on the topology of graph picture spaces abstract we study the space x d g of pictures of a graph g in complex projective d space the main result is that the homology groups with integer coefficients of x d g are completely determined by the tutte polynomial of g one application is a criterion in terms of the tutte polynomial for independence in the d parallel matroids studied in combinatorial rigidity theory for certain special graphs called orchards the picture space is smooth and has the structure of an iterated projective bundle we give a borel presentation of the cohomology ring of the picture space of an orchard and use this presentation to develop an analogue of the classical schubert calculus

control of an unstable reaction diffusion pde with long input delay abstract a smith predictor like design for compensation of arbitrarily long input delays is available for general controllable possibly unstable lti finite dimensional systems such a design has not been proposed previously for problems where the plant is a pde we present a design and stability analysis for a prototype problem where the plant is a reaction diffusion parabolic pde with boundary control the plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay with an unbounded input operator the predictor based feedback design extends fairly routinely within the framework of infinite dimensional backstepping however the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ode the unbounded character of the input operator requires that the stability be characterized in terms of the h 1 rather than the usual l 2 norm of the actuator state the analysis involves an interesting structure of interconnected pdes of parabolic and first order hyperbolic types where the feedback gain kernel for the undelayed problem becomes an initial condition in a pde arising in the compensator design for the problem with input delay space and time variables swap their roles in an interesting manner throughout the analysis

discrete chaos induced by heteroclinic cycles connecting repellers in banach spaces abstract this paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in banach spaces several criteria of chaos are established in general banach spaces and finite dimensional spaces respectively by employing the coupled expansion theory all the maps presented in this paper are proved to be chaotic in the sense of both li yorke and devaney or in the sense of both li yorke and wiggins or in the sense of li yorke an illustrative example is provided with computer simulations

finite spectrum assignment problem for systems with delay in state variables this correspondence is concerned with the finite spectrum assignment problem for linear systems with delay in state variables a systematic procedure for constructing the feedback law is presented on the basis of advanced elementary row operations on matrices with two variables s and z e sh

an observer of systems with delays in state variables the paper is concerned with the design of an observer of systems with delays in state variables and control it is shown on the basis of the finite spectrum assignment theory that if the system is spectrally observable there exists an observer which can estimate the state variables of the system as fast as desired a systematic design procedure of the observer is presented the advantage of the proposed method is that the observer is constructed without preliminary knowledge of the open loop spectrum in contrast with the design method based on the spectral decomposition

finite spectrum assignment of multi input systems with non commensurate delays we present a new framework for finite spectrum assignment for multi input systems with non commensurate delays using an algebraic approach over multidimensional polynomial matrices by focusing on the solvability of a bezout equation over multidimensional polynomial matrices we derive a necessary and sufficient condition for finite spectrum assignability under which a finite number of spectra can be assigned by a control law using a ring of entire functions i e laplace transforms of all exponential time functions with compact support furthermore using a solution to the bezout equation we present a design method for a controller that achieves finite spectrum assignment

on spectral controllability of multi input time delay systems abstract it is proved that under certain verifiable conditions spectral controllability of a linear system with commensurate state delays and with m inputs m 1 implies the existence of an n 1 vector b contained in the image of the control matrix b such that the system with b replaced by b is spectrally controllable both the necessity and the sufficiency of these conditions are shown a procedure to construct b is given and an example is discussed

finite spectrum assignment of systems with general delays this article presents a new framework for dealing with pure and distributed delays simultaneously and impartially as general delays in algebraic finite spectrum assignment the framework based on an important property common to general delays gives a sufficient condition for finite spectrum assignability of scalar systems with several general delays and a design method of a controller achieving finite spectrum assignment especially in algebraic control theory for delay systems pure and distributed delays have never been dealt with simultaneously and impartially the framework is significantly important also for multidimensional system theory because its potential applicability to the discussion on systems with general delays i e the possibility that especially distributed delay operators are used as variables for describing multidimensional systems is clarified for the first time

