riccati equation and em algorithm convergence for inertial navigation alignment this correspondence investigates the convergence of a kalman filter based expectation maximization em algorithm for estimating variances it is shown that if the variance estimates and the error covariances are initialized appropriately the underlying riccati equation solution and the sequence of iterations will be monotonically nonincreasing further the process noise variance estimates converge to the actual values when the measurement noise becomes negligibly small conversely when the process noise variance becomes negligible the measurement noise variance estimates asymptotically approach the true values an inertial navigation application is discussed in which performance depends on accurately estimating the process variances

comparison theorems for the matrix riccati equation on deduit une inegalite matricielle a partir d une inegalite differentielle on utilise ce resultat pour deduire des inegalites pour les solutions d equations de riccati matricielles

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

generalized riccati difference and differential equations abstract in this paper we investigate generalized riccati differential and difference equations obtained from standard riccati equations by adding a semidefinite perturbation term for such equations we give results on the monotonic dependence of the solutions on the coefficients and initial values as well as results on convergence of solutions

existence and uniqueness of open loop nash equilibria in linear quadratic discrete time games we present existence and uniqueness results for an equilibrium in an m person nash game with quadratic performance criteria and a linear difference equation as constraint describing the system dynamics under an open loop information pattern the approach used is the construction of a value function which leads to existence assertions in terms of solvability of certain symmetric and nonsymmetric riccati difference equations

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

irreducible modules over finite simple lie conformal superalgebras of type k we construct all finite irreducible modules over lie conformal superalgebras of type k we construct all finite irreducible modules over lie conformal superalgebras of type k

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

generalized riccati equations arising in stochastic games abstract we study a class of rational matrix differential equations that generalize the riccati differential equations the generalization involves replacing positive definite weighting matrices in the usual riccati equations with either semidefinite or indefinite matrices that arise in linear quadratic control problems and differential games both stochastic and deterministic the purpose of this paper is to prove some fundamental properties such as comparison monotonicity and existence theorems these properties are well known for classical riccati differential equations when certain matrices are assumed definite as applications we obtain conditions for the existence of solutions to the algebraic riccati equation and to equations with periodic coefficients

asymptotic optimality of the minimum variance fixed interval smoother this correspondence investigates the asymptotic performance of the discrete time and continuous time time varying minimum variance fixed interval smoothers comparison theorems are generalized to provide sufficient conditions for the monotonic convergence of the underlying riccati equations under these conditions the energy of the estimation errors asymptotically approach a lower bound and attain lscr2 l2 stability

coupled matrix riccati equations in minimal cost variance control problems we present an algorithm for the solution of a nontrivial coupled system of algebraic riccati equations appearing in risk sensitive control problems moreover we use comparison methods to derive non blowup conditions for the solutions of a corresponding terminal value problem for coupled systems of riccati differential equations

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

weyl titchmarsh m function asymptotics local uniqueness results trace formulas and borg type theorems for dirac operators we explicitly determine the high energy asymptotics for weyl titchmarsh matrices associated with general dirac type operators on half lines and on r we also prove new local uniqueness results for dirac type operators in terms of exponentially small differences of weyl titchmarsh matrices as concrete applications of the asymptotic high energy expansion we derive a trace formula for dirac operators and use it to prove a borg type theorem

a matrix cross ratio theorem for the riccati equation abstract the notion of the cross ratio and its invariance under bilinear mappings is an important result of classical complex analysis the cross ratio has been extended to functions of several complex variables by c l siegel in his study of the symplectic group we present in this paper a new result for a generalized cross ratio valid for solutions to the matrix riccati equation an application of this generalized cross ratio to singular filtering problems allows one to develop a singular perturbation expansion with matrix coefficients for the solution to the corresponding riccati equation

controllability observability and discrete time markovian jump linear quadratic control this paper is concerned with the controllability and observability of discrete time linear systems that possess randomly jumping parameters described by finite stale markov processes and the relationship between these properties and the solution of the infinite time jump linear quadratic jlq optimal control problem the solution of the markovian jlq problem with finite or infinite time horizons is known necessary and sufficient conditions for the existence of optimal constant control laws that lead to finite optimal expected costs as the time horizon becomes infinite are also known sufficient conditions for these steady state control laws to stabilize the controlled system are also available chizeck et al 1986 these conditions are not easy to test however various definitions of controllability and observability for stochastic systems exist in the literature these definitions are unfortunately not related to the steady state jlq control problem in a manner that is analogous to the role of deter

