on computing puiseux series for multiple imaginary characteristic roots of lti systems with commensurate delays obtaining the puiseux series of multiple imaginary characteristic roots mirs is a fundamental issue in the stability analysis of time delay systems however to the best of the authors knowledge this issue has not been fully investigated up to date this note focuses on the puiseux series expansion of mirs of linear time invariant systems including commensurate delays for an mir of any multiplicity we propose an algorithm for defining the structure of the puiseux series as well as the explicit computation of the corresponding coefficients by using the proposed method we can find all the puiseux series corresponding to all the root loci

some problems in the stability of networked control systems with periodic scheduling this article addresses three stability problems related to networked control systems ncss with periodic scheduling where control systems may have multiple samplings in a hyperperiod a hyperperiod is a periodically repeated scheduling sequence for all tasks in an ncs as expected the analysis of a system with multiple samplings is much richer than the case with single sampling for example a system with two samplings may be stable unstable even if it is unstable stable when sampled by either sampling in this context it is important to understand how network induced delays and multiple samplings affect the system s stability in this article three particular stability problems involving constant and or time varying parameters are investigated and the corresponding stability regions are derived numerical examples and various discussions complete the presentation

stability crossing surfaces for linear time delay systems with three delays in this article the stability of linear systems with time delays is studied the cases where the characteristic equations of the system include three delays are investigated using the geometrical relations in a normalised polynomial plane a graphical method is presented to visualise the stability domains in the space of the time delays in this space the points at which the characteristic equation has a zero on the imaginary axis the border between stability and instability regions are identified these points form several surfaces called the stability crossing surfaces this work extends the results of the previous works on the stability crossing curves defined in the two dimensional space plane of delays to a higher dimension and provides new geometric interpretations for the stability crossing conditions

invariance properties for a class of quasipolynomials when a time delay system involves multiple imaginary roots mirs the stability analysis will become much more complicated than that in the case with only simple imaginary roots sirs an mir may exhibit different splitting behaviors and to the best of the authors knowledge their properties have not been fully investigated in this paper we focus on characterizing the invariance properties for mirs with any multiplicity furthermore the proposed methodology makes it possible to also cover some degenerate cases already encountered and discussed in the literature in addition we propose an easily implemented frequency sweeping method making it possible to derive the asymptotic behavior without invoking the puiseux series

iterative approaches to convex feasibility problems in banach spaces abstract the convex feasibility problem cfp of finding a point in the nonempty intersection i 1 n c i is considered where n 1 is an integer and each c i is assumed to be the fixed point set of a nonexpansive mapping t i x x with x a banach space it is shown that the iterative scheme x n 1 n 1 y 1 n 1 t n 1 x n is strongly convergent to a solution of cfp provided the banach space x either is uniformly smooth or is reflexive and has a weakly continuous duality map and provided the sequence n satisfies certain conditions the limit of x n is located as q y where q is the sunny nonexpansive retraction from x onto the common fixed point set of the t i s

viscosity approximation methods for nonexpansive mapping sequences in banach spaces abstract let e be a real strictly convex banach space with a uniformly g teaux differentiable norm and k be a nonempty closed convex subset of e suppose that t n n 1 2 is a uniformly asymptotically regular sequence of nonexpansive mappings from k to itself such that f n 1 f t n 0 for arbitrary initial value x 0 k and fixed contractive mapping f k k define iteratively a sequence x n as follows x n 1 n 1 f x n 1 n 1 t n 1 x n n 0 where n 0 1 satisfies lim n n 0 and n 1 n suppose for any nonexpansive mapping t k k z t strongly converges to a fixed point z of t as t 0 where z t is the unique element of k which satisfies z t t f z t 1 t t z t then as n x n z our results extend and improve the corresponding ones of o hara et al j g o hara p pillay h k xu iterative approaches to finding nearest common fixed point of nonexpansive mappings in hilbert spaces nonlinear anal 54 2003 1417 1426 and j s jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 and h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291

cappell shaneson homotopy spheres are standard we show here that an infinite sequence of homotopy 4 spheres constructed by cappell shaneson are all diffeomorphic to s 4 this generalizes previous results of akbulut kirby and gompf

stability of time delay systems equivalence between lyapunov and scaled small gain conditions it is demonstrated that many previously reported lyapunov based stability conditions for time delay systems are equivalent to the robust stability analysis of an uncertain comparison system free of delays via the use of the scaled small gain lemma with constant scales the novelty of this note stems from the fact that it unifies several existing stability results under the same framework in addition it offers insights on how new less conservative results can be developed

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

moudafi s viscosity approximations with meir keeler contractions abstract in this paper we discuss moudafi s viscosity approximations with meir keeler contractions we first present very simple proofs of xu s theorems concerning moudafi s approximations we next prove that browder s and halpern s type convergence theorems imply moudafi s viscosity approximations using them we finally state several deduced theorems

viscosity approximation to common fixed points of a nonexpansive semigroup with a generalized contraction mapping abstract the purpose of this paper is to introduce and construct the implicit and explicit viscosity iterative processes by a generalized contraction mapping f and a nonexpansive semigroup t t t 0 and to prove that under suitable conditions these iterative processes converge strongly to a unique common fixed point of t t t 0 in reflexive banach spaces which admits a weakly sequentially continuous duality mapping

reversals in stability of linear time delay systems a finer characterization abstract in most of the numerical examples of time delay systems proposed in the literature the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root cir appears as the delay seen as a parameter increases this fact may lead to some misunderstandings i a multiple cir may at most affect the instability degree ii it cannot cause any stability reversals stability transitions from instability to stability as far as we know whether the appearance of a multiple cir can induce stability is still unclear in fact when a cir generates a stability reversal has not been specifically investigated in this paper we provide a finer analysis of stability reversals and some new insights into the classification the link between the multiplicity of a cir and the asymptotic behavior with the stabilizing effect based on these results we present an example illustrating that a multiple cir s asymptotic behavior is able to cause a stability reversal to the best of the authors knowledge such an example is a novelty in the literature on time delay systems

a new lkf approach to stability analysis of linear systems with uncertain delays in this paper we analyze the asymptotic stability of linear systems with multiple uncertain delays assuming the nominal delays to be constant with possible zero or nonzero values we apply the simple lyapunov krasovskii functional lkf with the derivative prescribed as a negative definite quadratic form of the current state of a system we combine this functional with some integral bound for the derivative of the functional along the solutions of an uncertain system what provides the novelty of approach this bound consists of an essential integral part and some bounded expression the fact is that the positiveness of the essential part provides itself the stability of a system the study is based on recent works where the functional was shown to admit a quadratic lower bound on some special set of functions though it is only known to have a local cubic one on the set of arbitrary functions the derived stability condition is much simpler than most ones obtained by lkfs and outlined in the literature it constitutes an algebraic inequality for delay perturbations based on the so called lyapunov matrix and converges to a necessary and sufficient condition in some sense

an exact method for the stability analysis of time delayed linear time invariant lti systems a general class of linear time invariant systems with time delay is studied recently they attracted considerable interest in the systems and control community the complexity arises due to the exponential type transcendental terms in their characteristic equation the transcendentality brings infinitely many characteristic roots which are cumbersome to elaborate as evident from the literature a number of methodologies have been suggested with limited ability to assess the stability in the parametric domain of time delay this study offers an exact structured and robust methodology to bring a closure to the question at hand ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions pockets in the domain of time delay but it also declares the number of unstable characteristic roots at any given pocket the method starts with the determination of all possible purely imaginary resonant characteristic roots for any positive time delay to achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation it is proven that the number of such resonant roots for a given dynamics is finite each one of these roots is created by infinitely many time delays which are periodically distributed an interesting property is also claimed next that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency these two unique findings facilitate a simple and practical stability method which is the highlight of the work

strong convergence theorems for finitely many nonexpansive mappings and applications abstract let e be a uniformly convex banach space which satisfies opial s condition or whose norm is frechet differentiable recently takahashi and shimoji w takahashi k shimoji convergence theorems for nonexpansive mappings and feasibility problems math comput modelling 32 2000 1463 1471 introduced an iterative scheme given by finitely many nonexpansive mappings in e and proved weak convergence theorems which are connected with the problem of image recovery in this paper we introduce a new iterative scheme which includes their iterative scheme as a special case under the assumption that e is a reflexive banach space whose norm is uniformly g teaux differentiable and which has a weakly continuous duality mapping we prove strong convergence theorems which are connected with the problem of image recovery using the established results we consider the problem of finding a common fixed point of finitely many nonexpansive mappings

the routh hurwitz method for stability determination of linear differential difference systems abstract the characteristic quasipolynomial for a linear differential difference system is replaced by a regular polynomial to which standard stability tests for finite dimensional systems can be applied the routh hurwitz stability test is then used to determine necessary and sufficient conditions for asymptotic stability of time delay systems independent of the length of the delay

critical frequencies and parameters for linear delay systems a lyapunov matrix approach abstract a new method based on recent results concerning the construction of the lyapunov matrix of delay systems reveals conditions under which the characteristic function has a root s 0 such that s 0 is also a root as the pure imaginary roots which are crucial in the stability analysis of time delay systems are of that type this method gives rise to a novel approach for the delay independent and delay dependent stability analyses of systems with delay that are multiple of a basic delay distributed delays and of a class of neutral type time delay systems a number of examples are given to illustrate the approach and to show its strength

an improved procedure in detecting the stability robustness of systems with uncertain delay a very efficient and reliable shortcut is explained here on the procedure which resolves the stability robustness of lti dynamics against an uncertain delay this procedure reduces the numerical complexity of the solution by an order of magnitude this note is in response to a number of user requests on the practicality of the mentioned methodology

strong convergence to common fixed points of families of nonexpansive mappings abstract we introduce iteration schemes for families of nonexpansive mappings in hilbert spaces and prove that the iterates converge strongly to common fixed points of the mappings

on a rational transfer function based approach to cal h _ infty filtering design for time delay linear systems this paper introduces a new procedure for h filter design of time delay linear systems a finite order lti system called comparison system is defined in such a way that its h norm is proven to be strongly related to the one of the time delay system differently from what can be viewed as a common feature of filter design methods available in the literature to date the one presented here treats the filtering design of time delay systems with classical numerical routines based on riccati equation and h theory of lti systems the proposed algorithm is simple efficient and easy to implement illustrative examples are solved and discussed in order to put in evidence the most relevant properties of the theoretical results furthermore a practical application is presented

parameter space stability analysis of lti time delay systems with parametric uncertainties a novel approach to stability analysis of linear time invariant lti time delay systems of retarded type with incommensurate time delays and polynomial dependence on parametric uncertainties is presented using a branch and bound algorithm which relies on taylor models and polynomials in bernstein form we first determine the stability crossing set in the delay parameter space and then evaluate stability for each disjoint region the novel approach can be used to either nonconservatively check stability of time delay systems with interval parameters or to map stable regions to a low dimensional parameter space while taking additional interval parameters into account the algorithm is applied to several examples with parametric uncertainties and or incommensurate time delays

delay dependent stability analysis by using delay independent integral evaluation for the stability analysis of time delay systems the available methods usually require the exact evaluation of some quantities the definite integral stability method originated from the argument principle or the cauchy theorem is effective because it only requires a rough estimation of the testing integral over a finite interval to judge stability however no general rule is given in the literature for properly choosing the upper limit of the testing integral in this paper two simple algorithms are presented for finding the parameter dependent critical upper limit and a parameter independent upper limit without any restriction on the number of time delays these results improve and complete the definite integral stability method as illustrated by the numerical examples the proposed algorithms work effectively and accurately

stability of linear systems with interval time delays excluding zero the stability of linear systems with multiple time invariant independent and uncertain delays is investigated each delay is assumed to reside within a known interval excluding zero a delay free sufficient comparison system is formed by replacing the delay elements with parameter dependent filters satisfying certain properties it is shown that robust stability of this finite dimensional parameter dependent comparison system guarantees stability of the original time delay system this result is novel in the sense that it does not require any a priori knowledge regarding stability of the time delay system for some fixed delay when the parameter dependent filters are formed in a particular manner using pade approximations an upper bound on the degree of conservatism of the comparison system may be obtained which is independent of the time delay system considered with this it is shown that the conservatism of this comparison system may be made arbitrarily small a linear matrix ineqaulity lmi formulation is presented for analysis of the stability of the parameter dependent comparison system in the single delay case an eigenvalue criterion is also available for stability analysis which incurs no additional conservatism

critical effects of the polarity change in delayed states within an lti dynamics with multiple delays this note concerns the polarity inversion of the delayed states in linear time invariant multiple time delay systems lti mtds to start with for such systems the assessment of asymptotic stability is complicated due to the infinite dimensionality introduced by the delays it is shown that the mentioned polarity inversion influences the respective delay dependent stability maps in interesting ways there are some intriguing invariant characteristics and correspondences between the original and inverted systems which form the primary contributions of this note these findings are shown to be the enabling features to relate the stability maps of the two systems the cluster treatment of characteristic roots ctcr paradigm is utilized to identify these features an example case study is provided to illustrate the claims

robust hurwitz stability test for linear systems with uncertain commensurate time delays the note deals with the robust hurwitz stability of a class of linear systems with uncertain commensurate time delays which is formulated as the robust hurwitz stability of a nonpolytopic family of quasi polynomials a np hard problem from the viewpoint of computational complexity on the basis of the edge theorem and sylvester resultant elimination a necessary and sufficient condition is developed for the robust stability of the whole family and it yields an effective testing procedure an illustrative example is given to show the effectiveness of the method

approximations of lipschitz maps via immersions and differentiable exotic sphere theorems abstract as our main theorem we prove that a lipschitz map from a compact riemannian manifold m into a riemannian manifold n admits a smooth approximation via immersions if the map has no singular points on m in the sense of f h clarke where dim m dim n as its corollary we have that if a bi lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense then they are diffeomorphic we have three applications of the main theorem the first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones the third one is that a compact n manifold m is a twisted sphere and there exists a bi lipschitz homeomorphism between m and the unit n sphere s n 1 which is a diffeomorphism except for a single point if m satisfies certain two conditions with respect to critical points of its distance function in the clarke sense moreover we have three corollaries from the third theorem the first one is that for any twisted sphere n of general dimension n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for a single point in particular there exists such a map between an exotic n sphere n of dimension n 4 and s n 1 the second one is that if an exotic 4 sphere 4 exists then 4 does not satisfy one of the two conditions above the third one is that for any grove shiohama type n sphere n there exists a bi lipschitz homeomorphism between n and s n 1 which is a diffeomorphism except for one of points that attain their diameters

complete stability robustness of third order lti multiple time delay systems a unique procedure is presented in this paper for a complete stability robustness of the third order lti multiple time delay systems lti mtds the uniqueness of the treatment is simply due to the fact that there is no comparable methodology presently in the literature the end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays the backbone of the method is a novel framework called the cluster treatment of characteristic roots ctcr ctcr is constructed over two fundamental propositions the first proposition claims the existence of a bounded number of so called kernel curves where the only imaginary characteristic roots occur the second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots for simplicity of conveyance and without loss of generality the number of time delays is taken as two in this document the new methodology is expandable to higher order dynamics with more time delays than two as the authors intend to demonstrate in future publications publications

a frequency sweeping framework for stability analysis of time delay systems for time delay systems the asymptotic behavior analysis of the critical imaginary roots w r t the infinitely many critical delays is an open problem in order to find a general solution we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency sweeping curves from a new analytic curve perspective as a consequence we will establish a frequency sweeping framework with three main results 1 a finer regularity singularity classification for time delay systems will be obtained 2 the general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w r t the infinitely many critical delays can be adequately studied 3 the complete stability problem can be fully solved moreover the frequency sweeping framework is extended to cover a broader class of time delay systems finally the geometric insights of frequency sweeping curves are investigated consequently some deeper properties on the asymptotic behavior of time delay systems and the link to frequency sweeping curves are found

free matrix based integral inequality for stability analysis of systems with time varying delay the free weighting matrix and integral inequality methods are widely used to derive delay dependent criteria for the stability analysis of time varying delay systems because they avoid both the use of a model transformation and the technique of bounding cross terms this technical note presents a new integral inequality called a free matrix based integral inequality that further reduces the conservativeness in those methods it includes well known integral inequalities as special cases using it to investigate the stability of systems with time varying delays yields less conservative delay dependent stability criteria which are given in terms of linear matrix inequalities two numerical examples demonstrate the effectiveness and superiority of the method

brief paper stabilization of linear strict feedback systems with delayed integrators the problem of compensation of input delays for unstable linear systems was solved in the late 1970s systems with simultaneous input and state delay have remained a challenge although exponential stabilization has been solved for systems that are not exponentially unstable such as chains of delayed integrators and systems in the feedforward form we consider a general system in strict feedback form with delayed integrators which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays and design a predictor feedback controller for this class of systems exponential stability is proven with the aid of a lyapunov krasovskii functional that we construct using the pde backstepping approach

tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials a vandermonde based approach a standard approach in analyzing dynamical systems consists in identifying and understanding the eigenvalues bifurcations when crossing the imaginary axis efficient methods for crossing imaginary roots identification exist however to the best of the author s knowledge the multiplicity of such roots was not deeply investigated in recent papers by the authors 1 2 it is emphasized that the multiplicity of the zero spectral value can exceed the number of the coupled scalar delay differential equations and a constructive approach vandermonde based allowing to an adaptive bound for such a multiplicity is provided namely it is shown that the zero spectral value multiplicity depends on the system structure number of delays and number of non zero coefficients of the associated quasipolynomial rather than the degree of the associated quasipolynomial 3 this technical note extends the constructive approach in investigating the multiplicity of crossing imaginary roots j omega where omega neq 0 and establishes a link with a new class of functional confluent vandermonde matrices a symbolic algorithm for computing the lu factorization for such matrices is provided as a byproduct of the proposed approach a bound sharper than the polya szego generic bound arising from the principle argument is established

stabilizing a first order system with an arbitrarily large time varying delay and an uncertain gain abstract handling delays and uncertain parameters in control systems is difficult and of long standing interest in this paper we consider the problem of stabilizing in a bounded input bounded output sense a first order linear time invariant plant with an arbitrarily large time varying delay and an arbitrarily large uncertain gain we propose the use of a mildly non linear periodic controller which is simple to implement we will derive a clear simple formula relating the maximum delay duration the location of the unstable plant pole and the bounds on the derivative of the delay for which our controller stabilizes the system

when is the achievable discrete time delay margin nonzero this technical note considers the delay margin problem for discrete time finite dimensional linear time invariant fdlti single input single output siso systems this problem has been studied extensively in continuous time it is well known that given a fdlti plant and controller forming a strictly proper stable feedback connection closed loop stability will be maintained for small delays hence the achievable delay margin using fdlti control is strictly greater than zero in this work we show that this is not the case for discrete time systems indeed we show that a fdlti siso discrete time plant has an achievable delay margin strictly greater than zero if and only if it has no negative real unstable poles

