positive solutions to a system of adjointable operator equations over hilbert c modules abstract we present necessary and sufficient conditions for the existence of a positive solution to the system of adjointable operator equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 over hilbert c modules we also derive a representation for a general positive solution to this system when the solvability conditions are satisfied the results of this paper extend some known results in the literature

the congruence class of the solutions of some matrix equations the congruence class of a least square solution for the following matrix equations ax b a xa d axb dand axxb ef is presented also we derive necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the singular value decomposition svd and generalized singular value decomposition gsvd

explicit solutions to the matrix inverse problem ax b abstract recently wu and cain l wu b cain linear algebra and its applications 236 1996 137 146 gave necessary and sufficient conditions for the matrix equationax b to have a solution with respect toa subject to the condition that the hermitian part ofa is non negative definite in this note we give an easier approach which also provides possible explicit expressions fora further necessary and sufficient conditions are derived when the hermitian part ofa is prescribed to be positive definite thus correcting a result in l wu b cain linear algebra and its applications 236 1996 137 146 theorem 5

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

common hermitian solutions to some operator equations on hilber c modules abstract we establish necessary and sufficient conditions for the existence of the general common hermitian solution to the equations a 1 x c 1 xb 1 c 2 a 3 xa 3 c 3 a 4 xa 4 c 4 for adjointable operators between hilbert c modules and present an expression for the common hermitian solution to the equations in terms of moore penrose inverse of operators when the solvability conditions are satisfied the findings of this paper extend some known results in the literature

on minima of a functional of the gradient necessary conditions in this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a pseudo wigner distribution pwd the decorrelation procedure is accomplished by a cascade recursive least squared crls principal component pc neural network the goal is to obtain a more eacient analysis of images by combining the advantages of using a high resolution joint representation given by the pwd with an eaective adaptive principal component analysis pca through the use of feedforward neural networks 1999 elsevier science b v all rights reserved

the common positive solution to adjointable operator equations with an application abstract we give necessary and sufficient conditions for the existence of the common positive solution to the equations a 1 x 1 c 1 x 1 b 1 d 1 a 2 x 2 c 2 x 2 b 2 d 2 a 3 x 1 a 3 a 4 x 2 a 4 c 5 for adjointable operators between hilbert c modules and present an expression of the positive solution to the system when the solvability conditions are satisfied as an application we consider the reflexive positive semidefinite solution to the system of complex matrix equations a x b x c d e x e f

controlled invariance and feedback laws we consider controlled invariance for cones translated cones polyhedra and various special polyhedral structures for any polyhedron if controlled invariance occurs then all and nothing but the admissible controls can be obtained by an inequative feedback controller for each special polyhedral structure we compare this feedback controller against piecewise affine linear and affine feedback controllers

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

on weak convergence implying strong convergence under extremal conditions it is well known that under suitable extreme point conditions for the limit function weak convergence in an l1 space can imply strong convergence t rzezuchowski bull austral math soc 39 1989 201 214 has shown that a similar phenomenon can occur under an extremal subset condition we generalize his result in several respects by means of the associated young measures similar to the analysis given in e j balder bull austral math soc 33 1986 363 368

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

on minima of a functional of the gradient upper and lower solutions abstract this paper studies a scalar minimization problem with an integral functional of the gradient under affine boundary conditions a new approach is proposed using a minimal and a maximal solution to the convexified problem we prove a density result any relaxed solution continuously depending on boundary data may be approximated uniformly by solutions of the nonconvex problem keeping continuity relative to data we also consider solutions to the nonconvex problem having lipschitz dependence on boundary data with the best lipschitz constant

causal state feedback parameterizations in robust model predictive control in this paper we investigate the problem of nonlinearity and non convexity typically associated with linear state feedback parameterizations in the robust model predictive control rmpc for uncertain systems in particular we propose two tractable approaches to compute an rmpc controller consisting of both a causal state feedback gain and a control perturbation component for linear discrete time systems involving bounded disturbances and norm bounded structured model uncertainties along with hard constraints on the input and state both the state feedback gain and the control perturbation are explicitly considered as decision variables in the online optimization while avoiding nonlinearity and non convexity in the formulation the proposed rmpc controller computed through lmi optimizations is responsible for steering the uncertain system state to a terminal invariant set numerical examples from the literature demonstrate the advantages of the proposed scheme

the most likely path of a differential inclusion can be used in alternative to stochastic processes as a model for systems whose evolution is non deterministic compared with the rich mathemati cal theory which is currently available for random processes however dif ferential inclusions have remained a rather thin subject concerned mainly with the existence of solutions and with the topological properties of the set of trajectories the present paper is an attempt to widen the scope of this theory introducing the notion of likelihood for a solution of a differential inclusion and formulating a corresponding class of estimation prediction and filtering problems to do this a simple approach first comes to mind define some probabil ity measure on the set of trajectories of 1 1 and formulate the prediction and filtering problems accordingly unfortunately there seems to be no canonical probability distribution which is supported precisely on the solution set of 1 1 the choice of any particular distribution would thus require considerable additional information about the system to be modeled for this reason we consider a new definition of likelihood which is entirely independent of probability theory relying solely on the metric structure if 9 denotes the family of all solutions of 1 1 and u e 9 a rough estimate of how many trajectories u e b remain close to u is given by fl d u f iv t u t l 0 as e 0 the i mum of the quantities 1 2 is a well defined number 155

continuous time analysis eigenstructure assignment and h sub 2 synthesis with enhanced linear matrix inequalities lmi characterizations this note describes a new framework for the analysis and synthesis of control systems which constitutes a genuine continuous time extension of results that are only available in discrete time in contrast to earlier results the proposed methods involve a specific transformation on the lyapunov variables and a reciprocal variant of the projection lemma in addition to the classical linearizing transformations on the controller data for a wide range of problems including robust analysis and synthesis multichannel h sub 2 stateand output feedback syntheses the approach leads to potentially less conservative linear matrix inequality lmi characterizations this comes from the fact that the technical restriction of using a single lyapunov function is to some extent ruled out in this new approach moreover the approach offers new potentials for problems that cannot be handled using earlier techniques an important instance is the eigenstructure assignment problem blended with lyapunov type constraints which is given a simple and tractable formulation

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

existence of minimizers for non quasiconvex integrals keywords quasiconvexity existence of minimizers reference caa article 1995 002doi 10 1007 bf00380915 record created on 2008 11 25 modified on 2017 05 12

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

brief nonlinear model predictive control with polytopic invariant sets ellipsoidal invariant sets have been widely used as target sets in model predictive control mpc these sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope the choice of inclusion polytope has a significant effect on the size of the target ellipsoid but the optimal inclusion polytope cannot in general be computed systematically this paper shows that use of polytopic invariant sets overcomes this difficulty allowing larger stabilizable sets without loss of performance in the interests of online efficiency consideration is focused on interpolation based mpc

optimized robust control invariance for linear discrete time systems theoretical foundations this paper introduces the concept of optimized robust control invariance for discrete time linear time invariant systems subject to additive and bounded state disturbances a novel characterization of two families of robust control invariant sets is given the existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear convex case and a standard linear quadratic program when the constraints are polyhedral or polytopic the solution of the same optimization problem yields the corresponding feedback control law that is in general set valued a procedure for selection of a point valued nonlinear control law is provided

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

positive solutions to operator equations axb c abstract the operator equation axb c has been studied by several authors but under the extra condition that the operators a and b have closed range in this article we present different results regarding the existence of solution and also the existence of positive solution to axb c without this extra hypothesis

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

low complexity polytopic invariant sets for linear systems subject to norm bounded uncertainty we propose a novel algorithm to compute low complexity polytopic robust control invariant rci sets along with the corresponding state feedback gain for linear discrete time systems subject to norm bounded uncertainty additive disturbances and state input constraints using a slack variable approach we propose new results to transform the original nonlinear problem into a convex lmi problem whilst introducing only minor conservatism in the formulation through numerical examples we illustrate that the proposed algorithm can yield improved maximal minimal volume rci set approximations in comparison with the schemes given in the literature

on efficient computation of low complexity controlled invariant sets for uncertain linear systems a method is presented for determining invariant low complexity polytopic sets and associated linear feedback laws for linear systems with polytopic uncertainty conditions based on the relationship between 2 and norms are used to define an initial invariant low complexity polytope as the solution of a semi definite program the problem of computing a maximal controlled invariant low complexity polytopic set is then formulated as a bilinearly constrained problem and a relaxation of this problem is derived as an iterative sequence of convex programs the proposed method scales linearly with the state dimension which allows the possibility of determining low complexity robust controlled invariant sets for high order systems

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

products of projections and self adjoint operators abstract let h be a hilbert space and l h be the algebra of all bounded linear operators from h to h our goal in this article is to study the set p l h of operators in l h that can be factorized as the product of an orthogonal projection and a self adjoint operator we describe p l h and present optimal factorizations in different senses for an operator in this set

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

on the hermitian solutions to a system of adjointable operator equations we establish necessary and sufficient conditions for the existence of a hermitian solution to the system of equations a1x1 c1 x1b1 d1 a2x2 c2 x2b2 d2 a3x1a3 a4x2a4 c5 for adjointable operators between hilbert c modules and provide an expression for the general hermitian solution to the system we present necessary and sufficient conditions for the existence of a unique hermitian solution to the systems a1x1 c1 x1b1 d1 and a3x1a3 c5 of adjointable operators between hilbert c modules several examples are given to explain the use of the theory some of the findings of this paper extend some known results in the literature

a version of olech s lemma in a problem of the calculus of variations this paper studies the solutions of the minimum problem for a functional of the gradient under linear boundary conditions a necessary and sufficient condition based on the facial structure of the epigraph of the integrand is provided for the continuous dependence of the solutions on boundary data

an interval approach to compute invariant sets this paper proposes an original interval based method to compute an outer approximation of all invariant sets such as limit cycles of a continuous time nonlinear dynamic system which are included inside a prior set of the state space contrary to all other existing approaches our method has the following properties first it is guaranteed a solution cannot be lost second it is applicable to a large class of systems without any specific assumption such as the knowledge of a lyapunov function or any partial linearity and third there is no need to integrate the system

about the von neumann regularity of triangular block matrices abstract necessary and sufficient conditions are given for the von neumann regularity of triangular block matrices with von neumann regular diagonal blocks over arbitrary rings this leads to the characterization of the von neumann regularity of a class of triangular toeplitz matrices over arbitrary rings some special results and a new algorithm are derived for triangular toeplitz matrices over commutative rings finally the drazin invertibility of some companion matrices over arbitrary rings is considered as an application

on the matrix equation ax b with applications to the generators of a controllability matrix abstract given two matrices a and b over an arbitrary field for which ax b is consistent we determine all possible ranks for solutions x of ax b given one solution x we furthermore determine all possible ranks of other solutions of this matrix equation we then apply the results to determine when a given n nm matrix a with first m columns b occurs as a controllability matrix a b ab a n 1 b for some a we determine all matrices a that generate a and find all their possible ranks and some bounds on their sparsity structure

invariant approximations of the minimal robust positively invariant set this note provides results on approximating the minimal robust positively invariant mrpi set also known as the 0 reachable set of an asymptotically stable discrete time linear time invariant system it is assumed that the disturbance is bounded persistent and acts additively on the state and that the constraints on the disturbance are polyhedral results are given that allow for the computation of a robust positively invariant outer approximation of the mrpi set conditions are also given that allow one to a priori specify the accuracy of this approximation

the skew symmetric orthogonal solutions of the matrix equation ax b abstract an n n real matrix x is said to be a skew symmetric orthogonal matrix if xt x and xtx i using the special form of the c s decomposition of an orthogonal matrix with skew symmetric k k leading principal submatrix this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew symmetric orthogonal solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided furthermore the procrustes problem of skew symmetric orthogonal matrices is considered and the formula solutions are provided finally an algorithm is proposed for solving the first and third problems numerical experiments show that it is feasible

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

common hermitian and positive solutions to the adjointable operator equations ax c xb d let a be a c algebra for any hilbert a modules h and k let l h k be the set of adjointable operators from h to k let h k l be hilbert a modules a c l h k and b d l l h in this paper we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions x l h to the equations ax c xb d and obtain the formulae for the general forms of these solutions some results known for finite matrices and hilbert space operators are extended to the adjointable operators acting on hilbert c modules

the re positive definite solutions to the matrix inverse problem ax b abstract an n n complex matrix a is termed re positive definite re pd if the real part of x ax is positive for every nonzero complex n vector x this paper is concerned with constructing complex re pd matrices and solving the following matrix inverse problem given complex matrices x and b find the set of all complex re pd matrices a such that ax b