unimodular congruence of the laplacian matrix of a graph abstract let g be a graph with vertices 1 2 n associated with g there is an integral quadratic form q x on the n tuple of indeterminates x x 1 x n given by q x x i x j 2 where the sum is taken over all edges i j of g in this paper we prove that the quadratic forms q 1 q 2 associated with graphs g 1 g 2 are congruent by a unimodular matrix if and only if g 1 and g 2 are cycle isomorphic

on common diagonal lyapunov solutions abstract several recent results regarding common diagonal lyapunov solutions are further explored here the first one attributed to redheffer and revisited by shorten and narendra reduces the diagonal stability of a matrix to common diagonal lyapunov solutions on two matrices of order one less we present a shorter purely matrix theoretic proof of this result along with its extensions the second one concerns two different necessary and sufficient conditions due to oleng narendra and shorten for a pair of 2 2 matrices to share a common diagonal lyapunov solution we show that these two conditions are connected directly to each other

some methods of construction of a common lyapunov solution to a finite set of complex systems abstract in this work two methods of determining a common lyapunov solution for a finite number of complex matrices are proposed the first one is an extension to the complex case of buyukkoroglu s result dedicated to real matrices of order three whereas the second one extending and completing a very recent paper of gumus and xu can be applied to a finite set of complex matrices of arbitrary order as special cases some known results as well as new ones concerning the common lyapunov solution problem for complex triangular systems are derived numerical examples are presented to illustrate and to compare the results

on the existence of a common quadratic lyapunov function for a rank one difference abstract suppose that a and b are real hurwitz matrices and that their difference a b is rank one then a and b have a common quadratic lyapunov function if and only if the product ab has no real negative eigenvalue this result is due to shorten and narendra who showed that it follows as a consequence of the kalman yacubovich popov lemma and the solution of the lur e problem here we present a new and independent proof based on results from convex analysis and the theory of moments

duality and the signed laplacian matrix of a graph abstract we give a necessary and sufficient condition for a bijection between the edge sets of two graphs to be a dual bijection the condition involves unimodular congruence of augmented signed laplacian matrices for the two graphs

on discrete time diagonal and d stability abstract this paper introduces four discrete time analogs of different types of matrix stability diagonal simultaneous vertex and d stability the last three being defined in terms of a certain associated polytope of matrices the diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence d stability of the vertices it is shown by a counterexample that d stability as in the continuous time case is not equivalent to diagonal stability several important classes for which this equivalence is true are identified it is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope and it is conjectured that this equivalence holds without the requirement of simultaneity some other conjectures relating the four types of stability are made and it is shown that in the 2 x 2 case all four are equivalent

stabilization of positive switched linear systems and its application in consensus of multiagent systems in this note we study the stabilization problem of positive switched linear system psls with disturbance by introducing a class of weak common linear copositive lyapunov functions we design time dependent switching rules under which the system is asymptotically stable both continuous time and discrete time psls are taken into consideration and the obtained results are also extended to nonpositive switched systems finally the theoretical results are applied to consensus problem of multiagent systems with switching topologies and disturbance

co positive lyapunov functions for the stabilization of positive switched systems in this paper exponential stabilizability of continuous time positive switched systems is investigated for two dimensional systems exponential stabilizability by means of a switching control law can be achieved if and only if there exists a hurwitz convex combination of the metzler system matrices in the higher dimensional case it is shown by means of an example that the existence of a hurwitz convex combination is only sufficient for exponential stabilizability and that such a combination can be found if and only if there exists a smooth positively homogeneous and co positive control lyapunov function for the system in the general case exponential stabilizability ensures the existence of a concave positively homogeneous and co positive control lyapunov function but this is not always smooth the results obtained in the first part of the paper are exploited to characterize exponential stabilizability of positive switched systems with delays and to provide a description of all the switched equilibrium points of an affine positive switched system

common diagonal lyapunov function for third order linear switched system consider the stability problem for the following linear switched system differential inclusion x a x a a 1 a 2 a n here a i i 1 2 n are n i n dimensional hurwitz stable real matrices in this study for this system we investigate the problem of the existence and construction of a common diagonal lyapunov function of the form v x x t d x where d is a positive diagonal matrix in the case of n 3 i e third order system we suggest a simple elimination algorithm which gives a common d in the case of existence highlights we consider n dimensional linear switched systems construction of a common diagonal lyapunov function for the systems is considered a simple elimination algorithm for third order system is given for the general case we apply kelly s cutting plane method to the given problem

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

a general stability test for switched positive systems based on a multidimensional system analysis the main goal of this technical note is to present a new sufficient condition for the stability of switched positive linear systems this condition is general in two senses first of all because it applies to switching systems of order composed of subsystems for any and secondly because it applies to both discrete and continuous time systems furthermore it is easy to verify as it amounts to check for the stability of a certain matrix the new condition is derived from a multidimensional system analysis where the relation between the stability of multidimensional and switched systems is investigated a comparison with other existing tests is also presented showing that the one proposed here allows to infer about the stability of a system in cases where other tests fail or do not apply at all

stability of acyclic matrices abstract necessary and sufficient conditions for stability of acyclic matrices whose principal minors are nonnegative are given also a characterization is given of additive d stable acyclic matrices

on the simultaneous diagonal stability of a pair of positive linear systems in this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic lyapunov function cqlf for a pair of positive linear time invariant lti systems

the α scalar diagonal stability of block matrices abstract we generalize two results kraaijevanger s 1991 characterization of diagonal stability via hadamard products and the block matrix version of the closure of the positive definite matrices under hadamard multiplication we restate our generalizations in terms of p matrices and scalar diagonally stable matrices

sufficient conditions for schur and hurwitz diagonal stability of complex interval matrices abstract let c a j b be a complex interval matrix where a b r n n are real interval matrices and j 2 1 denote by d the set of positive definite diagonal n n matrices the schur respectively hurwitz diagonal stability of c abbreviated as sds respectively hds is defined in terms of the stein respectively lyapunov inequality associated with a generic matrix c c the inequality must have a common solution q d for all c c the paper develops two types of results for the sds respectively hds analysis which exploit different properties of c each type of results presents i a sufficient condition relying on the solvability of a matrix norm inequality sds analysis or a matrix measure inequality hds analysis ii a construction procedure for a concrete matrix that satisfies the inequality at i and concomitantly yields a common solution q d for some particular cases both necessary and sufficient conditions can be derived the schur and hurwitz approaches are developed in parallel for concision reasons as well as for emphasizing the similarities numerical examples illustrate the applicability of the theoretical results

on a theorem of redheffer concerning diagonal stability abstract an important problem in system theory concerns determining whether or not a given lti system x ax is diagonally stable more precisely this problem is concerned with determining conditions on a matrix a such that there exists a diagonal matrix d with positive diagonal entries i e a positive diagonal matrix satisfying a t d da q 0 while this problem has attracted much attention over the past half century two results of note stand out i a result based on theorems of the alternative derived by barker berman and plemmons and ii algebraic conditions derived by redheffer this paper is concerned with the second of these conditions our principal contribution is to show that redheffer s result can be obtained from the kalman yacubovich popov lemma we then show that this method of proof leads to natural generalisations of redheffer s result and we use these results to derive new conditions for diagonal and hurwitz stability for special classes of matrices

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

stronger than lyapunov notions of matrix stability or how flowers help solve problems in mathematical ecology abstract persistent patterns of interactions in a multi component system e g intra and inter species relations in a community of n interacting species may imply a number of formalizations as special stronger than lyapunov notions of matrix stability like d stability qualitative stability volterra lyapunov stability and others a variety of these notions each having a certain motivation with regard to uncertainties inherent in model applications constitute a hierarchical topology sometimes very intricate and not yet well understood in a formal space of real n n matrices as visible forms of this hierarchy matrix flowers are suggested where petals correspond to subsets of particular stability kinds whose visible inclusion intersection represent logical implication junction the flowers are constructed under a few simple conventions and in the absence of ready characterizations to draw a petal often poses a challenging mathematical problem whose solution may reveal a new biological knowledge of a general nature particular petals concern the topics of strong and weak interactions in a community key species in its structure diffusion instability in its spatial dynamics where the flower plays a heuristic role in formulating a new problem or and stimulating a new application

strong stability of discrete time systems the paper introduces a new notion of stability for internal state space autonomous system descriptions in discrete time referred to as strong stability which extends a parallel notion introduced in the continuous time case this is a stronger notion of stability compared to alternative definitions asymptotic lyapunov which prohibits systems described by natural coordinates to have overshooting responses for arbitrary initial conditions in state space three finer notions of strong stability are introduced and necessary and sufficient conditions are established for each one of them the class of discrete time systems for which strong and asymptotic stability coincide is characterized and links between the skewness of the eigen frame and the violation of strong stability property are obtained connections between the notions of strong stability in the continuous and discrete domains are briefly discussed finally strong stabilization problems under state and output feedback are studied the results of the paper are illustrated with a numerical example

extension of diagonal stability and stabilization for continuous time fractional positive linear systems abstract this paper considers an extension of diagonal stability for continuous fractional positive linear systems fpls with the fractional order 0 1 based on diagonal stability of metzler matrix an extension that involves the combination of a diagonal positive definite matrix and a skew symmetric anti diagonal matrix related a hurwitz and metzler matrix is established combining this extension and the linear matrix inequalities lmis criteria of stability for fractional order systems fos a necessary and sufficient condition for extension diagonal stability of fpls is presented a state feedback controller is given which ensures the stabilization and positivity of the closed loop systems numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods

on quadratic lyapunov functions a topological structure as a subset of 0 2 spl pi sup l spl times spl ropf sub sup n 1 is proposed for the set of quadratic lyapunov functions qlfs of a given stable linear system a necessary and sufficient condition for the existence of a common qlf of a finite set of stable matrices is obtained as the positivity of a certain integral the structure and the conditions are considerably simplified for planar systems it is also proved that a set of block upper triangular matrices share a common qlf if each set of diagonal blocks share a common qlf

stability analysis of positive descriptor systems abstract in this paper we discuss stability properties of positive descriptor systems in the continuous time as well as in the discrete time case we present characterisations of positivity and establish generalised stability criteria for the case of positive descriptor systems we show that if the spectral projector onto the finite deflating subspace of the matrix pair e a is nonnegative then all stability criteria for standard positive systems take a comparably simple form in the positive descriptor case furthermore we provide sufficient conditions that guarantee entry wise nonnegativity along with positive semidefiniteness of solutions of generalised projected lyapunov equations as an application of the framework established throughout this paper we exemplarily generalise two criteria for the stability of two switched standard positive systems under arbitrary switching to the descriptor case

the lyapunov order for real matrices abstract the real lyapunov order in the set of real n n matrices is a relation defined as follows a b if for every real symmetric matrix s sb b t s is positive semidefinite whenever sa a t s is positive semidefinite we describe the main properties of the lyapunov order in terms of linear systems theory nevenlinna pick interpolation and convexity

common lyapunov solutions for two matrices whose difference has rank one abstract real stable matrices a and b with rank of a b equal to one have a common lyapunov solution if and only if their product ab has no real negative eigenvalue this was proved by shorten and narendra r n shorten k s narendra on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form ieee trans automat control 48 4 2003 618 621 whose proof is based on the fundamental results of kalman on lure s problem in this paper we give an alternative proof of this result and its generalization to the general regular inertia case and to the case when the matrices a and b are complex

an iterative method of lyapunov function construction for differential inclusions abstract the problem of choosing the lyapunov function from the class of quadratic forms for differential inclusions is reduced to that of searching saddle points of a suitable function an iterative algorithm is used for this purpose

multiplicative perturbations of stable and convergent operators abstract some familiar classes of stable hilbert space operators are studied to determine how they overlap and where the unitary similarity classes of their members lie analogous but less familiar classes of convergent operators are examined with the same aim the classes considered are often sets of products m a where m is a given set of diagonal or hermitian matrices and a is a single matrix the a s for which m a is a set of stable or convergent operators are sometimes characterized

on linear copositive lyapunov functions and the stability of switched positive linear systems we consider the problem of common linear copositive lyapunov function existence for positive switched linear systems in particular we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time invariant systems several applications of this result are also given

a novel methodology to solve the cqlf problem for a finite set of stable second order systems a new methodology to provide conclusive information about the existence non existence of a common quadratic lyapunov function cqlf for a finite set of stable second order systems is presented despite the high complexity of the cqlf problem even in the case of n second order systems the results presented in this paper have a very simple and intuitive theoretical support including topics such as classical intersection of convex sets and properties of convex linear combinations illustrative examples to show the performance of the proposed methodology are provided

common solution to the lyapunov equation for 2 2 complex matrices abstract in this work we solve the problem of a common solution to the lyapunov equation for 2 2 complex matrices we show that necessary and sufficient conditions for the existence of a common solution to the lyapunov equation for 2 2 complex matrices a and b is that matrices a i i b i i and a i i 1 b i i have no negative real eigenvalues for all r we show how these results relate to a special class of 4 4 real matrices

a characterization of lyapunov diagonal stability using hadamard products abstract several new characterizations of lyapunov diagonal stability are presented one of the characterizations is used to derive a simple necessary and sufficient condition for lyapunov diagonal stability of a 3 3 matrix

on easily verifiable conditions for the existence of common linear copositive lyapunov functions this technical note addresses the existence of a common linear copositive lyapunov function clclf for continuous time switched positive linear systems two necessary and sufficient conditions for the existence of a clclf are provided one condition shows that the existence of a clclf for a family of n n dimensional metzler matrices can only be determined by n 1 lower dimensional hurwitz matrices the other condition further shows that the existence of a clclf is also determined by n 1 characteristic constants which gives an easily verifiable computation algorithm without checking a large amount of metzler hurwitz matrices

absolute stability analysis in cellular neural networks with variable delays and unbounded delay abstract in this paper the existence and uniqueness of the equilibrium point and absolute stability of a class of cellular neural networks with globally lipschitz continue activation functions are investigated the neural networks contain both variable and unbounded delays the necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained by constructing proper vector liapunov functions nonlinear integrodifferential inequalities involving both variable delays and unbounded delay which are corresponding to the neural networks are established by using m matrix theory and qualitative property of the integrodifferential inequalities the sufficient conditions for absolute stability global asymptotic stability are obtained

a pair of matrices sharing common lyapunov solutions a closer look abstract let a b be a pair of matrices with regular inertia if ha a h and hb b h are both positive definite for some hermitian matrix h then all matrices in conv a a 1 b b 1 have identical regular inertia this in turn implies that both conv a b and conv a b 1 consist of non singular matrices in general neither of the converse implications holds in this paper we seek situations where they do hold in particular when a and b are real 2 2 matrices several aspects of the above statements for n n matrices are discussed a connection to the characterization of the convex hull of matrices with regular inertia is introduced differences between the real and the complex case are indicated

matrix theoretic conditions for the realizability of sliding manifolds systems of the type s ksgn s s rm where sgn is the vector sign function usually determine the realizability of sliding modes in control systems with high frequency gain k a new condition on k is established for the global stability of this system using a nonsmooth liapunov function this condition is used to establish necessary and sufficient stability conditions for second order systems

notes on ω and τ matrices abstract results on and matrices are surveyed the question whether an matrix with positive leading principal minors is a matrix is answered negatively other open questions are discussed new conjectures and questions are introduced

positivity of principal minors sign symmetry and stability the relation between positivity of principal minors sign symmetry and stability of matrices is studied it is shown that for sign symmetric matrices having positive principal minors is equivalent to stability to d stability and to having a positive scaling into a stable matrix the relation between spectra of matrices some of whose powers have positive principal minors and matrices whose corresponding powers have positive sums of principal minors of each order is studied as well it is shown that for matrices of order less than 4 these two classes share the same spectra the relation of these classes and stability is studied in particular for sign symmetric matrices and for anti sign symmetric matrices

on structural properties of the lyapunov matrix equation for optimal diagonal solutions we revisit the classical problem of finding a positive diagonal solution p to the lyapunov equation at p pa 0 that minimizes the lyapunov exponent the maximum eigenvalue of at p pa for a isin rnxn with the aim of identifying structural properties of the lyapunov matrix equation at the optimum using eigenvalue sensitivity notions together with optimization machinery we are able to obtain an explicit characterization of the minimum lyapunov exponent that provides such structural insight

not all gkk τ matrices are stable abstract hermitian positive definite totally positive and nonsingular m matrices enjoy many common properties in particular a positivity of all principal minors b weak sign symmetry c eigenvalue monolonicity d positive stability the class of gkk matrices is defined by properties a and b whereas the class of nonsingular matrices by a and c it was conjectured that a b d d carlson j res nat bur standards sect b 78 1974 1 2 a c d g m engel and h schneider linear and multilinear algebra 4 1976 155 176 a b a property stronger than d r varga numerical methods in linear algebra 1978 pp 5 15 a b c d d hershkowitz linear algebra appl 171 1992 161 186 we describe a class of unstable gkk matrices thus disproving all four conjectures

diagonal riccati stability and positive time delay systems abstract we consider a class of algebraic riccati equations arising in the study of positive linear time delay systems we show that this class admits diagonal positive definite solutions this implies that exponentially stable positive linear time delay systems possess lyapunpov krasovskii functionals of a simple quadratic form we also show that for this class of equations the existence of positive definite solutions is equivalent to a simple spectral condition on the coefficient matrices

convex invertible cones of matrices a unified framework for the equations of sylvester lyapunov and riccati abstract convex cones of matrices which are closed under matrix inversion were defined and various connections with the algebraic lyapunov equation of general inertia were studied in n cohen i lewkowicz linear algebra and its appl 250 1997 105 131 here in the price of doubling the size of the matrices involved we introduce a unified framework for the equations of sylvester lyapunov and riccati this enables one to extend the convex invertible cone structure to all these three equations and explore related properties in particular the use of the matrix sign function for solving these equations is examined

on common α scalar lyapunov solutions abstract we present two new characterizations regarding common scalar solutions for systems of lyapunov matrix inequalities so as to bridge and broaden relevant existing results as applications of such results we further explore the cases involving a set of conformally partitioned block triangular matrices and a set of z matrices respectively to obtain new results concerning these cases which are useful in the areas of control and systems theory and engineering

on the kalman yacubovich popov lemma and common lyapunov solutions for matrices with regular inertia in this paper we extend the classical lefschetz version of the kalman yacubovich popov kyp lemma to the case of matrices with general regular inertia we then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common lyapunov solution cls extending a recent result on cls existence for pairs of hurwitz matrices

on schur d stable matrices abstract it is shown that vertex stability implies schur d stability for real 2 2 matrices and real n n tridiagonal matrices additional results describing the class of n n complex schur d stable matrices are given

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

characterization of acyclic d stable matrices abstract necessary and sufficient conditions for d stability of acyclic matrices are given special cases of this characterization are recent results on tridiagonal matrices

a result on common quadratic lyapunov functions in this note we define strong and weak common quadratic lyapunov functions cqlfs for sets of linear time invariant lti systems we show that the simultaneous existence of a weak cqlf of a special form and the nonexistence of a strong cqlf for a pair of lti systems is characterized by easily verifiable algebraic conditions these conditions are found to play an important role in proving the existence of strong cqlfs for general lti systems

localization of the spectra of p and p0 matrices abstract it is well known that if s 1 n is a spectrum of a p matrix then arg i n we show that if the number of elements of s in the right half plane or in the left half plane is given then the above bound can be improved namely there exists such that arg i n when s has exactly one element in the right half plane it is shown that barg i 2 3 independently of n

on two classes of matrices with positive diagonal solutions to the lyapunov equation abstract we characterize real indecomposable quasi jacobi matrices of class d i e those which satisfy the lyapunov equation pa a p q with p diagonal and both p and q positive definite the subclass d 2 of class d when also q is diagonal is also characterized in the case of general indecomposable real matrices

an alternative proof of the barker berman plemmons bbp result on diagonal stability and extensions abstract we revisit the theorem of barker berman and plemmons on the existence of a diagonal quadratic lyapunov function for a stable linear time invariant lti dynamical system g p barker a berman r j plemmons positive diagonal solutions to the lyapunov equations linear and multilinear algebra 5 3 1978 249 256 we use recently derived results to provide an alternative proof of this result and to derive extensions

stability and stabilizability of continuous time linear compartmental switched systems in this paper we introduce continuous time linear compartmental switched systems and investigate their stability and stabilizability properties by their nature these systems are always stable necessary and sufficient conditions for asymptotic stability for arbitrary switching functions and sufficient conditions for asymptotic stability under certain dwell time conditions on the switching functions are proposed finally stabilizability is thoroughly investigated and proved to be equivalent to the existence of a hurwitz convex combination of the subsystem matrices a condition that for positive switched systems is only sufficient for stabilizability