stabilization of linear systems with non commensurate delays a stabilization method for linear systems with non commensurate delays in both state and control variables is presented this method is obtained by combining the algebraic stabilization method with the finite spectrum assignment method the present method can stabilize systems that algebraic methods cannot

occurrence of chaos in higher dimensional discrete time systems in this paper the occurrence of chaos is considered in two types of lotka volterra systems one is the higher dimensional version of the logistic type system x n 1 ax n 1 x n and the other is the higher dimensional version of the exponential type system x n 1 x n exp a 1 x n some simple methods of detecting chaos are presented in these systems by means of these systems two classes of examples of the higher dimensional benhabib day model with chaotic consumer behavior are presented

chaos and asymptotical stability in discrete time neural networks abstract this paper aims to theoretically prove by applying marotto s theorem that both transiently chaotic neural networks tcnn and discrete time recurrent neural networks drnn have chaotic structure a significant property tcnn and drnn is that they have only one bounded fixed point when absolute values of the self feedback connection weights in tcnn and the difference time in drnn are sufficiently large we show that this unique fixed point can actually evolve into a snap back repeller which generates chaotic structure if several conditions are satisfied on the other hand by using the lyapunov functions we also derive sufficient conditions on asymptotical stability for symmetrical versions of both tcnn and drnn under which tcnn and drnn asymptotically converge to a fixed point furthermore related bifurcations are also considered in this paper since both tcnn and drnn are not special but simple and general the obtained theoretical results hold for a wide class of discrete time neural networks to demonstrate the theoretical results of this paper better several numerical simulations are provided as illustrating examples

finite spectrum assignment problem for systems with multiple commensurate delays in state variables abstract this paper is concerned with the finite spectrum assignment problem for systems with multiple commensurate delays in state variables the necessary and sufficient condition for finite spectrum assignability is clarified a systematic design procedure which can straightforwardly yield a control law is presented

feedback controllers for a wind tunnel model involving a delay analytical design and numerical simulation an application of a finite spectrum assignment method for time delay systems to a feedback control of mach number in a wind tunnel is presented the linearized model of mach number control is a system of three state equations with a delay in one of the state variables the proposed feedback is a linear combination of state variables and weighted integrals of some of the state variables over a period equal to time delay the spectrum of the closed loop system is finite and consists of three eigenvalues that can be placed arbitrarily four possible variants of the feedback control law are presented the calculation of feedback coefficients is very simple which makes it possible to update the controller s parameters on line with the changes of the operating point an extensive numerical simulation of the system dynamics and the feedback controllers was done and led to several conclusions in particular a very good settling time under problem constraints has been obtained and several nice features of the proposed feedback controller have been demonstrated

finite spectrum assignment and observer for multivariable systems with commensurate delays this paper is concerned with control of multivariable systems with commensurate delays the purpose of this paper is to enlarge the class of finite spectrum assignable systems to spectrally controllable multivariable systems with commensurate delays by overcoming an infinite number of eigenvalues which are contained in the system and make control very complex it is proved that if the system is spectrally controllable there is a delayed feedback matrix such that the closed loop system is spectrally controllable through a single input combining this result with previous ones about single input systems it is verified that spectral controllability is equivalent to finite spectrum assignability for multivariable systems with commensurate delays an observer for the system with delayed outputs is presented by using these results the multivariable system can be regulated as desired without preliminary knowledge of open loop eigenvalues and a stability test of transcendental functions whenever the system is spectrally controllable and spectrally observable

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

period doubling and quadrupling for some second order difference schemes may exhibit stable n cycles that bifurcate into stable 2n cycles as a increases creating period doubling bifurcation sequences each beginning with a cycle of odd period these dynamics were first described by may i 21 and have since been investigated by many others see 3 61 for example a number of universal features of this bifurcation process have been observed primarily for quadratic maps 7 lo by comparison much less is known concerning the dynamics of higher order nonlinear schemes although there have been many numerical investigations of specific second order equations see i l 181 for example the full range of dynamics possible for such problems has not yet been determined in this work we investigate some general second order nonlinear difference schemes sufficient conditions are derived for the existence stability and bifurcations of periodic cycles for certain classes of such problems firstly the correspondence between periodic cycles of 1 and those of the second order problem x k l f xk l 3

snapback repellers and homoclinic orbits for multi dimensional maps abstract marotto extended li yorke s theorem on chaos from one dimension to multi dimension through introducing the notion of snapback repeller in 1978 due to a technical flaw he redefined snapback repeller in 2005 to validate this theorem this presentation provides two methodologies to facilitate the application of marotto s theorem the first one is to estimate the radius of repelling neighborhood for a repelling fixed point this estimation is of essential and practical significance as combined with numerical computations of snapback points secondly we propose a sequential graphic iteration scheme to construct homoclinic orbit for a repeller this construction allows us to track the homoclinic orbit applications of the present methodologies with numerical computation to a chaotic neural network and a predator prey model are demonstrated

observers for discrete time systems with multiple delays a systematic approach to design observers for discrete time systems with multiple delays under general conditions is presented the main feature of the proposed technique is that necessary and sufficient conditions for asymptotic stability are derived while the observer s order is independent from the number of delays and is equal to the state dimension to illustrate the efficiency of the proposed technique two numerical examples are provided

finite spectrum assignment of linear systems with a class of non commensurate delays this paper is concerned with finite spectrum assignment of linear systems with non commensurate delays it is proved that if a feedforward pass between states variables does not have multi non commensurate delays in parallel but only one or no delay in series then the system is finite spectrum assignable even though the feedback pass has multi non commensurate delays in parallel

stability of symbolic embeddings for difference equations and their multidimensional perturbations abstract in this paper we study complexity of solutions of a high dimensional difference equation of the form x i m x i 1 x i x i 1 x i n 0 i z where is a c 1 function from r l m n 1 to r l our main result provides a sufficient condition for any sufficiently small c 1 perturbation of to have symbolic embedding that is to possess a closed set of solutions that is invariant under the shift map such that the restriction of the shift map to is topologically conjugate to a subshift of finite type the sufficient condition can be easily verified when depends on few variables including the logistic and henon families to prove the result we establish a global version of the implicit function theorem for perturbed equations the proof of the main result is based on the brouwer fixed point theorem and the proof of the global implicit function theorem is based on the contraction mapping principle and other ingredients our novel approach extends results in 2 3 8 15 21

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

the c1 persistence of heteroclinic repellers in rn abstract in this paper we show that if f is a c 1 map from r n into itself and has heteroclinic repellers then g also has heteroclinic repellers with f g c 1 being small enough and exhibits devaney s chaos the results demonstrate c 1 structural stability of heteroclinic repellers in euclidean spaces in the end we give some examples to illustrate our theoretical results

recursive predictor design for state and output feedback controllers for linear time delay systems this paper presents a recursive method to design state and output feedback controllers for mimo block feedforward linear systems with delays in the inputs outputs and interconnections between the blocks the resulting controller is of predictor type which means that it contains finite integrals over past state and input values the method is a generalization of the well known model reduction approach for systems with input delay a recursive procedure replaces delay terms with non delay ones step by step from the top of the cascade structure down controller gains are computed for the proxy system without delays while the construction guarantees the same closed loop poles for the delay system and the proxy one the observer is designed by applying the duality argument and the separation principle is also shown to apply

an eigenvalue based approach for the stabilization of linear time delay systems of neutral type an eigenvalue based approach for the stabilization of linear neutral functional differential equations is presented which extends the recently developed continuous pole placement method for delay equations of retarded type the approach consists of two steps first the stability of the associated difference equation is determined and a procedure is applied to compute the supremum of the real parts of its characteristic roots which corresponds to computing the radius of the essential spectrum of the solution operator of the neutral equation no restrictions are made on the dimension of the system and the number of delays also the effect of small delay perturbations is explicitly taken into account as a result of this first step the stabilization problem of the neutral equation is reduced to a problem involving only a finite number of characteristic roots as a second step stabilization is achieved by shifting the rightmost or unstable characteristic roots to the left half plane in a quasi continuous way by applying small changes to the controller parameters and meanwhile monitoring other characteristic roots with a large real part a numerical example is presented

finite dimensional approximate observers for a class of transportation systems finite dimensional approximate observers are designed for linear functionals of the state of distributed parameter transportation systems a systematic design procedure is presented in which the observer parameters can be optimized for maximal accuracy of observation the maximal attainable accuracy can be improved by increasing either the observer s dynamical order or the number of point and or distributed measurements the eigenvalue spectrum of the observation error dynamics is freely assignable

heteroclinic cycles and chaos in a class of 3d three zone piecewise affine systems abstract little seems to be known about heteroclinic cycles and chaos in three dimensional piecewise smooth dynamical systems with two or more discontinuous boundaries this article presents a new class of three dimensional three zone piecewise affine systems with two discontinuous boundaries and provides some criteria for the existence of heteroclinic cycles in the following cases i one saddle point and two focus points ii two saddle points and one focus point and iii three saddle points moreover sufficient conditions for the existence of chaos are established finally two numerical examples are provided to show the feasibility of our theoretical approach

new coprimeness over multivariable polynomial matrices and its application to control delay systems abstract this paper presents a new notion of coprimeness over multivariable polynomial matrices where a single variable is given priority over the remaining variables from a characterization through a set of common zeros of the minors it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable an application of the presented coprimeness to control systems with non commensurate delays and finite spectrum assignment is also presented because the presented coprimeness is stronger than minor coprimeness non commensurate delays are difficult to deal with in algebraic control theory the rational ratio condition between delays which can reduce non commensurate delays to commensurate delays proves to be both powerful and practical concept in algebraic control theory for delay systems

dynamic compensation of siso systems over a principal ideal domain the polynomial equation approach and some concepts from algebraic geometry are employed in compensator design for a class of linear single input single output siso systems with coefficients from a principal ideal domain pid

output stabilization of time varying input delay systems using interval observation technique the output stabilization problem for a linear system with an unknown bounded time varying input delay is considered the interval observation technique is applied in order to obtain guaranteed interval estimate of the system state the procedure of the interval observer synthesis uses lower and upper estimates of the unknown delay and requires to solve a special silvester s equation the interval predictor is introduced in order to design a linear stabilizing feedback the control design procedure is based on linear matrix inequalities lmi the theoretical results are supported by numerical simulations and compared with a control design scheme based on a luenberger like observer

a necessary condition for spectral controllability of delay systems on the basis of finite laplace transforms abstract this paper is concerned with spectral controllability of systems with multiple commensurate delays in the state variables a necessary condition for spectral controllability is presented on the basis of finite laplace transforms it is closely related to the finite spectrum assignment problem physical meanings of it are also discussed

an internal state decomposition approach to a discrete time h2 control problem with irrational input non minimum phase property abstract an internal state decomposition approach to a class of discrete time h 2 control problems with irrational non minimum phase property at the control input is proposed this is a generalization of the state variable transformation to derive the discrete time smith predictor for a single input delay system the h 2 optimal controller is derived using the orthogonality principle in h 2 space as a direct consequence of the state decomposition the characterization of the optimal cost by the previous and alternative truncation operators is also discussed

