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

uncontrollability for linear autonomous multi input dynamical systems depending on parameters linear multi input dynamical systems smoothly depending on parameters are considered a set of parameter values corresponding to uncontrollable systems an uncontrollability set is studied the typical generic structure of the uncontrollability set is described a constructive method of perturbation analysis of the uncontrollability set is developed formulae of first order approximations for the uncontrollability set and generalized eigenvalues uncontrollable modes are derived and used for numerical construction of the uncontrollability set the method is based on the versal deformation theory for matrix pairs under feedback equivalence as an example the uncontrollability set is found for a three parameter two degree of freedom mechanical system

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

a knowledge based system for robustness analysis of large scale economic systems abstract the paper gives a conceptual framework for robustness analysis of large scale economic systems and its realization through interactive computer aided software the mismatch between the economic system and the corresponding mathematical model is discussed the computer aided system combines algorithmic and expert system techniques an important feature of the present system is the modularization of the software package which allows a distributed problem solving approach a fourth order macroeconomic model with typical parameters which demonstrates the margin of power of the governmental body can exercise on the various sectoral activities is used to illustrate some of the concepts presented in this paper

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

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

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

on hardy s inequality abstract in this paper some new inequalities on the constant e are deduced as an application we build a strengthened hardy s inequality

pole mobility and minimal norm stabilization of siso systems under bounded state feedback this paper examines the fundamental system properties of pole mobility and stabilizability under bounded feedback for the case of single input single output siso systems where the constraints on the feedback are defined in terms of the l 2 norm it is shown that the bounded gain assumption implies a bounded norm condition on the coefficient vector associated with the closed loop characteristic polynomial classical and new results on the root distribution of bounded coefficient polynomials are reviewed first with these results the closed loop pole mobility of siso systems under bounded state feedback is studied it is shown that the assignable closed loop poles are always within bounded regions and alternative estimates for these regions are given exact boundaries are also established by utilizing computational methods a common computational scheme is also developed to calculate the distances of stable polynomials from instability and the distance of unstable polynomials from stability finally f

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

vibrations of an elastic cylindrical shell near the lowest cut off frequency a new asymptotic approximation of the dynamic equations in the two dimensional classical theory of thin elastic shells is established for a circular cylindrical shell it governs long wave vibrations in the vicinity of the lowest cut off frequency at a fixed circumferential wavenumber the latter corresponds to the eigenfrequency of in plane vibrations of a thin almost inextensible ring it is stressed that the well known semi membrane theory of cylindrical shells is not suitable for tackling a near cut off behaviour the dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two dimensional shell equations asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set ups including the assumption on vanishing of shear and circumferential mid surface deformations used in the semi membrane theory the obtained results may be of interest for dynamic modelling of elongated cylindrical thin walled structures such as carbon nanotubes

note on the carleman s inequality and hardy s inequality in this article using the properties of power mean and induction new strengthened carleman s inequality and hardy s inequality are obtained we also give an answer to the conjectures proposed by x yang in the literature yang 2001 5

asymptotic orders of reachability in perturbed linear systems a framework for studying asymptotic orders of reachability in perturbed linear time invariant systems is developed the systems of interest are defined by matrices that have asymptotic expansions in powers of a perturbation parameter epsilon about the point 0 the reachability structure is exposed by means of the smith form of the reachability matrix this approach is used to provide insight into the kinds of inputs needed to reach weakly reachable target states into the structure of high gain feedback for pole placement and into the types of inputs that steer trajectories arbitrarily close to almost a b invariant subspaces and almost a b controllability subspaces

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

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

vibrations of circular cylindrical shells theory and experiments in the present paper a method for analysing linear and nonlinear vibrations of circular cylindrical shells having different boundary conditions is presented the method is based on the sanders koiter theory displacement fields are expanded in a mixed double series based on harmonic functions and chebyshev polynomials simply supported and clamped clamped boundary conditions are analysed as well as connections with rigid bodies in the latter case experiments are carried out comparisons with experiments and finite element analyses show that the technique is computationally efficient and accurate in modelling linear vibrations of shells with different boundary conditions an application to large amplitude of vibration shows that the technique is effective also in the case of nonlinear vibration comparisons with the literature confirm the accuracy of the approach the method proposed is a general framework suitable for analysing vibration of circular cylindrical shells both in the case of linear and nonlinear vibrations r 2007 elsevier ltd all rights reserved

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

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

three dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single walled carbon nanotubes abstract within the framework of three dimensional 3d nonlocal elasticity theory the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported single walled carbon nanotubes swcnts non embedded or embedded in an elastic medium using the multiple time scale method eringen s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation the interactions between the swcnt and its surrounding medium are modeled as a two parameter pasternak foundation model after performing a series of mathematical processes including nondimensionalization asymptotic expansion and successive integration etc the authors obtain recurrent sets of motion equations for various order problems the nonlocal classical shell theory cst is derived as a first order approximation of the current 3d nonlocal elasticity problem and the equations of motion for higher order problems retain the same differential operators as those of the nonlocal cst although with different nonhomogeneous terms the current asymptotic solutions for the natural frequency parameters of non embedded or embedded swcnts and their corresponding through thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature

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

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

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

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

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

low frequency linear vibrations of single walled carbon nanotubes analytical and numerical models abstract low frequency vibrations of single walled carbon nanotubes with various boundary conditions are considered in the framework of the sanders koiter thin shell theory two methods of analysis are proposed the first approach is based on the rayleigh ritz method a double series expansion in terms of chebyshev polynomials and harmonic functions is considered for the displacement fields free and clamped edges are analysed this approach is partially numerical the second approach is based on the same thin shell theory but the goal is to obtain an analytical solution useful for future developments in nonlinear fields the sanders koiter equations are strongly simplified neglecting in plane circumferential normal strains and tangential shear strains the model is fully validated by means of comparisons with experiments molecular dynamics data and finite element analyses obtained from the literature several types of nanotubes are considered in detail by varying aspect ratio chirality and boundary conditions the analyses are carried out for a wide range of frequency spectrum the strength and weakness of the proposed approaches are shown in particular the model shows great accuracy even though it requires minimal computational effort

some inequalities involving the constante and an application to carleman s inequality some inequalities involving the constanteare considered a strengthened carleman s inequality is proved

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

approximation formulas for the constant e and an improvement to a carleman type inequality abstract we give an explicit formula for the determination of the coefficients c j appearing in the expansion x 1 j 1 q c j x j x 1 2 1 x e o 1 x q 1 for x and q n 1 2 we also derive a pair of recurrence relations for the determination of the constants l and l in the expansion 1 1 x x e 1 l 1 l x l 2 l 1 as x based on this expansion we establish an inequality for 1 1 x x as an application we give an improvement to a carleman type inequality

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

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

approximation for constant e and its applications abstract a new approximation for the constant e is obtained and its application to carleman s inequality and hardy s inequality are obtained the results of this paper generalize some early results on this topic

estimating distances from quadruples satisfying stability properties to quadruples not satisfying them abstract we consider quadruples of matrices e a b c defining generalized linear multivariable time invariant dynamical systems e x t ax t bu t y t cx t with a e square matrices and b c rectangular matrices using geometrical techniques we present upper bounds and lower bounds for the distances between a quadruple and the nearest non structurally stable uncontrollable and or unobservable one in terms of the singular values of matrices associated to the quadruple

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