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

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

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

new conditions on global stability of cohen grossberg neural networks in this letter we discuss the dynamics of the cohen grossberg neural networks we provide a new and relaxed set of sufficient conditions for the cohen grossberg networks to be absolutely stable and exponentially stable globally we also provide an estimate of the rate of convergence

p matrix properties injectivity and stability in chemical reaction systems in this paper we examine matrices which arise naturally as jacobians in chemical dynamics we are particularly interested in when these jacobians are p matrices up to a sign change ensuring certain bounds on their eigenvalues precluding certain behavior such as multiple equilibria and sometimes implying stability we first explore reaction systems and derive results which provide a deep connection between system structure and the p matrix property we then examine a class of systems consisting of reactions coupled to an external rate dependent negative feedback process and characterize conditions which ensure that the p matrix property survives the negative feedback the techniques presented are applied to examples published in the mathematical and biological literature

global stability of cellular neural networks with constant and variable delays this paper gives new conditions ensuring global asymptotic stability and global exponential stability for cellular neural networks with constant delay and variable delay respectively these conditions are derived by using the essence of piecewise linearity of the output function of cellular neural networks and by constructing lyapunov functions and functionals furthermore these conditions are significantly weaker than those given in existing literature

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

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

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

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

robust decoupling of discrete systems using spl lscr sup 1 optimization we address a control design problem related to decoupling namely robust decoupling the idea of robust decoupling is defined and shown to be equivalent to robust performance of a hadamard weighted spl lscr sup 1 performance measure a necessary and sufficient condition to determine if a compensator satisfies robust performance of a hadamard weighted spl lscr sup 1 performance measure in the presence of structured uncertainty is given this condition can be used to determine if a compensator robustly decouples a plant finally an iterative scheme for synthesizing a robust decoupling controller is discussed

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

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

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

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

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

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

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

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

on the use of hadamard weightings in the partial decoupling of h based designs abstract in this paper a solution to the hadamard weighted h hwh 1 block problem is proposed the solution is based on a method by safonov and le 1988 the robust stability of designs based on the hwh method is also discussed

a second order overlapping schwarz method for a 2d singularly perturbed semilinear reaction diffusion problem an overlapping schwarz domain decomposition is applied to a semilinear reaction diffusion equation posed in a smooth two dimensional domain the problem may exhibit multiple solutions its diffusion parameter e2 is arbitrarily small which induces boundary layers the schwarz method invokes a boundary layer subdomain and an interior subdomain the narrow subdomain overlap being of width o e ln h where h is the maximum side length of mesh elements and the global number of mesh nodes does not exceed o h 2 we employ finite differences on layer adapted meshes of bakhvalov and shishkin types in the boundary layer subdomain and lumped mass linear finite elements on a quasiuniform delaunay triangulation in the interior subdomain for this iterative method we present maximum norm error estimates for e 0 1 it is shown in particular that when e c ln h 1 one iteration is sufficient to get second order convergence with in the case of the shishkin mesh a logarithmic factor in the maximum norm uniformly in e numerical results are presented to support our theoretical conclusions

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

scaling a canonical design problem for networks motivated by decentralized control and numerical methods applications we explore the scaling problem that is we consider the design of a diagonal matrix k so as to i place the eigenvalues of the product kg in the olhp and ii optimize a scalar performance measure c kg of interest for a square topology matrix g we first illustrate our methodology for scaling design which combines optimization machinery with graph algebra notions for a class of topology matrices and a specific performance measure in particular for symmetric positive definite topologies and for a dominant eigenvalue ratio performance measure we find that the optimally scaled system kg has a special structure the right eigenvectors associated with the largest and smallest eigenvalues have corresponding entries of identical magnitude and hence differ only in sign pattern using this structure we develop both explicit and algorithmic methods for computing the optimal scaling and cost after illustrating our methodology with this

a structured approximation problem with applications to frequency weighted model reduction the authors consider the problem of finding necessary and sufficient conditions for the existence of a q with no more than k poles in the closed left half plane such that the norm of a certain matrix

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

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

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

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

on a new method for h_ 2 based decomposition this note proposes a simple linear matrix inequality lmi system for the design of static precompensators to reduce the interactions of a multivariable system the approach is based on minimizing the h2 norm of a modified system and the lmis are derived from the well known technique of pseudodiagonalisation the approach is applied to two complex real life benchmark problems with high levels of interaction it is shown that its performance is significantly better than previously proposed lmi optimization techniques for designing static precompensators

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived