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 test and stability conditions for delay differential systems in this paper a stability test method is demonstrated and explicit stability conditions for the first order retarded delay differential system with constant coefficients in the delay interval are obtained this paper shows that under suitable conditions a class of first order delay systems can be alternately stable as the delay h

statistical mechanics of stochastic neural networks relationship between the self consistent signal to noise analysis thouless anderson palmer equation and replica symmetric calculation approaches we study the statistical mechanical aspects of stochastic analog neural network models for associative memory with correlation type learning we take three approaches to derive the set of the order parameter equations for investigating statistical properties of retrieval states the self consistent signal to noise analysis scsna the thouless anderson palmer tap equation and the replica symmetric calculation on the basis of the cavity method the scsna can be generalized to deal with stochastic networks we establish the close connection between the tap equation and the scsna to elucidate the relationship between the onsager reaction term of the tap equation and the output proportional term of the scsna that appear in the expressions for the local fields

asynchronous iterative water filling for gaussian frequency selective interference channels this paper considers the maximization of information rates for the gaussian frequency selective interference channel subject to power and spectral mask constraints on each link to derive decentralized solutions that do not require any cooperation among the users the optimization problem is formulated as a static noncooperative game of complete information to achieve the so called nash equilibria of the game we propose a new distributed algorithm called asynchronous iterative water filling algorithm in this algorithm the users update their power spectral density psd in a completely distributed and asynchronous way some users may update their power allocation more frequently than others and they may even use outdated measurements of the received interference the proposed algorithm represents a unified framework that encompasses and generalizes all known iterative water filling algorithms e g sequential and simultaneous versions the main result of the paper consists of a unified set of conditions that guarantee the global converge of the proposed algorithm to the unique nash equilibrium of the game

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

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

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

joint sensing and power allocation in nonconvex cognitive radio games quasi nash equilibria in this paper we propose a novel class of nash problems for cognitive radio cr networks composed of multiple primary users pus and secondary users sus wherein each su player competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration the detection thresholds and the vector power allocation over a multichannel link in addition to power budget constraints several deterministic or probabilistic interference constraints can be accommodated in the proposed general formulation such as constraints on the maximum individual aggregate probabilistic interference tolerable from the pus to keep the optimization as decentralized as possible global interference constraints when present are imposed via pricing the prices are thus additional variables to be optimized the resulting players optimization problems are nonconvex and there are price clearance conditions associated with the nonconvex global interference constraints to be satisfied by the equilibria of the game which make the analysis of the proposed game a challenging task none of classical results in the game theory literature can be successfully applied to deal with the nonconvexity of the game we introduce a relaxed equilibrium concept the quasi nash equilibrium qne and study its main properties performance and connection with local nash equilibria quite interestingly the proposed game theoretical formulations yield a considerable performance improvement with respect to current centralized and decentralized designs of cr systems which validates the concept of qne

dual methods for nonconvex spectrum optimization of multicarrier systems the design and optimization of multicarrier communications systems often involve a maximization of the total throughput subject to system resource constraints the optimization problem is numerically difficult to solve when the problem does not have a convexity structure this paper makes progress toward solving optimization problems of this type by showing that under a certain condition called the time sharing condition the duality gap of the optimization problem is always zero regardless of the convexity of the objective function further we show that the time sharing condition is satisfied for practical multiuser spectrum optimization problems in multicarrier systems in the limit as the number of carriers goes to infinity this result leads to efficient numerical algorithms that solve the nonconvex problem in the dual domain we show that the recently proposed optimal spectrum balancing algorithm for digital subscriber lines can be interpreted as a dual algorithm this new interpretation gives rise to more efficient dual update methods it also suggests ways in which the dual objective may be evaluated approximately further improving the numerical efficiency of the algorithm we propose a low complexity iterative spectrum balancing algorithm based on these ideas and show that the new algorithm achieves near optimal performance in many practical situations