stationary bifurcation control for systems with uncontrollable linearization stationary bifurcation control is studied under the assumption that the critical zero eigenvalue is uncontrollable for the linearized system the development facilitates explicit construction of feedback control laws that render the bifurcation supercritical thus the bifurcated equilibria in the controlled system are guaranteed stable both pitchfork bifurcation and transcritical bifurcation are addressed the results obtained for pitchfork bifurcations apply to general non linear models smooth in the state and the control for transcritical bifurcations the results require the system to be affine in the control

finite conformal modules over n 2 3 4 superconformal algebras in this paper we continue the study of representation theory of formal distribution lie superalgebras initiated by cheng and kac asian j math 1 181 193 1997 2 153 156 1998 erratum we study finite verma type conformal modules over the n 2 n 3 and the two n 4 superconformal algebras and also find explicitly all singular vectors in these modules from our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the n 2 n 3 and the two n 4 superconformal algebras

bifurcation and normal form of nonlinear control systems part ii the normal forms and invariants of control systems with a parameter are found bifurcations of equilibrium sets are classified the changes of properties such as controllability of the linearization or stabilizability near a bifurcation point of a control system are studied

on a certain asymptotic behavior of time invariant riccati matrix differential equations the main result of this paper reads as follows given real n times n matrices a b c and c 0 such that b c and c 0 are symmetric such that b and c 0 are nonnegative definite such that the pair a b is completely controllable i e rank b a cdots a n 1 b n and such that the triple a b c 0 is strongly observable i e dot x ax bu c 0 x equiv 0 on some nondegenerate interval implies x t equiv 0 then for any t 0 0 and any symmetric matrix q 0 the solution q t lambda of the riccati matrix differential equation dot q a t q qa qbq c lambda c 0 0 quad q 0 q 0 exists on 0 t 0 if lambda leqq lambda 0 and it satisfies lim lambda to infty q t 0 lambda infty i e all eigenvalues of the symmetric matrix q tend to infty the result is that the new notion of strong observability is even necessary for the assertion this result on riccati matrix equations is motivated by the follo

a comparison theorem for matrix riccati difference equations abstract difference equations of the form x t f t x t 1 f t f t x t 1 g t i g t x t 1 g t 1 g t x t 1 f t q t and their associated hermitian matrices h t f q f gg t are studied solution of different riccati equations can be compared if the difference of their corresponding hermitian matrices is semidefinite for all t an application to the discrete time lq optimal control problem is given

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

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

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

linear feedback stabilization of nonlinear systems with an uncontrollable critical mode abstract linear feedback stabilization of nonlinear systems is studied for systems whose linearization at an equilibrium point possesses a simple critical mode that is uncontrollable the results complement previous work on the synthesis of nonlinear stabilizing control laws the present work addresses continuous time systems for which the linearization has either a simple zero eigenvalue or a pair of simple pure imaginary eigenvalues both the stability analysis and stabilizing control design employ results on stability of bifurcations of parameterized systems

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

existence of algebraic matrix riccati equations arising in transport theory we consider the existence of positive solutions of a certain class of algebraic matrix riccati equations with two parameters c 0 c 1 and 0 1 here c denotes the fraction of scattering per collision and is an angular shift equations of this class are induced via invariant imbedding and the shifted gauss legendre quadrature formula from a simple transport model by establishing the existence of positive solutions of such equations the problem of the convergence of some iterative schemes for solving them can be completely solved

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

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

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

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

local feedback stabilization and bifurcation control ii stationary bifurcation abstract local feedback stabilization and bifurcation control of nonlinear systems are studied for the case in which the critical linearized system possesses a simple zero eigenvalue sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality and for local stabilizability of bifurcated equilibria these conditions involve assumptions on the controllability of the critical mode for the linearized system explicit stabilizing feedback controls are constructed the projection method of analysis of stationary bifurcations is employed this work complements an earlier study by the same authors systems control lett 7 1986 11 17 of stabilization and bifurcation control in the hopf bifurcation case of two pure imaginary eigenvalues of the linearized system at criticality

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

brief paper local l2 gain of bifurcation stabilization local l 2 gain analysis of a class of stabilizing controllers for nonlinear systems with both stationary and hopf bifurcations is studied in particular a family of lyapunov functions is first constructed for the corresponding critical system and simplified sufficient conditions to compute the l 2 gain are derived by solving the hamilton jacobi bellman hjb inequalities local robust analysis for a class of bifurcation stabilizing controllers can then be conducted through computing the local l 2 gain achieved by these controllers at the critical situation the results obtained in this paper provide useful guidance for selecting a robust controller from a given class of stabilizing controllers in terms of l 2 gain as an example application to an axial flow compressor control is discussed in detail

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

on stabilizing properties of solutions of the riccati difference equation monotonicity and stabilizing properties of solutions of the riccati difference equation rde are discussed the author considers the problem of selecting an initial condition for the rde in such a way that the update of the kalman filter gain can be stopped at any time and the resulting frozen filter is asymptotically stable the author also considers the case in which the initial condition of the rde may be less than the asymptotic solution the results are relevant to control and observer design including the stability of finite time horizon discrete time predictive control

generalized ricci curvature bounds for three dimensional contact subriemannian manifolds measure contraction property is one of the possible generalizations of ricci curvature bound to more general metric measure spaces in this paper we discover necessary and sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property

representations of simple finite lie conformal superalgebras of type w and s we construct all finite irreducible modules over lie conformal superalgebras of type w and s

a note on matrix riccati functional differential equations on etablit des solutions des equations differentielles fonctionnelles matricielles de riccati en fonction des matrices fondamentales

feedback strategies for discrete time linear quadratic two player descriptor games this paper deals with discrete time linear quadratic descriptor two player games the two frameworks of nash and stackelberg strategies with feedback information structure are considered the aim is to provide sufficient conditions ensuring for any initial state the existence and the uniqueness of a tri trajectory gathering the state and the controls of the two players which reaches the equilibrium the provided results are mainly based on a discrete time matrix block formulation the provided algorithms consist in an iterative backward in time procedure issued from dynamic programming numerical examples illustrate this approach and the particular case of explicit systems is recovered for both kinds of strategies

local feedback stabilization and bifurcation control i hopf bifurcation abstract local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple nonzero imaginary eigenvalues the conditions show in particular that generically these nonlinear critical systems can be stabilized locally even if the critical modes are uncontrollable the analysis also yields a direct method for computing stabilizing feedback controls use is made of bifurcation formulae which require only a series expansion of the vector field

time varying discrete riccati equation some monotonicity results abstract using a fre chet derivative based approach some monotonicity and comparison results concerning the solutions of the time varying discrete time riccati equation are obtained connections with the existence of the semistabilizing solution are made explicit as well

on the solution of discrete time markovian jump linear quadratic control problems abstract a necessary and sufficient condition for the existence of a positive semidefinite solution of the coupled algebraic discrete time riccati like equation occurring in markovian jump control problems is derived by verifying a simple matrix inequality it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence this leads to a straightforward numerical algorithm for the computation of the solution an example is given to illustrate the proposed method

on a class of rational matrix differential equations arising in stochastic control we prove a monotonicity and a comparison theorem for the solutions of a rational matrix differential equation appearing in stochastic control and derive existence and convergence results for the solutions of this differential equation moreover in the time invariant case we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing solution

a galois correspondence between sets of semidefinite solutions of continuous time algebraic riccati equations abstract the sets of negative semidefinite solutions t i of two algebraic riccati equations r i x a i x xa i xb i b i c i c i i x h i i x i 1 2 are compared under the hypothesis that h1 h2 if x 1 and x 2 are the greatest solutions in t 1 and t 2 respectively then x 1 x 2 a more general result will be proved which allows the comparison of other solutions of t 1 and t 2 besides the extremal ones and which in the case of stabilizability leads to a galois connection between t 1 and t 2 the comparison results are based on one hand on a decomposition of the equations r i x 0 into lyapunov matrix equations and genuine riccati equations which induce a corresponding decomposition of the solutions in t i and the other hand on a parametrization of the riccati components by ai invariant subspaces

classification of finite irreducible conformal modules over lie conformal superalgebras of block type abstract we introduce a class of infinite lie conformal superalgebras s p of block type and classify their finite irreducible conformal modules for any nonzero parameter p in particular we show that such a conformal module admits a nontrivial extension of a finite conformal module over ns if p 1 where ns is a neveu schwarz conformal subalgebra of s p as a byproduct we also obtain the classification of finite irreducible conformal modules over a series of finite lie conformal superalgebras s n for n 1