the solution by iteration of nonlinear equations involving psi strongly accretive operators let e be an arbitrary real banach space and let k c e be a proximal subset of e suppose t k i e is a lipschitz strongly accretive operator such that for any given g e the equation tx f has a solution x e d t it is proved that modified iteration processes of the mann and ishikawa types converge strongly to x related results deal with the solution of the equation x tx f 0 for a strongly accretive or strongly dissipative t and the iterative approximation of fixed points to a strongly pseudocontractive t we also discuss the case where the operator t is only uniformly continuous miramare trieste december 1996 permanent address department of mathematics and computer science nnamdi azikiwe university p m b 5025 awka anambra state nigeria

c cosine functions and the abstract cauchy problem ii for a bounded linear injectioncon a banach spacexand a closed linear operatora d a x xwhich commutes withcwe prove that 1 the abstract cauchy problem u t au t t r u 0 cx u 0 cy has a unique strong solution for everyx y d a if and only if 2 a1 a d a2 generates ac1 cosine function onx1 d a with the graph norm if and only if in caseahas nonempty resolvent set 3 agenerates ac cosine function onx herec1 c x1 under the assumption thatais densely defined andc 1ac a statement 3 is also equivalent to each of the following statements 4 the problemv t av t c x ty t0 cg r dr t r v 0 v 0 0 has a unique strong solution for everyg l1locandx y x 5 the problemw t aw t cg t t r w 0 cx w 0 cy has a unique weak solution for everyg l1locandx y x finally as an application it is shown that for any bounded operatorbwhich commutes withcand has range contained in the range ofc a bis also a generator

a consensus protocol under directed communications with two time delays and delay scheduling this paper studies a consensus protocol over a group of agents driven by second order dynamics the communication among members of the group is assumed to be directed and affected by two rationally independent time delays one in the position and the other in the velocity information channels these delays are unknown but considered to be constant and uniform throughout the system the stability of the consensus protocol is studied using a simplifying factorisation procedure and deploying the cluster treatment of characteristic roots ctcr paradigm this effort results in a unique depiction of the exact stability boundaries in the domain of the delays the ctcr requires the knowledge of the potential stability switching loci exhaustively within this domain the creation of these loci is an important contribution of this work it is done in a new surrogate coordinate system called the spectral delay space the relative stability of the system that is the speed to reach consensus is also investigated fo

picard iterations for solution of nonlinear equations in certain banach spaces abstract let e be a real uniformly smooth banach space and let a d a e e be locally lipschitzian and strongly quasi accretive it is proved that a picard recursion process converges strongly to the unique solution of the equation ax f f r a with the convergence being at least as fast as a geometric progression related results deal with the convergence of picard iterations to the fixed point of locally lipschitzian strong hemicontractions t and to the solutions of nonlinear equations of the forms x ax f and x ax f where a is an accretive type map

matrices of operators and regularized cosine functions this paper is concerned with matrices of abstract differential operators which are parabolic in the sense of shilov or correct in the sense of petrovskij we show that they generate regularized cosine functions with suitable regularizing operators under sharper conditions the results then are applied to matrices of partial differential operators on many function spaces finally the wellposedness of the associated second order systems is discussed

iterative solution of nonlinear equations involving set valued uniformly accretive operators abstract let e be a real normed linear space and let a e 2 e be a uniformly continuous and uniformly quasi accretive multivalued map with nonempty closed values such that the range of i a is bounded and the inclusion f ax has a solution x e it is proved that ishikawa and mann type iteration processes converge strongly to x further if t e 2 e is a uniformly continuous and uniformly hemicontractive set valued map with bounded range and a fixed point x e it is proved that both the mann and ishikawa type iteration processes converge strongly to x the strong convergence of these iteration processes with errors is also proved

