functional differential equations with unbounded delay in a banach space reference l contains an extensive survey of work in the area of equations with unbounded delay a short history is included along with discussions of applications of such equations recent theoretical results important outstanding problems and a large bibliography an attempt has been made in this paper to make the notation consistent with the notation used in l let b denote a banach space and let t be a fixed positive number this paper deals with functional differential equations with unbounded delay of the form

asymmetricalgebraicriccatiequation ahomeomorphicparametrizationofthesetof solutions abstract in this paper asymmetric algebraic riccati equations are analyzed in particular we derive a new parametrization of the set of solutions generalizing on the symmetric case the proposed parametrization is obtained in terms of pairs of invariant subspaces of two related feedback matrices moreover the connection is clarified between the new parametrization and the classical homeomorphic one based on graph invariant subspaces of the pseudo hamiltonian matrix associated with the equation we finally show that also the newly introduced parametrization is given by a homeomorphic map

perturbation expansions for invariant subspaces abstract this paper applies a technique previously described by the author for obtaining perturbation expansions and k th order perturbation estimations for invariant subspaces of a matrix

jordan type inequalities involving the bessel and modified bessel functions in this work two general forms of jordan s inequalities for the bessel and modified bessel functions are established and proved by using l hospital s rule for monotonicity and some properties of the spherical bessel and the modified spherical bessel functions of the first kind the applications of the results above give two new infinite series for the bessel and modified bessel functions

uniform asymptotic stability of abstract functional differential equations abstract this paper considers a class of abstract functional differential equations by using razumikhin technique we establish several stability criteria in terms of two measures which improve some earlier results found in the literature several examples are also worked out to demonstrate the advantages of our results

on impulsive functional differential inclusions with hille yosida operators in banach spaces abstract we consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a banach space assuming that its linear part is a non densely defined hille yosida operator we assume that the multivalued nonlinearity of upper caratheodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness we apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results the application to an optimization problem for an impulsive feedback control system is given

existence of a periodic solution for some partial functional differential equations with infinite delay abstract this paper deals with the existence of periodic solutions for some partial functional differential equations with infinite delay we suppose that the linear part is nondensely defined and satisfies the hille yosida condition in the nonlinear case we give several criteria to ensure the existence of a periodic solution in the nonhomogeneous linear case we prove the existence of a periodic solution under the existence of a bounded solution

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

application of vibrational control to linear quadratic control problems abstract the application of the theory of vibrational control to linear quadratic control problems is developed the solution of the matrix riccati differential equation rde and the optimal trajectory are found approximately as power series of a small parameter the minimizing effect of vibrations on the maximal solution of rde and the cost functional is studied the region of attraction of the maximal solution of rde for the case of hamiltonian matrix with imaginary axis eigenvalues is found special attention is paid to the application of vibrations to the linear quadratic problem of stabilization with respect to a part of variables and transfer of the other variables to a given position a problem of vibrational stabilization and optimal control of a carriage with an inverted pendulum is solved as an example

the solution set of the algebraic riccati equation and the algebraic riccati inequality abstract we present a parametrization of the solution set of the algebraic riccati equation and the algebraic riccati inequality of optimal control where we assume only sign controllability of the underlying system furthermore we characterize the existence of maximal and minimal solutions of both the equation and the inequality by checkable conditions

perturbation bounds and characterisation of the solution of the associated algebraic riccati equation abstract the paper deals with the associated algebraic matrix riccati equation aamre closely related to the standard algebraic matrix riccati equation arising in the theory of linear quadratic optimisation and filtering the sensitivity of the aamre relative to perturbations in its coefficients is studied both linear local norm wise and component wise and non linear non local perturbation bounds are obtained the conditioning of the aamre is determined in particular a full characterisation of the solution of aamre in terms of neutral subspaces of certain hermitian matrix is given which is a counterpart of the characterisation of the solutions to the standard riccati equation in terms of the invariant subspaces of the corresponding hamiltonian matrix a reliable method to obtain all solutions to aamre is briefly outlined

perturbation analysis of singular subspaces and deflating subspaces perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author the perturbation expansions are then used to derive fr echet derivatives condition numbers and r th order perturbation bounds for the subspaces vaccaro s result on second order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper besides new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations the results of this paper reveal an important fact each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers

necessary conditions for constant solutions of coupled riccati equations in nash games abstract the asymptotic behaviour of solutions as t for coupled matrix riccati equations occurring in open loop linear quadratic nash games is studied in this paper a general formula representing all possible solutions is given necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained two examples illustrate these results

some results for impulsive fractional differential inclusions with infinite delay and sectorial operators in banach spaces this paper is concerned with impulsive fractional differential inclusions with sectorial operators nonlocal conditions and infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory we derive some existence results of pc mild solutions when the multivalued map f is convex as well as nonconvex furthermore the compactness of the set of solutions and continuous dependence results are also obtained as an application an impulsive neutral partial functional differential equation is also investigated at last

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

semi fredholm operators and periodic solutions for linear functional differential equations abstract we deal with the inhomogeneous linear periodic equation with infinite delay of the formdx dt ax t b t xt f t whereais the generator of ac0 semigroup on a banach space assuming that it has a bounded solution we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space b and in the concrete phase space b ucg the key of our approach is the employment of the perturbation theory of semi fredholm operators to show that the period map satisfies the condition of the fixed point theorem by chow and hale funkcial ekvac 17 1974 31 38

optimal direct velocity feedback we present a novel approach to the problem of direct velocity feedback dvf optimization of vibrational structures which treats simultaneously small as well as large gains for that purpose we use two different approaches the first one is based on the gains optimization using the lyapunov equation in the scope of this approach we present a new formula for the optimal gain and we present a relative error for modal approximation in addition we present a new formula for the solution of the corresponding lyapunov equation for the case with multiple undamped eigenfrequencies which is a generalization of existing formulae the second approach studies the behavior of the eigenvalues of the corresponding quadratic eigenvalue problem since this approach leads to the parametric eigenvalue problem we consider small and large gains separately for the small gains which are connected to a modal damping approximation we present a standard approach based on gerschgorin discs for the large gains we present a new approach which allows us to approximate all eigenvalues very accurately and efficiently

invariant manifolds and projective combinations of solutions of the riccati differential equation abstract in this paper we show how families of solutions of the general riccati differential equation rde can be generated via projective combinations of a given number of reference solutions our approach is based upon the extension of the domain of the equation to the grassmannian manifold and the application of the radon lemma in this context we briefly discuss the relevance of our results to the study of the invariant manifolds of the equation and compare them to existing results concerning representation formulas for solutions of rde the results of the paper have been motivated by the recent characterization of solutions of the rde given in m pavon d d alessandro families of solution of matrix riccati differential equations siam j control optim 35 1 1997 194 204 which extends the classical results on the algebraic riccati equation due to willems coppel and shayman

totally isotropic subspaces for pairs of hermitian forms and applications to riccati equations abstract we prove a result on the existence of a common totally isotropic subspace for a pair of forms on v v where v is a finite dimensional space real complex or quaternionic and where one of the forms is hermitian while the other is skew hermitian applications to algebraic matrix riccati equations are given

uncontrollable dissipative systems observability and embeddability the theory of dissipativity is well developed for controllable systems a more appropriate definition of dissipativity in the context of uncontrollable systems is in terms of the existence of a storage function namely a function such that along every system trajectory its rate of change at each time instant is at most the power supplied to the system at that time however even when the supplied power is expressible in terms of just the external variables the dissipativity property for uncontrollable systems crucially hinges on whether or not the storage function depends on variables unobservable hidden from the external variables this paper investigates the key aspects of both cases and also proposes another intuitive definition of dissipativity these three definitions are compared we show that drawbacks of one definition are addressed by another dealing first with observable storage functions under the conditions that no two uncontrollable poles add to zero and that dissipativity is strict as f

existence and continuous dependence results for nonlinear differential inclusions with infinite delay this paper is concerned with nonlinear functional differential inclusions with infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory existence and continuous dependence results are obtained for integral solutions without the assumption of compactness on the associated nonlinear semigroup

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

existence and continuous dependence results for semilinear functional differential inclusions with infinite delay we consider local and global existence results for dierential inclusions with infinite delay in a banach space of the form y 0 t 2 ay t f t yt t ae yae 2 b and continuous dependence of the solutions set on the initial functions

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

fractional semilinear differential inclusions in this paper we first present a version of filippov s theorem for fractional semilinear differential inclusions of the form cd ay ay t f t y t a e email protected 0 b y 0 y email protected x where cd a is the caputo fractional derivative and f is a set valued map after several existence results the compactness of solutions sets is also investigated some results from topological fixed point theory together with notions of measure of noncompactness are used

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

a general refinement of jordan type inequality in this work a general form of jordan s inequality p 2 n x a n 1 p 2 4x 2 n email protected email protected p 2 n x email protected n 0na email protected 2 email protected 2 n 1 p 2 4x 2 n 1 is established where email protected 0 p 2 p 2 n x email protected n 0 na n p 2 4x 2 n a email protected a email protected 3 a n 1 2n 12 n 1 p 2a n 116n n 1 p 2a n 1 and n 0 is a natural number the applications of the above result give the general improvement of the yang le inequality and a new infinite series sinx email protected n 0 a n p 2 4x 2 n for 0 x email protected 2

families of solutions of algebraic riccati equations abstract we consider homogeneous algebraic riccati equations in the general situation when the matrix of the dynamics can be mixed we show that in this case the equation may have infinitely many families of solutions an analysis of these families is carried over and an explicit formula is derived when the matrix of the dynamics is unmixed this formula provides an explicit parametrization of the whole set of the solutions we finally show that there are situations in which the equation has spurious solutions that do not belong to any of the families and derive sufficient conditions under which on the contrary the union of the families covers the whole set of solutions

perturbation of invariant subspaces the questions of how perturbation of an operator affects its invariant subspaces have raised a substantial amount of interest especially in the w x finite dimensions see e g 5 10 12 and the references given there it turns out that linear perturbation of the operator gives rise to an analytic perturbation of the projection on its invariant subspace under some reasonable conditions although the question of estimating the norms of the terms of the taylor series expansion of this projection valued function does have some applications in numerical analysis it seems to be studied w x less frequently for instance in 12 the matrix algebra approach combined with complex analysis in several variables was used to get existence and analyticity of the perturbed projection the terms were obtained as well using specially developed formulas for their direct computation however we find it hard to see that these estimates are good enough to ensure the convergence of this series let alone to give an estimate of its radius of convergence in this article we propose a different approach to the subject which achieves the two goals at one stroke and we believe with better results namely we study the problem in terms of operator valued analytic functions thus reducing the problem to solving an algebraic riccati type equation in order to solve it we extend in sections 2 and 3 somewhat the usual methods to our case see the comments about the meaning of usual in the last paragraph of the introduction the main result given in section 4 is the estimates of the radius of convergence and of the terms the estimates are clearly good enough to yield an estimated radius of

functional inequalities involving special functions ii this paper is motivated by an open problem of anderson vamanamurthy and vuorinen we prove a chain of inequalities for hypergeometric functions generalized and normalized bessel functions kummer functions and for some general power series some particular cases and refinements are given

remarks on semilinear partial functional differential equations with infinite delay abstract in this paper we will be concerned with the abstract semilinear equation x t ax t f t xt t x where a represents the infinitesimal generator of a linear c0 semigroup in a banach space and x denotes the segment of the solution x due to the initial function taken from up to t conditions for existence continuous dependence and regularity of solutions are given which are independent of the topology of the phase space as well as conditions which guarantee the compactness of bounded orbits even if exp at is not compact for t 0

perturbation analysis of the matrix sign function perturbation expansions and new perturbation bounds for the matrix sign function are derived by using a simple technique the perturbation expansions are then used to derive frechet derivatives th

geometry of the algebraic riccati equation part ii we prove that the set of real symmetric solutions of the algebraic riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related n times n matrix by characterizing the structure of this variety we obtain a detailed description of the geometric properties of the solution set of the algebraic riccati equation

matrix riccati inequality existence of solutions abstract we consider the various criteria including hamiltonian and frequency domain ones for solvability of the matrix riccati inequality arising from control problems some criteria are given in terms of symplectic algebra which is also used as an important technical tool

successive approximations for solutions of functional integral equations in a banach space x where r 0 j o b or icj b and the integral is in the sense of bochner this equation is a unification and extension of a nonlinear volterra integral equation and a functional differential equation of retarded type indeed if in 1 1 the function f satisfies that f f s a for t e j then it is reduced to a functional differential equation in a banach space de balasi and myjak 21 gave a global convergence version of successive approximations to solutions for such reduced equations under the assumption that g is lipschitz continuous and bounded if in addition r 0 then 1 1 is reduced to an ordinary differential equation for this vidossich 161 discussed the global convergence of picard s successive approximations to solutions under some less restrictive conditions on g the main purpose of this paper is to extend vidossich s results to the solutions of eq 1 1 our conditions made on g are similar to what miller 4 and artzstein 1 made for the integral equation 1 1 with r 0 and what vidossich 61 made for the ordinary differential equation referred above

existence and uniqueness of unmixed solutions of the discrete time algebraic riccati equation a solution x of a discrete time algebraic riccati equation is called unmixed if the corresponding closed loop matrix x has the property that the common roots of det si x and det i s x if any are on the unit circle a necessary and sufficient condition is given for existence and uniqueness of an unmixed solution such that the eigenvalues of x lie in a prescribed subset of c

the regular indefinite linear quadratic problem with linear endpoint constraints this paper deals with the infinite horizon linear quadratic problem with indefinite cost given a linear system a quadratic cost functional and a subspace of the state space we consider the problem of minimizing the cost functional over all inputs for which the state trajectory converges to that subspace our results generalize classical results on the zero endpoint version of the linear quadratic problem and more recent results on the free endpoint version of this problem

existence of solutions and kamke s theorem for functional differential equations in banach spaces in this paper we shall consider the cauchy problem for functional differential equations fdes for a simplicity cp 1 or cp f 6 cp where f 52 e 52 c co co x 99 is uniformly continuous here e is a banach space with norm 1 ie and g is an abstract phase space satisfying the fundamental axioms see section 1 introduced by hale and kato 6 also refer to 8 173 the purpose of this paper is to establish an existence theorem of solu tions for cp l and kamke s theorem in fdes it is well known that the compactness condition which is described by means of the measure of non compactness intrdoduced by kuratowski is useful in showing the existence of solutions for ordinary and functional differential equations in banach spaces cf l 3 4 10 12 13 241 we note that these results on the existence of solutions are closely related to the property of the phase spaces eand c r o e o r a in this paper we shall prove an existence theorem of solutions for cp l under the condition

acausal models and balanced realizations of stationary processes abstract we study acausal realizations of stationary or stationary increment processes in particular we characterize the family of models whose corresponding spectral factors have a fixed zero structure acausal models with a fixed zero structure are related to each other by a certain group of state feedback transformations which is naturally parametrized by the solution set of a homogeneous algebraic riccati equation each feedback transformation reflects some of the eigenvalues of the generator matrix a of the representation to a mirror image with respect to the imaginary axis dually acausal models with a fixed pole structure are parametrized by a dual riccati equation and by a corresponding family of output injection transformations from a general standpoint the results of this study clarify the role played by dual pairs of riccati equations in spectral factorization and may be relevant to other problem areas than stochastic modelling one natural application of the concepts discussed in the paper is to stochastic balancing balancing of models with an essentially arbitrary eigenvalue location can be accomodated very naturally in this framework a balancing algorithm involving the solution of a dual pair of riccati equations is discussed

comparison theorems and uniqueness of mild solutions to semilinear functional differential equations in banach spaces let e be a banach space over the real field r with norm 1 ie and let 6 5 cbe be the phase space satisfying the fundamental axioms introduced by hale and kato l if x 00 t a e o asm then for any t e 00 a we define a mapping x oo o e by 8 x t i oo 0 i 0 we denote by t t a linear semigroup on e of class c for brevity c semigroup on e and let a be the infinitesimal generator of t t the purpose of this paper is concerned with the existence and uniqueness of mild solutions to the initial value problem for a semilinear functional differential equation in e for brevity zp o 9

a further refinement of a jordan type inequality and its application abstract by introducing taylor polynomials a new sharpened and generalized version of jordan s inequality is established the result is then used to obtain a substantially more refined inequality of jordan type moreover an application of the results presented here toward the improvement of the yang le inequality is also considered in this paper

a symmetry group for the matrix riccati equation we describe a symmetry group for the matrix riccati differential equation and prove a reduction theorem which simplifies the calculation of the elements of this group we also find necessary and sufficient conditions for two equilibrium points to be symmetry equivalent