geometric significance of toeplitz kernels abstract let l 2 be the lebesgue space of square integrable functions on the unit circle we show that the injectivity problem for toeplitz operators is linked to the existence of geodesics in the grassmann manifold of l 2 we also investigate this connection in the context of restricted grassmann manifolds associated to p schatten ideals and essentially commuting projections

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

self adjoint block operator matrices with non separated diagonal entries and their schur complements in this paper self adjoint 2 2 block operator matrices a in a hilbert space h1 h2 are considered for an interval which does not intersect the spectrum of at least one of the diagonal entries of a we prove angular operator representations for the corresponding spectral subspace l a of a and we study the supporting subspace in this angular operator representation of l a which is the orthogonal projection of l a to the corresponding component h1 or h2 our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding schur complement of a in the endpoints of

non symmetric perturbations of self adjoint operators abstract we investigate the effect of non symmetric relatively bounded perturbations on the spectrum of self adjoint operators in particular we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates these results extend and improve classical perturbation results by kato and by gohberg kre n further we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator in the latter case we can even allow for perturbations with relative bound 1 the generality of our results is illustrated by several applications massive and massless dirac operators point coupled periodic systems and two channel hamiltonians with dissipation

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

solution of a nonsymmetric algebraic riccati equation from a two dimensional transport model abstract for the steady state solution of an integral differential equation from a two dimensional model in transport theory we shall derive and study a nonsymmetric algebraic riccati equation b xf f x xb x 0 where f i s pd b b i s p d and b b i s pd with a nonnegative matrix p positive diagonal matrices d and nonnegative parameters f b b 1 f and s s 1 f we prove the existence of the minimal nonnegative solution x under the physically reasonable assumption f b s p d d 1 and study its numerical computation by fixed point iteration newton s method and doubling we shall also study several special cases e g when b 0 and p is low ranked then x s 2 uv is low ranked and can be computed using more efficient iterative processes in u and v numerical examples will be given to illustrate our theoretical results

extended hilbert space approach to few body problems a general formulation of the quantum scattering theory for a system of few particles which have an internal structure is given due to freezing out the internal degrees of freedom in the external channels a certain class of energy dependent potentials is generated by means of potential theory a modified faddeev equation is derived both in external and internal channels the fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem

banach lie groupoids and generalized inversion abstract we study a few basic properties of banach lie groupoids and algebroids adapting some classical results on finite dimensional lie groupoids as an illustration of the general theory we show that the notion of locally transitive banach lie groupoid sheds fresh light on earlier research on some infinite dimensional manifolds associated with banach algebras

brief paper adaptive optimal control for continuous time linear systems based on policy iteration in this paper we propose a new scheme based on adaptive critics for finding online the state feedback infinite horizon optimal control solution of linear continuous time systems using only partial knowledge regarding the system dynamics in other words the algorithm solves online an algebraic riccati equation without knowing the internal dynamics model of the system being based on a policy iteration technique the algorithm alternates between the policy evaluation and policy update steps until an update of the control policy will no longer improve the system performance the result is a direct adaptive control algorithm which converges to the optimal control solution without using an explicit a priori obtained model of the system internal dynamics the effectiveness of the algorithm is shown while finding the optimal load frequency controller for a power system

on the newtonkleinman method for strongly stabilizable infinite dimensional systems we consider the newtonkleinman method for strongly stabilizable infinite dimensional systems under certain assumptions the maximal self adjoint solution to the associated control algebraic riccati equation is constructed the constructed solution is also the maximal solution to the corresponding control algebraic riccati inequality

spectral inclusion for unbounded block operator matrices abstract in this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators a admitting a block operator matrix representation for diagonally dominant and off diagonally dominant block operator matrices we show that the recently introduced quadratic numerical range w 2 a contains the eigenvalues of a and that the approximate point spectrum of a is contained in the closure of w 2 a this provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non convex set w 2 a several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems both in the non self adjoint case and in the self adjoint case applications to dirac operators and to two channel hamiltonians are given

on collectionwise normality of locally compact normal spaces we prove that by adjoining supercompact many cohen or random reals to a model of zfc set theory in the resulting model every normal locally compact space is collectionwise normal in the same models countably paracompact locally compact t 3 spaces are expandable local compactness in the above theorems can be weakened to being of point countable type a condition that is implied by both cech completeness and first countability

an extremal problem in fourier analysis with applications to operator theory a certain minimal extrapolation problem for fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators in this paper we estimate the value of the constant in the fourier transform problem by an analytic reformulation

malykhin s problem abstract we construct a model of zfc where every separable frechet topological group is metrizable this solves a 1978 problem of v i malykhin

extensions of banach lie poisson spaces abstract the extension of banach lie poisson spaces is studied and linked to the extension of a special class of banach lie algebras the case of w algebras is given particular attention semidirect products and the extension of the restricted banach lie poisson space by the banach lie poisson space of compact operators are given as examples

essentially commuting projections abstract let h h h be a fixed orthogonal decomposition of a hilbert space with both subspaces of infinite dimension and let e e be the projections onto h and h we study the set p c c of orthogonal projections p in h which essentially commute with e or equivalently with e i e p e p e e p is compact by means of the projection onto the calkin algebra one sees that these projections p p c c fall into nine classes four discrete classes which correspond to p being 0 1 e or e and five essential classes which we describe below the discrete classes are respectively the finite rank projections finite co rank projections the sato grassmannian of h and the sato grassmannian of h thus the connected components of each of these classes are parametrized by the integers via de rank the co rank or the fredholm index respectively the essential classes are shown to be connected we are interested in the geometric structure of p c c being the set of selfadjoint projections of the c algebra b c c of operators in b h which essentially commute with e in particular we study the problem of existence of minimal geodesics joining two given projections in the same component we show that the hopf rinow theorem holds in the discrete classes but not in the essential classes conditions for the existence and uniqueness of geodesics in these latter classes are found

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

the block numerical range of matrix polynomials block numerical ranges of matrix polynomials especially the quadratic numerical range are considered the main results concern spectral inclusion boundedness of the block numerical range an estimate of the resolvent in terms of the quadratic numerical range geometrical properties of the quadratic numerical range and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions as an application we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials

closing the gap in a subspace perturbation bound abstract this paper sharpens an operator inequality from perturbation theory suppose x and y are self adjoint operators on a hilbert space kx x and ky y are two closed sets bounded apart by 0 e is the spectral projector for x belonging to kx and f is the spectral projector for y belonging to ky it is known that 2 x y ef for all operators as described and for any unitary invariant norm in this paper we show 2 is best possible in this inequality we then discuss an analogous bound which applies when x and y are normal

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

infinite dimensional linear systems a distributional approach we introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems the class of distributional resolvent linear systems includes all linear time invariant systems that have been studied in the control literature we show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory

the restricted grassmannian banach lie poisson spaces and coadjoint orbits we investigate some basic questions concerning the relationship between the restricted grassmannian and the theory of banach lie poisson spaces by using universal central extensions of lie algebras we find that the restricted grassmannian is symplectomorphic to symplectic leaves in certain banach lie poisson spaces and the underlying banach space can be chosen to be even a hilbert space smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established several pathological properties of the restricted algebra are pointed out

removal of the resolvent like energy dependence from interactions and invariant subspaces of a total hamiltonian the spectral problem a v z z is considered where the main hamiltonian a is a self adjoint operator of sufficiently arbitrary nature the perturbation v z b a z 1b depends on the energy z as resolvent of another self adjoint operator a the latter is usually interpreted as a hamiltonian describing an internal structure of physical system the operator b is assumed to have a finite hilbert schmidt norm the conditions are formulated when one can replace the perturbation v z with an energy independent potential w such that the hamiltonian h a w has the same spectrum more exactly a part of spectrum and the same eigenfunctions as the initial spectral problem the hamiltonian h is constructed as a solution of the non linear operator equation h a v h it is established that this equation is closely connected with the problem of searching for invariant subspaces of the hamiltonian h b aa b the orthogonality and expansion theorems are proved for eigenfunction systems of the hamiltonian h a w

hopf rinow theorem in the sato grassmannian abstract let u 2 h be the banach lie group of unitary operators in the hilbert space h which are hilbert schmidt perturbations of the identity 1 in this paper we study the geometry of the unitary orbit u p u u u 2 h of an infinite projection p in h this orbit coincides with the connected component of p in the hilbert schmidt restricted grassmannian gr res p also known in the literature as the sato grassmannian corresponding to the polarization h p h p h it is known that the components of gr res p are differentiable manifolds here we give a simple proof of the fact that gr res 0 p is a smooth submanifold of the affine hilbert space p b 2 h where b 2 h denotes the space of hilbert schmidt operators of h also we show that gr res 0 p is a homogeneous reductive space we introduce a natural metric which consists in endowing every tangent space with the trace inner product and consider its levi civita connection this connection has been considered before for instance its sectional curvature has been computed we show that the levi civita connection coincides with a linear connection induced by the reductive structure a fact which allows for the easy computation of the geodesic curves we prove that the geodesics of the connection which are of the form t e t z p e t z for z a p co diagonal anti hermitic element of b 2 h have minimal length provided that z 2 note that the condition is given in terms of the usual operator norm a fact which implies that there exist minimal geodesics of arbitrary length also we show that any two points p 1 p 2 gr res 0 p are joined by a minimal geodesic if moreover p 1 p 2 1 the minimal geodesic is unique finally we replace the 2 norm by the k schatten norm k 2 and prove that the geodesics are also minimal for these norms up to a critical value of t which is estimated also in terms of the usual operator norm in the process minimality results in the k norms are also obtained for the group u 2 h

banach lie poisson spaces and reduction the category of banach lie poisson spaces is introduced and studied it is shown that the category of w algebras can be considered as one of its subcategories examples and applications of banach lie poisson spaces to quantization and integration of hamiltonian systems are given the relationship between classical and quantum reduction is discussed

weakly coupled bound state of 2 d schrodinger operator with potential measure abstract we consider a self adjoint two dimensional schrodinger operator h which corresponds to the formal differential expression where is a finite compactly supported positive radon measure on r 2 from the generalized kato class and 0 is the coupling constant it was proven earlier that ess h 0 we show that for sufficiently small the condition d h 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion c o 1 exp 4 r 2 0 with a certain constant c 0 we also obtain a formula for the computation of c the asymptotic expansion of the corresponding eigenfunction is provided the statements of this paper extend the results of simon 41 to the case of potentials measures also for regular potentials our results are partially new

rational eigenvalue problems and applications to photonic crystals we establish new analytic results for a general class of rational spectral problems they arise e g in modelling photonic crystals whose capability to control the flow of light depends on specific

linear sigma additivity and some applications we show that countable increasing unions preserve a large family of well studied covering properties which are not necessarily additive using this together with infinite combinatorial methods and simple forcing theoretic methods we explain several phenomena settle problems of just miller scheepers and szeptycki 1996 gruenhage and szeptycki 2005 tsaban and zdomskyy 2008 and tsaban 2006 2007 and construct topological groups with very strong combinatorial properties

newton s method for concave operators with resolvent positive derivatives in ordered banach spaces abstract we prove a non local convergence result for newton s method applied to a class of nonlinear equations in ordered real banach spaces the key tools in our approach are special notions of concavity and the spectral theory of resolvent positive operators

normal nonmetrizable moore space from continuum hypothesis or nonexistence of inner models with measurable cardinals abstract assuming the continuum hypothesis a normal nonmetrizable moore space is constructed this answers a question raised by f b jones in 1931 using an axiom well known at that time for the construction a consequence of the continuum hypothesis that also follows from the nonexistence of an inner model with a measurable cardinal is used hence it is shown that to prove the consistency of the statement that all normal moore spaces are metrizable one must assume the consistency of the statement that measurable cardinals exist

comparison theorems for infinite dimensional riccati equations abstract comparison theorems are proved for the algebraic riccati equation in the framework of infinite dimensional hilbert spaces these theorems extend well known results for the finite dimensional case

a class of iterative methods for solving nonsymmetric algebraic riccati equations arising in transport theory in this paper we consider the nonsymmetric algebraic riccati equation arising in transport theory an important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation we propose a class of iterative methods to solve the vector equation the convergence analysis shows that the sequence of vectors generated by iterative methods with two kinds of specific iterative matrices is monotonically increasing and converges to the minimal positive solution of the vector equation numerical experiments show that the new methods outperform the modified simple iterative method and newton s method

