a small gain theorem for switched interconnected nonlinear systems and its applications a small gain theorem for switched interconnected nonlinear systems is set up which provides a tool for analyzing the behavior of switched systems also a method exploiting an average dwell time and the small gain technique is proposed based on the small gain theorem this technical note investigates the problem of global stabilization of switched nonlinear systems in strict feedback form whose all zero dynamics are input to state stable iss or only some are iss and others are not a sufficient condition for the global stabilization problem under a class of switching signals with average dwell time is derived by constructing state feedback controllers for subsystems the result is also applied to a class of switched nonlinear network control systems

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 the least eigenvalue of cacti with pendant vertices abstract a connected graph is a cactus if any two of its cycles share at most one common vertex we determine the unique graph with minimum least eigenvalue among the cacti with fixed numbers of vertices and pendant vertices

the least eigenvalue of graphs whose complements have only two pendent vertices abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of a g are referred to as the eigenvalues of g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices

the least eigenvalue of a graph with a given domination number abstract in this paper we characterize the unique graph whose least eigenvalue achieves the minimum among all graphs with n vertices and domination number thus we can obtain a lower bound on the least eigenvalue of a graph in terms of the domination number

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

an approximation theory for strongly stabilizing solutions to the operator lq riccati equation the linear quadratic lq control problem is considered for a class of infinite dimensional systems with bounded input and output operators that are not exponentially stabilizable but only strongly stabilizable a sufficient condition for the existence of a minimizing control and of a stabilizing solution to the associated lq riccati equation is given the main contribution of this paper is the convergence of the stabilizing solutions of a sequence of finite dimensional riccati equations to the strongly stabilizing solution of the infinite dimensional riccati equation the result is applied to a model of propagation of sound waves in a one dimensional wave guide

the least eigenvalue of a graph with cut vertices in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices during the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound

an extension of lasalle s invariance principle for switched systems in this paper we address invariance principles for a certain class of switched nonlinear systems we provide an extension of lasalle s invariance principle for these systems and state asymptotic stability criteria we also present some related results on the compactness of the trajectories of these switched systems

the least eigenvalue of the complements of trees let tnc be the set of the complements of trees of order n in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in tnc

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

adaptive fuzzy tracking control of switched uncertain nonlinear systems with unstable subsystems this paper is concerned with the problem of adaptive fuzzy tracking control for a class of switched uncertain nonlinear systems where the solvability of the fuzzy tracking control problem for subsystems is unnecessary fuzzy logic systems are utilized to approximate the unknown nonlinear functions and a new adaptive fuzzy tracking control technique for the problem under study is set up based on the technique above we design a switching law and construct adaptive fuzzy controllers of subsystems explicitly by a recursive design algorithm to guarantee that all the signals in the resulting closed loop system remain bounded and the system output of the closed loop system converges to a small neighborhood of the reference signal multiple lyapunov functions are exploited to get out of the conservativeness caused by adoption of a common lyapunov function for all subsystems which is commonly required when applying the backstepping like recursive design scheme a mass spring damper system is provided to demonstrate the effectiveness of the proposed design method

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

graphs with least eigenvalue 2 ten years on abstract the authors monograph spectral generalizations of line graphs was published in 2004 following the successful use of star complements to complete the classification of graphs with least eigenvalue 2 guided by citations of the book we survey progress in this area over the past decade some new observations are included

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

convergence rate of nonlinear switched systems abstract this paper is concerned with the convergence rate of the solutions of nonlinear switched systems we first consider a switched system which is asymptotically stable for a class of switching signals but not for all switching signals we show that solutions corresponding to that class of switching signals converge arbitrarily slowly to the origin then we consider analytic switched systems for which a common weak quadratic lyapunov function exists under two different sets of assumptions we provide explicit exponential convergence rates for switching signals with a fixed dwell time

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

sharp lower bounds of the least eigenvalue of planar graphs let g be a simple graph with n 3 vertices and orientable genus g and non orientable genus h we define the euler characteristic g of a graph g by g max 2 2g 2 h let g be the least eigenvalue of the adjacency matrix a of g in this paper we obtain the following lower bounds of g g 2 n g in particular if g is the planar graph then g 2n 4 the equality holds if and only if g k2 n 2 further we have same result of series parallel graph

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

analyzing the stability of switched systems using common zeroing output systems this paper introduces the notion of common zeroing output systems czos to analyze the stability of switched systems the concept of czos allows one to verify weak zero state detectability it characterizes a common behavior of any individual subsystem when the output signal for each subsystem is approaching zero heuristically speaking it removes the effect of switching behavior and thus enables one to analyze stability properties in systems with complex switching signals with the help of czos the krasovskii lasalle theorem can be extended to switched nonlinear time varying systems with both arbitrary switching and more general restricted switching cases for switched nonlinear time invariant systems the needed detectability condition is further simplified leading to several new stability results particularly when a switched linear time invariant system is considered it is possible to generate a recursive method which combines a krasovskii lasalle result and a nested matrosov result to find a czos if it exists the power of the proposed czos is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions

interval observer design for continuous time switched systems under known switching and unknown inputs this paper deals with unknown input estimation for switched linear systems in an unknown but bounded error ubbe framework based on a known switching signal and under the fulfilment of the relati

lower bounds on the spectra of symmetric matrices with nonnegative entries if we normalize a symmetric n n matrix with nonnegative entriesso that its largest entry is 1 then its spectrum is bounded below by n2 the lower bound is achieved in all even dimensions for and only for adjacency matrices of complete bipartite graphs with equal parts

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

the convergence of general products of matrices and the weak ergodicity of markov chains abstract we determine a sufficient condition for the convergence to 0 of general products formed from a sequence of real or complex matrices our result is applied to obtain a condition for the weak ergodicity of an inhomogeneous markov chain we make some remarks comparing coefficients of ergodicity and we give a method for constructing these

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

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

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

invariance principles for switching systems via hybrid systems techniques invariance principles and sufficient conditions for asymptotic stability for switching systems are given multiple lyapunov like functions are used and dwell time persistent dwell time and weak dwell time switching signals are considered the invariance principles are derived from general invariance principles for hybrid systems asymptotic stability is concluded under observability assumptions or common bounds on the lyapunov like functions

uniform asymptotic stability of switched nonlinear time varying systems and detectability of reduced limiting control systems this paper is concerned with the study of both local and global uniform asymptotic stability for switched nonlinear time varying nltv systems through the detectability of output maps with this aim the notion of reduced limiting control systems for switched nltv systems whose switchings verify time state dependent constraints and the concept of weak zero state detectability for those reduced limiting systems are introduced necessary and sufficient conditions for the global uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property these sufficient conditions in conjunction with the existence of multiple weak lyapunov functions yield a criterion for the global uniform asymptotic stability of families of trajectories of the switched system this criterion can be seen as an extension of the classical krasovskii lasalle theorem an interesting feature of the results is that no dwell time assumptions are made moreover they can be used for establishing the global uniform asymptotic stability of the switched nltv system under arbitrary switchings the effectiveness of the proposed results is illustrated by means of various interesting examples including the stability analysis of a semiquasi z source inverter

the least eigenvalue of unicyclic graphs with n vertices and k pendant vertices abstract let u n k be the set of unicyclic graphs with n vertices and k pendant vertices in this paper we determine the unique graph with the minimal least eigenvalue among all graphs in u n k the work is related with that of guo s g guo the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices linear algebra appl 408 2005 78 85 which determined the unicyclic graph with the maximal spectral radius in u n k we can observe that the extremal graph on the least eigenvalue is different from that on the spectral radius

the least eigenvalue of graphs with given connectivity abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of g are those of a g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity

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

minimizing the least eigenvalues of unicyclic graphs with application to spectral spread the spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph let u k denote the set of connected unicyclic graphs of order n and girth k and let un denote the set of connected unicyclic graphs of order n in this paper we determine the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in u k we finally characterize the unique graph with minimum least eigenvalue respectively the unique graph with maximum spread among all graphs in un

an invariance principle for nonlinear hybrid and impulsive dynamical systems in this paper we develop an invariance principle for dynamical systems possessing left continuous flows specifically we show that left continuity of the system trajectories in time for each fixed state point and continuity of the system trajectory in the state for every time in some dense subset of the semi infinite interval are sufficient for establishing an invariance principle for hybrid and impulsive dynamical systems as a special case of this result we state and prove new invariant set stability theorems for a class of nonlinear impulsive dynamical systems namely state dependent impulsive dynamical systems these results provide less conservative stability conditions for impulsive systems as compared to classical results in the literature and allow us to address the stability of limit cycles and periodic orbits of impulsive systems

generalizations of the projection methods with application to sor theory for hermitian positive semidefinite linear systems ann n complex matrixb is calledparacontracting if b 2 1 and 0 x n i b bx 2 x 2 we show that a productb b k b k 1 b1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofb in terms of the subdominant singular values of theb i s and in terms of the angles between certain subspaces our results here extend earlier results due to halperin and due to smith solomon and wagner we also determine necessary and sufficient conditions forn numbers in the interval 0 1 to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product in the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the sor iteration matrix associated with ann n hermitian positive semidefinite matrixa none of whose diagonal entries vanish

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

system theoretic properties of a class of spatially invariant systems in this paper we develop new readily testable criteria for system theoretic properties such as stability controllability observability stabilizability and detectability for a class of spatially invariant systems our approach uses the well established theory developed to solve infinite dimensional systems the theoretical results are illustrated by several examples

bicyclic graphs for which the least eigenvalue is minimum abstract the spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph in this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n also we determine the unique graph with maximum spread in this class for each n 28

optimal linear quadratic control of asymptotically stabilizable systems using approximations in this paper we study approximations to the infinite horizon quadratic optimal control problem for linear systems that may be only asymptotically stabilizable for linear systems this issue only arises with infinite dimensional systems we provide sufficient conditions which guarantee when approximations to the optimal feedback result in the cost converging to the optimal cost one technique for approximate solution of the optimal control problem is to use newton kleinman iterations for the associated riccati equation some new results in this direction are provided several important classes of systems lightly damped second order systems and a platoon type system are shown to be optimizable also finding an initial stabilizing control for the newton kleinman iteration can be non trivial the initial iterate for these classes is described

graphs for which the least eigenvalue is minimal ii let g be a connected graph whose least eigenvalue g is minimal among the connected graphs of prescribed order and size we show first that either g is complete or g is a simple eigenvalue in the latter case the sign pattern of a corresponding eigenvector determines a partition of the vertex set and we study the structure of g in terms of this partition we find that g is either bipartite or the join of two graphs of a simple form

the vertex edge independence number vertex edge cover number and the least eigenvalue of a graph in this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex edge independence number or vertex edge cover number and get some bounds for the vertex edge independence number vertex edge cover number of a graph in terms of the least eigenvalue of the graph

a kleinman newton construction of the maximal solution of the infinite dimensional control riccati equation abstract assuming only strong stabilizability we construct the maximal solution of the algebraic riccati equation as the strong limit of a kleinman newton sequence of bounded nonnegative operators as a corollary we obtain a comparison of the solutions of two algebraic riccati equations associated with different cost functions we show that the weaker strong stabilizability assumptions are satisfied by partial differential systems with collocated actuators and sensors so the results have potential applications to numerical approximations of such systems by means of a counterexample we illustrate that even if one assumes exponential stabilizability the kleinman newton construction may provide a solution to the riccati equation that is not strongly stabilizing

an approximation theorem for the algebraic riccati equation for an infinite dimensional linear quadratic control problem in hilbert space approximation of the solution of the algebraic riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems as an application strong onvergence of the approximating riccati operators in case of a previously developed spline approximation scheme for delay systems is established finally convergence of the transfer functions of the approximating systems is investigated

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

an invariance principle for nonlinear switched systems abstract in this paper we address the problem of extending lasalle invariance principle to switched system we prove an extension of the invariance principle relative to dwell time switched solutions and a second one relative to constrained switched systems

switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems this paper investigates the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary a new switched adaptive control technique for the problem studied is set up by exploiting the generalized multiple lyapunov functions method and the parameter separation technique different update laws are designed to reduce the conservativeness caused by adoption of a common update law for all subsystems also the proposed technique permits removal of a common restriction in which the parameterization in the switched systems is restricted to a linear parameterization as an application of the proposed design technique stabilization for a class of non triangular switched systems with nonlinear parameterization is achieved by design of adaptive controllers since no subsystem of such a switched system is assumed to be adaptively stabilizable the problem under study cannot be handled by the existing methods a two inverted pendulums as a practical example is also provided to demonstrate the effectiveness of the proposed design method

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples