green s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces by virtue of two systems of vector functions and the propagator matrix method greens functions for transversely isotropic piezoelectric functionally graded exponentially graded in the vertical direction and multilayered half spaces are derived it is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions for a point force and point charge density applied at any location of the functionally graded half space greens functions are expressed in terms of one dimensional infinite integrals to carry out the numerical integral involving bessel functions an adaptive gauss quadrature approach is introduced and modified the piezoelectric functionally graded greens functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature two piezoelectric functionally graded half space models are analyzed numerically one is a functionally graded pzt 4 half space and the other a coated functionally graded pzt 4 layer over a homogeneous batio3 half space the effects of different exponential factors on greens function components are clearly demonstrated which could be useful in the design and manufacturing of piezoelectric functionally graded structures 2004 elsevier ltd all rights reserved

an improved suboptimal model reduction for singular systems an improved suboptimal model reduction for singular systems is proposed the proposed method is based on silverman ho algorithm as in wang et al 2004 a full parametrization of reduced order models is provided unlike the technique of wang et al 2004 where in only partial parametrization is presented the proposed method is illustrated by a numerical example and is compared with the method of wang et al 2004

higher order weighted sobolev spaces on the real line for strongly degenerate weights application to variational problems in elasticity of beams abstract for one dimensional interval and integrable weight function w we define via completion a weighted sobolev space h w m p of arbitrary integer order m the weights in consideration may suffer strong degeneration so that in general functions u from h w m p do not have weak derivatives this contribution is focussed on studying the continuity properties of functions u at a chosen internal point x 0 to which we attribute a notion of criticality of order k and with respect to the weight w for non critical points x 0 we formulate a local embedding result that guarantees continuity of functions u or their derivatives conversely we employ duality theory to show that criticality of x 0 furnishes a smooth approximation of functions in h w m p admitting jump type discontinuities at x 0 the work concludes with demonstration of established results in the context of variational problem in elasticity theory of beams with degenerate width distribution

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

h suboptimal model reduction for singular systems the model reduction problem was investigated for singular systems in this paper to solve the problem the silverman ho algorithm was given and based on this algorithm an optimal model reduction method for the fast subsystem was presented to obtain reduced order stable models for singular systems sequentially the suboptimal model reduction algorithm was obtained for singular systems the advantage of the presented algorithm is that the impulsive nature of singular system is preserved in the reduced order models then some necessary and sufficient conditions for the existence of a stable reduced order system were given and these conditions can be verified numerically finally illustrative examples were given to show the effectiveness of the proposed approach

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

hankel norm model reduction with fixed modes a constrained hankel norm approximation problem is considered the reduced order model is required to retain as a subset of its poles some prescribed eigenvalues from the original system the constraint approximation problem is solved with an optimal hankel norm criterion and an l sup infinity error bound for the approximation error is provided the proposed method for model reduction can be described as a partial eigenvalue preservation method the reduced order model is allowed to have a set of free poles in addition to eigenvalues which are preserved from the original system because of physical considerations although the method can be used in conjunction with the dominant mode concept it is equally feasible to retain some eigenvalues of particular interest but otherwise nondominant and let the free poles take care of the dominant characteristics of the system

controller reduction with error performance continuous and discrete time cases this paper is concerned with the problem of controller reduction for linear systems both continuous and discrete time cases are considered with necessary and sufficient conditions obtained for the existence of desired reduced order controllers in solving this problem two approaches are presented the first approach is based on the projection lemma where the admissible controllers can be parameterized after a set of conditions are satisfied and the second one directly incorporates the controller matrices to be determined into a set of conditions by introducing new techniques and thus no parameterization procedure is needed these necessary and sufficient conditions are formulated in terms of linear matrix inequalities lmis plus some equality constraints since these conditions are not convex the cone complementarity linearization ccl idea is exploited to cast them into sequential minimization problems subject to lmi constraints which can be readily solved by standard numerical software a num

h sub 2 near optimal model reduction this note considers the problem of finding a stable reduced order model for a given stable model so that its h sub 2 model reduction cost differs by less than a prescribed error from the optimal cost which may or may not be achievable it is shown that this new version of the long standing h sub 2 optimal model reduction problem can be reduced to a well posed smooth constrained minimization problem whose global solution is guaranteed to exist in addition a globally convergent algorithm in the form of an ordinary differential equation is derived

internal positivity preserved model reduction this article studies model reduction of continuous time stable positive linear systems under the hankel norm h norm and h 2 norm performance the reduced order systems preserve the stability as well as the positivity of the original systems this is achieved by developing new necessary and sufficient conditions of the model reduction performances in which the lyapunov matrices are decoupled with the system matrices in this way the positivity constraints in the reduced order model can be imposed in a natural way as the model reduction performances are expressed in linear matrix inequalities with equality constraints the desired reduced order positive models can be obtained by using the cone complementarity linearisation iterative algorithm a numerical example is presented to illustrate the effectiveness of the given methods

hankel norm approximation of linear systems with time varying delay continuous and discrete cases this paper investigates the problem of hankel norm model reduction for linear systems with time varying delay in the state for a given stable system our attention is focused on the construction of reduced order model which guarantees the corresponding error system to be asymptotically stable and has a specified hankel norm error performance two different approaches are proposed to solve this problem one casts the model reduction into a convex optimization problem by using a linearization procedure and the other is based on the cone complementarity linearization idea which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints both continuous and discrete time cases are considered a numerical example is provided to show the effectiveness of the proposed theory

optimal hankel norm approximation for stable systems with first order stable weighting functions abstract given a stable rational transfer function g s and weighting function w s the problem of finding g s of macmillan degree k so as to minimise w g g h is considered this problem is solved for w s s s with no assumptions on the signs of and this gives rise to approximations where w g g can be accurately bounded from above and below in terms of the hankel singular values of wg when 0

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on some necessary conditions of optimality for a nonlocal variational principle the aim of this work is to deduce optimality conditions for a nonlocal variational principle in the one dimensional scalar case we consider the relaxation of the problem in terms of young measures this relaxation is a new problem where we perform variations to derive optimality conditions and those conditions yield explicit information about minimizers in the homogeneous case it also provides a method capable of finding minimizers for some specific problems

h_ infty model reduction in the stochastic framework this paper investigates the problems of h infty model reduction for both continuous and discrete stochastic systems in terms of certain linear matrix inequalities lmis and a coupling nonconvex rank constraint necessary and sufficient conditions for the existence of solutions to such problems are obtained an explicit parametrization of all reduced order models corresponding to a feasible solution is also proposed in particular when a zeroth order h infty approximation is desired conditions are obtained using lmis only without any rank constraints and a parametrization of all solutions is also presented finally an illustrative example is provided to demonstrate the effectiveness of the proposed approach

a state space algorithm for the superoptimal hankel norm approximation problem it has been demonstrated by n t young nato asi series f34 springer verlag berlin new york 1987 that given a stable matrix valued function g 0 s and a nonnegative integer k there exists a unique superoptimal approximation phi s with no more than k poles in the left half plane that minimizes the sequence s 1 infty g 0 phi s 2 infty g 0 phi cdots with respect to lexicographic ordering where s i infty g 0 phi sup omega s i g 0 phi j omega and s i cdot are the singular values in descending order of magnitude this paper presents a constructive state space algorithm that evaluates the superoptimal approximating matrix function the procedure recursively minimizes each frequency dependent singular value with the aid of all pass transformations constructed from the kth schmidt pairs of a sequence of hankel operators the algorithm may be stopped after an arbitrary number of say l leqq min m p steps the representation formula at the

structural optimization of thin elastic plates the three dimensional approach the natural way to find the most compliant design of an elastic plate is to consider the three dimensional elastic structures which minimize the work of the loading term and pass to the limit when the thickness of the design region tends to zero in this paper we study the asymptotics of such a compliance problem imposing that the volume fraction remains fixed no additional topological constraint is assumed on the admissible configurations we determine the limit problem in different equivalent formulations and we provide a system of necessary and sufficient optimality conditions these results were announced in bouchitte et al c r acad sci paris ser i 345 713 718 2007 furthermore we investigate the vanishing volume fraction limit which turns out to be consistent with the results in bouchitte and fragala arch rat mech anal 184 257 284 2007 siam j control optim 46 1664 1682 2007 finally some explicit computation of optimal plates are given

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

green s function for a two dimensional exponentially graded elastic medium the freespace green function for a twodimensional exponentially graded elastic medium is derived the shear modulus is assumed to be an exponential function of the cartesian coordinates x y i e

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

multiplicative hankel norm approximation of linear multivariable systems abstract in control system design specifications are commonly given in terms of log magnitude quantities i e decibels this description induces a multiplicative or relative error criterion for plant model reduction techniques from the theory of additive hankel norm approximation and spectral factorization are synthesized to produce a multiplicative approximant of a possibly unstable or non square plant it is shown that the reduced order model preserves the unstable poles and right half plane zeros of the original system explicit state space formulae are provided for the construction of the reduced order model and its error properties are discussed the method is illustrated by a numerical example

a banach lattice approach to convergent integrably bounded set valued martingales and their positive parts denote by cf x the set of all nonempty convex closed subsets of a separable banach space x let be a complete probability space and denote by l1 cf x the complete metric space of equivalence classes of a e equal integrably bounded cf x valued functions for any preassigned filtration i we describe the space of convergent integrably bounded cf x valued martingales in terms of the closure of i 1 l1 i cf x in l1 cf x in particular we provide a formula to calculate the join of two such martingales and the positive part of such a martingale our object is achieved by considering the more general setting of a near vector lattice s d endowed with a riesz metric d by means of radstrom s embedding theorem for such spaces a link is established between the space of convergent martingales in s and the space of convergent martingales in the radstrom completion r s of s this link provides information about the former space of martingales via known properties of measure free martingales in riesz normed vector lattices applicable to r s we also apply our general results to the spaces of convergent ck x valued martingales where ck x denotes the set of all nonempty convex compact subsets of x

improved lmi representations for the analysis and the design of continuous time systems with polytopic type uncertainty simple modifications to the representation of the bounded real lemma brl in the linear continuous time invariant case are introduced these modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic type uncertainties a different lyapunov function accompanies each of the vertices of the uncertainty polytope thus eliminating the need for quadratic stability or stabilizability the advantages of these new representations are demonstrated by way of two examples

regularized formulation for optimal design of axisymmetric plates abstract this paper presents a regularized mathematical formulation necessary conditions of optimality and solutions of problems of optimal design of solid elastic axisymmetric plates of prescribed total material volume single purpose design criteria of minimum static compliance maximum stiffness and maximum frequency of free transverse vibrations are considered the regularization which alleviates some anomalies and difficulties encountered earlier in plate optimization problems is based on a new compound plate model with two simultaneous design variables namely variable thickness of a solid part of the plate and variable concentration of a dense system of thin integral stiffeners attached to the solid plate part a numerical optimality criterion based method of solution is developed for the problem and several optimal designs are presented the results are compared with results obtained from optimal design formulations applied heretofore and substantiate the superiority of the new regularized formulation

a duality approach in the optimization of beams and plates we introduce a class of nonlinear transformations called resizing rules which associate with optimal shape design problems certain equivalent distributed control problems while preserving the state of the system this puts into evidence the duality principle that the class of system states that can be achieved under a prescribed force via modifications of the structure shape of the system can be obtained as well via the modifications of the force action under a prescribed structure we apply such transformations to the optimization of beams and plates and in the simply supported or cantilevered cases the obtained control problems are even convex in all cases we establish existence theorems for optimal pairs by assuming only boundedness conditions moreover in the simply supported case we also prove the uniqueness of the global minimizer a general algorithm that iterates between the original transformed problems is introduced and studied the applications also include the case of variational inequalities

green s functions of an exponentially graded transversely isotropic half space abstract by virtue of a complete set of displacement potential functions and hankel transform the analytical expressions of green s function of an exponentially graded elastic transversely isotropic half space is presented the given solution is analytically in exact agreement with the existing solution for a homogeneous transversely isotropic half space employing a robust asymptotic decomposition technique the green s function is decomposed to the closed form green s function corresponding to the homogeneous transversely isotropic half space and grading term with strong decaying integrands this representation is very useful for numerical methods which are based on boundary integral formulations such as boundary element method since the numerically evaluated part is not responsible for the singularity the high accuracy of the proposed numerical scheme is confirmed by some numerical examples

l2 and l2 l model reduction via linear matrix inequalities necessary and sufficient conditions are derived for the existance of a solution to the continuous time and discrete time suboptimal l2 and l2 l model reduction problems these conditions are expressed in terms of linear matrix inequalities lmis and a coupling non convex rank constraint in addition explicit parametrizations of all reduced order models that correspond to a feasible solution are presented in terms of contractive matrices

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

second order energies on thin structures variational theory and non local effects abstract for a given positive measure on r n we consider integral functionals of the kind f u r n f x u 2 u d u c 0 r n and we study their relaxation with respect to the l p topology p being the growth exponent of f to obtain the relaxed energy f we develop a suitable theory of second order intrinsic operators related to a cosserat vector field and to a curvature tensor our main theorem shows that the functional f is in general a non local one this unexpected feature occurs even in very simple examples when is the one dimensional hausdorff measure over a closed lipschitz curve in the plane

h model reduction of markovian jump linear systems abstract in this paper the h model reduction problem for linear systems that possess randomly jumping parameters is studied the development includes both the continuous and discrete cases it is shown that the reduced order models exist if a set of matrix inequalities is feasible an effective iterative algorithm involving linear matrix inequalities is suggested to solve the matrix inequalities characterizing the model reduction solutions using the numerical solutions of the matrix inequalities the reduced order models can be obtained an example is given to illustrate the proposed model reduction method

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

ℋ model reduction for linear parameter varying systems with distributed delay this paper is concerned with the model reduction for linear parameter varying lpv systems with both discrete and distributed delays for a given stable system our attention is focused on the construction of reduced order models which approximate the original system well in an norm sense first a sufficient condition is proposed for the asymptotic stability with an performance of the error system by using the parameter dependent lyapunov functional method then the decoupling technique is applied such that there does not exist any product term between the lyapunov matrices and the system matrices in the parametrised linear matrix inequality plmi constraints thus a new sufficient condition is obtained based on the new condition two different approaches are developed to solve the model reduction problem one is the convex linearisation approach and the other is the projection approach finally a numerical example is provided to demonstrate the effectiveness of the proposed design method

integral representation of convex functionals on a space of measures abstract in duality pairs such as m b c 0 and w 1 p w 0 1 p a convex integral functional on the space of functions has a polar which admits an integral representation this representation is the sum of a first term involving the absolutely continuous component of the measure and of a second one which is a positively homogeneous function of the singular part the duality is useful in plasticity theory in the sobolev case the study of non parametric integrands is new a description of the sub differential is obtained

on green s function for a three dimensional exponentially graded elastic solid the problem of a point force acting in an unbounded three dimensional isotropic elastic solid is considered kelvin solved this problem for homogeneous materials here the material is inhomogeneous it is functionally graded specifically the solid is exponentially graded which means that the lame moduli vary exponentially in a given fixed direction the solution for the green9s function is obtained by fourier transforms and consists of a singular part given by the kelvin solution plus a non singular remainder this grading term is not obtained in simple closed form but as the sum of single integrals over finite intervals of modified bessel functions and double integrals over finite regions of elementary functions knowledge of this new fundamental solution for graded materials permits the development of boundary integral methods for these technologically important inhomogeneous solids

green s function expansion for exponentially graded elasticity new computational forms are derived for the green s function of an exponentially graded elastic material in three dimensions by suitably expanding a term in the defining inverse fourier integral the displacement tensor can be written as a relatively simple analytic term plus a single double integral that must be evaluated numerically the integration is over a fixed finite domain the integrand involves only elementary functions and only low order gauss quadrature is required for an accurate answer moreover it is expected that this approach will allow a far simpler procedure for obtaining the first and second order derivatives needed in a boundary integral analysis the new green s function expressions have been tested by comparing with results from an earlier algorithm

classification of four dimensional estimation algebras of central importance in nonlinear filtering theory is the classification of finite dimensional estimation algebras through which we can construct recursive filters we classify all t dimensional estimation algebras t spl les 4 for arbitrary state space dimension

on h2 model reduction of bilinear systems the h 2 model reduction problem for continuous time bilinear systems is studied in this paper by defining the h 2 norm of bilinear systems in terms of the state space matrices the h 2 model reduction error is computed via the reachability or observability gramian necessary conditions for the reduced order bilinear models to be h 2 optimal are given the gradient flow approach is used to obtain the solution of the h 2 model reduction problem the formulation allows certain properties of the original models to be preserved in the reduced order models the model reduction procedure developed can also be applied to finite dimensional linear time invariant systems a numerical example is employed to illustrate the effectiveness of the proposed method

a new model for thin plates with rapidly varying thickness abstract we study the bending of a thin plate with rapidly varying thickness for example one with rib like stiffeners or perforated by small holes we obtain a fourth order equation for the midplanc displacement using an asymptotic analysis based on three dimensional linear elasticity the coefficients of this equation represent the constitutive law relating bending moments to midplane curvature they are explicitly determined by the plate geometry our analysis distinguishes between three different cases in which the thickness varies on a length scale longer than on the order of or shorter than the mean thickness

optimality conditions for mass design problems and applications to thin plates we derive necessary and sufficient optimality conditions for a quite large class of structural design problems which can be formulated as follows under a given load and a total volume constraint minimize a suitable notion of compliance among all admissible mass distributions represented by positive measures with prescribed integral mean as a special case we focus attention on the optimization of thin plates we detail the corresponding optimality conditions and we show how they can be handled in order to determine analytically some optimal plates

brief paper weighted h model reduction for linear switched systems with time varying delay this paper is concerned with model reduction for continuous time linear switched systems with time varying delay for a given stable switched system our attention is focused on construction of a reduced order model such that the error system is exponentially stable with a prescribed weighted performance by applying the average dwell time approach and the piecewise lyapunov function technique delay dependent deley independent sufficient conditions are proposed in terms of linear matrix inequality lmi to guarantee the exponential stability and the weighted performance for the error system the model reduction problem is solved by using the projection approach which casts the model reduction problem into a sequential minimization problem subject to lmi constraints by employing the cone complementary linearization algorithm a numerical example is provided to illustrate the effectiveness of the proposed theory