perturbation of spectra and spectral subspaces we consider the problem of variation of spectral subspaces for linear self adjoint operators with emphasis on the case of off diagonal perturbations we prove a number of new optimal results on the shift of the spectrum and obtain sharp estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators respectively

complete second order linear differential operator equations in hilbert space and applications in hydrodynamics we study the cauchy problem for a complete second order linear differential operator equation in a hilbert space h of the form d 2 u dt 2 f ik du dt bu f u 0 u 0 u 1 0 u 1 problems of this kind arise e g in hydrodynamics where the coefficients f k and b are unbounded selfadjoint operators it is assumed that f is the dominating operator in the cauchy problem above i e d f d b d f d k we also suppose that f and b are bounded from below but the operator coefficients are not assumed to commute the main results concern the existence of strong solutions to the stated cauchy problem and applications of these results to the cauchy problem associated with small motions of some hydrodynamical systems

hierarchy of hamilton equations on banach lie poisson spaces related to restricted grassmannian abstract we consider the banach lie poisson space i r u l res 1 and its complexification c l res 1 where the first one of them contains the restricted grassmannian gr res as a symplectic leaf using the magri method we define an involutive family of hamiltonians on these banach lie poisson spaces the hierarchy of hamilton equations given by these hamiltonians is investigated the operator equations of ricatti type are included in this hierarchy for a few particular cases we give the explicit solutions

the palindromic generalized eigenvalue problem a x λax numerical solution and applications abstract in this paper we propose the palindromic doubling algorithm pda for the palindromic generalized eigenvalue problem pgep a x ax we establish a complete convergence theory of the pda for pgeps without unimodular eigenvalues or with unimodular eigenvalues of partial multiplicities two one or two for eigenvalue 1 some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the pda

on the solution of falkner skan equations abstract new existence and uniqueness proofs are given for the solutions of the equations governing the self similar compressible boundary layer falkner skan equations the properties of the solutions are studied and some bounds on important quantities are concluded the paper is restricted to favourable pressure gradients and to wall cooling

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

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

on applications of schauderʼs fixed point theorem for the solution of the non symmetric algebraic riccati equation abstract in this article we give sufficient conditions for the existence of contractive solutions for the non symmetric algebraic riccati equation the conditions are given in terms of the matrix of the coefficients only we also provide sufficient conditions for unique contractive solution which is a unique minimal norm solution the results are generalized to contractive solutions with respect to generalized unit matrix balls

riccati equation and the problem of decoherence the block operator matrix theory is used to investigate the problem of a single qubit we establish a connection between the riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question the model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed we show that the model defined in such a way can be reduced to a time independent problem

variational principles and eigenvalue estimates for unbounded block operator matrices and applications in this paper we establish variational principles eigenvalue estimates and asymptotic formulae for eigenvalues of three different classes of unbounded block operator matrices the results allow to characterise eigenvalues that are not necessarily located at the boundary of the spectrum applications to an example from magnetohydrodynamics and to dirac operators on certain manifolds are given

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

a generalized negative imaginary lemma and riccati based static state feedback negative imaginary synthesis abstract in this paper we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition then an algebraic riccati equation method is given to determine if a system is negative imaginary also a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property the controller synthesis procedure is based on the proposed negative imaginary lemma using this procedure the closed loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty such as in the case of unmodeled spill over dynamics in a lightly damped flexible structure a numerical example is presented to illustrate the usefulness of the results

a new concept for block operator matrices the quadratic numerical range abstract in this paper a new concept for 2 2 block operator matrices the quadratic numerical range is studied the main results are a spectral inclusion theorem an estimate of the resolvent in terms of the quadratic numerical range factorization theorems for the schur complements and a theorem about angular operator representations of spectral invariant subspaces which implies e g the existence of solutions of the corresponding riccati equations and a block diagonalization all results are new in the operator as well as in the matrix case

existence and uniqueness of contractive solutions of some riccati equations abstract in this paper we establish criteria for the existence and uniqueness of contractive solutions k of the riccati equation kbk ka dk c 0 under the assumption that the spectra of a and d are disjoint

an efficient schur method to solve the stabilizing problem an efficient solution to the stabilizing problem for both continuous and discrete constant linear systems is presented the real schur form reduction of the system state matrix is used

structured doubling algorithms for weakly stabilizing hermitian solutions of algebraic riccati equations in this paper we propose structured doubling algorithms for the computation of the weakly stabilizing hermitian solutions of the continuous and discrete time algebraic riccati equations respectively assume that the partial multiplicities of purely imaginary and unimodular eigenvalues if any of the associated hamiltonian and symplectic pencil respectively are all even and the c dare and the dual c dare have weakly stabilizing hermitian solutions with property p under these assumptions we prove that if these structured doubling algorithms do not break down then they converge to the desired hermitian solutions globally and linearly numerical experiments show that the structured doubling algorithms perform efficiently and reliably

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

normal not paracompact spaces this is the published version also available here http www dx doi org 10 1090 s0273 0979 1982 15020 0 first published in bull ams in 1982 published by the american mathematical society

coprime factorizations and well posed linear systems we study the basic notions related to the stabilization of an infinite dimensional well posed liner system in the sense of salamon and weiss we first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in h infty the converse is also true every function with a doubly coprime factorization in h infty is the transfer function of a jointly stabilizable and detectable well posed linear system we show further that a stabilizable and detectable system is stable if and only if its input output map is stable finally we construct a dynamic possibly non well posed stabilizing compensator the notion of stability that we use is the natural one for the quadratic cost minimization problem and it does not imply exponential stability

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

on hermitian solutions of the symmetric algebraic riccati equation the structure of the set of hermitian solutions of the matrix quadratic equation xdx xa a x c 0 is studied under the conditions that c c d is positive semidefinite and a d is stabilizable new features e g nonexistence of the minimal solution appear in contrast with the known case when a d is controllable

on iterative methods for the quadratic matrix equation with m matrix in this paper we consider newton s method and bernoulli s method for a quadratic matrix equation arising from an overdamped vibrating system by introducing m matrix to this equation we provide a sufficient condition for the existence of the primary solution moreover we show that newton s method and bernoulli s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition

low rank approximation to the solution of a nonsymmetric algebraic riccati equation from transport theory abstract we consider the solution of the large scale nonsymmetric algebraic riccati equation xcx xd ax b 0 from transport theory juang 1995 with m d c b a r 2 n 2 n being a nonsingular m matrix in addition a d are rank 1 updates of diagonal matrices with the products a 1 u a u d 1 v and d v computable in o n complexity for some vectors u and v and b c are rank 1 the structure preserving doubling algorithm by guo et al 2006 is adapted with the appropriate applications of the sherman morrison woodbury formula and the sparse plus low rank representations of various iterates the resulting large scale doubling algorithm has an o n computational complexity and memory requirement per iteration and converges essentially quadratically as illustrated by the numerical examples

monotonicity of maximal solutions of algebraic riccati equations abstract a monotonicity result for the maximal solution of the equation xbb x a x xa q 0 q q a b stabilizable is proved

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

on the tangential interpolation problem for functions the aim of this paper is to solve a matrix valued version of the nevanlinna pick interpolation problem for hi functions we reduce this prob lem to a nevanlinna pick interpolation problem for schur functions and obtain a linear fractional transformation which describes the set of all solutions

design of consensus and adaptive consensus filters for distributed parameter systems this work establishes an abstract framework that considers the distributed filtering of spatially varying processes using a sensor network it is assumed that the sensor network consists of groups of sensors each of which provides a number of state measurements from sensing devices that are not necessarily identical and which only transmit their information to their own sensor group a modification to the local spatially distributed filters provides the non adaptive case of spatially distributed consensus filters which penalize the disagreement amongst themselves in a dynamic manner a subsequent modification to this scheme incorporates the adaptation of the consensus gains in the disagreement terms of all local filters both the well posedness of these two consensus spatially distributed filters and the convergence of the associated observation errors to zero in appropriate norms are presented their performance is demonstrated on three different examples of a diffusion partial differential equation with point measurements

efficient methods for solving a nonsymmetric algebraic riccati equation arising in stochastic fluid models we consider the nonsymmetric algebraic riccati equation xm12x xm11 m22x m21 0 where m11 m12 m21 m22 are real matrices of sizes n n n m m n m m respectively and m mij 2i j 1 is an irreducible singular m matrix with zero row sums the equation plays an important role in the study of stochastic fluid models where the matrix m is the generator of a markov chain the solution of practical interest is the minimal nonnegative solution this solution may be found by basic fixed point iterations newton s method and the schur method however these methods run into difficulties in certain situations in this paper we provide two efficient methods that are able to find the solution with high accuracy even for these difficult situations

if all normal moore spaces are metrizable then there is an inner model with a measurable cardinal this is the published version also available here http www dx doi org 10 1090 s0002 9947 1982 0664048 8 first published in trans amer math soc in 1982 published by the american mathematical society

