new solutions for effective elastic moduli of microcracked solids abstract a new energy balance equation is proposed to evaluate the effective moduli of solids with randomly dispersed cracks in this study it is assumed that the potential energy released by embedding a circular or spherical rve with microcracks into an infinite matrix is equal to that induced by introducing its effective medium into the identical infinite matrix new non interacting and self consistent solutions for the effective moduli of linear elastic isotropic solids with randomly oriented microcracks were evaluated on the basis of the present energy balance equation the linearization of the two present solutions leads to the dilute solution in the limit of small crack density comparison studies with various existing solutions are also presented in detail

bisimulation relations for dynamical control and hybrid systems the fundamental notion of bisimulation equivalence for concurrent processes has escaped the world of continuous and subsequently hybrid systems inspired by the categorical framework of joyal nielsen and winskel we develop novel notions of bisimulation equivalence for dynamical systems as well as control systems we prove that these notions can be captured by the abstract notion of bisimulation as developed by joyal nielsen and winskel this is the first unified notion of system equivalence that transcends discrete and continuous systems furthermore this enables the development of a novel and natural notion of bisimulation for hybrid systems which is the final goal of this paper

addition formulas and the rayleigh identity for arrays of elliptical cylinders we apply the rayleigh method to solve the problem where a uniform electrostatic field is imposed upon a rectangular array of elliptical cylinders embedded in a matrix of unit dielectric constant this new formulation overcomes geometric restrictions inherent in previous methods and is shown in principle and in various examples to converge for all possible geometries of the array and inclusion also presented are forms of both the interior and exterior addition formulas for harmonic functions in elliptical coordinates that possess optimal regions of convergence

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

algebras of almost periodic functions with bohr fourier spectrum in a semigroup hermite property and its applications abstract it is proved that the unital banach algebra of almost periodic functions of several variables with bohr fourier spectrum in a given additive semigroup is an hermite ring the same property holds for the wiener algebra of functions that in addition have absolutely convergent bohr fourier series as applications of the hermite property of these algebras we study factorizations of wiener hopf type of rectangular matrix functions and the toeplitz corona problem in the context of almost periodic functions of several variables

solving quadratic distance problems an lmi based approach the computation of the minimum distance of a point to a surface in a finite dimensional space is a key issue in several system analysis and control problems the paper presents a general framework in which some classes of minimum distance problems are tackled via linear matrix inequality lmi techniques exploiting a suitable representation of homogeneous forms a lower bound to the solution of a canonical quadratic distance problem is obtained by solving a one parameter family of lmi optimization problems several properties of the proposed technique are discussed in particular tightness of the lower bound is investigated providing both a simple algorithmic procedure for a posteriori optimality testing and a structural condition on the related homogeneous form that ensures optimality a priori extensive numerical simulations are reported showing promising performances of the proposed method

spectral convergence of self adjoint operators and generalized wave operators abstract it is well known that if the difference of self adjoint operators a and b is of trace class then the generalized wave operators w b a s lim t exp itb exp ita p ac a exist we give a sufficient condition for s lim n w b n a n w b a when sequences of operators are considered the condition involves the convergence of the spectral density of a n examples of operators with this property are given this investigation is also related to the absolute continuity of self adjoint operators

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

oscillatory riemann hilbert problems and the corona theorem the paper is devoted to the riemann hilbert problem with matrix coefficient g l r 2 2 having detg 1 in hardy spaces hp 2 1 p on half planes c under the assumption of existence of a non trivial solution of corresponding homogeneous riemann hilbert problem in h 2 we study the solvability of the non homogeneous riemann hilbert problem in hp 2 1 p and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for g as well as explicit formulas for its factors in terms of solutions of two associated corona problems in c and c a separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in h are obtained making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2 2 matrix functions whose diagonal entries do not possess factorizations such matrices arise e g in the theory of convolution type equations on finite intervals

