a spectral sequence for polyhedral products abstract the purpose of this paper is to exhibit fine structure for polyhedral products z k x a and polyhedral smash products z k x a where moment angle complexes are special cases in which x a d 2 s 1 there are three main parts to this paper 1 one part gives a natural filtration of the polyhedral product together with properties of the resulting spectral sequence in theorem 2 15 applications of this spectral sequence are given 2 the second part gives a homological decomposition of z k x a in case x a consists of cw pairs by using part 1 3 applications to the ring structure of z k x a are given for cw pairs x a satisfying suitable freeness conditions

discrete time lq optimal control problems for infinite markov jump parameter systems optimal control problems for discrete time linear systems subject to markovian jumps in the parameters are considered for the case in which the markov chain takes values in a countably infinite set two situations are considered the noiseless case and the case in which an additive noise is appended to the model the solution for these problems relies in part on the study of a countably infinite set of coupled algebraic riccati equations icare conditions for existence and uniqueness of a positive semidefinite solution to the icare are obtained via the extended concepts of stochastic stabilizability ss and stochastic detectability sd which turn out to be equivalent to the spectral radius of certain infinite dimensional linear operators in a banach space being less than one for the long run average cost ss and sd guarantee existence and uniqueness of a stationary measure and consequently existence of an optimal stationary control policy furthermore an extension of a lyapunov equation result is derived for the countably infinite markov state space case

optimal interpolation for linear stochastic systems let y satisfying dy cxdt ddw y 0 0 represent the noisy observation of the gauss markov process x generated by dx axdt bdw x 0 x 0 on the finite interval 0 t in this paper we solve the following interpolation problem determine the best least squares estimate of x t t in 0 t given the increments of y on the intervals 0 t 1 and t 2 t where 0 t 1 t 2 t and the statistics of x 0 several alternative expressions for the optimal interpolator are derived some of which are phrased in terms of two kalman bucy estimates as a by product we also solve the interpolation problem for the missing increments of y

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

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

estimating the plant state from the compensator state for the system dot w sw where w x over z this note states and proves necessary and sufficient conditions under which a z is an estimate of a linear function of x i e z estimates tx or b a linear function of z is an estimate of x i e vz estimates x the mapping t has been used previously by the authors to compute the estimator controller form of a compensator whose state vector z had the same dimension as the plant state vector x

on the relation between additive and multiplicative decompositions of rational matrix functions in this paper a general frame to study relations between additive and multiplicative representations of rational matrix functions is presented various extensions of the positive real lemma as well as of other classical factorization results both in the continuous and discrete time cases are established in the case of square factorizations the map between solutions to an asymmetric algebraic riccati equation and pairs of factors is shown to be a homeomorphism in this framework we also derive a geometric characterization of non square factorizations

the polyhedral product functor a method of decomposition for moment angle complexes arrangements and related spaces this article gives a natural decomposition of the suspension of generalized moment angle complexes or partial product spaces which arise as polyhedral product functors described below the geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of goresky macpherson concerning complements of certain subspace arrangements as well as davis januszkiewicz and buchstaber panov concerning moment angle complexes since the stable decompositions here are geometric they provide corresponding homological decompositions for generalized moment angle complexes for any homology theory

convergence of the time invariant riccati differential equation and lq problem mechanisms of attraction the nature of the attraction of the solution of the time invariant matrix riccati differential equation towards the stabilizing solution of the algebraic riccati equation is studied this is done on an explicit formula for the solution when the system is stabilizable and the hamiltonian matrix has no eigenvalues on the imaginary axis various aspects of this convergence are analysed by displaying explicit mechanisms of attraction and connections are made with the literature the analysis ultimately shows the exponential nature of the convergence of the solution of the riccati differential equation and of the related finite horizon lq optimal state and control trajectories as the horizon recedes computable characteristics are given which can be used to estimate the quality of approximating the solution of a large finite horizon lq problem by the solution of an infinite horizon lq problem

computing the estimator controller form of a compensator a technique is presented for finding the estimator and controller gains to implement a desired compensator in estimator controller form the technique is derived by mapping the states of the compensator to those of the estimator controller via a linear transformation using the technique involves finding the real invertible solutions of a non symmetric steady state matrix riccati equation the ricatti equation may have zero two or more real invertible solutions from each solution a set of estimator and controller gains can be computed multiple solutions correspond to the freedom to choose which closed loop system poles are associated with the controller and which are associated with the estimator this technique can be applied to multi input multi output systems in an example the technique is applied to the despin control system of a dual spin satellite

two sided linear prediction and reconstruction of a stationary time series the problem of two sided linear prediction and reconstruction of a stationary process y on a finite interval is solved

singular perturbation modelling of large scale systems with multi time scale property the problem is investigated of singular perturbation modelling which plays an important role in cases of the application of the singular perturbation method to the design and analysis of large scale systems in both continuous and discrete time versions modelling concepts based on structural properties are studied and computer oriented algorithms are developed the results of two time scale systems are extended to multi time scale cases furthermore in order to verify the validity of these studies some numerical examples are given

on the phase portrait of the matrix riccati equation arising from the periodic control problem a comprehensive description is provided of the properties of the matrix riccati differential equation in which the coefficient matrices are t periodic for some t 0 the main results include 1 a classification of all the periodic equilibria t periodic solutions 2 necessary and sufficient conditions for convergence to the uniformly asymptotically stable completely unstable periodic equilibrium as t to infty t to infty 3 necessary and sufficient conditions for a solution to escape in finite forward or backward time 4 a description of every almost periodic solution 5 a proof that every solution which does not escape in finite time is asymptotically almost periodic and an explicit formula for the limiting almost periodic solution

regulators for optimal periodic processes a feedback controller called a periodic regulator is developed for implementing an optimal periodic process to a physical system under state deviations and constant parameter variation the objective of the periodic regulator is to bring the perturbed plant back to a neighboring optimal periodic path in such a manner that the infinite time second variation of the cost is minimized the transition matrix associated with the variational problem evaluated over one period is symplectic and generically possesses two unity eigenvalues coupled in the same jordan box the primary eigenvector associated with the unity eigenvalue corresponds to the orbital velocity direction it is shown that there exists a locally stable subspace in the neighborhood of the periodic orbit moreover a simple condition is given for defining the state variation which guarantees that it is contained in this stable subspace a controller including feedback of the parameter variation is derived which provides first order convergence to a new optimal periodic trajectory finally the periodic regulator developed is applied to a simple two state periodic optimal control problem to demonstrate its convergence properties

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

roth s theorems for matrix equations with symmetry constraints abstract this note deals with the consistency of complex matrix equations ax yb c and ax xb c under the constraints y x and x x

new method for optimal control and filtering of weakly coupled linear discrete stochastic systems abstract the algebraic regulator and filter riccati equations of weakly coupled discrete time stochastic linear control systems are completely and exactly decomposed into reduced order continuous time algebraic riccati equations corresponding to the subsystems that is the exact solution of the global discrete algebraic riccati equation is found in terms of the reduced order subsystem nonsymmetric continuous time algebraic riccati equations in addition the optimal global kaiman filter is decomposed into local optimal filters both driven by the system measurements and the system optimal control inputs as a result the optimal linear quadratic gaussian control problem for weakly coupled linear discrete systems takes decomposition and parallelism between subsystem filters and controllers

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

geometric aspects of the riccati difference equation in the nonsymmetric case abstract this paper provides a collection of geometric results concerning the nonsymmetric riccati difference equation it gives representation formulas which express all the solutions in terms of a suitable number of known ones further results include geometric properties of the difference of two solutions and the generalization of some of the author s earlier results to the nonsymmetric case

pole placement by performance criterion modification a procedure is presented that modifies the state weighting matrix q and introduces a degree of relative stability into the original performance criterion to shape the resulting optimal dynamics by positioning the closed loop eigenvalues along the real axis of the optimal system in the linear quadratic regulator problem it is based on the results of the algebraic riccati equation are which establish the invariance of certain eigenspaces of the associated hamiltonian matrix with respect to certain perturbations of q and the degree of relative stability

sdp based approximation of stabilising solutions for periodic matrix riccati differential equations numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions the most successful method require

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

robust control with pole shifting via performance index modification robust controllers that ensure closed loop eigenvalue placement are designed for linear systems with parametric uncertainties by using an appropriate state weighting matrix choice and a relative stability degree an lmi based algorithm is proposed to provide guaranteed cost controllers that ensure i the closed loop nominal eigenvalues shifting by a pre specified amount ii a prescribed degree of stability for all system uncertainties of a given class a certain level of closed loop performance is also ensured by minimizing the associated guaranteed cost

decomposing euclidean space with a small number of smooth sets let the cardinal invariant fin denote the least number of continuously smooth n dimensional surfaces into which n l dimensional euclidean space can be decomposed it will be shown to be consistent that fin is greater than sn l these cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones

geometric properties and invariant manifolds of the riccati equation this short paper considers the general riccati matrix differential equation it reviews and extends results on the characterization and existence of equilibrium solutions establishes that the riccati equation has at most one stable equilibrium solution as t rightarrow infty or t rightarrow infty and confines the region of attraction to this unique stable equilibrium solution by identifying and utilizing linear and nonlinear invariant manifolds of the riccati equation

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

the homotopy type of the polyhedral product for shifted complexes abstract we prove a conjecture of bahri bendersky cohen and gitler if k is a shifted simplicial complex on n vertices x 1 x n are pointed connected c w complexes and c x i is the cone on x i then the polyhedral product determined by k and the pairs c x i x i is homotopy equivalent to a wedge of suspensions of smashes of the x i s earlier work of the authors dealt with the special case where each x i is a loop space new techniques are introduced to prove the general case these have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes we discuss connections between polyhedral products and toric topology combinatorics and classical homotopy theory