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

snapback repellers as a cause of chaotic vibration of the wave equation with a van der pol boundary condition and energy injection at the middle of the span a wave equation on a one dimensional interval i has a van der pol type nonlinear boundary condition at the right end at the left end the boundary condition is fixed at exactly the midpoint of the interval i energy is injected into the system through a pair of transmission conditions in the feedback form of anti damping we wish to study chaotic wave propagation in the system a cause of chaos by snapback repellers has been identified these snapback repellers are repelling fixed points possessing homoclinic orbits of the non invertible map in 2d corresponding to wave reflections and transmissions at respectively the boundary and the middle of the span points existing literature f r marotto j math anal appl 63 199 223 1978 on snapback repellers contains an error we clarify the error and give a refined theorem that snapback repellers imply chaos numerical simulations of chaotic vibration are also illustrated

on controllability and observability of time delay systems this paper deals with controllability and observability properties of time delay systems in the state space r n times l p in particular we prove the equivalence of spectral controllability and approximate null controllability moreover it is shown that the necessary condition for approximate f controllability obtained recently by manitius is also sufficient and a verifiable and matrix type criterion for f controllability is derived for systems with commensurate delays finally we introduce the dual observability notion of approximate controllability and prove that the control system is exactly null controllable if and only if the transposed delay system tis continuously finally observable

on necessary and sufficient conditions for regulation of linear systems over rings necessary and sufficient conditions are given for regulation of linear systems over rings using observers and causal dynamic state feedback systems with the polynomial fractional representation property the results are then used to obtain stabilizability conditions for systems over integers delay differential systems systems over polynomial rings and to obtain conditions to make a 2 d system nonrecursive

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

continuous measures in one dimension families of unimodal maps satisfying 1 t 1 1 1 1 witht 1 1 and t 1 1 2 t x isc2 inx2 and and symmetric inx 3 t0 0 0 t1 0 1 with frac d d lambda t 0 0 are considered the results of guckenheimer 1982 are extended to show that there is a positive measure of for whicht has a finite absolutely continuous invariant measure

brief finite spectrum assignment for linear systems with non commensurate time delays this paper studies finite spectrum assignment for linear scalar systems with non commensurate time delays based on a practically important class of finite laplace transforms the finite spectrum assignability can be reduced to the solvability of a bezout equation over a multivariable polynomial ring with coefficients in the class of finite laplace transforms it is shown that in the non commensurate delay case spectral canonicity is not sufficient for the finite spectrum assignability

observers design for linear time delay systems abstract in this contribution we propose a simple and useful approach to design observers for discrete time systems with delays in the state and output variables the main feature is that the necessary and sufficient conditions for the existence of such observer are derived the stability analysis is performed by the lyapunov approach where the obtained conditions are expressed in terms of a modified riccati equation numerical examples are provided to show efficiency of the proposed observer

output feedback stabilization of delay systems via generalization of the transformation method the left characteristic matrix equation l c m e of a delay system has been employed to develop a state feedback stabilization theory for delay systems analogously the right characteristic matrix equation r c m e has led to the development of an observer theory for this class of systems the characteristic matrix equation approach however only permits the relocation of integral multiples of n poles where n represents the system order this restriction is removed in this work through a generalization of the l c m e and r c m e in a manner which permits the stabilization of an arbitrary but finite number of unstable modes the resulting state feedback controller and state estimator permit the output feedback stabilization of the most general linear autonomous delay system under the minimal assumptions of spectral stabilizability and detectability

on the spectral controllability of delay differential equations a computable necessary and sufficient condition is proved for the spectral controllability of a class of linear hereditary systems and a counterexample is given to a previously published result

a simple proof for persistence of snap back repellers abstract in this article we show that if f has a snap back repeller then any small c 1 perturbation of f has a snap back repeller and hence has li yorke chaos and positive topological entropy by simply using the implicit function theorem we also give some examples

geometry of graph varieties first published in transactions of the american mathematical society in volume 355 2003 4151 4169 published by the american mathematical society

on matrix fraction representations for linear systems over commutative rings this paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory these representations are then used in establishing new results for various classes of systems including split systems the relevance of these results to the regulation of various nonclassical classes of linear systems for example delay differential systems two dimensional systems etc is also discussed

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

further study of spectral controllability of systems with multiple commensurate delays in state variables the paper presents a necessary and sufficient condition for spectral controllability of systems with multiple commensurate delays in state variables on the basis of finite laplace transforms

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3