delayless observers for systems with delay the design of minimal order observers for linear systems having a time delay in the state variables is considered necessary and sufficient conditons are established for the existence of delayless observers for these systems

proper stable bezout factorizations and feedback control of linear time delay systems this paper deals with the existence and construction of proper stable bezout factorizations of transfer function matrices of linear time invariant systems with commensurate time delays existence of factorizations is characterized in terms of spectral controllability or spectral observability of the co canonical or canonical realization of the transfer function matrix an explicit procedure for computing proper stable bezout factorizations is given in terms of a specialized ring of pure and distributed time delays this procedure is utilized to construct finite dimensional stabilizing compensators and to construct feedback systems which assign the characteristic polynomial of the closed loop system

brief paper global stabilization of semilinear systems using switching controls this paper considers feedback stabilization of distributed semilinear systems using switching controls the semilinear processes together with the switched controllers constitute a type of switched nonlinear system three kinds of stabilizabilities namely weak strong and exponential stabilizabilities are investigated one by one sets of sufficient conditions are obtained for each case a necessary condition for strong stabilizability is given the stabilizing control is characterized using a minimization problem applications to parabolic and hyperbolic like equations are considered

criteria for the assessment of random number generators abstract for any finite sequence of pseudo random numbers special simulation problems can be constructed such that this sequence yields useless results in order to study the quality of the result of a simulation the assessment of a pseudorandom number generator based on theoretical considerations is a great advantage several criteria for the assessment of random number generators are known the most important criteria are discussed

stabilisation of the gradient of distributed bilinear systems abstractthe aim of this paper is to investigate the gradient stabilisation of infinite dimensional bilinear systems then under sufficient conditions we establish exponential strong and weak grad

stabilization of second order evolution equations with unbounded feedback with time dependent delay we consider abstract second order evolution equations with unbounded feedback with time varying delay existence results are obtained under some realistic assumptions we prove the exponential decay under some conditions by introducing an abstract lyapunov functional our abstract framework is applied to the wave to the beam and to the plate equations with boundary delays

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

feedback boundary stabilization of wave equations with interior delay in this paper we consider a boundary stabilization problem for the wave equation with interior delay we prove an exponential stability result under some lions geometric condition the proof of the main result is based on an identity with multipliers that allows us to obtain a uniform decay estimate for a suitable lyapunov functional

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

exponential stability of nondissipative linear system in banach space and application to unbounded bilinear systems abstract we consider linear control systems of the form d z t d t a z t b z t where is a real parameter a is the infinitesimal generator of a linear c 0 semigroup of contractions s 0 t on a banach space x and the linear perturbation b is a relatively bounded operator with respect to a we provide necessary and sufficient conditions for exponential stability of the above system our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators applications to hyperbolic like equations are considered

stabilization of distributed control systems with delay abstract in this paper the asymptotic stabilization of linear distributed parameter control systems with delay is considered specifically we are concerned with the class of control systems described by the equation x t a x t l x t b u t where a is the infinitesimal generator of a strongly continuous semigroup on a banach space x assuming appropriate conditions we will show that the usual spectral controllability assumption implies the feedback stabilization of the system applications to systems described by partial differential equations with delay are given

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

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

weak and strong stabilisation of bilinear systems in a banach space abstractin this paper we study the feedback stabilisation of homogeneous bilinear systems on a banach state space sufficient conditions for weak and strong stabilisation are given a decay rate o

on the global finite time stabilization of bilinear systems by homogeneous feedback laws applications to some pde s abstract in this paper the problem of global finite time stabilization of bilinear control systems by means of homogeneous feedback law is investigated we prove under some reasonable assumptions on the operators a and b that continuous bounded and discontinuous unbounded feedbacks stabilize globally in finite time the closed loop system for illustrative the examples of heat schrodinger and transport equations are considered where homogeneous stabilizing feedback is built for these systems

on feedback stabilizability of linear systems with state and input delays in banach spaces the feedback stabilizability of a general class of well posed linear systems with state and input delays in banach spaces is studied in this paper using the properties of infinite dimensional linear systems a necessary condition for the feedback stabilizability of delay systems is presented which extends the well known results for finite dimensional systems to infinite dimensional ones this condition becomes sufficient as well if the semigroup of the delay free system is immediately compact and the control space is finite dimensional moreover under the condition that the banach space is reflexive a rank condition in terms of eigenvectors and control operators is proposed when the delay free state space and control space are all finite dimensional a very compact rank condition is obtained finally the abstract results are illustrated with examples

stabilisation and polynomial decay estimate for distributed semilinear systems in this article we propose a family of feedback controls that guarantee weak and strong stabilisability for distributed semilinear systems in the case of strong stabilisation an optimal decay of the state is estimated applications to wave equations are provided

feedback stabilisation and polynomial decay estimate for time delay bilinear systems in this paper we are concerned with the question of weak and strong stabilisation for distributed bilinear systems with time delay using continuous feedback controls in the case of strong stabili

stationary measures and invariant subsets of homogeneous spaces iii let g be a real lie group be a lattice in g and g be a compactly generated closed subgroup of g if the zariski closure of the group ad g is semisimple with no compact factor we prove that every g orbit closure in g is a finite volume homogeneous space we also establish related equidistribution properties

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

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

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

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

exponential and weak stabilization of constrained bilinear systems this paper considers feedback stabilization for the infinite dimensional bilinear system frac dz t dt az t v t bz t where a is the infinitesimal generator of a linear c 0 semigroup of contractions on a hilbert space h and b h rightarrow h is a linear bounded operator sufficient conditions for exponential and weak stabilization are given a minimization problem is also considered applications to parabolic and hyperbolic equations are presented

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

feedback stabilisation and polynomial decay estimate for distributed bilinear parabolic systems with time delay in this paper we examine the feedback stabilisation for distributed bilinear parabolic systems described by partial differential equation with time delay the stabilisation of a such system reduce

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

spectral decomposition and invariant manifolds for some functional partial differential equations we study the integrodifferential convolution equationddt x x ax x fon 0 x on 0 as well as a nonlinear perturbation of the corresponding homogeneous equation hereais the generator of an analytic semigroup on a hilbert spaceh and and are operator valued dominated measures with values inl h andl d a h respectively under the assumption that the operator given by the laplace transform of the left hand side of the equation is boundedly invertible on some right half plane and on a line in the left half plane parallel to the imaginary axis we decompose the solutions into components with different exponential growth rates we construct projectors onto the stable and unstable subspaces which are then used for the construction of stable and unstable manifolds for the nonlinear equation which can have a fully nonlinear character the results are applied to two equations of parabolic type moreover the spectrum of the generator of the translation semigroup in various weighted spaces is determined including the stable and unstable subspaces of our problem

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

stabilization and decay estimate for distributed bilinear systems abstract in this paper we consider the problem of feedback stabilization for the distributed bilinear system y t ay t u t by t here a is the infinitesimal generator of a linear c0 semigroup of contractions on a hilbert space h and b h h is a linear bounded operator a sufficient condition for feedback stabilization is given and explicit decay estimate is established applications to vibrating systems are presented

a direct approach to compensator design for distributed parameter systems we present a direct approach to finite order compensator design for distributed parameter systems i e one that is not based on reduced order modelling instead we use a parametrization around an initial compensator which displays both controller order and closed loop stability in a convenient way the main result is an existence theorem which holds for a wide class of linear time invariant systems parabolic delay damped hyperbolic the most important assumptions are bounded inputs and outputs finitely many unstable modes completeness of eigenvectors an example is included to illustrate the feasibility of our method for purposes of design

the exact determination of rectangle discrepancy for linear congruential pseudorandom numbers l algorithme presente peut etre utilise dans les cas de grands generateurs avec un temps de calcul tres raisonnable

feedback stabilization of linear retarded systems in banach spaces feedback stabilizability is studied for linear retarded systems in banach spaces under the assumptions that the control is finite dimensional and the corresponding instantaneous free system generates a compact semigroup the rank condition for exponential stabilizability is established based on the stabilizability result due to bhat and wonham the condition is expressed in terms of eigenvectors and controllers and the equivalent spectral mode controllability and observability conditions are given an application is given to a class of parabolic partial functional differential equations with mixed boundary conditions

stabilization of linear distributed control systems with unbounded delay in this paper we study the asymptotic stabilization of linear distributed parameter control systems with unbounded delay assuming that the semigroup of operators associated with the uncontrolled and nondelayed equation is compact and that the phase space is a uniform fading memory space we characterize those systems that can be stabilized using a feedback control as consequence we conclude that every system of this type is stabilizable with an appropriated finite dimensional control