a general iterative algorithm for semigroups of nonexpansive mappings with generalized contractive mapping the purpose of this paper is to study the strong convergence of the implicit and composite viscosity iteration schemes to a unique common fixed point of nonexpansive semigroups t which is a solution of some variational inequality under certain conditions as a application we apply the proposed iterative algorithm to the minimization problem of finding a optimality condition the results of this paper extend some recent results

inverted pendulum stabilization characterization of codimension three triple zero bifurcation via multiple delayed proportional gains the paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin the controller that we propose uses multiple delayed measurements instead of derivative terms doing so we increase the performances of the closed loop in presence of system uncertainties and or noisy measurements the problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally on one hand we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension three triple zero bifurcation on the other hand we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium the aim of this analysis is to characterize the possible local bifurcations finally the proposed control scheme is numerically illustrated and discussed

stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters this paper presents a new method for assessing the bounded input bounded output stability of a class of fractional delay systems with commensurate orders and multiple commensurate delays of retarded type in the proposed method first by mapping the principal sheet of the riemann surface and a pseudo delay transformation an auxiliary polynomial is generated then this auxiliary polynomial is employed to find roots of the characteristic equation on the imaginary axis the properties of the root path close to these roots are used to identify intervals of delay values in which the system is stable the obtained results are illustrated via some numerical examples

brief paper stability independent of delay using rational functions this paper is concerned with the problem of assessing the stability of linear systems with a single time delay stability analysis of linear systems with time delays is complicated by the need to locate the roots of a transcendental characteristic equation in this paper we show that a linear system with a single time delay is stable independent of delay if and only if a certain rational function parameterized by an integer k and a positive real number t has only stable roots for any finite t 0 and any k 2 we then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number d in the open interval 0 1 the paper is closed by showing counterexamples where stability of the roots of the rational function when k 1 is not sufficient for stability of the associated linear system with time delay we also introduce a variation of an existing frequency sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized nyquist criterion the results are illustrated by numerical examples

an augmented model for robust stability analysis of time varying delay systems stability analysis of linear systems with time varying delay is investigated in order to highlight the relations between the variation of the delay and the states redundant equations are introduced to construct a new modelling of the delay system new types of lyapunov krasovskii functionals are then proposed allowing to reduce the conservatism of the stability criterion delay dependent stability conditions are then formulated in terms of linear matrix inequalities finally several examples show the effectiveness of the proposed methodology

viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings abstract let x be a reflexive and smooth real banach space which has a weakly sequentially continuous duality mapping in this paper we consider the following viscosity approximation scheme x n 1 n 1 f x n 1 n 1 t n 1 x n where f is a generalized contraction mapping for infinitely many nonexpansive self mappings t 1 t 2 t 3 in x we establish a strong convergence result which generalizes some results in the literature

updated semi discretization method for periodic delay differential equations with discrete delay an updated version of the semi discretization method is presented for periodic systems with a single discrete time delay the delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined stability charts are constructed for the damped and delayed mathieu equation for different time period time delay ratios the convergence of the method is investigated by examples stability charts are constructed for 1 and 2 degree of freedom milling models the codes of the algorithm are also attached in the appendix copyright 2004 john wiley sons ltd

improved delay dependent stability criteria for linear systems with multiple time varying delays this paper concentrates on the stability analysis of continuous time linear systems with multiple time varying delays the time delays are assumed to be bounded and have or not lower and upper boun

consensus analysis with large and multiple communication delays using spectral delay space concept in this study we consider the consensus problem for a group of second order agents interacting under a fixed undirected communication topology communication lines are affected by two rationally independent delays the first delay is assumed to be in the position information channels whereas the second delay is in the velocity information exchange the delays are assumed to be large and uniform throughout the entire network the stability analysis of such systems becomes quickly intractable as the number of agents increases and the delays enlarge to resolve this dilemma we first reduce the complexity of the problem dramatically by decomposing the characteristic equation of the system into a set of second order factors then we assess the stability of the resulting subsystems exactly and exhaustively in the domain of the time delays using the cluster treatment of characteristic roots ctcr paradigm ctcr requires the determination of all the potential stability switching loci in the domain of the d

on characterizations of meir keeler contractive maps let x d be a metric space a map t x x is called meir keeler contractive if 0 0 such that d x y d tx ty we introduce l functions and characterize meir keeler contractive maps as maps that satisfy d tx ty d x y for some l function this characterization makes it easy to compare such maps with those satisfying the boyd wong s condition

a numerical method for stability windows and unstable root locus calculation for linear fractional time delay systems this paper aims to provide a numerical algorithm able to locate all unstable poles and therefore the characterization of the stability as a function of the delay for a class of linear fractional order neutral systems with multiple commensurate delays we start by giving the asymptotic position of the chains of poles and the conditions for their stability for a small delay when these conditions are met the root continuity argument and some simple substitutions allow us to determine the locations where some roots cross the imaginary axis providing therefore the complete characterization of the stability windows the same method can be extended to provide the position of all unstable poles as a function of the delay

a delay dependent stability criterion for systems with uncertain time invariant delays this paper provides a new stability criterion for systems with time invariant uncertain delays based on an improved upper bound for the inner product of two vectors a new delay dependent robust stability criterion is derived which is shown by an example less conservative than existing stability criteria

new results on eigenvalue distribution and controller design for time delay systems this paper considers the eigenvalue distribution of a linear time invariant lti system with commensurate time delays and its application to proportional integral derivative pid controller design for a delay plant via dominant eigenvalue assignment a new result on the root distribution of a quasi polynomial is first produced by applying part of pontryagin s conclusions this result which gives a necessary and sufficient condition can be directly used to judge the number of the right half plane eigenvalues of the characteristic equation of a time delay system based on the proposed result necessary and sufficient conditions on dominant eigenvalue assignment for pid control of time delay systems are presented and an algorithm is then provided to determine the pid controller gains the proposed approaches can assign some one or two eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones which enables us to design the controller according to the desired performance indexes for a standard first order system or a standard second order system in addition to stability the method is effective for the closed loop characteristic equation being retarded type or neutral type the controller gains to achieve the control objective can be characterized by a straightforward computation further a result on degradation to proportional integral pi control of time delay systems is given

iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive banach spaces abstract let e be a real reflexive banach space which admits a weakly sequentially continuous duality mapping from e to e and k be a nonempty closed convex subset of e suppose that t n n 1 2 is a uniformly asymptotically regular sequence of nonexpansive mappings from k into itself such that f n 1 f t n 0 for arbitrary initial value x 0 k and fixed contractive mapping f k k define iteratively a sequence x n as follows x n 1 n 1 f x n 1 n 1 t n 1 x n n 0 where n 0 1 satisfies lim n n 0 and n 1 n we prove that x n converges strongly to p f as n where p is the unique solution in f to the following variational inequality i f p j p u 0 for all u f t our results extend and improve the corresponding ones given by o hara et al j g o hara p pillay h k xu iterative approaches to finding nearest common fixed point of nonexpansive mappings in hilbert spaces nonlinear anal 54 2003 1417 1426 j s jung iterative approaches to common fixed points of nonexpansive mappings in banach spaces j math anal appl 302 2005 509 520 h k xu viscosity approximation methods for nonexpansive mappings j math anal appl 298 2004 279 291 and o hara et al j g o hara p pillay h k xu iterative approaches to convex feasibility problem in banach space nonlinear anal available online 20 october 2005 doi 10 1016 j na 2005 07 36

on joint design of intentionally introduced delay and controller gain for stabilization of second order oscillatory systems this paper focuses on stabilization of second order oscillatory systems using both feedback control and intentionally introduced time delay the argument principle is employed as a key technique to divide the parameter space of time delay and controller gain into several regions every admissible value in these regions moves the poles of the closed loop system towards the left of the complex plane such that stability improvement is achieved the concept of potent asymptotic stability is first introduced referring to the property that all the poles of the closed loop system are stable and lie on the left of open loop system poles in the complex plane analytical characterizations on parameter pairs of time delay and controller gain that result in potent asymptotic stability are established all the poles and corresponding root locus with respect to gain are examined numerical examples and simulations are given to illustrate the usefulness and merits of the theoretical results

exact region of stability for a linear orthogonal cutting model in this work stability of chatter in orthogonal cutting is investigated in order to identify the critical spindle speeds the process is modeled by a second order linear delay differential equation and the characteristic equation is analyzed in formulating the continuous stability boundaries explicitly for any given range of the spindle speeds moreover a simple algorithm is introduced for checking whether the system is in the stable zone without determining the whole stability diagram numerical simulations and comparison with some other methods are presented to justify the theoretical results

iterative approaches to common fixed points of nonexpansive mappings in banach spaces abstract the iteration scheme for families of nonexpansive mappings essentially due to halpern bull amer math soc 73 1967 957 961 is established in a banach space the main theorem extends a recent result of o hara et al nonlinear anal 54 2003 1417 1426 to a banach space setting for the same iteration scheme with finitely many mappings a complementary result to a result of jung and kim bull korean math soc 34 1997 93 102 also bauschke j math anal appl 202 1996 150 159 is obtained by imposing other condition on the sequence of parameters our results also improve results in c r acad sci ser a b paris 284 1977 1357 1359 j math anal appl 211 1997 71 83 arch math 59 1992 486 491 in framework of a hilbert space

quasiconformal 4 manifolds for any pseudo group of homeomorphism of euclidean space one can define the corresponding category of manifolds the most familiar examples in topology are the full pseudo group of homeomorphisms giving rise to the theory of topological manifolds and the subgroup of smooth differomorphisms giving rise to the theory of c manifolds in this paper we discuss an intermediate category quasiconformal homeomorphisms and manifolds recall that a homeomorphism 9 d r n from a domain d in r n to its image 9 d is k quasiconformal if for all x in d

quasi polynomial based robust stability of time delay systems can be less conservative than lyapunov krasovskii approaches in this article stability of uncertain time delay unforced systems is analyzed through a generalized characteristic quasi polynomial which is exactly rewritten as a polytope whose interpolating functions exhibit mutual dependence it is shown that an adequate combination of kharitonov like results and polya like relaxations leads to progressively better results at the price of more demanding computational burden stability methodologies are developed and proved advantageous when compared with traditional lyapunov based approaches both for structured and unstructured uncertainty

stability criteria for linear time invariant systems with point delays based on one dimensional routh hurwitz tests this brief deals with the asymptotic stability of a class of linear time invariant systems subject to point constant uncommensurate delays results are obtained dependent on and independent of the delays which may be tested be performing a finite set of 1 d routh hurwitz tests on a corresponding set of auxiliary delay free linear time invariant systems plus some supplementary conditions related to either commutation pairwise of the associate dynamics matrices or their norm closeness or some simple testable conditions if the matrices have particular forms and there is only one single delay the obtained results are applicable to the case when a finite set of matrices describing the dynamics of a set of delay free auxiliary dynamical systems are hurwitz and either commute pair wise or their norms are sufficiently close to each other some extensions are given to robust stability of point time delay time invariant systems with commensurate delays parametrized by an uncertain parameter vector whose components are in real intervals of known limits

stability crossing set for systems with three delays this article describes the stability crossing set for linear time delay systems of arbitrary order with three delays the crossing frequency set which consists of all frequencies where a pair of zeros of the characteristic quasipolynomial may cross the imaginary axis is partitioned to grashof sets and non grashof sets of various types it was found that the general characteristics of the stability crossing set is completely determined by the partition structure of the crossing frequency set with the exception of degenerate cases this article provides a method of explicit and complete parameterization and geometric characterization of the stability crossing set of linear systems with three delays with the well known method of finding the crossing directions this provides a powerful method of finding parameter regions of stable systems extension to systems with additional fixed delays is also discussed

stability robustness analysis of multiple time delayed systems using building block concept an intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays it is based on a holographic mapping which is implemented over the domain of the delays this mapping considerably alleviates the problem which is otherwise known to be notoriously complex it creates a dramatic reduction in the dimension of the problem from infinity to manageably small number ultimately the process is reduced to studying the problem within a finite dimensional cube with edges of length 2pi in the new domain what we call the building block in essence the mapping collapses the entire set of potential stability switching points onto a small upperbounded number of building hypersurfaces we further demonstrate that these building hypersurfaces can be implicitly defined and they are completely isolated within the above mentioned cube it is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture we seek as a consequence this concept yields a very practical and efficient procedure for the stability assessment of such systems this novel perspective serves very well for the preparatory steps of the authors earlier contribution in the area cluster treatment of characteristic roots ctcr we elaborate on this combination which forms the main contribution of the paper several example case studies are also provided

optimal sign inverting control for time delayed systems a concept study with experiments an intriguing control logic sign inverting control sic is considered for control systems with delayed feedback it starts with a nominal control law formulated for non delayed case and simply inverts the sign of the control gains for some surprising benefits when used with the delays this operation sounds paradoxical as the sign inversion potentially harms the stability of the non delayed dynamics however sic with large delays may yield some complementary benefits to the nominal control logic from delay robustness perspective the main question we address in this paper is how to select the nominal control law so that such a contribution can be a feasible b optimal in some sense a structured methodology is proposed to achieve this starting with a linear quadratic regulator based controller a single scaling factor on the corresponding control gains is used for one dimensional optimisation experimental validation of the concept of this optimal sic procedure is also reported on a single axis

stability analysis for lti systems with multiple time delays using the bounds of its imaginary spectra abstract the stability of linear time invariant lti systems with multiple independent time delays and the cluster treatment of characteristic roots ctcr paradigm are investigated from a new perspective any delay composition that results in an imaginary characteristic root lies either on a small number of kernel hypersurfaces kh or their infinitely many offspring hypersurfaces oh the complete description of kh is the only prerequisite for the ctcr based stability assessment procedure as the number of delays increases however the determination of these kh becomes computationally costly instead we present a practical procedure to extract the 2 d cross sections of the kh set in the domain of the two arbitrarily selected delays first we determine the exact upper and lower bounds of the imaginary spectra in this 2 d cross section of interest this process starts with a half angle tangent representation of the characteristic equation followed by the dixon resultant operation based on the full knowledge of kh the ctcr paradigm creates the complete stability map in the domain of these two arbitrarily selected delays we demonstrate the effectiveness of this methodology over an example case study

