszul duality equivalence established by bezrukavnikov and yun

quadratic stability and stabilization of bimodal piecewise linear systems abstract this paper deals with quadratic stability and feedback stabilization problems for continuous bimodal piecewise linear systems first we provide necessary and sufficient conditions in terms of linear matrix inequalities for quadratic stability and stabilization of this class of systems later these conditions are investigated from a geometric control point of view and a set of sufficient conditions in terms of the zero dynamics of one of the two linear subsystems for feedback stabilization are obtained

a jurdjevic quinn type theorem for stochastic nonlinear control systems we consider stochastic nonaffine nonlinear control systems x sub t x sub 0 spl int sub 0 sup t f x sub s u ds spl int sub 0 sup t g x sub s u d spl omega sub s written in the sense of ito spl omega being a standard wiener process for which we give a sufficient condition for global stabilization by a bounded smooth state feedback that is explicitly given this condition generalizes the well known jurdjevic quinn result for deterministic affine control systems

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

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

stability of switched systems with partial state reset in this note we consider switched systems and switched systems with state reset in particular we focus on the case of partial reset i e where only some state components may undergo the action of a reset first we consider switched systems with pre specified partial reset and investigate under which conditions such systems are stable in a second stage we consider the problem of stabilization by partial reset which consists in finding a suitable partial reset for a given switched system that makes this system stable under arbitrary switching

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

feedback stabilization for multiinput switched nonlinear systems two subsystems case in this note based on the control lyapunov function approach a new method for solving the feedback stabilization problem of switched systems that are switching arbitrarily between two multiinput nonlinear subsystems is introduced necessary and sufficient conditions for the existence of globally uniformly asymptotically stabilizing state feedback controllers are derived additionally a universal formula for constructing implementable stabilizing controllers is given when the presented conditions hold one numerical example is given to illustrate the results

controllability of switched bilinear systems the controllability of switched bilinear systems sblss is considered three kinds of controllabilities including weak controllability approximate controllability and global controllability are investigated one by one sets of easily verifiable sufficient conditions are obtained for each case which are applicable to a large class of switched bilinear systems

on exponential almost sure stability of random jump systems this paper is concerned with a class of random jump systems represented by transition operators which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case a series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios the stability criteria obtained are further extended to markov jump linear systems with infinite states and hence a unified approach to describing the almost sure stability of mjlss is addressed under this context all the results in the work are developed for both the continuous and discrete time systems

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

fredholm theory for band dominated and related operators a survey abstract this paper presents the fredholm theory on l p spaces for band dominated operators and important subclasses such as operators in the wiener algebra it particularly closes several gaps in the previously known results for the case p and addresses the open questions raised by chandler wilde and lindner in s n chandler wilde m lindner limit operators collective compactness and the spectral theory of infinite matrices mem amer math soc 210 989 2011 the main tools are provided by the limit operator method and an algebraic framework for the description and adaption of fredholmness and convergence a comprehensive overview of this approach is given

light leaves and lusztig s conjecture we define a map f with domain a certain subset of the set of light leaves certain morphisms between soergel bimodules introduced by the author in an earlier paper and range the set of prime numbers using results of soergel we prove the following property of f if the image p f l of some light leaf l under f is bigger than the coxeter number of the corresponding weyl group then there is a counterexample to lusztig s conjecture in characteristic p we also introduce the double leaves basis which is an improvement of the light leaves basis that has already found interesting applications in particular it forms a cellular basis of soergel bimodules that allows us to produce an algorithm to find the bad primes for lusztig s conjecture

brief paper componentwise ultimate bound and invariant set computation for switched linear systems we present a novel ultimate bound and invariant set computation method for continuous time switched linear systems with disturbances and arbitrary switching the proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties the method provides ultimate bounds and invariant sets in the form of polyhedral and or mixed ellipsoidal polyhedral sets is completely systematic once the aforementioned transformation is obtained and provides a new sufficient condition for practical stability we show that the transformation required by our method can easily be found in the well known case where the subsystem matrices generate a solvable lie algebra and we provide an algorithm to seek such transformation in the general case an example comparing the bounds obtained by the proposed method with those obtained from a common quadratic lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases

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

controllability analysis and control design for switched boolean networks with state and input constraints this paper investigates the controllability analysis and the control design for switched boolean networks sbns with state and input constraints by using the semi tensor product method and presents a number of new results on their controllability optimal control and stabilization first the constrained incidence matrix is proposed for sbns based on which several necessary and sufficient conditions are obtained for the controllability second using the results on the controllability two algorithms are established to design proper switching sequence and controls that minimize the cost functional in a fixed designed shortest termination time third the constrained sbn is converted to an equivalent unconstrained one based on which some necessary and sufficient conditions are presented for the stabilization of the constrained sbn with open loop and closed loop controls respectively finally a practical example of the apoptosis network is studied by using the new results obtained in this paper the s

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

periodical stabilization of switched linear systems periodical stabilization problems for switched linear systems are investigated in this paper for autonomous switched systems if there exists a stable convex combination of the subsystems then a periodically switching signal can be constructed such that the overall system is asymptotically stable based on this fact for switched control systems corresponding sufficient conditions are presented under which constant switching direct observer based state feedback controller can be designed such that the corresponding closed loop systems are asymptotically stable under some periodically switching signal some numerical examples are given to illustrate our results

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

some results on quadratic stability of switched systems with interval uncertainty in this paper we present some results on the quadratic stability of switched systems with uncertain parameters we show that the quadratic stability of a class of switched uncertain systems may be readily verified using simple algebraic conditions

stabilization and destabilization of hybrid systems of stochastic differential equations this paper aims to determine whether or not a stochastic feedback control can stabilize or destabilize a given nonlinear hybrid system new methods are developed and sufficient conditions on the stability and instability for hybrid stochastic differential equations are provided these results are then used to examine stochastic stabilization and destabilization

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

stability and controllability of planar conewise linear systems abstract this paper presents a fairly complete treatment of stability and controllability of piecewise linear systems defined on a conic partition of r 2 this includes necessary and sufficient conditions for stability and controllability as well as establishing that controllability implies stabilizability by piecewise linear state feedback a key tool in the approach is the study of the poincare map

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

kazhdan lusztig combinatorics in the moment graph setting abstract motivated by a question on the graded rank of the stalks of the canonical sheaf on a bruhat graph we lift some equalities concerning parabolic kazhdan lusztig polynomials to this moment graph setting our proofs hold also in positive characteristic under some technical assumptions