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

decomposing borel sets and functions and the structure of baire class 1 functions all spaces considered are metric separable and are denoted usually by the letters x y or z w stands for the set of all natural numbers if a metric separable space is additionally complete we call it polish if it is a continuous image of ww or equivalently of a polish space it is called souslin the main subject of the present paper is the structure of baire class 1 functions recent developments in banach space theory in particular discoveries of rosenthal and bourgain fremlin and talagrand see r for a survey of these developments stimulated investigations into the structure of baire class 1 functions the interested reader may consult papers by haydon odel rosenthal hor kechris louveau kl rosenthal r2 and references quoted there these investigations have already provided new applications in banach space theory see for example ri in our study of baire class 1 functions we will be interested in two decomposition properties one of them defined by lusin the other one by jayne and rogers first however we want to consider a more general problem of determining how difficult it is to represent a borel set as a union of simpler borel sets or the graph of a borel function as a union of the graphs of simpler borel functions using effective descriptive set theory in particular louveau s theorem we show that if a c x is borel x polish then a e e or there is a continuous injection q ww a such that q 1 b is meager for any b c a which is e2 this gives a new proof of j stern s result that if a borel set a is the union of cov m sets in e then a is itself e cov m is the smallest cardinality of a family of meager sets covering ill we prove similar results for functions put for f x y and a family of functions g

periodic riccati difference equations approaching equilibria by implicit systems a parameterization of solutions to the periodic riccati difference equation rde is given by using behavioral tools the moving equilibria of the periodic rde are encompassed among the trajectories of the implicit ar model associated to the symplectic pencil of the corresponding hamiltonian equations this allows a parameterization of solutions of periodic rdes even in the case the dynamic matrix is singular

the exact slow fast decomposition of the algebraic ricatti equation of singularly perturbed systems the algebraic riccati equation for singularly perturbed control systems is completely and exactly decomposed into two reduced order algebraic riccati equations corresponding to the slow and fast time scales the pure slow and pure fast algebraic riccati equations are asymmetric ones but their o epsilon perturbations are symmetric it is shown that the newton method is very efficient for solving the obtained asymmetric algebraic riccati equations the method presented is very suitable for parallel computations due to the complete and exact decomposition of the riccati equation this procedure might produce new insight into the two time scale optimal filtering and control problems

maximal solution of a certain class of periodic riccati differential equations this paper deals with a generalized version of the well known periodic riccati differential equation which arises in certain optimal filtering problems for linear periodic systems a monotonicity property for symmetric periodic equilibrium solutions is established and it is proved that there exists a maximal symmetric periodic equilibrium solution

families of solutions of matrix riccati differential equations the j c willems coppel shayman geometric characterization of solutions of the algebraic riccati equation are is extended to asymmetric riccati differential equations with time varying coefficients the coefficients do not need to satisfy any definiteness periodicity or system theoretic condition more precisely given any two solutions x 1 t and x 2 t of such equation on a given interval t 0 t 1 we show how to construct a family of solutions of the same equation of the form x t i pi t x 1 t pi t x 2 t where pi is a suitable matrix valued function even when specialized to the case of x 1 and x 2 equilibrium solutions of a symmetric equation with constant coefficients our results considerably extend the classical ones as no further assumption is made on the pair x 1 x 2 and on the coefficient matrices

decompositions of the polyhedral product functor with applications to moment angle complexes and related spaces this article gives a natural decomposition of the suspension of a generalized moment angle complex or partial product space which arises as the polyhedral product functor described below the introduction and application of the smash product moment angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor one direct consequence is an analysis of the associated cohomology for the special case of the complements of certain subspace arrangements the geometrical decomposition implies the homological decomposition in earlier work of others as described below because the splitting is geometric an analogous homological decomposition for a generalized moment angle complex applies for any homology theory implied therefore is a decomposition for the stanley reisner ring of a finite simplicial complex and natural generalizations

upper and lower bounds for the solution of the general matrix riccati differential equation on a time scale we obtain upper and lower bounds for the solution of the general matrix riccati differential equation on a time scale t r t a t b t r t r t b t r t c t r t where a t and c t are symmetric n n matrices while b t v t t t and r t are n n matrices and denotes the transpose of the matrix we use the quasilinearization technique to obtain these bounds we also study the monotonicity of the successive approximations

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

stabilization of generalized linear systems via the algebraic riccati equation abstract this paper presents results from a study to examine the stabilization properties of linear time delay systems and systems depending on a parameter using techniques associated with the algebraic riccati equation are these classes of systems can be viewed as members of a broader class of systems called generalized linear systems gls techniques of spectral factorization for the gls play an essential role in the methods for performing this stabilization necessary and sufficient conditions for the existence of a polynomial solution to the generalized are are derived numerical examples are given in order to illustrate and clarify the results

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

estimates of asymptotic trajectory bounds in digital implementations of linear feedback control systems estimates are established for trajectory bounds which are easy to calculate applicable to general controller topologies and dependent on parameters i e eigenvalues and eigenvectors of the idealized linear system i e the control system model neglecting quantizers the approach utilizes the comparison principle in the lyapunov theory and the properties of minimal matrix norms the applicability of the results are demonstrated by means of a specific example

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence