on stability of lms methods and characteristic roots of delay differential equations we investigate the use of linear multistep lms methods for computing characteristic roots of systems of linear delay differential equations ddes with multiple fixed discrete delays these roots are important in the context of stability and bifurcation analysis we prove convergence orders for the characteristic root approximations and analyze under what condition for the steplength the discrete integration scheme retains certain delay independent stability properties of the original equations unlike existing results we concentrate on the recovery of both stability and instability we illustrate our findings with a number of numerical test results

on a class of compactly epi lipschitzian sets the paper is devoted to the study of the so called compactly epi lipschitzian sets these sets are needed for many aspects of generalized differentiation particulary for necessary optimality conditions stability of mathematical programming problems and calculus rules for subdifferentials and normal cones we present general conditions under which sets defined by general constraints are compactly epi lipschitzian this allows us to show how the compact epi lipschitzness properties behave under set intersections

extended kronecker summation for cluster treatment of lti systems with multiple delays a new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant lti dynamics with multiple delays these hypersurfaces as they have very recently been introduced in a concept paper r sipahi and n olgac automatica 41 2005 pp 1413 1422 form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots ctcr in fact these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots to determine the kernel and offspring we use the extraordinary features of the extended kronecker summation operation in this paper the end result is that the infinite dimensional problem reduces to a finite dimensional one and preferably into an eigenvalue problem following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces we demonstrate these concepts via some example case studies one of the examples treats a 3 delay system for this case another interesting perspective called the building block is also utilized to display the kernel in three dimensional space in the domain of spectral delays

semi discretization method for delayed systems summary the paper presents an ecient numerical method for the stability analysis of linear delayed systems the method is based on a special kind of discretization technique with respect to the past eect only the resulting approximate system is delayed and also time periodic but still it can be transformed analytically into a high dimensional linear discrete system the method is applied to determine the stability charts of the mathieu equation with continuous time delay copyright 2002 john wiley sons ltd

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

on the comparison of semi analytical methods for the stability analysis of delay differential equations abstract this paper provides the first comparison of the semi discretization spectral element and legendre collocation methods each method is a technique for solving delay differential equations ddes as well as determining regions of stability in the dde parameter space we present the necessary concepts assumptions and equations required to implement each method to compare the relative performance between the methods the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example the delayed damped mathieu equation and a helicopter rotor control problem for each study we present one or more stability diagrams in the parameter space and one or more convergence plots the spectral element method is demonstrated to have the quickest convergence rate while the legendre collocation method requires the least computational time the semi discretization method on the other hand has both the slowest convergence rate and requires the most computational time

absolute continuity of the best sobolev constant abstract let q inf u l p p u l q p u w 0 1 p 0 be the best sobolev constant of the immersion w 0 1 p l q where is a bounded and smooth domain of r n 1 p n and 1 q p n p n p we prove that the function q 1 p q is absolutely continuous

weighted eigenvalue problems for quasilinear elliptic operators with mixed robin dirichlet boundary conditions abstract we investigate the existence of principal eigenvalues type problems with weights for the quasilinear operator p v p with mixed weighted robin dirichlet boundary conditions in a bounded regular domain we also give some results on the existence of nonprincipal eigenvalues

the fuzzy intersection rule in variational analysis with applications the fuzzy intersection rule for frechet normal cones in asplund spaces was established by mordukhovich and the author using the extremal principle which appears more convenient to apply in some applications in this paper we present a complete discussion of this rule in various aspects we show that the fuzzy intersection rule is another characterization of the asplund property of the space various applications are considered as well in particular a complete set of fuzzy calculus rules for general lower semicontinuous functions are established

on the fuzzy intersection rule abstract the paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory new forms of the rule and further applications are presented

metric regularity tangential distances and generalized differentiation in banach spaces abstract in this paper we establish new generalized differentiation rules in general banach spaces regarding normal cones to set images under functions coderivatives of compositions of set valued mappings as well as calculus results for normal compactness of sets and their images in addition to the metric regularity of mappings our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations basic properties as well as relations to similar notions some related results are also established

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

generalized sequential normal compactness in banach spaces abstract normal compactness conditions are important properties of sets set valued mappings in variational analysis which are generalized versions of the classical lipschitzian property and are essential for the calculus of generalized differentiation theory in this paper we propose the notion called the generalized sequential normal compactness and establish its basic properties and calculus in general banach spaces