distributed computation of equilibria in misspecified convex stochastic nash games the distributed computation of nash equilibria is assuming growing relevance in engineering where such problems emerge in the context of distributed control accordingly we present schemes for computing equilibria of two classes of static stochastic convex games complicated by a parametric misspecification a natural concern in the control of large scale networked engineered system in both schemes players learn the equilibrium strategy while resolving the misspecification 1 monotone stochastic nash games we present a set of coupled stochastic approximation schemes distributed across agents in which the first scheme updates each agent s strategy via a projected stochastic gradient step whereas the second scheme updates every agent s belief regarding its misspecified parameter using an independently specified learning problem we proceed to show that the produced sequences converge in an almost sure sense to the true equilibrium strategy and the true parameter respectively surprisingly convergence in the equilibrium strategy achieves the optimal rate of convergence in a mean squared sense with a quantifiable degradation in the rate constant 2 stochastic nash cournot games with unobservable aggregate output we refine 1 to a cournot setting where we assume that the tuple of strategies is unobservable while payoff functions and strategy sets are public knowledge through a common knowledge assumption by utilizing observations of noise corrupted prices iterative fixed point schemes are developed allowing for simultaneously learning the equilibrium strategies and the misspecified parameter in an almost sure sense

optimal wideband spectrum sensing framework for cognitive radio systems an optimal wideband spectrum sensing framework which identifies secondary transmission opportunities over multiple nonoverlapping narrowband channels is presented the framework which is referred to as multiband sensing time adaptive joint detection improves the overall secondary user performance while protecting the primary network and keeping the harmful interference below a desired low level considering a periodic sensing scheme the detection problem is formulated as a joint optimization problem to maximize the aggregate achievable secondary throughput capacity given a bound on the aggregate interference imposed on the primary network it is demonstrated that the problem can be solved by convex optimization if certain practical constraints are applied simulation results attest that the proposed wideband spectrum sensing framework achieves superior performance compared to contemporary frameworks an efficient iterative algorithm which solves the optimization problem with much lower complexity compared to other numerical methods is presented it is established that the iteration complexity and the complexity per iteration of the proposed algorithm increases linearly as the number of optimization variables i e the number of narrowband channels increases the algorithm is evaluated via simulation and is shown to obtain the optimal solution very quickly and efficiently

existence of generalized equilibria we consider the problem of the existence of equilibria generalized equilibria and 8xed points of a correspondence f de8ned on a compact subset m of rn with values in rn when the set m is neither assumed to be convex nor smooth 1 the general framework of the article is the following let f be a correspondence from m to rn that is a map from m to the set of all the subsets of rn the correspondence f is said to be upper semicontinuous u s c if the set x m f x v is open in m for every open set v rn let n be a normal cone correspondence i e at this stage n is a correspondence from m to rn such that for every x m n x

stability analysis and stability conditions in the delay intervals for second order delay systems abstract in tills paper the stability and the stability conditions of a number of second order retarded delay linear differential systems are discussed the paper shows especially that the system which has stationary oscillation when without delay can become asymptotically stable in the large in certain suitable delay intervals

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

distributed algorithms for aggregative games on graphs we consider a class of nash games termed as aggregative games being played over a networked system in an aggregative game a player s objective is a function of the aggregate of all the players decisions every player maintains an estimate of this aggregate and the players exchange this information with their local neighbors over a connected network we study distributed synchronous and asynchronous algorithms for information exchange and equilibrium computation over such a network under standard conditions we establish the almost sure convergence of the obtained sequences to the equilibrium point we also consider extensions of our schemes to aggregative games where the players objectives are coupled through a more general form of aggregate function finally we present numerical results that demonstrate the performance of the proposed schemes

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

spatial neuron model with two parameter ornstein uhlenbeck input current abstract we consider a new and extended spatial neuron model in which the neuronal electrical depolarization from resting level satisfies a cable partial differential equation the synaptic input current is also a function of space and time and satisfies a first order linear partial differential equation driven by a two parameter random process a natural choice for these random input processes is to make them two parameter poisson processes for both excitation and inhibition for such inputs the mean subthreshold voltage is found in the case of an infinite cable and of finite cables assuming uniform amplitudes and rates exact expressions are obtained in the case of particular boundary conditions we then consider a diffusion approximation and show that the membrane current is a two parameter ornstein uhlenbeck process whose statistical properties are derived using representations for the voltage in terms of stochastic integrals in the plane we find in the case of finite space intervals the mean variance and covariance of the subthreshold voltage for large times the voltage process is shown to be covariance stationary in time and the corresponding spectral density is found and compared with the result for a purely two parameter white noise driven cable the limiting white noise case is obtained from the extended model as the decay parameter becomes infinite finally we develop useful simulation methods for the solution of the extended spatial model using properties of stochastic integrals involving eigenfunctions to obtain one dimensional representations which are easily implemented

variational convergence of bifunctions motivating applications it is shown that a number of variational and equilibrium problems can be cast as finding the maxinf points or minsup points of bivariate functions for short bifunctions these problems include linear and nonlinear complementarity problems fixed points variational inequalities inclusions noncooperative games and walras and nash equilibrium problems one appeals to the theory of lopsided convergence for bifunctions to derive stability results for each one of these problems

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

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

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

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

optimal linear precoding strategies for wideband noncooperative systems based on game theory part i nash equilibria in this two part paper we propose a decentralized strategy based on a game theoretic formulation to find out the optimal precoding multiplexing matrices for a multipoint to multipoint communication system composed of a set of wideband links sharing the same physical resources i e time and bandwidth we assume as optimality criterion the achievement of a nash equilibrium and consider two alternative optimization problems 1 the competitive maximization of mutual information on each link given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies and 2 the competitive maximization of the transmission rate using finite order constellations under the same constraints as above plus a constraint on the average error probability in this first part of the paper we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies then we prove that the optimal precoding multiplexing scheme for both games leads to a channel diagonalizing structure so that both matrix valued problems can be recast in a simpler unified vector power control game with no performance penalty thus we study this simpler game and derive sufficient conditions ensuring the uniqueness of the nash equilibrium interestingly although derived under stronger constraints incorporating for example spectral mask constraints our uniqueness conditions have broader validity than previously known conditions finally we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a pareto optimal centralized scheme to reach the nash equilibria of the game in part ii we propose alternative distributed algorithms along with their convergence conditions

nonconvex games with side constraints this paper develops an optimization based theory for the existence and uniqueness of equilibria of a noncooperative game wherein the selfish players optimization problems are nonconvex and there are side constraints and an associated price clearance to be satisfied by the equilibria a new concept of equilibrium for such a nonconvex game which we term a quasi nash equilibrium qne is introduced as a solution of the variational inequality vi obtained by aggregating the first order optimality conditions of the players problems while retaining the convex constraints if any in the defining set of the vi under a second order sufficiency condition from nonlinear programming a qne becomes a local nash equilibrium of the game uniqueness of a qne is established using a degree theoretic proof under a key boundedness property of the karush kuhn tucker multipliers of the nonconvex constraints and the positive definiteness of the hessians of the players lagrangian functions we establish the single valuedness of the players best response maps from which the existence of a nash equilibrium ne of the nonconvex game follows we also present a distributed algorithm for computing an ne of such a game and provide a matrix theoretic condition for the convergence of the algorithm an application is presented that pertains to a special multi leader follower game wherein the nonconvexity is due to the followers equilibrium conditions in the leaders optimization problems another application to a cognitive radio paradigm in a signal processing game is described in detail in g scutari and j s pang ieee trans inform theory submitted j s pang and g scutari joint ieee trans signal process submitted

stability and the matrix lyapunov equation for delay differential systems abstract asymptotic stability independent of the delay of linear differential delay systems with commensurate and non commensurate delays is analysed using 2 d two dimensional and n d n dimensional state space models sufficient conditions for stability are obtained in terms of the frequency dependent 1 d one dimensional lyapunov equations 2 d and n d lyapunov equations with constant parameters are derived using the discrete positive real lemma and the continuous bounded real lemma

real and complex monotone communication games noncooperative game theoretic tools have been increasingly used to study many important resource allocation problems in communications networking smart grids and portfolio optimization in this paper we consider a general class of convex nash equilibrium problems neps where each player aims at solving an arbitrary smooth convex optimization problem differently from most of current works we do not assume any specific structure for the players problems and we allow the optimization variables of the players to be matrices in the complex domain our main contribution is the design of a novel class of distributed asynchronous best response algorithms suitable for solving the proposed neps even in the presence of multiple solutions the new methods whose convergence analysis is based on variational inequality vi techniques can select among all the equilibria of a game those that optimize a given performance criterion at the cost of limited signaling among the players this is a major departure from existing best response algorithms whose convergence conditions imply the uniqueness of the ne some of our results hinge on the use of vi problems directly in the complex domain the study of these new kind of vis also represents a noteworthy innovative contribution we then apply the developed methods to solve some new generalizations of single input single output siso and multiple input multiple output mimo games in cognitive radio systems showing a considerable performance improvement over classical pure noncooperative schemes

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

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

design of cognitive radio systems under temperature interference constraints a variational inequality approach the concept of cognitive radio cr has recently received great attention from the research community as a promising paradigm to achieve efficient use of the frequency resource by allowing the coexistence of licensed primary and unlicensed secondary users in the same bandwidth in this paper we propose a novel nash equilibrium ne problem to model concurrent communications of cognitive secondary users who compete against each other to maximize their information rate the formulation contains constraints on the transmit power and possibly spectral masks as well as aggregate interference tolerable at the primary users receivers the coupling among the strategies of the players due to the interference constraints presents a new challenge for the analysis of this class of nash games that cannot be addressed using the game theoretical models proposed in the literature for this purpose we need the framework given by the more advanced theory of finite dimensional variational inequalities vi this provides us with all the mathematical tools necessary to analyze the proposed ne problem e g existence and uniqueness of the solution and to devise alternative distributed algorithms along with their convergence properties

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

joint sensing and power allocation in nonconvex cognitive radio games nash equilibria and distributed algorithms in this paper we propose a novel class of nash problems for cognitive radio cr networks modeled as gaussian frequency selective interference channels wherein each secondary user su competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration the detection thresholds and the vector power allocation the proposed general formulation allows us to accommodate several transmit power and deterministic probabilistic interference constraints such as constraints on the maximum individual and or aggregate probabilistic interference tolerable at the primary receivers to keep the optimization as decentralized as possible global coupling interference constraints are imposed by penalizing each su with a set of time varying prices based upon his contribution to the total interference the prices are thus additional variable to optimize the resulting players optimization problems are nonconvex moreover there are possibly price clearing conditions associated with the global constraints to be satisfied by the solution all this makes the analysis of the proposed games a challenging task none of classical results in the game theory literature can be successfully applied the main contribution of this paper is to develop a novel optimization based theory for studying the proposed nonconvex games we provide a comprehensive analysis of the existence and uniqueness of a standard nash equilibrium devise alternative best response based algorithms and establish their convergence some of the proposed algorithms are totally distributed and asynchronous whereas some others require limited signaling among the sus in the form of consensus algorithms in favor of better performance overall they are thus applicable to a variety of cr scenarios either cooperative or noncooperative which allows the sus to explore the existing tradeoff between signaling and performance

time to first spike in stochastic hodgkin huxley systems abstract the time to first spike is an experimentally observed quantity in laboratory experiments in the auditory somatic and visual sensory modalities the times of first spikes in the corresponding cortical neurons have been implicated as coding much of the information about stimulus properties we describe an analytical approach for determining the time to first spike from a given initial state which may be applied to a general nonlinear stochastic model neuron we illustrate with a standard hodgkin huxley model with a gaussian white noise input current whose drift parameter is and whose variance parameter is partial differential equations pdes of second order are obtained for the first two moments of the time taken for the depolarization to reach a threshold value from rest state as functions of the initial values simulation confirms that for small noise amplitudes a 2 component model is reasonably accurate for small values of the noise parameter including the deterministic case 0 perturbation methods are used to find the moments of the firing time and the results compare favorably with those from simulation the approach is accurate for almost all when is above threshold for action potentials in the absence of noise and over a considerable range of values of when is as small as 2 the same methods may be applied to models similar to hodgkin huxley which involve channels for additional or different ionic currents

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