on the uniqueness of the lyapunov scaling factors sufficient conditions are given for a lyapunov diagonally semistable matrix a to have a unique up to scalar multiplication lyapunov scaling factor suppose d is a lyapunov scaling factor of a a necessary and sufficient condition is given for the uniqueness of d the results extend work of hershkowitz and schneider

on global asymptotic stability of recurrent neural networks with time varying delays in this paper by constructing a new lyapunov functional and using m matrix and topological degree tool problem of the global asymptotic stability gas is discussed for a class of recurrent neural networks with time varying delays some simple and new sufficient conditions are obtained ensuring existence uniqueness of the equilibrium point and its gas of the neural networks some previous works are improved in addition this condition does not require the activation functions to be differentiable bounded and monotone nondecreasing and the weight connected matrices to be symmetric the neural network model considered in this paper include the delayed hopfield neural networks bidirectional associative memory networks and delayed cellular neural networks as its special cases

generalized joint spectral radius and stability of switching systems abstract this paper extends the notion of generalized joint spectral radius with exponents originally defined for a finite set of matrices to probability distributions we show that under a certain invariance condition the radius is calculated as the spectral radius of a matrix that can be easily computed extending the classical counterpart using this result we investigate the mean stability of switching systems in particular we establish the equivalence of mean square stability simultaneous contractibility in square mean and the existence of a quadratic lyapunov function also the stabilization of positive switching systems is studied numerical examples are given to illustrate the results

convex invertible cones and the lyapunov equation abstract convex cones of matrices which are closed under matrix inversion are defined and their structure is studied various connections with the algebraic lyapunov equation of general inertia are explored

dual graphs and knot invariants abstract let g be a signed plane graph and g d its signed dual graph methods from knot theory are used to show that the signed laplacian matrices l g and l g d are goeritz congruent there exists diagonal 0 1 matrices 1 and 2 and a unimodular matrix u such that 1 l g u 2 l g d u t for laplacian matrices of an unsigned plane graph and its dual l g is goeritz congruent to l g d

on spectral properties of signed laplacians with connections to eventual positivity signed graphs have appeared in a broad variety of applications ranging from social networks to biological networks from distributed control and computation to power systems in this paper we investigate spectral properties of signed laplacians for undirected signed graphs we find conditions on the negative weights under which a signed laplacian is positive semidefinite via the kron reduction and multiport network theory for signed laplacians that are indefinite we characterize their inertias with the same framework furthermore we build connections between signed laplacians generalized m matrices and eventually exponentially positive matrices

the relationship between hadamard and conventional multiplication for positive definite matrices abstract it is known that if a is positive definite hermitian then a a 1 i in the positive semidefinite ordering our principal new result is a converse to this inequality under certain weak regularity assumptions about a function f on the positive definite matrices a f a af a for all positive definite a if and only if f a is a positive multiple of a 1 in addition to the inequality a a 1 i it is known that a a 1 t i and stronger that min a b min ab t for a b positive definite hermitian we also show that min a b min ab and note that min ab and min ab t can be quite different for a b positive definite hermitian we utilize a simple technique for dealing with the hadamard product which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under hadamard multiplication and of the inequalities mentioned

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

a new characterization of simultaneous lyapunov diagonal stability via hadamard products abstract a well known characterization by kraaijevanger 14 for lyapunov diagonal stability states that a real square matrix a is lyapunov diagonally stable if and only if a s is a p matrix for any positive semidefinite s with nonzero diagonal entries this result is extended here to a new characterization involving similar hadamard multiplications for simultaneous lyapunov diagonal stability on a set of matrices among the main ingredients for this extension are a new concept called p sets and a recent result regarding simultaneous lyapunov diagonal stability by berman goldberg and shorten 2

diagonal riccati stability and the hadamard product we present an extension of a recent characterisation of diagonal riccati stability and using this extend a result of kraaijevanger on diagonal lyapunov stability to riccati stability of time delay systems we also describe a class of transformations that preserve the property of being diagonally riccati stable and apply these two results to provide novel stability results for classes of time delay systems