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

an eigenvalue perturbation approach to stability analysis part ii when will zeros of time delay systems cross imaginary axis this paper presents an application of the eigenvalue series developed in part i j chen et al siam j control optim 48 2010 pp 5564 5582 to the study of linear time invariant delay systems focusing on the asymptotic behavior of critical characteristic zeros on the imaginary axis we consider systems given in state space form and as quasi polynomials and we develop an eigenvalue perturbation analysis approach which appears to be both conceptually appealing and computationally efficient our results reveal that the zero asymptotic behavior of time delay systems can in general be characterized by solving a simple eigenvalue problem and additionally when described by a quasi polynomial by computing the derivatives of the quasipolynomial

on the first exterior p harmonic steklov eigenvalue abstract in this short paper we study the sobolev function property of the rayleigh s quotient q inf u e 1 p u u p u q u p as a function of q 1 p for p p n 1 n p when p 1 n as well as the asymptotic behavior of positive solutions with minimal energy of the following problem p u 0 in u subject to u p 2 u u q 2 u on u to the first p harmonic steklov eigenpair on an exterior region u r n when n 3

pseudospectral differencing methods for characteristic roots of delay differential equations in d breda s maset and r vermiglio ima j numer anal 24 2004 pp 1 19 and d breda the infinitesimal generator approach for the computation of characteristic roots for delay differential equations using bdf methods research report udmi rr17 2002 dipartimento di matematica e informatica universita degli studi di udine udine italy 2002 the authors proposed to compute the characteristic roots of delay differential equations ddes with multiple discrete and distributed delays by approximating the derivative in the infinitesimal generator of the solution operator semigroup by runge kutta rk and linear multistep lms methods respectively in this work the same approach is proposed in a new version based on pseudospectral differencing techniques we prove the spectral accuracy convergence behavior typical of pseudospectral schemes as also illustrated by some numerical experiments

the asymptotic stability of linear autonomous systems with commensurate time delays the asymptotic stability of linear systems with commensurate time delays is studied algorithms are given to check if a given system is asymptotically stable independent of delay or to determine the largest delay time the system can tolerate to preserve its asymptotic stability these algorithms are simple to apply since they require only to find the eigenvalues of two specially constructed matrices and some simple mathematical calculations

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

on the weak star extensibility the paper encompasses a complete study of the weak star extensibility of subspaces of banach spaces introduced by b s mordukhovich and b wang when studying the restrictive metric regularity in variational analysis many useful properties are presented in particular a new point of view and formulation of the property in the framework of banach space theory is established we also propose the concept of weak star extensible banach spaces that has interesting applications and is fairly broad in that it contains many important classes of banach spaces some further applications of this extensibility to the theory of linear operators between banach spaces are also explored

a fundamental control performance limit for a class of positive nonlinear systems abstract a fundamental performance limit is derived for a class of positive nonlinear systems the performance limit describes the achievable output response in the presence of a positive disturbance and subject to a sign constraint on the allowable input an explicit optimal input is derived which minimises the maximum output response whilst ensuring that the minimum output response does not fall below a pre specified lower bound the result provides a fundamental performance standard against which all control policies including closed loop schemes can be compared implications of the result are examined in the context of blood glucose regulation for type 1 diabetes

bounds of imaginary spectra of lti systems in the domain of two of the multiple time delays the stability of linear time invariant lti systems with independent multiple time delays and the cluster treatment of characteristic roots ctcr paradigm are investigated from a new perspective it is known that for such systems all the imaginary characteristic roots can be detected completely on a small set of hypersurfaces in the domain of the delays sipahi and olgac 2006a they are called kernel hypersurfaces kh the complete description of kh is the only prerequisite for the ctcr stability assessment procedure as the number of delays increases however their evaluation becomes infeasible instead we present a procedure to extract the 2 d cross sections of these hypersurfaces in the domain of any two of the delays by fixing the remaining delays in the 2 delay domain of interest the exact upper and lower bounds of the imaginary spectra are determined for this a combination of half angle tangent representation of the characteristic equation and the dixon resultant theory is used as the main contributions of this paper the complete kh are obtained by sweeping the root crossing frequency in this interval using this knowledge ctcr creates the cross section of the stability map in the domain of the two arbitrarily selected delays we demonstrate the effectiveness of this methodology over an example case study with three independent delays and two commensurate ones

