backward error for the discrete time algebraic riccati equation the normwise backward error of an approximate solution to the discrete time algebraic riccati equation is evaluated the results are illustrated by using simple numerical examples c elsevier sci

condition number and backward errors of nonsymmetric algebraic riccati equation abstract the normwise condition number and two kinds of backward errors are considered when nonsymmetric algebraic riccati equation nare has the minimal nonnegative solution based on the techniques for the symmetric case we apply the condition number theory developed by rice to define condition number for nare the explicit expression is derived in a uniform manner meanwhile two kinds of backward errors are defined and evaluated by the explicit formulas numerical experiments are listed to illustrate and compare the practical performance

on the convergence of inexact newton methods for discrete time algebraic riccati equations abstract in this paper we present a convergence analysis of the inexact newton method for solving discrete time algebraic riccati equations dares for large and sparse systems the inexact newton method requires at each iteration the solution of a symmetric stein matrix equation these linear matrix equations are solved approximatively by the alternating directions implicit adi or smith s methods we give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact newton sequences we show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d stability the theoretical results developed in this paper are an extension to the discrete case of the analysis performed by feitzinger et al 2009 8 for the continuous time algebraic riccati equations in the last section we give some numerical experiments

sensitivity analysis of the discrete time algebraic riccati equation abstract consider the discrete time algebraic riccati equation dare a t xa x a t xb s r b t xb 1 b t xa s t q 0 where a r n n b s r n m r rt r m m q qt r n n the available perturbation theory for the dare can only be applied to the case r 0 however in some control problems the matrix r can be singular in this paper we study perturbation properties of the dare without the restriction r 0 perturbation bounds and a relative condition number for the stabilizing solution of the dare are derived computable residual bounds for an approximate solution are also derived the theoretical results are illustrated by numerical examples

block arnoldi based methods for large scale discrete time algebraic riccati equations in the present paper we present block arnoldi based methods for the computation of low rank approximate solutions of large discrete time algebraic riccati equations dare the proposed methods are projection methods onto block or extended block krylov subspaces we give new upper bounds for the norm of the error obtained by applying these block arnoldi based processes we also introduce the newton method combined with the block arnoldi algorithm and present some numerical experiments with comparisons between these methods

residual bounds of approximate solutions of the discrete time algebraic riccati equation let tilde x be a hermitian matrix which approximates the unique hermitian positive semi definite solution x to the discrete time algebraic riccati equation dare where f in cal c n times n c 2 in cal c m times m is hermitian positive definite g 1 in cal c n times m c in cal c r times n the pair f g 1 is stabilizable and the pair c f is detectable assume that i g tilde x is nonsingular and i g tilde x 1 f is stable let g g 1g 2 1 g 1 rm h h c rm h c and let be the residual of the dare with respect to tilde x define the linear operator vec l by the main result of this paper is if where denotes any unitarily invariant norm and then

perturbation analysis of a quadratic matrix equation associated with an m matrix this paper is devoted to the perturbation analysis for the quadratic matrix equation x 2 e x f 0 where e is a diagonal matrix and f is an m matrix the quadratic matrix equation of this type arises in noisy wiener hopf problems for markov chains the solution of practical interest is a particular m matrix solution in this paper the perturbation bound of the m matrix solution is presented meanwhile the residual bound for an approximate solution to the m matrix solution is obtained the theoretical results are illustrated by using simple numerical examples

perturbation analysis of the maximal solution of the matrix equation x a x 1a p consider the nonlinear matrix equation x a x 1 a p where a p are n x n complex matrices with p hermitian positive definite and a denotes the conjugate transpose of a matrix a in this pape

existence and relaxation results for nonlinear second order multivalued boundary value problems in rn abstract in this paper we study second order differential inclusions with nonlinear boundary conditions our formulation is general and incorporates as special cases well known problems such as the dirichlet picard neumann and periodic problems we prove existence theorems under various sets of hypotheses for both the convex and nonconvex problems also we show the existence of extremal solutions and that the extremal solutions are dense in the solutions of the convex problem for thew1 2 t r n norm strong relaxation theorem finally we examine the dirichlet problem when the multivalued right hand side does not depend on the derivative of x and satisfies a general growth hypothesis and a sign type condition for this problem we prove existence results and a relaxation theorem

existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions abstract in this paper we consider a quasilinear second order differential inclusion with a convex valued multivalued term and nonlinear multivalued boundary conditions using the leray schauder fixed point theorem and techniques from multivalued analysis and from nonlinear analysis we prove the existence of a solution our formulation of the problem is general and includes as special cases the dirichlet the neumann the periodic problems as well as certain sturm liouville type problems

second order nonlinear multivalued boundary problems in hilbert spaces in this paper we consider second order differential inclusions in real hilbert space namely p t x t r t x t ax t f t x t a e on 0 t under the nonlinear boundary conditions using techniques from multivalued analysis and the theory of operators of monotone type we prove the existence of solutions for both the convex and nonconvex problems finally we present a special case of interest which fit into our framework illustrating the generality of our results

necessary and sufficient conditions for discrete and differential inclusions of elliptic type abstract this paper deals for the first time with the dirichlet problem for discrete p d discrete approximation problem on a uniform grid and differential p c inclusions of elliptic type in the form of euler lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings the results obtained are generalized to the multidimensional case with a second order elliptic operator

on the existence of periodic solutions for differential inclusions in this paper we study the existence of periodic solutions for differential inclusions we prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems also we show the existence of extreme solutions some feedback control systems are also considered

the asymptotic behavior of plurigenera for a normal isolated singularity we study here some invariants of normal isolated singularities which look like the kodaira dimensions and thereby prepare for the possible localization of the rough classification theory in algebraic geometry let x x be a normal isolated singularity on an analytic space x of dimension n 2 kn 511er and watanabe independently defined m genera y x x and 6re x x of x x by considering the space of meromorphic m pie n forms divided by those with natural growth conditions at the singularity kn w1 in this paper we introduce a third m genus d x x me 1 and investigate the basic properties of these m genera it turns out that all these three m genera ym x x 6m x x d x x have integral growth orders in m which are denoted by xr x x x6 x x and xa x x respectively the asymptotic behavior of the above three m genera are summarized as follows

existence theorems for nonlinear boundary value problems for second order differential inclusions abstract in this paper we consider a nonlinear two point boundary value problem for second order differential inclusions using the leray schauder principle and its multivalued analog due to dugundji granas we prove existence theorems for convex and nonconvex problems our results are quite general and incorporate as special cases several classes of problems which are of interest in the literature

boundary value problems for nonconvex differential inclusions abstract in this paper we consider multivalued boundary value problems of the sturm liouville type first we prove the existence of extremal trajectories and then we show that they are dense in the solution set of the original convex problem we also prove an existence theorem when the orientor field is l s c and nonconvex valued some feedback control systems are also considered

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

on strong vanishing property and plurigenera of isolated singularities abstract we introduce a new concept of strong vanishing property and investigate this property for minimally elliptic singularities a comparison theorem about strong vanishing property is also obtained we use zariski decomposition to give a formula of plurigenera for gorenstein singularities and use it to test whether a divisor is f nef