convex invertible cones and positive real analytic functions in this paper we study classical spaces of analytic functions which are convex cones and closed under the involution f 1 f these include the spaces of positive real positive real odd and strictly positive real functions these spaces are associated in the engineering literature with energy dissipation in the sense of lyapunov we discuss the geometric aspects of these spaces in analogy with similar spaces in matrix theory which are also related to the matrix lyapunov equation stability and in view of a general theory of convex invertible cones

asymptotic stability of systems with saturation constraints in the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in r sup n and of linear systems with partial state saturation constraints they also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems with saturation constraints on both states using the above results systems of the type considered herein are widely used in several areas of applications including control systems signal processing and artificial neural networks

a comprehensive review of stability analysis of continuous time recurrent neural networks stability problems of continuous time recurrent neural networks have been extensively studied and many papers have been published in the literature the purpose of this paper is to provide a comprehensive review of the research on stability of continuous time recurrent neural networks including hopfield neural networks cohen grossberg neural networks and related models since time delay is inevitable in practice stability results of recurrent neural networks with different classes of time delays are reviewed in detail for the case of delay dependent stability the results on how to deal with the constant variable delay in recurrent neural networks are summarized the relationship among stability results in different forms such as algebraic inequality forms m matrix forms linear matrix inequality forms and lyapunov diagonal stability forms is discussed and compared some necessary and sufficient stability conditions for recurrent neural networks without time delays are also discussed concluding remarks and future directions of stability analysis of recurrent neural networks are given

maximal lyapunov scaling factors and their applications in the study of lyapunov diagonal semistability of block triangular matrices abstract for a lyapunov diagonally semistable matrix a we introduce the concept of maximal lyapunov scaling factor that is a lyapunov scaling factor d of a such that the range of ad da t is of maximal dimension we investigate properties of maximal lyapunov scaling factors and apply them in proving a characterization for block triangular lyapunov diagonally semistable matrices in terms of lyapunov diagonal semistability of the diagonal blocks the latter result is used to prove that a block triangular matrix is lyapunov diagonally semistable if and only if every 2 2 block principal submatrix of it is lyapunov diagonally semistable

lyapunov diagonal semistability of real h matrices we characterize lyapunov diagonally stable real h matrices and those real h matrices which are lyapunov diagonally semistable but not lyapunov diagonally stable called lyapunov diagonally near stable the latter characterization is given in terms of the principal submatrix rank property defined here we apply our results to the numerical abscissas of real matrices one of our main tools is a slight strengthening of classical results of ostrowski which we derive from a fundamental theorem of wielandt

linear copositive lyapunov functions for continuous time positive switched systems continuous time positive systems switching among p subsystems are introduced and a complete characterization for the existence of a common linear copositive lyapunov function for all the subsystems is provided when the subsystems are obtained by applying different feedback control laws to the same continuous time single input positive system the above characterization leads to a very easy checking procedure

tensor conditions for the existence of a common solution to the lyapunov equation necessary and sufficient conditions for the existence of a common solution to the lyapunov equation for two stable complex matrices are derived these conditions are applied to the cases when a common weak solution to the lyapunov equation exists conditions for the existence of a common solution to the lyapunov equation for two complex 2 2 and two complex 3 3 matrices are derived

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

isospectral riemannian metrics and potentials pour une variete de riemann compacte m g on considere le spectre de l operateur de laplace beltrami et de l operateur de schrodinger v agissant sur l u 2 m g

strong stability of internal system descriptions this article introduces a new notion of stability for internal autonomous system descriptions that is referred to as strong stability and characterises the case of avoiding overshoots for all initial conditions within a given sphere this is a stronger notion of stability compared with alternative definitions asymptotic lyapunov which allows the analysis and design of control systems described by natural coordinates to have no overshooting response for arbitrary initial conditions for the case of linear time invariant systems necessary and sufficient conditions for strong stability are established in terms of the negative definiteness of the symmetric part of the state matrix the invariance of strong stability under orthogonal transformations is established and this enables the characterisation of the property in terms of the invariants of the schur form of the state matrix some interesting relations between strong stability and special forms of coordinate frames are examined and links between th