on stability crossing curves for general systems with two delays for the general linear scalar time delay systems of arbitrary order with two delays this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero the crossing set consisting of all the frequencies corresponding to all the points in the stability crossing curves are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms this crossing set forms a finite number of intervals of finite length the corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases these curves may be closed curves open ended curves and spiral like curves oriented horizontally vertically or diagonally the category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set the directions in which the zeros cross the imaginary axis are explicitly expressed an algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial and preservation of stability as a special case

a unique methodology for the stability robustness of multiple time delay systems abstract the stability robustness is considered for linear time invariant lti systems with rationally independent multiple time delays against delay uncertainties the problem is known to be notoriously complex primarily because the systems are infinite dimensional due to delays multiplicity of the delays in this study complicates the analysis even further and rationally independent feature of the delays makes the problem prohibitively challenging as opposed to the tds with commensurate time delays where time delays are rationally related a unique framework is described for this broadly studied problem and the enabling propositions are proven we show that this procedure analytically reveals all possible stability regions exclusively in the space of the delays as an added strength it does not require the delay free system under consideration to be stable our methodology offers a resolution to this question which has been studied from variety of directions in the past four decades none of these respectable investigations can however deliver an exact and exhaustive robustness declaration from this stand point the new method has a unique contribution

scharlemann s manifold is standard in his 1974 thesis martin scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f q s3 x si s2 x s2 and asked the question whether or not the manifold q itself is diffeomorphic to s3 x s1 s2 x s2 here we answer this question affirmatively

some special cases in the stability analysis of multi dimensional time delay systems using the matrix lambert w function this paper revisits a recently developed methodology based on the matrix lambert w function for the stability analysis of linear time invariant time delay systems by studying a particular yet common second order system we show that in general there is no one to one correspondence between the branches of the matrix lambert w function and the characteristic roots of the system furthermore it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system and that the principal branch can be used to find several roots and not the dominant root only as stated in previous works the results are first presented analytically and then verified by numerical experiments

brief a practical method for analyzing the stability of neutral type lti time delayed systems a new paradigm is presented for assessing the stability posture of a general class of linear time invariant neutral time delayed systems lti ntds the ensuing method which we name the direct method dm offers several unique features it returns the number of unstable characteristic roots of the system in an explicit and non sequentially evaluated function of time delay t consequently the direct method creates exclusively all possible stability intervals of t furthermore it is shown that this method inherently verifies a widely accepted necessary condition for the t stabilizability of a lti ntds in the core of the dm lie a root clustering paradigm and the strength of rekasius transformation in mapping a transcendental characteristic equation into an equivalent rational polynomial in addition we also demonstrate by an example that dm can tackle systems with unstable starting posture for t 0 only to stabilize for higher values of delay which is rather unique in the literature

iterative approaches to finding nearest common fixed points of nonexpansive mappings in hilbert spaces abstract the iteration scheme xn 1 n 1y 1 n 1 tn 1xn is first considered for infinitely many nonexpansive maps t1 t2 t3 in a hilbert space a result of shimizu and takahashi j math anal appl 211 1997 71 is generalized and it is shown that the sequence of iterates converges to py where p is some projection for this same iteration scheme with finitely many maps t1 t2 tn a complementary result to a result of bauschke j math anal appl 202 1996 150 is proved by introducing a new condition on the sequence of parameters n this condition improves lions condition c r acad sci paris ser a b 284 1977 1357 the iterates converge to py where p is the projection onto the intersection of the fixed point sets of the tis

an exact method for the stability analysis of linear consensus protocols with time delay this technical note presents a methodology for the stability analysis of linear consensus protocols with time delayed communications second order agent dynamics with a fixed and undirected communication topology and uniform delays are considered this class of group dynamics is very complex and is not fully explored to date the proposed technique takes advantage of the general structure of the control protocols in performing a state transformation that allows a decomposition of the characteristic equation into a set of factors these factors distribute the imprint of the delay in the characteristic equation in a much simpler form to achieve the stability analysis in parts the procedure also prepares the characteristic equation for the deployment of the cluster treatment of characteristic roots paradigm a recent method which declares the stability features of the system for various compositions of the time delay and other control parameters in order to show the effectiveness of this approach it is applied to different consensus protocols under the assumptions of fixed and undirected communication topologies and uniform communication time delays

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

gauge theory on asymptotically periodic 4 manifolds il existe une famille non denombrable de classes de diffeomorphismes des 4 varietes orientees homeomorphes a r 4