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

the solution by iteration of nonlinear equations in uniformly smooth banach spaces abstract letebe a uniformly smooth banach space and lett d t e ebe a strong pseudocontraction with an open domaind t ineand a fixed pointx d t we establish the strong convergence of the mann and ishikawa iterative processes with errors to the fixed point oft related results deal with the iterative solution of operator equations of the formsf txandf x tx 0 whentis a set valued strongly accretative operator our theorems include the cases in which the operatortis defined only locally explicit error estimates are also given

numerical stability analysis of a large scale delay system modeling a lateral semiconductor laser subject to optical feedback this paper highlights the use of advanced numerical tools to study the stability of large scale systems of delay differential equations ddes specifically we consider a model describing a semiconductor laser subject to conventional optical feedback and lateral carrier diffusion the symmetry of the governing rate equations allows external cavity mode solutions ecms to be computed as steady state solutions using the software package dde biftool branches of ecms are computed as a function of varying feedback strength the stability along these branches is computed by solving eigenvalue problems the size of which is governed by a step length heuristic in this paper we employ an improved heuristic which substantially reduces the size of these eigenvalue problems this approach makes the stability analysis of large scale systems of ddes computationally feasible

iteration processes for nonlinear lipschitzian strongly accretive mappings in lp spaces abstract suppose x l p or l p 1 p 2 let t x x be a lipschitzian and strongly accretive map with constant k 0 1 and lipschitz constant l satisfying p p 1 l 2 p 2 p 1 1 k define s x x by sx tx x for arbitrary x 0 x the sequence x n n 1 is defined by x n 1 1 n x n n sy n y n 1 n x n n sx n n 0 where n n 1 n n 1 are two real sequences satisfying i 0 n 2 1 p 2 p 1 1 k k n l 2 n p 2 p l 2 1 w n w l 2 1 2 n p 2 p 1 1 k k n l 2 n p 1 2 l 2 1 w n w l 2 1 2 n 1 1 for each n ii 0 n min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 for each n iii n n where w p 2 p 1 1 k p 2 p l 2 then x n 1 converges strongly to the unique solution of tx moreover if n 2 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 and n for each n and some 0 min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 then x n 1 q p n 2 x 1 q where q denotes the solution of tx and 1 4 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 0 1 a related result details with the iterative approximation of lipschitz strongly pseudocontractive maps in x

numerical stability analysis and computation of hopf bifurcation points for delay differential equations we present a numerical technique for the stability analysis and the computation of branches of hopf bifurcation points in nonlinear systems of delay differential equations with several constant delays the stability analysis of a steady state solution is done by a numerical implementation of the argument principle which allows to compute the number of eigenvalues with positive real part of the characteristic matrix the technique is also used to detect bifurcations of higher singularity hopf and fold bifurcations during the continuation of a branch of hopf points this allows to trace new branches of hopf points and fold points

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

efficient computation of characteristic roots of delay differential equations using lms methods we aim at the efficient computation of the rightmost stability determining characteristic roots of a system of delay differential equations the approach we use is based on the discretization of the time integration operator by a linear multistep lms method the size of the resulting algebraic eigenvalue problem is inversely proportional to the steplength we summarize theoretical results on the location and numerical preservation of roots furthermore we select nonstandard lms methods which are better suited for our purpose we present a new procedure that aims at computing efficiently and accurately all roots in any right half plane the performance of the new procedure is demonstrated for small and large scale systems of delay differential equations

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

ishikawa iteration process for nonlinear lipschitz strongly accretive mappings abstract let e lp p 2 and let t e e be a lipschitzian and strongly accretive mapping let s e e be defined by sx tx x it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process x0 e xn 1 1 n xn ns 1 n xn nsxn n 0 converges strongly to the unique solution of tx a related result deals with the iterative approximation of fixed points for lipschitz strongly pseudocontractive mappings in e a consequence of our result gives an affirmative answer to a problem posed by c e chidume j math anal appl 151 no 2 1990 453 461

inverse feedback shapers for coupled multibody systems the concept of inverse feedback shapers is fully developed in this report for the case of multibody systems with strongly coupled dynamics the presented results constitute a nontrivial extension of the previous papers by the authors on input shaping focusing on effectively decoupled controlled and flexible parts as the main technical result the target mode for the inverse shaper design is identified based on an input output transformation of the multibody system into the controlled subsystem the attached residual oscillatory dynamics and an internal feedback loop representing the dynamical coupling a state feedback controller applied for the selected generalized coordinate is considered throughout this paper