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

a kleinman newton construction of the maximal solution of the infinite dimensional control riccati equation abstract assuming only strong stabilizability we construct the maximal solution of the algebraic riccati equation as the strong limit of a kleinman newton sequence of bounded nonnegative operators as a corollary we obtain a comparison of the solutions of two algebraic riccati equations associated with different cost functions we show that the weaker strong stabilizability assumptions are satisfied by partial differential systems with collocated actuators and sensors so the results have potential applications to numerical approximations of such systems by means of a counterexample we illustrate that even if one assumes exponential stabilizability the kleinman newton construction may provide a solution to the riccati equation that is not strongly stabilizing

multipulse like orbits for a singularly perturbed nearly integrable system abstract we use topological shooting arguments to study the existence of multipulse like orbits for a singularly perturbed 3 dimensional dynamical system established as a model problem for the 1 dimensional complex ginzburg landau equations our main result improves an earlier result obtained by doelman and hek

separation and coding we construct a normal collectionwise hausdorff space which is not collectionwise normal with respect to copies of 0 1 we do this by developing a general theory of coding properties into topological spaces we construct a para lindelof regular space in which para lindelof is coded directly rather than a para lindelof and normal we construct a normal collectionwise hausdorff space which is not collectionwise normal in which collectionwise hausdorff is coded directly rather than obtained as a side effect to countable approximation we also show that the martin s axiom example of a normal space which is not collectionwise hausdorff is really just a kind of dual of bing s space

an exact line search method for solving generalized continuous time algebraic riccati equations we present a newton like method for solving algebraic riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of newton s method it avoids the problem of a disastrously large first step and accelerates convergence when newton steps are too small or too long the additional work to perform the line search is small relative to the work needed to calculate the newton step

the falkner skan equation i the creation of strange invariant sets abstract we study bifurcations in the falkner skan equation we show that for positive integer values of the parameter there are bifurcations which create an infinity of periodic and other interesting trajectories these bifurcations provide a new mechanism for the creation of strange invariant sets in ordinary differential equations and are co dimension one

katetov s problem in 1948 miroslav katetov showed that if the cube x 3 of a compact space x satisfies the separation axiom t 5 then x must be metrizable he asked whether x 3 can be replaced by x 2 in this metrization result in this note we prove the consistency of this implication

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

left separated spaces with point countable bases this is the published version also available here http www dx doi org 10 2307 2000207 first published in trans amer math soc in 1986 published by the american mathematical society

on parametrization of all the exact pole assignment state feedbacks for lti systems in this technical note we provide a parametrization of all the exact pole assignment state feedbacks for lti systems when the set of poles to be assigned contains sufficient number of real eigenvalues we also refute a recent conjecture concerning the explicit form of all the state feedbacks assigning a given set of poles to the closed loop

hamiltonians and riccati equations for linear systems with unbounded control and observation operators in this paper we construct infinitely many selfadjoint solutions of the control algebraic riccati equation using invariant subspaces of the associated hamiltonian we do this under the assumptions that the system operator is normal and has compact inverse and that the hamiltonian possesses a riesz basis of invariant subspaces

a representation of all solutions of the control algebraic riccati equation for infinite dimensional systems we obtain a representation of all self adjoint solutions of the control algebraic riccati equation associated to the infinite dimensional state linear system a b c under the following assumptions a generates a c0 group the system is output stabilizable strongly detectable and the dual riccati equation has an invertible self adjoint non negative solution

the falkner skan equation ii dynamics and the bifurcations of p and q orbits the falkner skan equation is a reversible three dimensional system of ordinary differential equations with two distinguished straight line trajectories which form a heteroclinic loop between fixed points at infinity we showed in the previous paper 1995 j differential equations119 336 394 that at positive integer values of the parameter there are bifurcations creating large sets of periodic and other interesting trajectories here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur