where is the nearest non regular pencil abstract this is a first step toward the goal of finding a way to calculate a smallest norm deregularizing perturbation of a given square matrix pencil minimal de regularizing perturbations have geometric characterizations that include a variable projection linear least squares problem and a minimax characterization reminiscent of the courant fischer theorem the characterizations lead to new computationally attractive upper and lower bounds we give a brief survey and illustrate strengths and weaknesses of several upper and lower bounds some of which are well known and some of which are new the ultimate goal remains elusive

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

nearest pair with more nonconstant invariant factors and pseudospectrum abstract let a b c n n c n m suppose that the number of nonconstant i e 1 invariant factors of the polynomial matrix i n 0 a b is less than k for all complex number denote by n k 1 i n 0 a b the greatest n k 1 th singular value of the matrix i n 0 a b the minimum absolute value of the real function of complex variable n k 1 i n 0 a b gives the distance from a b to the set of pairs with more or equal number of nonconstant invariant factors when k 1 this specializes in the formula of eising for the distance from a controllable pair a b to the nearest uncontrollable pair the complex numbers lying in the sublevel set c n i n 0 a b e of the function n i n 0 a b are the uncontrollable modes of all the pairs that are within an e tolerance of a b all the results of this paper are an immediate consequence of the singular value decomposition of a matrix and of the interpretation of the singular values as the distances to the nearest matrices of lower ranks

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

an information based neural approach to constraint satisfaction a novel artificial neural network approach to constraint satisfaction problems is presented based on information theoretical considerations it differs from a conventional mean field approach in the form of the resulting free energy the method implemented as an annealing algorithm is numerically explored on a testbed of k sat problems the performance shows a dramatic improvement over that of a conventional mean field approach and is comparable to that of a state of the art dedicated heuristic gsat walk the real strength of the method however lies in its generality with minor modifications it is applicable to arbitrary types of discrete constraint satisfaction problems

the structured distance to uncontrollability under multi perturbations an approach using multi valued linear operators abstract in this paper we develop a unifying approach for computing the distance to uncontrollability of linear control systems by using multi valued linear operators in representing and estimating the system s equations and matrices we are able to derive computable formulas of the distance from a controllable linear system to the nearest uncontrollable system under the assumption that the system s matrices are subjected to structured multi perturbations and measured by arbitrary operator norms in the case of spectral norms the obtained results unify and extend some previous works as well as a recent interesting result in m karrow d kressner on the structured distance to uncontrollability systems control lett 58 2009 128 132 some illustrating examples are given

a counterexample for two conjectures about stability one measure of the stability of a matrix a is the distance to the nearest unstable matrix b recently van loan presented an algorithm to compute b which depended on a conjecture about the location of its eigenvalues we provide a counterexample to this conjecture which shows that the algorithm may overestimate the distance to b by an arbitrary amount the same counterexample invalidates another conjecture and algorithm of the author

computing the radius of controllability for state space systems abstract in this paper we discuss the problem of computing the nearest uncontrollable system to a given control system represented by a matrix pair a b in order to do so we construct a sequence of structured matrices from given system matrices a and b controllability of the pair a b is equivalent to a condition on the null space dimension of an appropriate matrix in this sequence we show that the dimension of the reachability space is also related to the above condition further it is shown that the nearest structured low rank approximation slra of this structured matrix corresponds to a nearest uncontrollable system to the pair a b

computing rank deficiency of rectangular matrix pencils abstract rectangular matrix pencils arise in many contexts in linear control theory for example the pbh test for controllability and the computation of transmission zeros in this paper we examine a method to find the distance from an arbitrary given pencil to a nearby rank deficient one in light of the fact that rectangular pencils are generically full rank we propose an experimental computational method that exhibits quadratic convergence to a local minimum of this distance function this partially answers the question of the existence of a rank deficient pencil in a neighborhood of a given pencil we use the algorithm to illustrate some limitations of previous algorithms to measure this distance

algebraic riccati equations and the distance to the nearest uncontrollable pair a connection is established between nearness to unstabilizability of a stabilizable pair a b of matrices and nearness to singularity of the symmetric positive definite solution to an associated algebraic riccati equation from this result computable upper and lower bounds are derived for the distance of a b to the nearest uncontrollable pair numerical tests confirm the validity of the method and potential applications are discussed

an efficient mean field approach to the set covering problem a mean field feedback artificial neural network ann algorithm is developed and explored for the set covering problem a convenient encoding of the inequality constraints is achieved by means of a multilinear penalty function an approximate energy minimum is obtained by iterating a set of mean field equations in combination with annealing the approach is numerically tested against a set of publicly available test problems with sizes ranging up to 5 103 rows and 106 columns when comparing the performance with exact results for sizes where these are available the approach yields results within a few percent from the optimal solutions comparisons with other approximate methods also come out well in particular given the very low cpu consumption required typically a few seconds arbitrary problems can be processed using the algorithm via a public domain server 2001 elsevier science b v less

robust stability of linear systems described by higher order dynamic equations the stability radius of higher order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied different perturbation norms are considered the aim is to derive robustness criteria that are expressed directly in terms of the original data previous results on robust stability of hurwitz and schur polynomials are extended to monic matrix polynomials for disturbances acting via a uniform input matrix computable formulas are obtained whereas for perturbations with multiple input matrices structured singular values are involved

the structured distance to non surjectivity and its application to calculating the controllability radius of descriptor systems abstract the classical eckart young formula for square matrices identifies the distance to singularity of a matrix the main purpose of this paper is to get generalizations of this formula we characterize the distance to non surjectivity of a linear operator w l x y in finite dimensional normed spaces x y under the assumption that the operator w is surjective i e w x y and subjected to structured perturbations of the form w m n as an application of these results we shall derive formulas of the controllability radius for a descriptor controllable system e a b e x a x b u t 0 under the assumption that systems matrices e a b are subjected to structured perturbations and to multi perturbations

an ode based method for computing the distance of coprime polynomials to common divisibility the problem of computing the distance of two real coprime polynomials to the set of polynomials with a nontrivial greatest common divisor gcd appears in computer algebra signal processing and control theory it has been studied in the literature under the names approximate common divisor varepsilon gcd and distance to uncontrollability existing solution methods use different types of local optimization methods and require a user defined initial approximation in this paper we propose a new method that allows us to include constraints on the coefficients of the polynomials moreover the method proposed in the paper is more robust to the initial approximation than the newton type optimization methods available in the literature our approach consists of two steps 1 reformulate the problem as a problem of determining the structured distance to singularity of an associated sylvester matrix and 2 integrate a system of odes which describes the gradient associated to the functional to be minim

stability radii of higher order positive difference systems in this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices it is shown that the real and complex stability radii coincide moreover explicit formulas are derived for these stability radii and illustrated by some examples

a global minimum search algorithm for estimating the distance to uncontrollability abstract let a r n n and b r n m we suggest a new search algorithm for estimating the distance a b of a controllable pair a b to the set of uncontrollable pairs by estimating the global minimum of the function min a i b c where min denotes the smallest singular value of a matrix using simple properties of this function due to ralph byers one first observes that provided rank b n the minimization problem can be transformed to a minimization problem in the bounded region x z x a 2 z a 2 in the two dimensional real plane the algorithm then progressively partitions this region into simplexes and by determining whether their vertices x j z j satisfy z j min y r min a x j iy i b it computes after a finite number of steps upper and lower bounds for a b the difference between the two is small if a b is small while the lower bound is larger if a b is large thus ensuring safe decisions an error analysis numerical examples and an operation count are all presented only simple modifications of the search region are necessary to extend the applicability of the algorithm to the case when rank b n

structured low rank approximation this paper concerns the construction of a structured low rank matrix that is nearest to a given matrix the notion of structured low rank approximation arises in various applications ranging from signal enhancement to protein folding to computer algebra where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system the task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix this paper addresses some of the theoretical and numerical issues involved in the problem two procedures for constructing the nearest structured low rank matrix are proposed the procedures are flexible enough that they can be applied to any lower rank any linear structure and any matrix norm in the measurement of nearness the techniques can also be easily implemented by utilizing available optimization packages the special case of symmetric toeplitz structure using the frobenius matrix norm is used to exemplify the ideas throughout the discussion the concept rather than the implementation details is the main emphasis of the paper

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

higher order continuous time implicit systems consistency and weak consistency impulse controllability geometric concepts and invertibility properties abstract the paper provides a simple and fully algebraic distributional setup for a higher order linear implicit system with arbitrary constant coefficients on the continuous nonnegative time axis the distributional system equation exhibits impulses depending on arbitrary points in the state space and for these points several concepts of weak consistency of increasing degree are introduced in such a way that weak consistency of degree 0 coincides with the standard concept of consistency whereas weak consistency of the highest degree is related to impulse controllability all weakly consistent subspaces are expressed in the consistent subspace and for the latter space a numerically stable algorithm is presented also we derive for every concept of weak consistency a condition in terms of the system coefficients only for the statement that the associated subspace is as large as possible in particular we get a condition for impulse controllability that generalizes the celebrated condition for impulse controllability of a first order system further we state two conditions for control solvability and we specify when distributional state and or input trajectories are unique finally we define and characterize various subspaces for a higher order system in combination with an output equation and link these spaces to two of our four invertibility concepts namely those in the strong sense and we establish that the above composite system is left right invertible in the strong sense if and only if the corresponding system matrix is left right invertible as a rational matrix even if the transfer matrix does not exist

stability radii of positive higher order difference system in infinite dimensional spaces abstract in this paper we study stability radii of positive higher order difference system under affine perturbations of the coefficient operators it is shown that the real and complex stability radii coincide moreover explicit formulas are derived for these stability radii and illustrated by some examples the results obtained are extensions of the recent work in d hinrichsen p h a ngoc n k son stability radii of positive higher order difference system systems control lett 49 2003 377 388 to banach spaces

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

learning by dilution in a neural network a perceptron with n random weights can store of the order of n patterns by removing a fraction of the weights without changing their strengths the critical storage capacity as a function of the concentration of the remaining bonds for random outputs and for outputs given by a teacher perceptron is calculated a simple hebb like dilution algorithm is presented which in the teacher case reaches the optimal generalization ability

the characteristic finite volume element method for the nonlinear convection dominated diffusion problem in modern numerical simulation of prospecting and exploiting oil gas resources and environmental science it is important to consider a numerical method for nonlinear convection dominated diffusion problems based on actual conditions such as the three dimensional characteristics of large scale science engineering computation we present a kind of characteristic finite volume element method some techniques such as calculus of variations commutating operators the theory of prior estimates and techniques are adopted suboptimal order error estimate in l 2 norm and optimal order error estimate in h 1 norm are derived to determine the errors for the approximate solution numerical results are presented to verify the performance of the scheme

extended hopfield models for combinatorial optimization the extended hopfield neural network proposed by abe et al 1992 for solving combinatorial optimization problems with equality and or inequality constraints has the drawback of being frequently stabilized in states with neurons of ambiguous classification as active or inactive we introduce in the model a competitive activation mechanism and we derive a new expression of the penalty energy allowing us to reduce significantly the number of neurons with intermediate level of activations the new version of the model is validated experimentally on the set covering problem our results confirm the importance of instituting competitive activation mechanisms in hopfield neural network models

on the stability radius of a generalized state space system the concept of distance to instability of a system matrix is generalized to system pencils which arise in descriptor semistate systems difficulties arise in the case of singular systems because the pencil can be made unstable by an infinitesimal perturbation it is necessary to measure the distance subject to restricted or structured perturbations in this paper a suitable measure for the stability radius of a generalized state space system is defined and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one for systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles

the structured distance to the nearest system without property mathcal p for a system matrix m this paper explores the smallest frobenius norm additive structured perturbation delta m for which a system property mathcal p e g controllability observability stability etc fails to hold i e delta m is the structured perturbation with smallest frobenius norm such that there exists a property matrix r in mathcal p for which m delta m r drops rank the frobenius norm is used because of its direct dependence on the magnitude of each entry in the perturbation matrix necessary conditions on a locally minimum norm structured rank reducing perturbation delta m and associated property matrix r are set forth and proven an iterative algorithm is also set forth that computes a locally minimum norm structured perturbation and associated property matrix satisfying the necessary conditions algorithm convergence is proven using a discrete lyapunov function

the structured controllability radius of linear delay systems in this article we shall deal with the problem of calculation of the controllability radius of a delay dynamical systems of the form x t a 0 x t a 1 x t h 1 a k x t h k bu t by using multi valued linear operators we are able to derive computable formulas for the controllability radius of a controllable delay system in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

on the real stability radius of sparse systems abstract in this paper we study robust stability of sparse lti systems using the stability radius sr as a robustness measure we consider real perturbations with an arbitrary and pre specified sparsity pattern of the system matrix and measure their size using the frobenius norm we formulate the sr problem as an equality constrained minimization problem using the lagrangian method for optimization we characterize the optimality conditions of the sr problem thereby revealing the relation between an optimal perturbation and the eigenvectors of an optimally perturbed system further we use the sylvester equation based parametrization to develop a penalty based gradient newton descent algorithm which converges to the local minima of the optimization problem finally we illustrate how our framework provides structural insights into the robust stability of sparse networks

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

distance to uncontrollability for convex processes the classical study of controllability of linear systems assumes unconstrained control inputs the distance to uncontrollability measures the size of the smallest perturbation to the matrix description of the system rendering it uncontrollable and is a key measure of system robustness we extend the standard theory of this measure of controllability to the case where the control input must satisfy given linear inequalities specifically we consider the control of differential inclusions concentrating on the particular case where the control input takes values in a given convex cone

on the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control according to the methodology of 6 many measures of distance arising in problems in numerical linear algebra and control can be bounded by a factor times the reciprocal of an appropriate condition number where the distance is thought of as the distance between a given problem to the nearest ill posed problem in this paper four major problems in numerical linear algebra and control are further considered the computation of system hessenberg form the solution of the algebraic riccati equation the pole assignment problem and the matrix exponential the distances considered here are the distance to uncontrollability and the distance to instability

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

an algorithm for computing the distance to uncontrollability in this paper we present an algorithm to compute the distance to uncontrollability the problem of computing the distance is an optimization problem of minimizing sigma x y over the complete plane this new approach is based on finding zero points of grad sigma x y we obtain the explicit expression of the derivative matrix of grad sigma x y the newton s method and the bisection method are applied to approach these zero points numerical results show that these methods work well

cascade decomposition of nonlinear systems foliations and ideals of transitive lie algebras abstract we relate the cascade decomposition of nonlinear systems to the joint notions of foliations and ideals of transitive lie algebras

solving optimization problems with mean field methods abstract a brief review is given for the use of feed back artificial neural networks ann to obtain good approximate solutions to combinatorial optimization problems the key element is the mean field approximation mft which differs from conventional methods and feels its ways towards good solutions rather than fully or partly exploring different possible solutions the methodology which is illustrated for the graphs bisection and knapsack problems is easily generalized to potts systems the latter is related to the deformable templates method which is illustrated with the track finding problem the mean field approximation is based on a variational principle which also turns out to be very profitable when computing correlations in polymers

more on pseudospectra for polynomial eigenvalue problems and applications in control theory definitions and characterizations of pseudospectra are given for rectangular matrix poly nomials expressed in homogeneous form p da d d 1 a d 1 da 0 it is shown that problems with infinite pseudo eigenvalues are elegantly treated in this framework for such problems stereographic projection onto the riemann sphere is shown to provide a convenient way to visualize pseudospectra lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system with equality for standard state space systems are obtained in terms of pseudospectra showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory how and why to incorporate linear structure into pseudospectra is also discussed by example

controllability radii of linear neutral systems under structured perturbations abstractin this paper we shall deal with the problem of calculation of the controllability radii of linear neutral systems of the form x t a0x t a1x t h a 1x t h bu t we will derive the definition of exact controllability radius approximate controllability radius and euclidean controllability radius for this system by using multi valued linear operators the computable formulas for these controllability radii are established in the case where the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

decay rate estimations for linear quadratic optimal regulators let u t fx t be the optimal control of the open loop system x t ax t bu t in a linear quadratic optimization problem by using different complex variable arguments we give several lower and upper estimates of the exponential decay rate of the closed loop system x t a bf x t main attention is given to the case of a skew hermitian matrix a given an operator a for a class of cases we find a matrix b that provides an almost optimal decay rate we show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems a bj j 1 n and illustrate this on an example of a concrete mechanical system at the end of the article we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem

on the distance to singular descriptor dynamical systems with impulsive initial conditions in this paper we study the problem of computing the distance between a given singular descriptor system e a and a nearest descriptor system that has impulsive initial conditions the link between existence of impulsive initial conditions and zeros at infinity for the associated matrix pencil se a is well known much of the literature focusses on the case when only one of e and a is perturbed we give a closed form expression of the distance to a nearest descriptor system having impulsive solutions via rank 1 perturbations when both e and a are perturbed next for the case of perturbations without rank restrictions we propose and evaluate the bounds for the distance in the context of structured perturbations we formulate and obtain an explicit expression for the distance when e and a are hermitian and are perturbed by hermitian matrices for a suitable class of systems we also show that upper and lower bounds are within a factor of sqrt 2 we finally construct examples and compare the bounds obtained from our results with those from the literature as well as with computed values of the distance obtained via three numerical optimization techniques such as the structured low rank approximation the broyden fletcher goldfarb shanno algorithm and direct optimization tools like globalsearch

the structured controllability radii of higher order systems abstract in this paper we shall deal with the problem of calculation of the controllability radius of higher order dynamical systems of the form ex k t a k 1 x k 1 t a 1 x 1 t a 0 x t bu t where the leading coefficient matrix e need not assumed to be non singular by using multi valued linear operators we are able to derive computable formulas of the controllability radius under the assumption that the system s coefficient matrices are subjected to structured affine perturbations some examples are provided to illustrate the obtained results

statistical mechanics of the knapsack problem the knapsack problem is an np complete combinatorial optimization problem with inequality constraints using the replica method of statistical physics we study the space of its solutions for a large problem size it turns out that this problem is closely related to the theory of the binary perceptron

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

an efficient tabu search approach for the 0 1 multidimensional knapsack problem abstract in this paper we describe a new approach to tabu search ts based on strategic oscillation and surrogate constraint information that provides a balance between intensification and diversification strategies new rules needed to control the oscillation process are given for the 0 1 multidimensional knapsack 0 1 mkp based on a portfolio of test problems from the literature our method obtains solutions whose quality is at least as good as the best solutions obtained by previous methods especially with large scale instances these encouraging results confirm the efficiency of the tunneling concept coupled with surrogate information when resource constraints are present

an information based neural approach to generic constraint satisfaction a novel artificial neural network heuristic inn for general constraint satisfaction problems is presented extending a recently suggested method restricted to boolean variables in contrast to conventional ann methods it employs a particular type of non polynomial cost function based on the information balance between variables and constraints in a mean field setting implemented as an annealing algorithm the method is numerically explored on a testbed of graph coloring problems the performance is comparable to that of dedicated heuristics and clearly superior to that of conventional mean field annealing

real perturbation values and real quadratic forms in a complex vector space abstract a sequence of real numbers connected to a complex matrix is introduced it is shown how these real perturbation values can be computed and that they have several properties similar to the singular values the so called real pseudospectra and real stability radii can be computed using the real perturbation values the main result concerns the signature of real quadratic forms in complex vector spaces

radius of approximate controllability of linear retarded systems under structured perturbations abstract in this paper we shall deal with the problem of calculation of the radius of approximate controllability in the banach state space k n l 2 h k 0 k n for linear retarded systems of the form x t a 0 x t a 1 x t h 1 a k x t h k b u t by using multi valued linear operators we are able to derive computable formulas for this radius when the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

computation of the real controllability radius and minimum norm perturbations of higher order descriptor and time delay lti systems a linear time invariant lti system s controllability radius measures the norm of the smallest parametric perturbation such that the perturbed system is uncontrollable and is of practical importance in this note we study the real controllability radii of i higher order systems ii descriptor systems and iii time delay systems where the perturbations are restricted to the set of real values and the spectral norm is considered formulas for these radii are presented using a framework involving generalized real perturbation values which has certain computational advantages over other formulations found in the literature in particular the formulas are readily more computable especially for higher dimensional systems and a minimum norm perturbation can also easily be obtained numerical examples are presented

a formula for computation of the real stability radius this paper presents a readily computable formula for the real stability radius with respect to an arbitrary stability region in the complex plane

knapsack packing networks a knapsack packing neural network of 4n units with both low order and conjunctive asymmetric synapses is derived from a non hamiltonian energy function parallel simulations of randomly generated problems of size n in 5 10 20 are used to compare network solutions with those of simple greedy fast parallel enumerative algorithms

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a tutorial on the geometric analysis of linear time invariant implicit systems abstract a background of recent work in nonregular and regular implicit systems of the form e x ax bu is provided then the output nulling unknown input and composite subspaces are defined for singular systems with recursions given for their computation inner subspaces that is those in the domain of e and a are distinguished from outer subspaces those in the codomain of e and a the notions of composite preimage and composite image are introduced the result is a framework that unifies some of the recent work in the geometric theory of implicit systems system properties are discussed for both regular and nonregular systems and duality is investigated

on the structured distance to uncontrollability this article is concerned with the structured distance to uncontrollability of a linear time invariant system and relates this concept to a variation of the value the developed framework is applied to derive computational expressions for the class of real perturbations as well as for hermitian symmetric and skew symmetric perturbations in a relatively simple manner examples demonstrate that the structured distance can differ from the standard unstructured distance to uncontrollability by an arbitrary amount it is also shown how systems of higher order can be addressed

real controllability stabilizability radius of lti systems in this note we present a simple and readily computable formula for the calculation of the real controllability stabilizability and observability detectability radii for linear time invariant systems

matching hierarchical structures using association graphs it is well known that the problem of matching two relational structures can be posed as an equivalent problem of finding a maximal clique in a derived association graph however it is not clear how to apply this approach to computer vision problems where the graphs are hierarchically organized i e are trees since maximal cliques are not constrained to preserve the partial order we provide a solution to the problem of matching two trees by constructing the association graph using the graph theoretic concept of connectivity we prove that in the new formulation there is a one to one correspondence between maximal cliques and maximal subtree isomorphisms this allows us to cast the tree matching problem as an indefinite quadratic program using the motzkin straus theorem and we use replicator dynamical systems developed in theoretical biology to solve it such continuous solutions to discrete problems are attractive because they can motivate analog and biological implementations the framework is also extended to the matching of attributed trees by using weighted association graphs we illustrate the power of the approach by matching articulated and deformed shapes described by shock trees

numerical linear algebra aspects of control design computations the interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems control and estimation theory is discussed the impact of finite precision finite range arithmetic including the implications of the proposed ieee floating point standard s on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning basic tools from numerical linear algebra such as linear equations linear least squares eigenproblems generalized eigenproblems and singular value decomposition are then outlined a selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials frequency response system balancing and matrix riccati equations the implementation of such algorithms as robust mathematical software is then discussed a number of issues are addressed including characteristics of reliable mathematical software availability and evaluation language implications fortran ada etc and the overall role of mathematical software as a component of computer aided control system design

the multidimensional 0 1 knapsack problem an overview abstract the multidimensional 0 1 knapsack problem is one of the most well known integer programming problems and has received wide attention from the operational research community during the last four decades although recent advances have made possible the solution of medium size instances solving this np hard problem remains a very interesting challenge especially when the number of constraints increases this paper surveys the main results published in the literature the focus is on the theoretical properties as well as approximate or exact solutions of this special 0 1 program

a potts neuron approach to communication routing a feedback neural network approach to communication routing problems is developed with emphasis on multiple shortest path problems with several requests for transmissions between distinct start and endnodes the basic ingredients are a set of potts neurons for each request with interactions designed to minimize path lengths and to prevent overloading of network arcs the topological nature of the problem is conveniently handled using a propagator matrix approach although the constraints are global the algorithmic steps are based entirely on local information facilitating distributed implementations in the polynomially solvable single request case the approach reduces to a fuzzy version of the bellman ford algorithm the method is evaluated for synthetic problems of varying sizes and load levels by comparing to exact solutions from a branch and bound method or to approximate solutions from a simple heuristic with very few exceptions the potts approach gives legal solutions of very high quality the computational demand scales merely as the product of the numbers of requests nodes and arcs

a new low order least squares nonconforming characteristics mixed finite element method for burgers equation in this paper a new low order least squares nonconforming characteristics mixed finite element method mfem is considered for two dimensional burgers equation by use of two typical characters of the elements for approximating the velocity and flux variables a the consistency errors of the elements can be estimated as order o h 2 with respect to the mesh size h in the broken h 1 norm one order higher than their interpolation errors b the elements interpolation operators satisfy certain orthogonalities which lead to o h 2 order error estimates in l 2 norm and some superclose results the accuracy of corresponding variables in l 2 norm is improved by one order compared with other mfems in the existing literature it seems that the results provided herein have never been derived before

between controllable and uncontrollable the distance between a system a b and the set of uncontrollable systems is the minimum of the smallest singular value of i a b with respect to

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

controllability and observability of second order descriptor systems we analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case we show that performing a classical transformation to first order form may destroy some controllability and observability properties as an example we demonstrate that the loss of impulse controllability in constrained multibody systems is due to the representation as a first order system to avoid this problem we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties

computing the distance to an uncontrollable system algorithms for computing the distance from a particular controllable system to the closest uncontrollable system are presented the authors give an interpretation of the problem and propose solutions in the form of several linearly convergent iterative algorithms the algorithms are presented for the real and complex perturbation problems i e when the real system under investigation is subject to either real or complex perturbations

adaptive and self averaging thouless anderson palmer mean field theory for probabilistic modeling we develop a generalization of the thouless anderson palmer tap mean field approach of disorder physics which makes the method applicable to the computation of approximate averages in probabilistic models for real data in contrast to the conventional tap approach where the knowledge of the distribution of couplings between the random variables is required our method adapts to the concrete set of couplings we show the significance of the approach in two ways our approach reproduces replica symmetric results for a wide class of toy models assuming a nonglassy phase with given disorder distributions in the thermodynamic limit on the other hand simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional tap approaches

the semidiscrete finite volume element method for nonlinear convection diffusion problem abstract in this work a semidiscrete finite volume element method for the nonlinear convection diffusion problem is considered under some regular assumptions we obtain the l2 and h1 norm error estimates of numerical solution furthermore we investigate two grid finite volume element method for the considered equations compared with the standard method the two grid method is of the same order as the standard method in the h1 norm as long as the mesh sizes satisfy h o h 1 3 however the two grid method involves much less work than the standard method finally some numerical results are provided to verify the established theoretical analysis

rank preservation of matrices with structured uncertainties and its applications in robust control theory matrix rank is determined by the nonsingularity of its submatrices for matrices in which entries are quadratic functions of some uncertain parameters this paper derives sufficient conditions on parameters to that ensure the matrices preserve to some degrees the ranks they have when the parameters are all zero the rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion and suitable tools such as the spl mu analysis method are used to solve the problem applications in robust control theory including tests for robust controllability observability minimum phaseness coprimeness and schur stability are given together with illustrative examples

the observability radius of networks this paper studies the observability radius of network systems which measures the robustness of a network to perturbations of the edges we consider linear networks where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes we allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics to comply with the network setting our work considers perturbations with a desired sparsity structure thus extending the classic literature on the observability radius of linear systems the paper proposes two sets of results first we propose an optimization framework to determine a perturbation with smallest frobenius norm that renders a desired mode unobservable from the existing sensor nodes second we study the expected observability radius of networks with given structure and random edge weights we provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks showing that line networks are inherently more robust than star networks

a new shift invariant representation for periodic linear systems abstract this paper presents a new family of shift invariant representations for periodic linear discrete time systems this representation has the advantages over the standard representation 5 of preserving the time step of the original system and of preserving causality under arbitrary feedback the disadvantage is that the realization is not minimal an application in attitude determination is presented

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

a statistical analysis of the knapsack problem we investigate the dependence of the multi knapsack objective function on the knapsack capacities and on the number of capacity constraints p in the case when all n objects are assigned the same profit value and the weights are uniformly distributed over the unit interval a rigorous upper bound to the optimal profit is obtained employing the annealed approximation and then compared with the exact value obtained through the lagrangian relaxation method the analysis is restricted to the regime where n goes to infinity and p remains finite

estimating the distance to uncontrollability a fast method and a slow one abstract for a linear control system a b the distance to uncontrollability is characterized by min ec n a i b where n a i b is the smallest singular value of the augmented matrix a i b two methods are developed to estimate the distance to uncontrollability giving both a lower bound and an upper bound one method is fast requiring only one spectral decomposition of a and computations of three smallest singular values and being used for well conditioned a the other is slow requiring computations of a large number of the smallest singular values but it produces bounds as tight as possible and also a region containing global minimizers newton s method can be used to compute the global minimizers if one really wants

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

solving overdetermined eigenvalue problems we propose a new interpretation of the generalized overdetermined eigenvalue problem left bf a lambda bf b right bf v approx 0 for two m times n m n matrices bf a and bf b its stability analysis and an efficient algorithm for solving it usually the matrix pencil bf a lambda bf b does not have any rank deficient member therefore we aim to compute lambda for which bf a lambda bf b is as close as possible to rank deficient i e we search for lambda that locally minimize the smallest singular value over the matrix pencil bf a lambda bf b practically the proposed algorithm requires cal o mn 2 operations for computing all the eigenpairs we also describe a method to compute practical starting eigenpairs the effectiveness of the new approach is demonstrated with numerical experiments a matlab based implementation of the proposed algorithm can be found at http www mat univie ac at neum software oeig

the feedback search algorithm estimating the distance of semi explicit systems to r uncontrollability abstract we show that the distance of a special descriptor system termed semi explicit from the nearest r uncontrollable system can be estimated by the global minimum of a specific real valued univariate function with variable we study this function for both real and complex and then develop a novel global optimization algorithm called the feedback search algorithm the algorithm is initially developed for one variable and then it is extended to two variables we implement the algorithm and apply it to the above function when varies on the real axis and also on the complex plane in addition to this we apply the algorithm to another relevant function from the literature for the numerical applications several random examples as well as examples from the literature were used

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

measuring how far a controllable system is from an uncontrollable one the main goal of this note is to explore some of the properties of a controllable system which in a precise sense to be defined is near an uncontrollable pair to do this we first discuss the notion of the distance between a controllable system and the nearest uncontrollable one in both real and complex cases 2 5 when this distance is relatively small we show to what extent this distance corresponds to certain properties of the uncontrollability matrix and the reachability grammian

formulae for the distance between controllable and uncontrollable linear systems in this paper under some special well defined norms the formulae for the distance between controllable and uncontrollable linear systems of the general form eucx ax bu are presented where e a and b are constant complex matrices and e may be singular these results generalize and improve the existing ones