an efficient method for locating and computing periodic orbits of nonlinear mappings abstract the accurate computation of periodic orbits of nonlinear mappings and the precise knowledge of their properties are very important for studying the behavior of many dynamical systems of physical interest in this paper we present an efficient numerical method for locating and computing to any desired accuracy periodic orbits stable unstable and complex of any period the method described here is based on the topological degree of the mapping and is particularly useful since the only computable information required is the algebraic signs of the components of the mapping this method always converges rapidly to a periodic orbit independently of the initial guess and is particularly useful when the mapping has many periodic orbits stable and unstable close to each other all of which are desired for the application we illustrate this method first on a two dimensional quadratic mapping used in the study of beam dynamics in particle accelerators to compute rapidly and accurately its periodic orbits of periods p 1 5 16 144 1296 10368 and then obtain periodic orbits of its four dimensional complex version for periods which also reach up to the thousands

design of proportional integral retarded pir controllers for second order lti systems this note presents a design technique for the delay based controller called proportional integral retarded pir which solves the regulation problem of a general class of stable second order lti systems using spectral analysis the technique yields a tuning strategy for the pir by placing a triple real dominant root for the closed loop system this result ultimately guarantees a desired exponential decay rate sigma d while achieving the pir tuning as an explicit function of sigma d and system parameters

controller design for delay systems via eigenvalue assignment on a new result in the distribution of quasi polynomial roots this paper considers the eigenvalue distribution of a linear time invariant lti system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment first a new result on the root distribution for a class of quasi polynomials is developed based on the extension of the hermite biehler theorem then such result is applied to proportional integral pi controller parameter design for a first order plant with time delay through pole placement the complete region of pi gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed loop system with delay plant are assigned to desired positions in the complex plane furthermore on the basis of the previous result this paper also extended the pi control to the proportional integral derivative pid control it is worth pointing out that this work aims to improve the performance of the closed loop system on the premise of guaranteeing the stability

a computing method on stability intervals of time delay for fractional order retarded systems with commensurate time delays abstract this paper investigates the stability intervals of time delays for fractional order retarded time delay systems by the orlando formula the existence of the crossing frequencies is brought to verify the stability related to the commensurate time delay for each crossing frequency the corresponding critical time delays are determined by the generalized eigenvalues of two matrices constructed by the crossing frequency the commensurate fractional order and the coefficients of the characteristic function the root tendency rt is defined to provide a method to analyze the number of the unstable roots for a given crossing frequency and critical time delay based on the rt values and the number of the unstable roots for fractional order systems with no time delay a computing method on the stability intervals of time delay is proposed in this paper finally a numerical example is offered to validate the effectiveness of this method

modification of mikhaylov criterion for neutral time delay systems the main goal of the technical note is to extend the mikhaylov criterion to the case of neutral time delay systems the modification consists in determining the vertex angle that bounds the argument oscillations of the mikhaylov hodograph at high frequency ranges utilizing the strong stability concept the presented stability criterion is examined from the viewpoint of potential fragility with respect to arbitrarily small delay changes to facilitate the more demanding argument assessment the mikhaylov hodograph is converted to poincareacute like mapping

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

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

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

a universal map of three dominant pole assignment for pid controller tuning a novel dimensionless parameterisation of the three pole assignment for control plants with delay is presented a second order model with input delay is chosen as an approximation of the plant prop

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

lyapunov matrix based necessary and sufficient stability condition by finite number of mathematical operations for retarded type systems abstract a new necessary and sufficient exponential stability condition for systems with multiple delays is presented it is given in terms of a symmetric block matrix uniquely determined by the delay lyapunov matrix a notable feature is that the stability test requires a finite number of mathematical operations

locating all the zeros of an analytic function in one complex variable based on the argument principle we propose an adaptive multilevel subdivision algorithm for the computation of all the zeros of an analytic function f c c within a bounded domain we illustrate the reliability of this method by several numerical examples

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

evolution equations for abstract differential operators we study in this paper the wellposedness and regularity of solutions of evolution equations associated with abstract differential operators on a banach space the results can be applied to many partial differential equations on different function spaces

further remarks on the effect of multiple spectral values on the dynamics of time delay systems application to the control of a mechanical system a question of ongoing interest for linear time delay systems is to determine conditions on the equation parameters that guarantee the exponential stability of solutions in recent works a new interesting property of time delay systems was emphasized as a matter of fact the multiple spectral values for time delay systems was characterized by using a birkhoff vandermonde based approach then a multiplicity induced stability criteria exhibited for reduced order systems scalar delay equations and a special class of second order systems this work further explores such a criteria and shows their applicability to the control of a mechanical system

reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method spectral discretization methods are well established methods for the computation of characteristic roots of time delay systems in this paper a method is presented for computing all characteristic roots in a given right half plane in particular a procedure for the automatic selection of the number of discretization points is described this procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions first a region that contains all desired characteristic roots is estimated second the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate finally the characteristic roots approximations obtained from solving the discretized eigenvalue problem are corrected up to the desired precision by a local method the effectiveness and robustness of the procedure are illustrated with several examples and compared with dde biftool

stability and stabilization through envelopes for retarded and neutral time delay systems this paper deals with a new approach to develop an envelope that engulfs all poles of a time delay system through linear matrix inequalities we are able to determine envelopes for retarded and neutral time delay systems the envelopes proposed are not only tighter than the ones in the literature but with our procedure they can also be applied to verify the stability of the system and design state feedback controllers which cope with design requirements regarding alpha stability

dynamic similarity approach to control system design delayed pid control loop abstractthe paper deals with a novel approach to dimensional analysis based design of control systems the case of delayed pid control loop is used to show the benefits of the dynamic similarity concept in providing the control with response resulting from a constrained integral absolute error iae optimisation the particular aim is to achieve this response in disturbance rejection ability of the control loop to reach a generalised description in evaluating the control loop dynamics a dimensionless model is applied resulting from the dimensional analysis of the control loop parameters by means of originally introduced similarity numbers of laggardness and swingability explicit relationships between the dominant pole coordinates and the pid controller gains are obtained the proposed optimum setting of control loop constrained in damping is referred to the ultimate oscillations of the plant

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

iterative solution of nonlinear equations of the φ strongly accretive type abstract letq 1 and letebe a realq uniformly smooth banach space lett e ebe a lipschitz strongly accretive operator and suppose the equationtx f f e has a solution it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process formula converges strongly to the unique solution of the equationtx f a related result deals with the approximation of fixed points of hemicontractive operators a class of operators which is much more general than the important class of strongly pseudocontractive operators

ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations abstract the purpose of this paper is to revise the definitions of ishikawa and mann iterative processes with errors by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationtx fand the convergence problem of ishikawa and mann iterative sequences for strongly pseudo contractive mappings without the lipschitz condition the results presented in this paper improve and extend the corresponding results in 4 5 7 10 12 15 16 in the more general setting in particular the open problem mentioned by chidume in 5 has been given an affirmative answer

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

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

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

strong convergence of averaged approximants for lipschitz pseudocontractive maps abstract let t be a lipschitzian pseudocontractive self mapping of a closed convex and bounded subset k of a banach space e which is both uniformly convex and q uniformly smooth such that the set f t of fixed points of t is nonempty then f t is a sunny nonexpansive retract of k if u is the sunny nonexpansive retraction of k onto f t is any point of k and an n 0 a real sequence in 0 1 then the sequence xn n 0 in k defined by x n a n 1 a n 1 n 1 j 0 n 1 a j i a j t x n for n 0 1 2 converges strongly to u no compactness assumption is made on k

optimization of the mathcal h _2 norm for single delay systems with application to control design and model approximation we propose a novel approach for the optimization of the mathcal h 2 norm for time delay systems grounded in its characterization in terms of the delay lyapunov matrix we show how the partial derivatives of the delay lyapunov matrix with respect to system or controller parameters can be semianalytically computed by solving a delay lyapunov equation with inhomogeneous terms it allows us to obtain the gradient of the mathcal h 2 norm and in turn to use it in a gradient based optimization framework we demonstrate the potential of the approach on two classes of problems the design of robust controllers and the computation of approximate models of reduced dimension thereby a major advantage is the flexibility in the former class of applications the order or structure of the controller can be prescribed including recently proposed delay based controllers for the latter class of applications approximate models described by both ordinary and delay differential equations e g inhering the structure of the original system can be synthesized