combination of sign inverting and delay scheduling control concepts for multiple delay dynamics abstract in this paper we treat a novel combination of two controversial control concepts sign inverting control sic and delay scheduling ds for systems with multiple independent and large delays sic suggests the inversion of the control polarity and ds prolongs the existing delays the combined scheme functions with a single requirement that the union of the control schemes provides a larger stable operating region than each of its components does in the domain of the delays the critical knowledge that is needed to execute such a unified control strategy is the crisp description of these stable regions for each time delayed control scheme this need can be fulfilled using the recent cluster treatment of characteristic roots ctcr paradigm which establishes the stable regions exhaustively and non conservatively the resulting options in selecting operating modes render more robust control performance against much larger delay variations than each of the schemes we also investigate the disturbance rejection speeds within these enlarged stable regions in order to improve the control performance even further such multi faceted paradoxical combinations provide previously unexplored tools to control designers experimental validations of these novel concepts are presented on a simple setup with a single axis manipulator

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

on the dirichlet problem involving the equation δpu λus 1 abstract we study with varying in r the behavior of the l s norm of the unique positive solution of the following dirichlet problem p u u s 1 in u 0 when s p in particular we prove that this norm is asymptotic to c p 1 p s for some positive constant c where p is the first eigenvalue of the p laplacian on

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

on extensions of c0 valued operators abstract we study pairs of banach spaces x y with y x for which the thesis of sobczyk s theorem holds namely such that every bounded c 0 valued operator defined in y extends to x we are mainly concerned with the case when x is a c k space and y c l is a banach subalgebra of c k the main result of the article states that if k is a compact line and l is countable then every bounded c 0 valued operator defined in c l extends to c k

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

stability of linear time periodic delay differential equations via chebyshev polynomials this paper presents a new technique for studying the stability properties of dynamic systems modeled by delay differential equations ddes with time periodic parameters by employing a shifted chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period the dynamic system can be reduced to a set of linear difference equations for the chebyshev expansion coefficients of the state vector in the previous and current intervals this defines a linear map which is the infinite dimensional floquet transition matrix u two different formulas for the computation of the approximate u whose size is determined by the number of polynomials employed are given the first one uses the direct integral form of the original system in state space form while the second uses a convolution integral variation of parameters formulation additionally a variation on the former method for direct application to second order systems is also shown an error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance an extension of the method to the case where the delay and parametric periods are commensurate is also shown stability charts are produced for several examples of time periodic ddes including the delayed mathieu equation and a model for regenerative chatter in impedance modulated turning the results indicate that this method is an effective way to study the stability of time periodic ddes

structure and stability of restrictively metrically regular mappings abstract restrictive metric regularity rmr was first proposed by b s mordukhovich and b wang 2004 in 13 in this paper we study the structure of rmr mappings and then establish stability properties of such mappings under perturbations

mapping based algorithm for large scale computation of quasi polynomial zeros a method for computing all zeros of a retarded quasi polynomial that are located in a large region of the complex plane is presented the method is based on mapping the quasi polynomial and on utilizing asymptotic properties of the chains of zeros first the asymptotic exponentials of the chains are determined based on the distribution diagram of the quasi polynomial secondly large regions free of zeros are defined finally the zeros are located as the intersection points of the zero level curves of the real and imaginary parts of the quasi polynomial which are evaluated over the areas of the region outside those free of zeros

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

stability in linear delay equations abstract for linear autonomous differential difference equations of retarded or neutral type necessary and sufficient conditions are given for the zero solution to be stable hyperbolic for all values of the delays

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

p laplacian boundary value problems on exterior regions abstract this paper treats some variational principles for solutions of inhomogeneous p laplacian boundary value problems on exterior regions u r n with dimension n 3 existence uniqueness results when p 1 n are provided in a space e 1 p u of functions that contains w 1 p u functions in e 1 p u are required to decay at infinity in a measure theoretic sense various properties of this space are derived including results about equivalent norms traces and an l p imbedding theorem also an existence result for a general variational problem of this type is obtained

on weak extensible banach spaces abstract we study the stability properties of the class of weak extensible spaces introduced by wang zhao and qiang showing among other things that weak extensibility is equivalent to having a weak sequentially continuous dual ball in short w sc for duals of separable spaces or twisted sums of w sc spaces this shows that weak extensibility is not a 3 space property solving a question posed by wang zhao and qiang we also introduce a restricted form of weak extensibility called separable weak extensibility and show that separably weak extensible banach spaces have the gelfand phillips property although they are not necessarily w sc spaces

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