absolute value equations abstract we investigate existence and nonexistence of solutions for np hard equations involving absolute values of variables ax x b where a is an arbitrary n n real matrix by utilizing an equivalence relation to the linear complementarity problem lcp we give existence results for this class of absolute value equations aves as well as a method of solution for special cases we also give nonexistence results for our ave using theorems of the alternative and other arguments

a dynamic model to solve the absolute value equations abstract in this paper the analytic solution of the absolute value equations ave is investigated as far as we know there are some numerical methods to obtain the solution of the ave however there is not a study on exact solution of the ave here we try to obtain the exact solution of the ave based on a dynamical system model constructed by the projection function finally the simulation results show the effectiveness and the accuracy of the method

banach spaces of universal disposition in this paper we present a method to obtain banach spaces of universal and almost universal disposition with respect to a given class m of normed spaces the method produces among others the only separable banach space of almost universal disposition with respect to the class f of finite dimensional spaces gurari space g or the only under ch banach space with density character the continuum which is of universal disposition with respect to the class s of separable spaces kubis space k we moreover show that k is isomorphic to an ultrapower of the gurari space and that it is not isomorphic to a complemented subspace of any c k space other properties of spaces of universal disposition are also studied separable injectivity partially automorphic character and uniqueness

hermitian solutions of the discrete algebraic riccati equation hermitian solutions of the discrete algebraic riccati equation play an important role in the least squares optimal control problem for discrete linear systems in this paper we describe the set of hermitian solutions in various ways in terms of factorizations of rational matrix functions which take hermitian values on the unit circle in terms of certain invariant subspaces of a matrix which is unitary in an indefinite scalar product and in terms of all invariant subspaces of a certain matrix these results are inspired by known results for the algebraic riccati equation arising in the least squares optimal control problem for continuous linear systems

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

a separable frechet space of almost universal disposition abstract the gurari space is the unique separable banach space g which is of almost universal disposition for finite dimensional banach spaces which means that for every e 0 for all finite dimensional normed spaces e f for every isometric embedding e e g there exists an e isometric embedding f f g such that f e e we show that g n with a special sequence of semi norms is of almost universal disposition for finite dimensional graded frechet spaces the construction relies heavily on the universal operator on the gurari space recently constructed by garbulinska wegrzyn and the third author in addition we consider a non graded sequence of semi norms on g n with which the space g n is of almost universal disposition for finite dimensional frechet spaces with a fixed sequence of semi norms in both cases this yields in particular that g n is universal in the class of all separable frechet spaces

uniformly exhaustive submeasures and nearly additive set functions every uniformly exhaustive submeasure is equivalent to a measure from this we deduce that every vector measure with compact range in an f space has a control measure we also show that co or any e space is a t space i e cannot be realized as the quotient of a nonlocally convex f space by a one dimensional subspace

h sub spl infin differential riccati equations convergence properties and finite escape phenomena the paper studies the transient and asymptotic behavior of the solution of the sign indefinite differential riccati equation dre arising in finite horizon h sub spl infin filtering and control problems differently from the sign definite h sub spl infin dre the solution of the h sub spl infin dre can have finite escape times even for nonnegative initial conditions sufficient and necessary conditions for boundedness and convergence are derived in correspondence to a fixed value of the h sub spl infin norm attenuation level spl gamma finally a stepwise spl gamma switching strategy is devised to guarantee boundedness as well as asymptotic performance

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

amalgamation and ramsey properties of lp spaces abstract we study the dynamics of the group of isometries of l p spaces in particular we study the canonical actions of these groups on the space of isometric embeddings of finite dimensional subspaces of l p 0 1 into itself and we show that for every real number 1 p with p 4 6 8 they are e transitive provided that is small enough we achieve this by extending the classical equimeasurability principle of plotkin and rudin we define the central notion of a fraisse banach space which underlies these results and of which the known separable examples are the spaces l p 0 1 p 4 6 8 and the gurarij space we also give a proof of the ramsey property of the classes l p n n p 2 viewing it as a multidimensional borsuk ulam statement we relate this to an arithmetic version of the dual ramsey theorem of graham and rothschild as well as to the notion of a spreading vector of matousek and rodl finally we give a version of the kechris pestov todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space x with a ramsey property of the collection of finite dimensional subspaces of x

an efficient dynamic model for solving a portfolio selection with uncertain chance constraint models this paper presents a neural network model for solving maximization programming model with chance constraint in which the security returns are uncertain variables the main idea is to replace the portfolio selection models when the uncertain returns are chosen as some special cases such as linear uncertain variables trapezoidal uncertain variables and normal uncertain variables with a linear programming lp problem according to the saddle point theorem optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the original problem it is also shown that the proposed neural network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

the lie algebra of homeomorphisms of the circle we define and study an infinite dimensional lie algebrahomeo which is shown to be naturally associated to thetopologicallie grouphomeo of all orientation preserving homeomorphisms of the circle roughly we rely on the universal decorated teichmuller theory developed before as motivation to provide frechet coordinates on the homogeneous space given byhomeo modulo the group of real fractional linear transformations whose corresponding vector fields on the circle we then extend by the usual lie algebrasl2of real traceless two by two matrices in order to definehomeo surprisingly homeo turns out to be equal to the algebra of all vector fields on the circle which are piecewisesl2 in the obvious sense it is evidently important to consider the relationship between our new frechet coordinates and the usual trigonometric functions on the circle and we undertake here both natural infinitesimal calculations we finally apply some further previous work in order to give sufficient conditions on the fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory

solution and asymptotic behavior of coupled riccati equations in jump linear systems a new necessary and sufficient condition for the existence of a positive semidefinite solution of coupled riccati equations occurring in jump linear systems is derived by verifying a riccati inequality it is shown that such a solution exists in addition two numerical algorithms are given to compute it an example is given to illustrate the proposed method

finite escapes and convergence properties of guaranteed cost robust filters the design of robust filters for linear time varying systems subject to parameter perturbations is considered the design is based on the minimization of an upper bound for the error covariance the computation of the filter matrices calls for the solution of a pair of differential riccati equations parameterized by a scalar design parameter the main results of the paper concern the asymptotic properties of the filter when it is applied to time invariant systems differently from the standard kalman filter the robust filter may abruptedly cease to exist owing to the presence of finite escape times in the associated riccati equations nevertheless provided that the parameter perturbations do not affect the unstable dynamics convergence to a stable steady state filter can always be ensured by properly adjusting the scalar design parameter since such an adjustment might deteriorate the steady state performance a time varying strategy for the tuning of the scalar parameter is developed in order to guarantee both convergence and performance

convergence of the dre solution to the are strong solution in this paper we use the boundary solutions to a linear matrix inequality lmi associated with kalman filtering to investigate the convergence of the solution of a differential riccati equation dre to the so called strong solution of an algebraic riccati equation are we furthermore extend our results to kalman filtering with indefinite input noise covariances

on the convergence to the strong solution of periodic riccati equations the convergence of the solution of the differential and difference periodic riccati equation with non negative initial condition is studied in particular with reference to the riccati equation for optimal filtering we consider detectable but not necessarily stabilizable systems and no restrictive assumption is imposed on the modulus of the unreachable characteristic multipliers then a new condition for the convergence to the periodic strong solution is provided as a corollary it follows that for any positive definite initial condition the solution of the periodic riccati equation converges to the periodic strong solution the new condition encompasses as particular cases the convergence results already available in the literature

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

an efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side our goal in this work is to give an optimum correction of the infeasible absolute value equations ave in order to make the mentioned system feasible we apply the minimal correction using the l 2 norm by changing just the right hand vector we will show that this problem can be formulated as an unconstrained optimization problem with a quadratic objective function we propose an extension of newton s method for solving unconstrained objective optimization some examples are provided to illustrate the efficiency and validity of our proposed method

a miscellany of results of an equation of count j f riccati a collection of results on the riccati equation is presented the questions addressed are the existence of strong solutions of the algebraic riccati equation and the convergence of solutions of the riccati difference equation to those of the algebraic equation the results derived utilize detestability conditions only

quasi banach spaces of almost universal disposition we show that for each p 0 1 p 0 1 there exists a separable p banach space gpgp of almost universal disposition that is having the following extension property for each e 0e 0 and each isometric embedding g x yg x y where y is a finite dimensional p banach space and x is a subspace of gpgp there is an e isometry f y gpf y gp such that x f g x x f g x for all x xx x such a space is unique up to isometries does contain an isometric copy of each separable p banach space and has the remarkable property of being locally injective amongst p banach spaces we also present a nonseparable generalization which is of universal disposition for separable spaces and separably injective no separably injective p banach space was previously known for p 1p 1