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

stabilization of time delay systems using finite dimensional compensators for linear time invariant systems with one or more noncommensurate time delays necessary and sufficient conditions are given for the existence of a finite dimensional stabilizing feedback compensator in particular it is shown that a stabilizable time delay system can always be stabilized using a finite dimensional compensator the problem of explicitly constructing finite dimensional stabilizing compensators is also considered

finite spectrum assignment for multivariable delay systems in the frequency domain a frequency domain approach to arbitrary finite spectrum assignment for multivariable delay systems is developed and a practical example is presented for illustration

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

convergence of spherical averages for actions of free groups let x v be a probability space and suppose a free group fm with m generators acts on x v by measure preserving transformations let ai am be a set of free generators for fm and let t1 tm x x be transformations corresponding to the generators write t i t 1 for i 1 m and set a m 1 1 m we also have the action fm on l1 x v defined by tg p o tg 1 g e fm consider the set wa of all finite words over the alphabet a

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

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

statistical independence properties of pseudorandom vectors produced by matrix generators abstract the uniformity test and the serial test are standard statistical tests for equidistribution and statistical independence respectively in sequences of uniform pseudorandom numbers we introduce analogues of these tests for sequences of uniform pseudorandom vectors and we apply these tests to matrix generators for uniform pseudorandom vector generation the results show that these generators behave well under these tests provided that the matrix in the generation algorithm is chosen carefully the essential condition is that a certain figure of merit attached to the matrix be large we also prove a general theorem which guarantees the existence of matrices with large figure of merit

random walks on random coset spaces with applications to furstenberg entropy we determine the range of furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces

on stability independent of delay for linear systems a new criterion for asymptotic stability of solutions of certain linear differential difference equations independent of the delay duration is given in terms of solutions of a complex lyapunov matrix equation

partial stabilisation of semilinear systems using bounded controls this paper studies a partial stabilisation problem for distributed semilinear systems this consists of steering only a part of the system s state to its equilibrium constrained controls are proposed to force the desired part of the state to approach the equilibrium necessary and sufficient conditions for weak and strong partial stabilisation are given applications are presented

regulation of linear systems over rings by dynamic output feedback abstract a general algebraic solution to the problem of regulation of linear systems over arbitrary commutative rings by dynamic output feedback is given which extends the theory of regulation for such systems furthermore the notion of row column regularity for polynomial matrices over a class of rings bezout domains is introduced this concept is shown to simplify the regulation problem considerably for these rings compared to the general case

robustness with respect to delays for exponential stability of distributed parameter systems in this paper we address the question of whether the open loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop when the input operator is admissible and the class of possible feedbacks consists of compact operators we find that if a feedback can improve the exponential growth rate then it can do so robustly furthermore we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly even if such feedbacks can improve the exponential growth rate in the absence of delays

stabilization of hereditary distributed parameter control systems abstract in this paper we study the asymptotic stabilization of semilinear distributed parameter control systems with delay assuming that the semigroup of operators associated with the uncontrolled and undelayed equation is compact we prove that a well known rank condition which implies the feedback stabilization for lumped systems can be extended to this type of systems

exponential stabilization of distributed semilinear systems by optimal control this paper studies exponential stabilization of distributed semilinear systems the paper i gives a constrained feedback control that ensure the exponential stabilizability and ii shows that this control is the unique solution of an appropriate minimization problem examples of hyperbolic partial equations are provided

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

semigroups and linear partial differential equations with delay abstract we prove the equivalence of the well posedness of a partial differential equation with delay and an associated abstract cauchy problem this is used to derive sufficient conditions for well posedness exponential stability and norm continuity of the solutions applications to a reaction diffusion equation with delay are given

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

on non exact controllable systems in this paper we are concerned with linear control systems of first or second order defined on banach spaces we established some negative facts for these systems fundamentally related with linear compact operators in particular we show several sufficient conditions of a general character for the non exact controllability or these systems we also show that it is not possible to uniformly stabilize a first order control system with states in a banach space using a compact control feedback this generalizes gibson s 1980 result for systems on hilbert spaces

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

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16

remarks concerning the asymptotic stability and stabilization of linear delay differential equations abstract this note concerns the asymptotic stability for all values of the delays of controlled and uncontrolled linear delay differential equations in the case of uncontrolled systems it is shown by example that structural considerations must be accounted for in the controlled case although arbitrary pole placement may not be possible there is an algorithm which sometimes reduces the spectrum of the feedback system to one which is finite and is contained in the left half plane