canonical forms for time delay systems polynomial transformations are considered for linear systems with time delays a technique for state reduction is given as well as schemes for obtaining other useful canonical forms the results are achieved using a combination of module theoretic ideas polynomial ring and functional analysis

delay independent stability analysis of linear time delay systems based on frequency discretization this paper studies strong delay independent stability of linear time invariant systems it is known that delay independent stability of time delay systems is equivalent to some frequency dependent linear matrix inequalities to reduce or eliminate conservatism of stability criteria the frequency domain is discretized into several sub intervals and piecewise constant lyapunov matrices are employed to analyze the frequency dependent stability condition applying the generalized kalman yakubovich popov lemma new necessary and sufficient criteria are then obtained for strong delay independent stability of systems with a single delay the effectiveness of the proposed method is illustrated by a numerical example

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

algebraic properties and the finite rank problem for toeplitz operators on the segal bargmann space abstract we study three different problems in the area of toeplitz operators on the segal bargmann space in c n extending results obtained previously by the first author and y l lee and by the second author we first determine the commutant of a given toeplitz operator with a radial symbol belonging to the class s y m 0 c n of symbols having certain growth at infinity we then provide explicit examples of zero products of non trivial toeplitz operators these examples show the essential difference between toeplitz operators on the segal bargmann space and on the bergman space over the unit ball finally we discuss the finite rank problem we show that there are no non trivial rank one toeplitz operators t f for f s y m 0 c n in all these problems the growth at infinity of the symbols plays a crucial role

commuting toeplitz operators on the segal bargmann space abstract consider two toeplitz operators t g t f on the segal bargmann space over the complex plane let us assume that g is a radial function and both operators commute under certain growth condition at infinity of f and g we show that f must be radial as well we give a counterexample of this fact in case of bounded toeplitz operators but a fast growing radial symbol g in this case the vanishing commutator t g t f 0 does not imply the radial dependence of f finally we consider toeplitz operators on the segal bargmann space over c n and n 1 where the commuting property of toeplitz operators can be realized more easily

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

commuting toeplitz operators with quasi homogeneous symbols on the segal bargmann space abstract let t t z l z k with l k n 0 be a toeplitz operator with monomial symbol acting on the segal bargmann space over the complex plane we determine the symbols of polynomial growth at infinity such that t and t z l z k commute on the space of all holomorphic polynomials by using polar coordinates we represent as an infinite sum r e i j j r e i j then we are able to reduce the above problem to the case of quasi homogeneous symbols j e i j we obtain the radial part j r in terms of the inverse mellin transform of an expression which is a product of gamma functions and a trigonometric polynomial if we allow operator symbols of higher growth at infinity we point out that in some of the cases more than one toeplitz operator t j e i j exists commuting with t

easily testable necessary and sufficient algebraic criteria for delay independent stability of a class of neutral differential systems this paper analyzes the eigenvalue distribution of neutral differential systems and the corresponding difference systems and establishes the relationship between the eigenvalue distribution and delay independent stability of neutral differential systems by using the complete discrimination system for polynomials easily testable necessary and sufficient algebraic criteria for delay independent stability of a class of neutral differential systems are established the algebraic criteria generalize and unify the relevant results in the literature moreover the maximal delay bound guaranteeing stability can be determined if the systems are not delay independent stable some numerical examples are provided to illustrate the effectiveness of our results c 2007 elsevier b v all rights reserved

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

commutants of toeplitz operators with radial symbols on the fock sobolev space abstract in the setting of the fock space over the complex plane bauer and lee have recently characterized commutants of toeplitz operators with radial symbols under the assumption that symbols have at most polynomial growth at infinity their characterization states if one of the symbols of two commuting toeplitz operators is nonconstant and radial then the other must be also radial we extend this result to the fock sobolev spaces

lmi characterization of the strong delay independent stability of linear delay systems via quadratic lyapunov krasovskii functionals abstract in this note is proposed an analogue for linear delay systems of the characterization of asymptotic stability of the rational systems by the solvability of associated lyapunov equation it is shown that strong delay independent stability of delay system is equivalent to the feasibility of certain linear matrix inequality lmi related to quadratic lyapunov krasovskii functionals

stability and stabilization of delay differential systems the delay systems considered here are represented by linear delay differential equations the system parameters and the delays are assumed to be imperfectly known the instantaneous state vector is perturbed by a bounded external disturbance vector the problem addressed is that of characterizing conditions that guarantee that the trajectory of the instantaneous state vector remains in a domain defined by a set of symmetrical linear constraints it is shown that the positive invariance property can be used to solve this problem and that positive invariance of a compact domain of the instantaneous state space implies delay independent asymptotic stability of the associated deterministic system the possible use of these results for the control of a multiple delay mimo differential model is then presented finally an example is given

on stabilization by state feedback for neutral differential difference equations in this paper we consider a class of feedback operators for a linear differential difference equation of the neutral type we show that if a system has unstable neutral root chains any stabilizing feedback operator in this class must contain a derivative of the delayed state component under the assumption of exact controllability we construct a stabilizing feedback operator

mellin transform monomial symbols and commuting toeplitz operators abstract in this paper we give necessary and sufficient conditions for a symbol that produces a toeplitz operator on the bergman space that commutes with another such operator whose symbol is a monomial the result is stated in terms of the mellin transform of the symbol as a corollary we show that a toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol

stability of x t ax t bx t tau a stability criterion for linear time delay systems described by a differential difference equation of the form dx t ax t bx t tau is proposed the result obtained includes information on the size of the delay and therefore can be a delay dependent stability condition its relation to existing delay independent stability criteria is also discussed

a simplified derivation of the zeheb walach 2 d stability test with applications to time delay systems zeheb and walach gave a stability test for n d systems for 2 d systems a simpler derivation is presented here using the results of decarlo et al it is then shown how the method may also be used to test for stability independent of delay of retarded time delay systems

stability conditions for linear retarded functional differential equations abstract the purpose of this note is to study the exponential stability for the linear retarded functional differential equation x t 1 0 d x t r where the delay function r 0 is continuous and is of bounded variation on the interval 1 0 it is shown that the spectral limit function for the equation above has a continuous dependence on the pair r the set of all functions of bounded variation for which the equation above is exponentially stable for every delay function r the so called region of stability globally in the delays is a cone therefore for a fixed r the set of all which make our equation exponentially stable that is the region of stability for the delay function r contains a cone a discussion of the characterization of these regions of stability as well as of the largest cone contained in each region of stability for a fixed delay function r is given some remarks are made with respect to a similar question for the equation x t ax t 1 0 d x t r where a is a real n by n matrix is bounded variation on 1 0 and r as before several examples illustrate the results obtained

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

frequency sweeping tests for stability independent of delay in this paper we study the stability properties of linear time invariant delay systems given in a state space form we consider specifically the notion of asymptotic stability independent of delay systems with both commensurate and noncommensurate delays are investigated we present for each class of systems a necessary and sufficient condition in terms of structured singular values and further we demonstrate how these conditions may be extended to study stability independent of delay for uncertain systems our results consist of several frequency sweeping tests that can be systematically implemented and that should complement the previous work

algebraic properties of toeplitz operators on generalized fock spaces on cd abstract we study two problems involving algebraic properties of toeplitz operators on generalized fock spaces on c d with weights of the form z 2 s e z 2 m m 1 s 0 we determine the commutant of a given toeplitz operator with a radial symbol which satisfies certain growth conditions we also discuss the equation t f t g 0 when f or g is radial

stabilization in the presence of an uncertain arbitrarily large delay handling delays in control systems is difficult and is of long standing interest it is well known that given a finite dimensional linear time invariant lti plant and a finite dimensional lti stabilizing controller closed loop stability will be maintained under a small delay in the feedback loop however in some situations there is a large delay perhaps arising from a slow communications link or a large but variable computational delay e g in a system which uses image processing while there are techniques available to design a controller to handle a known delay there is no general theory for designing a controller to handle an arbitrarily large uncertain delay here we prove constructively that given a finite dimensional lti plant and an upper bound on the admissible time delay there exists a linear periodic controller which robustly stabilizes the plant

computation of delay intervals for stability of time delay systems abstract this paper addresses the problem of computing the delay intervals for stability of time delay systems time delays are frequently a source of instability and quite often difficult to estimate the major features of the method are 1 the delay uncertainty is represented by a relative error i e h h 0 1 2 the time delay system is recast by a representation that is compatible with robust control analysis which requires a norm bounded modeling uncertainty and stable transfer function and 3 a new tool i e singular phase value is proposed to find the exact bounds on the delay that sufficiently stabilize the time delay system the method can be used to find the first delay interval for stability but it can also be extended to find other delay intervals using standard robust control tools therefore the proposed method is suitable for computer implementation with a simple modification the proposed method can also be used to test the robust stability of various controller schemes under closed loop conditions

stability criteria for linear time invariant systems with multiple delays two new asymptotic stability criteria for linear time invariant systems with multiple delays are established by using a quadratic inequality lemma together with frequency domain techniques one criterion is for delay dependent stability and another is for delay independent asymptotic stability each of them has only one tuning parameter matrix p 0 as the lyapunov type stability criteria for the corresponding linear time invariant systems without delays do both of the established stability criteria are given in the form of lmi problems and hence are very easy to use in applications in addition it is remarked that the established delay independent asymptotic stability criterion with one tuning parameter matrix p 0 improves and simplifies the existing criterion with more than one tuning parameter matrix in the literature

further results on stability of x t ax t bx t t abstract in this paper it is shown that the restricted region where unstable characteristic roots of time delay systems x t ax t bx t exist could be more accurately estimated and then by applying this key observation an improved delay dependent stability criterion is proposed a simple example is worked out to illustrate the result

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

quantitative robustness measure of uncertain time delay systems via nonsingularity analysis the objective of this paper is to derive an upper bound for the delay time constant such that a linear uncertain time delay system still keeps its asymptotic stability if it is nominally stable and the delay time constant remains smaller than the upper bound the problem is solved by using matrix measure and nonsingularity analysis of matrices and the results are shown to be less conservative than previous results

control of linear systems with fixed noncommensurate point delays the first solution is given to the fundamental open problem of stabilizability and detectability necessary sufficient conditions for internal stabilization by feedback of retarded and a large class of neutral delay differential systems with several fixed noncommensurate point delays using causal compensators observers and state feedback or dynamic output feedback which are also the same type of neutral or retarded delay differential systems with fixed point delays only our results are rank conditions on the system matrices zi f g and zif t h t evaluated at points in the complex plane and are the weakest possible generally applicable sufficient such rank conditions for stabilization of neutral systems in the light of what is known on the stability of such systems these conditions are necessary for most practical purposes the class of systems we consider includes all retarded delay differential systems with noncommensurate fixed point delays in the case of retarded systems these rank conditions are necessary and sufficient conditions for stabilization via compensators which are causal retarded delay differential systems with fixed point delays only these constitute the first full solution of these previously unsolved problems of stabilizability and detectability which together are necessary and sufficient conditions for internal stabilization by feedback for delay differential systems even in the retarded single fixed point delay case an application of our results to a problem of practical importance in control of linear systems with no delays provides a stabilization criterion interesting in itself

commuting toeplitz operators with pluriharmonic symbols on the fock space abstract in the setting of the bergman space over the disk or the ball it has been known that two toeplitz operators with bounded pluriharmonic symbols can semi commute only in the trivial cases in this paper we study the analogues on the fock space over the multi dimensional complex space as is the case in various other settings we are naturally led to the problem of characterizing a certain type of fixed points of the berezin transform for such fixed points we obtain a complete characterization by means of eigenfunctions of the laplacian we also obtain other characterizations in particular it turns out that there are many nontrivial cases on the fock space for semi commuting toeplitz operators with pluriharmonic symbols all in all our results reveal that the situation on the fock space appears to be much more complicated than that on the classical bergman space setting which partly is caused by the unboundedness of the operator symbols some of our results are restricted to the one variable case and the corresponding several variable case is left open

on stability of quasipolynomials with annular uncertainties and interval commensurate delays the paper concerns the hurwitz stability of a family of quasipolynomials with commensurate delays each coefficient of the quasipolynomials belongs to a prescribed annulus in the complex plane and the delay belongs to a prescribed real interval a computationally tractable robust stability criterion is the main result of the paper

observability and related structural results for linear hereditary systems new results on the theory of observability and deterministic observers for linear time invariant systems with delays are presented the results have been established only for the case of point delays which are commensurable however most of the results can be generalized to the situation with distributed delays and to the ease with non commensurate delays

feedback stabilization of linear autonomous time lag systems a stabilization theory which employs well established finite dimensional control system tools is developed for the stabilization of linear autonomous time lag systems the main ideas include 1 a set whose elements are matrices each of which is a left zero of the system characteristic quasi polynomial matrix and 2 a linear transformation which reduces the delay system to a delay free system whose state matrix is a direct sum of n elements of the matrix set where n is some positive integer from the definition of this matrix set it is shown that each of its elements inherits its spectrum from that of the delay system so that by design the system unstable poles may be embedded in the spectrum of the delay free system under the assumption of spectral stabilizability it is then shown how to obtain a stabilizing feedback control law on the basis of the delay free system numerical examples are presented to confirm the theory

criteria for stability in approximate delay systems abstract this paper introduces some results about stability for a class of linear point delayed systems and how to compute these results this work uses some theorems on stability for delayed systems due to mori 1 3 our method consists basically in finding extensions of such results to the case of an approximate representation of the linear delayed system i e under which circumstances stability for an approximate representation of the delayed system implies stability for the exact real delayed system sufficient conditions for asymptotic stability of linear delayed systems are greatly simplified and straightforwardly computed under taylor series representation of the system state equations conditions under which asymptotic stability for the approximate system implies asymptotic stability for the real system are also outlined in the paper

on computing the maximal delay intervals for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems the authors consider delay systems of both retarded and neutral types expressed in state space forms the author s main goal is to provide a computation oriented method for computing the maximal delay intervals over which the systems under consideration maintain stability the author s results show that this can be accomplished by computing the generalized eigenvalues of certain frequency dependent matrices based on these results the author also states a necessary and sufficient condition concerning stability independent of delay for each of the retarded and neutral systems the author s results can be readily implemented and appear suitable for analyzing systems with high dimensions and many delay units

absolute stability in linear delay differential systems ill posedness and robustness the author considers matrix delay differential systems which are polynomial in several delay operators using a necessary and sufficient criterion for stability independent of delay or absolute stability the author shows that system stability for all values of the delay vector lying in a sector will imply absolute stability we then show that absolute stability is ill posed with respect to arbitrarily small perturbations of the delay ratios if a certain extended delay differential system which is formed from the original is not also absolutely stable

the commutants of certain toeplitz operators on weighted bergman spaces abstract for 1 let a 2 be the corresponding weighted bergman space of the unit ball in c n for a bounded measurable function f let t f be the toeplitz operator with symbol f on a 2 this paper describes all the functions f for which t f commutes with a given t g where g z z 1 l 1 z n l n for strictly positive integers l 1 l n or g z z 1 s 1 z n s n h z for non negative real numbers s 1 s n and a bounded measurable function h on 0 1

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

estimates for periodic zakharov shabat operators abstract we consider the periodic zakharov shabat operators on the real line the spectrum of this operator consists of intervals separated by gaps with the lengths g n 0 n z let n be the corresponding effective masses and let h n be heights of the corresponding slits in the quasi momentum domain we obtain a priori estimates of sequences g g n n z n n z h h n n z in terms of weighted l p norms at p 1 the proof is based on the analysis of the quasi momentum as the conformal mapping

a lyapunov theory for linear time delay systems some relationships between a lyapunov based stability test and linear control system models with time delays are explored the main result is a generalization of the result that if a b is a reachable pair of matrices then square a is a stability matrix if and only if there is a positive definite matrix k such that ak ka bb

delay independent stability of higher order systems global asymptotic stability of the equilibrium point of a delayed system given by a higher order delayed differential equation of retarded type with several time varying delays is considered for this purpose the given system is transformed to a multi variable first order system and then the previously obtained stability criterion is applied to this several illustrative examples are presented to show the stability region in the parameter spaces and they are compared to the cases with constant delays

brief paper a discrete delay decomposition approach to stability of linear retarded and neutral systems this paper is concerned with stability of linear time delay systems of both retarded and neutral types by using some new simple quadratic lyapunov krasovskii functionals these lyapunov krasovskii functionals consist of two parts one part comes from some existing lyapunov krasovskii functionals employed in han q l 2005a absolute stability of time delay systems with sector bounded nonlinearity automatica 41 2171 2176 han q l 2005b a new delay dependent stability criterion for linear neutral systems with norm bounded uncertainties in all system matrices international journal of systems science 36 469 475 the other part is constructed by uniformly dividing the discrete delay interval into multiple segments and choosing proper functionals with different weighted matrices corresponding to different segments then using these new simple quadratic lyapunov krasovskii functionals some new discrete delay dependent stability criteria are derived for both retarded systems and neutral systems it is shown that these criteria for retarded systems and neutral systems are always less conservative than the ones in han 2005a and han 2005b cited above respectively numerical examples also show that the results obtained in this paper significantly improve the estimate of the discrete delay limit for stability over some other existing results

zero sets of multiparameter functions and stability of multidimensional systems new theorems which can be used to find the zero set of a multiparameter rational function of a complex variable where the distinguished boundary of the parameters closed domain of definition is composed of piecewise smooth jordan arcs or curves are stated and proved the results are used to derive a new simplified procedure for a multi dimensional stability test presented for the continuous analog case as well as for the discrete digital case now proofs to huang s and strintzis theorems on multidimensional stability are also provided as an application of the theorems on zero sets

asymptotics of spectral quantities of zakharov shabat operators abstract in this paper we provide asymptotic expansions of various spectral quantities of zakharov shabat operators on the circle with new error estimates these estimates hold uniformly on bounded subsets of potentials in sobolev spaces

commutants of separately radial toeplitz operators in several variables if is a bounded separately radial function on the unit ball the toeplitz operator t is diagonalizable with respect to the standard orthogonal basis of monomials on the bergman space given such a we characterize bounded functions for which t commutes with t several examples are given to illustrate our results

a note on dynamic output feedback for linear systems over rings it is shown that two previous regulation techniques one from systems over rings the other from systems over fields can be combined with a slight modification to obtain a dynamic output feedback compensator for regulating stabilizable and detectable systems over rings the advantage of the resulting technique is that it is a more direct and explicit output feedback technique than the available alternative technique of using observers and dynamic state feedback configuration and as such it brings more insight into regulation of linear systems over rings

a method for computing the interval of delay values for which a differential delay system is stable a test for determining the interval of delay values for which delay differential system is asymptotically stable is presented i e an explicit method is given for calculating h such that for 0

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

polynomial two parameter eigenvalue problems and matrix pencil methods for stability of delay differential equations several recent methods used to analyze asymptotic stability of delay differential equations ddes involve determining the eigenvalues of a matrix a matrix pencil or a matrix polynomial constructed by kronecker products despite some similarities between the different types of these so called matrix pencil methods the general ideas used as well as the proofs differ considerably moreover the available theory hardly reveals the relations between the different methods in this work a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem the polynomial two parameter eigenvalue problem of which the quadratic two parameter eigenvalue problem is a special case this framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods we also recognize a few new matrix pencil variants to determine dde stability finally the recognition of the new types of eigenvalue problem opens a door to efficient computation of dde stability

conformal spectral theory for the monodromy matrix for any n x n monodromy matrix we define the lyapunov function which is analytic on an associated n sheeted riemann surface on each sheet the lyapunov function has the standard properties of the lyapunov function for the hill operator the lyapunov function has real or complex branch points which we call resonances we determine the asymptotics of the periodic anti periodic spectrum and of the resonances at high energy we show that the endpoints of each gap are periodic anti periodic eigenvalues or resonances real branch points moreover the following results are obtained 1 we define the quasimomentum as an analytic function on the riemann surface of the lyapunov function various properties and estimates of the quasimomentum are obtained 2 we construct the conformal mapping with imaginary part given by the lyapunov exponent and we obtain various properties of this conformal mapping which are similar to the case of the hill operator 3 we determine various new trace formulae for potentials and the lyapunov exponent 4 we obtain a priori estimates of gap lengths in terms of the dirichlet integral we apply these results to the schrodinger operators and to first order periodic systems on the real line with a matrix valued complex self adjoint periodic potential

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

stability conditions for the general linear difference differential equation with constant coefficients and one constant delay abstract for the application low degree cases of the equations of the title are of special interest lag independent stability conditions are derived for all systems up to degree two inclusive some necessary stability conditions for large lags are expounded in the non neutral case left half planes depending on the lag and containing all zeros of the characteristic function are obtained a sufficient lag independent condition for the whole class considered is found special attention is devoted to the neutral case for that case a collection of disks covering all zeros of the characteristic function is given the result is illustrated by the example of the general neutral characteristic function with coalescing zeros finally zero free regions around the zeros of the coefficient polynomials are derived to derive the results conformal mappings are studied