difference and differential riccati equations a note on the convergence to the strong solution the convergence of the solutions of the differential difference riccati equations to the strong solution of the corresponding algebraic riccati equation are is addressed detectability only is required in the analysis and no assumption is made on the eigenvalues on the real imaginary axis on the unit circle in the discrete time case in particular from the results it follows that under the sole assumption of detectability a positive definite initial condition guarantees convergence to the strong solution even in the presence of unreachable eigenvalues on the imaginary axis or on the unit circle

on the necessity of diffusive couplings in linear synchronization problems with quadratic cost we show that diffusive couplings are necessary for minimization of cost functionals integrating quadratic synchronization error and quadratic input signals this holds for identical linear systems with eigenvalues either on the imaginary axis or in the open left half plane whilst for eigenvalues in the open right half plane we present a counterexample in which the strong solution to the associated algebraic riccati equation is not diffusive for nonidentical systems satisfying the internal model principle for synchronization we show that a certain part of the coupling must be diffusive for equally chosen weights in the cost functional we show that the dimension of the associated algebraic riccati equation can be reduced significantly

fraisse limits in functional analysis we provide a unified approach to fraisse limits in functional analysis including the gurarij space the poulsen simplex and their noncommutative analogs we obtain in this general framework many known and new results about the gurarij space and the poulsen simplex and at the same time establish their noncommutative analogs particularly we construct noncommutative analogs of universal operators in the sense of rota

a dynamical model for solving degenerate quadratic minimax problems with constraints this paper presents a new neural network model for solving degenerate quadratic minimax dqm problems on the basis of the saddle point theorem optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle the equilibrium point of the proposed network is proved to be equivalent to the optimal solution of the dqm problems it is also shown that the proposed network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the original problem several illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

a generalization of the gauss seidel iteration method for solving absolute value equations based on the gauss seidel splitting we present a new matrix splitting iteration method called generalized gauss seidel ggs iteration method for solving the large sparse absolute value equation ave a x x b where a r n n and b r n and investigate its convergence properties moreover by preconditioning ave a preconditioned variant of the ggs pggs method is presented numerical experiments illustrate the efficiency of both ggs and pggs iterations

a generalized newton method for absolute value equations associated with circular cones in this paper we study a class of absolute value equations associated with circular cone ccave for short which is a generalization of the absolute value equations discussed recently in the literature analogously to the fact that circular cone is the very generalization of the second order cone we show that the ccave is equivalent to a class of circular cone linear complementarity problems and hence generalize the well known equivalence between absolute value equations and linear complementarity problems useful properties of the generalized differential of the absolute value function over the circular cone are investigated which helps to propose a generalized newton method for solving the ccave the convergence of this method is established under mild conditions as well as the efficiency of which is illustrated by some preliminary numerical results

a splitting theorem for the space of smooth functions we will show using purely linear functional analytic methods that each exact complex0 f c s0 p0 c s1 p1 c s2 p2 wherepiare matrices of convolution operators in particular linear differential operators with constant coefficients splits fromp1on in the category of topological vector spaces moreover we characterize when these complexes split completely the obtained result covers all known particular cases as well as some more general cases independently of the analytic nature ofpi the result is a consequence of studying splitting of short exact sequences0 f jx qg 0 wheref x gare graded frechet spaces i e frechet spaces equipped with fixed reduced spectra representing them andj qare consistent with the graded structures we characterize splitting of in case i forgare spacesc where is an open subset in rn or ii gis the space of all sequences

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

observability and nonlinear filtering this paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability we consider a general class of hidden markov models in continuous time with compact signal state space and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process we demonstrate that observability implies stability of the filter i e the filtered estimates become insensitive to the initial measure at large times for the special case where the signal is a finite state markov process and the observations are of the white noise type a complete necessary and sufficient characterization of filter stability is obtained in terms of a slightly weaker detectability condition in addition to observability the role of controllability is explored finally the results are partially extended to non compact signal state spaces

monotonicity and stabilizability properties of solutions of the riccati difference equation propositions lemmas theorems fallacious conjectures and counterexamples abstract the problem considered is that of selecting an initial covariance matrix for the kalman filter to ensure that the closed loop filter at every subsequent time instant is exponentially asymptotically stable as a time invariant filter sufficient conditions are derived based on monotonicity properties of the solution of the riccati difference equation the results have application in observer design and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included