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface abstract the paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface the rayleigh type representative unit cell approach has been used the micro geometry of composite is modeled by a periodic structure with a unit cell containing multiple elliptic inclusions the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions an appropriate choice of the potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the local gradient and flux fields the convergence of solution has been verified and the parametric study of the model has been performed the obtained accurate statistically meaningful results illustrate a substantial effect of imperfect interface on the effective behavior of composite

computing output feedback controllers to enlarge the domain of attraction in polynomial systems the problem of computing controllers to enlarge the domain of attraction da of equilibrium points of polynomial systems is considered in order to deal with such a problem a technique for computing static nonlinear output feedback controllers which maximize the largest estimate of the da leda induced by a given polynomial lyapunov function is proposed the main contribution of the note is to show that a lower bound of the maximum achievable leda and a corresponding controller can be obtained through linear matrix inequality optimizations moreover a necessary condition for tightness of this lower bound is presented which is also a sufficient condition to establish the tightness of the lower bound of the leda for a given controller

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

interaction of two elliptic inclusions abstract the paper presents the solution of a boundary value problem involving two interacting elliptical inhomogeneities in an infinite elastic body the loading cases considered include a far field biaxial tension and a thermally induced residual field that is modeled by uniform eigenstrains sustained by the inhomogeneities the mathematical formulation is based upon the papkovich neuber displacement approach certain interesting aspects of the solution and the effects of perfectly bonded and slipping interfaces are discussed in some detail

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

on the gap between positive polynomials and sos of polynomials this note investigates the gap existing between positive polynomials and sum of squares sos of polynomials which affects several analysis and synthesis tools in control systems based on polynomial sos relaxations and about which almost nothing is known in particular a matrix characterization of the pns that is the positive homogeneous forms that are not sos is proposed which allows to show that any pns is the vertex of an unbounded cone of pns moreover a complete parametrization of the set of pns is introduced

a converse sum of squares lyapunov result with a degree bound although sum of squares programming has been used extensively over the past decade for the stability analysis of nonlinear systems several fundamental questions remain unanswered in this paper we show that exponential stability of a polynomial vector field on a bounded set implies the existence of a lyapunov function which is a sum of squares of polynomials in particular the main result states that if a system is exponentially stable on a bounded nonempty set then there exists a sum of squares lyapunov function which is exponentially decreasing on that bounded set furthermore we derive a bound on the degree of this converse lyapunov function as a function of the continuity and stability properties of the vector field the proof is constructive and uses the picard iteration our result implies that semidefinite programming can be used to answer the question of stability of a polynomial vector field with a bound on complexity

a globally asymptotically stable polynomial vector field with rational coefficients and no local polynomial lyapunov function abstract we give an explicit example of a two dimensional polynomial vector field of degree seven that has rational coefficients is globally asymptotically stable but does not admit an analytic lyapunov function even locally

a passivity based approach to stability of spatially distributed systems with a cyclic interconnection structure a class of distributed systems with a cyclic interconnection structure is considered these systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns in this paper a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified for these systems global stability results hold if the ldquosecantrdquo criterion is satisfied in the linear case it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic lyapunov function which extends a recent diagonal stability result to partial differential equations for reaction diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established all of the derived results remain true for both linear and nonlinear positive diffusion terms similar results are shown for compartmental systems

brief paper estimating the domain of attraction for non polynomial systems via lmi optimizations this paper proposes a strategy for estimating the domain of attraction da for non polynomial systems via lyapunov functions lfs the idea consists of converting the non polynomial optimization arising for a chosen lf in a polynomial one which can be solved via lmi optimizations this is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst case remainders in truncated taylor expansions moreover a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen lf and another condition is provided for establishing whether such an estimate has been found the proposed strategy can readily be exploited with variable lfs in order to search for optimal estimates lastly it is worth remarking that no other method is available to estimate the da for non polynomial systems via lmis

projective free algebras of continuous functions on compact abelian groups abstract it is proved that the wiener algebra of functions on a connected compact abelian group whose bohr fourier spectra are contained in a fixed subsemigroup of the additive dual group is projective free the semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite the projective free property is proved also for the algebra of continuous functions with the same condition on their bohr fourier spectra as an application the connected components of the set of factorable matrices are described the proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration

convex programs for temporal verification of nonlinear dynamical systems a methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently conditions that must be satisfied by a barrier certificate can be formulated as a convex program and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region the dual of this problem i e the reachability problem concerns proving the existence of a trajectory starting from the initial set that reaches another given set using insights from the linear programming duality appearing in the discrete shortest path problem we show in this paper that reachability of continuous systems can also be verified through convex programming several convex programs for verifying safety and reachability as well as other temporal properties such as eventuality avoidance and their combinations are formulated some examples are provided to illustrate the application of the proposed methods finally we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates

sum of squares certificates for stability of planar homogeneous and switched systems we show that existence of a global polynomial lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field under a mild condition implies existence of a polynomial lyapunov function that is a sum of squares sos and that the negative of its derivative is also a sum of squares this result is extended to show that such sos based certificates of stability are guaranteed to exist for all stable switched linear systems for this class of systems we further show that if the derivative inequality of the lyapunov function has an sos certificate then the lyapunov function itself is automatically a sum of squares these converse results establish cases where semidefinite programming is guaranteed to succeed in finding proofs of lyapunov inequalities finally we demonstrate some merits of replacing the sos requirement on a polynomial lyapunov function with an sos requirement on its top homogeneous component in particular we show that this is a weaker algebraic requirement in addition to being cheaper to impose computationally

almost periodic factorization of certain block triangular matrix functions let formula math where c j e c m m 0 and a for rational a v such matrices g are periodic and their wiener hopf factorization with respect to the real line r always exists and can be constructed explicitly for irrational a v a certain modification called an almost periodic factorization can be considered instead the case of invertible co and commuting c 1 c 0 1 c 1 c 0 1 was disposed of earlier it was discovered that an almost periodic factorization of such matrices g does not always exist and a necessary and sufficient condition for its existence was found this paper is devoted mostly to the situation when co is not invertible but the c j commute pairwise j 0 1 the complete description is obtained when m 3 for an arbitrary m certain conditions are imposed on the jordan structure of c j difficulties arising for m 4 are explained and a classification of both solved and unsolved cases is given the main result of the paper existence criterion is theoretical however a significant part of its proof is a constructive factorization of g in numerous particular cases these factorizations were obtained using maple the code is available from the authors upon request

almost periodic factorization of block triangular matrix functions revisited abstract let g be an n n almost periodic ap matrix function defined on the real line r by the ap factorization of g we understand its representation in the form g g g where g 1 g 1 is an ap matrix function with all fourier exponents of its entries being non negative respectively non positive and x diag ei 1x ei nx 1 n r this factorization plays an important role in the consideration of systems of convolution type equations on unions of intervals in particular systems of m equations on one interval of length lead to ap factorization of matrices 0 1 g x e i x i m 0 f x e i x i m we develop a factorization techniques for matrices of the form 0 1 under various additional conditions on the off diagonal block f the cases covered include f with the fourier spectrum f lying on a grid f h z and the trinomial f f with

a numerical simulation for effective elastic moduli of plates with various distributions and sizes of cracks abstract a fast convergent numerical model is developed to calculate the effective moduli of plates with various distributions and sizes of cracks in which the crack line is divided into m parts to obtain the unknown traction on the crack line when m 1 the model reduces to kachanov s approximation method int j solids struct 23 1987 23 six types of crack distributions and three kinds of crack sizes are considered which are four regular equilateral triangle equilateral hexagon rectangle and diamond and two random distributions random location and orientation and parallel orientation and random location and one two and random crack sizes some typical examples are also analyzed using the finite element method fem to validate the present model then the effective moduli associated with the crack distributions and sizes are calculated in detail the present results for the regular distributions show some very interesting phenomena that have not been revealed before and for the two random distributions as the effective moduli depend on samples due to the randomness the effect of the sample size and number are analyzed first then effective moduli for plates with the three sizes of cracks are calculated it is found that the effect of crack sizes on the effective moduli is significant for high crack densities and small for low crack densities and the random crack size leads to the lowest effective moduli the present numerical results are compared with several popular micromechanics models to determine which one can provide the optimum estimation of the effective moduli of cracked plates with general crack densities furthermore some existing numerical results are analyzed and discussed

interacting elliptic inclusions by the method of complex potentials abstract an accurate series solution has been obtained for a piece homogeneous elastic plane containing a finite array of non overlapping elliptic inclusions of arbitrary size aspect ratio location and elastic properties the method combines standard muskhelishvili s representation of general solution in terms of complex potentials with the superposition principle and newly derived re expansion formulae to obtain a complete solution of the many inclusion problem by exact satisfaction of all the interface conditions a primary boundary value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well posed set of linear algebraic equations a properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm the theory developed is rather general and can be applied to solve a variety of composite mechanics problems the advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber fiber interactions can be considered in this way the numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix inclusion interface

sufficient conditions for the projective freeness of banach algebras let r be a unital semi simple commutative complex banach algebra and let m r denote its maximal ideal space equipped with the gelfand topology sufficient topological conditions are given on m r for r to be a projective free ring that is a ring in which every finitely generated projective r module is free several examples are included notably the hardy algebra h x of bounded holomorphic functions on a riemann surface of finite type and also some algebras of stable transfer functions arising in control theory

solids with periodically distributed cracks abstract a systematic method is presented for estimating the overall properties of solids with periodically distributed cracks in view of the periodicity the displacement strain and stress fields of the cracked solid can be expressed in fourier series elastic solids with periodically distributed flat voids are considered first the results for cracks are then obtained by letting the thickness aspect ratio of the void approach zero this limiting process is performed with care the only approximation involved is the distribution of the homogenization eigenstrains which is assumed to be piecewise constant the estimate of overall elastic moduli crack opening displacements and stress intensity factors eventually reduces to the calculation of several infinite series the formulation is valid for elliptic as well as two dimensional line slit like cracks and cracks with arbitrary shapes it fully includes the interaction effects

stress intensity factor and effective stiffness of a solid containing aligned penny shaped cracks abstract the stress state and effective elastic moduli of an isotropic solid containing equally oriented penny shaped cracks are evaluated accurately the geometric model of a cracked body is a spatially periodic medium whose unit cell contains a number of arbitrarily placed aligned circular cracks a rigorous analytical solution of the boundary value problem of the elasticity theory has been obtained using the technique of triply periodic solutions of the lame equation by exact satisfaction of the boundary conditions on the cracks surfaces the primary problem is reduced to solving an infinite set of linear algebraic equations an asymptotic analysis of the stress field has been performed and the exact formulae for the stress intensity factor sif and effective elasticity tensor are obtained the numerical results are presented demonstrating the effect of the crack density parameter and arrangement type on sif and overall elastic response of a solid and comparison is made with known approximate theories

unitary colligations reproducing kernel hilbert spaces and nevanlinna pick interpolation in several variables abstract recently j agler studied the class s d of scalar valued analytic functions of d complex variables f for which f t 1 t d has norm at most 1 for any collection of d commuting contractions t 1 t d on a hilbert space h among other results he obtained a characterization of such functions in terms of a positivity property and in terms of a representation as the transfer function of a certain type of d variable linear system as well as a nevanlinna pick interpolation theorem for this class of functions in this note we examine the system theory aspects and uniqueness of the transfer function representation and give a simpler proof of the nevanlinna pick interpolation theorem for the class s d and obtain a d variable version of the toeplitz corona theorem by using ideas of arov and grossman introduced for 1 variable problems as a bonus we obtain a collection of linear fractional maps which parametrize the set of all s d solutions of an interpolation problem

corona problem and flows abstract using a version of ambrose s theorem we generalize the corona theorem to the setting of bounded analytic functions induced by flows the structure of the maximal ideal space is also investigated in particular an open dense subset just like the open unit disc in the classical case is determined

guest editorial special issue on positive polynomials in control the 16 papers in this special section focus on positive polynomials in control these papers propose new contributions to the characterization of positive polynomials and illustrate novel applications in the areas of uncertain systems nonlinear systems time delay systems hybrid systems and model predictive control

the corona theorem and the existence of canonical factorization of triangular ap matrix functions abstract using the corona theorem canonical factorizability criteria for triangular almost periodic matrix valued functions of the form formula are obtained this allows the authors to obtain effective criteria for the existence of canonical factorization as well as to construct explicit factorizations for new classes of almost periodic matrix functions in a companion paper

paper maximal lyapunov functions and domains of attraction for autonomous nonlinear systems in this paper we present various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system these methods are based on the concept of a maximal lyapunov function which is introduced in this paper a partial differential equation characterizing a maximal lyapunov function is derived and the relationships of this equation as compared to zubov s partial differential equation are discussed an iterative procedure is given for solving the new partial differential equation this procedure yields lyapunov function candidates that are rational functions rather than polynomials the method is applied to four two dimensional examples and one three dimensional example and it is shown that the estimates obtained using this method are in many cases substantially better than those obtained using known methods

hierarchically consistent control systems large scale control systems typically possess a hierarchical architecture in order to manage complexity higher levels of the hierarchy utilize coarser models of the system resulting from aggregating the detailed lower level models in this layered control paradigm the notion of hierarchical consistency is important as it ensures the implementation of high level objectives by the lower level system in this paper we define a notion of modeling hierarchy for continuous control systems and obtain characterizations for hierarchically consistent linear systems with respect to controllability objectives as an interesting byproduct we obtain a hierarchical controllability criterion for linear systems from which we recover the best of the known controllability algorithms from numerical linear algebra

analysis of two dimensional finite solids withmicrocracks a general method is presented for solving the plane elasticity problem of finite plates with multiple microcracks the method directly accounts for the interactions between different microcracks and the effect of outer boundary of a finite plate analysis is based on a superposition scheme and series expansions of the complex potentials by using the traction free conditions on each crack surface and resultant forces relations along outer boundary a set of governing equations is formulated the governing equations are solved numerically on the basis of a boundary collocation procedure the effective young s moduli for randomly oriented cracks and parallel cracks are evaluated for rectangular plates with microcracks the numerical results are compared with those from various micromechanics models and experimental data these results show that the present method provides a direct and efficient approach to deal with finite solids containing multiple microcracks

effect of crack orientation statistics on effective stiffness of mircocracked solid this paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics to do so the representative unit cell approach has been used the microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks a sufficient number is taken to account for the microstructure statistics the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions a proper choice of potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields the convergence study has been performed the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks comparison has been made with the selected simple micromechanical models namely non interaction approximation differential scheme and modified differential scheme it is found that among these models the differential scheme provides the best fit of the numerical data

a hybrid micro macro bem with micro scale inclusion crack interactions local analysis schemes capable of detailed representations of micro features of a problem are integrated with a macro scale bem technique capable of handling finite geometries and realistic boundary conditions this paper focuses on micro scale interactions among cracks and inclusions as well as their ramifications on macro scale damage evaluations the micro scale effects are introduced into the macro scale bem computations through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows complete decomposition of the real problem into two sub problems one residing entirely at the micro level while the other resides at the macro level this allows for investigations of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the structural component under consideration as a first attempt dilute inclusion densities with strong inclusion crack and crack crack interactions are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can be easily extended to investigate the effects of other micro features e g interfaces short or continuous fibers in the context of linear elasticity on macro scale failure modes observed in structural components

factorization of matrix functions with subgroup supported fourier coefficients abstract it is proved that if a wiener class matrix function on a connected compact abelian group admits a factorization with respect to a connected compact abelian supergroup then it admits a factorization with respect to the original group in particular all factorization indices are characters of the original group an illustrative example is considered

one sided invertibility of matrices over commutative rings corona problems and toeplitz operators with matrix symbols abstract conditions are established under which fredholmness coburn s property and one or two sided invertibility are shared by a toeplitz operator with matrix symbol g and the toeplitz operator with scalar symbol det g these results are based on one sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems

fredholmness of toeplitz operators and corona problems a meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left invertible in a space of meromorphic functions the fredholmness of toeplitz operators with symbol g l r 2 2 is shown to be equivalent to that of a toeplitz operator with scalar symbol detg provided that the riemann hilbert problem g m m admits a solution such that the meromorphic corona problems with data m are solvable the fredholm properties are characterized in terms of m and the corresponding meromorphic left inverses partial index estimates for the symbols and fredholmness criteria are established for several classes of toeplitz operators

system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

a new approach to factorization of a class of almost periodic triangular symbols and related riemann hilbert problems the factorization of almost periodic triangular symbols g associated to finite interval convolution operators is studied for two classes of operators whose fourier symbols are almost periodic polynomials with spectrum in the group z z z 0 1 1 q the factorization problem is solved by a method that is based on the calculation of one solution of the riemann hilbert problem g in l r and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the riemann hilbert problem some unexpected but interesting results are obtained concerning the fourier spectrum of the solutions of g in particular it is shown that a solution exists with fourier spectrum in the additive group z z whether this group contains z or not possible application of the method to more general classes of symbols is considered in the last section of the paper

caratheodory toeplitz and nehari problems for matrix valued almost periodic functions in this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated we present nece ssary and sufficient conditions for the existence of extensions in terms of toeplit z and hankel operators on besicovitch spaces and lebesgue spaces furthermore when a solution exists a special extension the band extension is constructed which enjoys a maximum entropy property a linear fractional parameterization of the set of all extensions is also provided the techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

polynomial almost periodic solutions for a class of riemann hilbert problems with triangular symbols abstract let g a e i b c e i with 0 1 such that 1 1 q and a b c c 0 in this paper the existence of almost periodic polynomial app solutions to the equation g h e l l with h h e h and l h is studied the natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group z z z due to the difficulty in dealing with the problem in that generality solutions are sought with spectrum in the group z z several interesting and totally new results are obtained it is shown that if 1 z z no polynomial solutions exist i e almost periodic polynomial solutions exist only if z z z z z keeping to this setting it is shown that app solutions exist if and only if the function g satisfies the simple spectral condition 1 2 the proof of this result is nontrivial and has a number theoretic flavour explicit formulas for the solution to the above problem are given in the final section of the paper the derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of app solutions

green s tensors and lattice sums for electrostatics and elastodynamics we present expressions for the green9s tensors in terms of lattice sums for general two dimensional arrays in both elastostatics and elastodynamics we represent the lattice sums in rapidly convergent forms by fourier transform methods we compare the static lattice sums with recently published highly accurate values for square arrays we also establish rayleigh identities for elastostatic and elastodynamic problems generalizing previous work in electrostatics and electromagnetism the rayleigh identities may be used in studies of phononic band gaps

constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry using an abbreviation e to denote the function ei x on the real line r let g e 0fe where f is a linear combination of the functions e e e e with some 0 the criterion for g to admit a canonical factorization was established recently by avdonin bulanova and moran 2007 1 we give an alternative approach to the matter proving the existence when it does take place via deriving explicit factorization formulas the non existence of the canonical factorization in the remaining cases then follows from the continuity property of the geometric mean

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

on the problem of linear elasticity for an infinite region containing a finite number of non intersecting spherical inhomogeneities abstract the problem of linear elasticity for an infinite number of non intersecting spherical and more generally ellipsoidal inhomogeneities is attacked the approach taken does not misrepresent geometry of inhomogeneities although the continuity conditions at the intefaces are only approximately satisfied the principal idea of the approach is to extend the method of kachanov 1985 int j fracture28 r11 r19 1987 int j solids structures23 23 43 for interacting cracks to the realm of the eshelby equivalent inclusion method the application to a test problem for two spherical cavities suggests that the approach can be useful for predictions of the overall response of composite materials and interfacial stress concentrations

invertibility of toeplitz operators and corona conditions in a strip abstract a toeplitz operator with symbol g such that det g 1 is invertible if there is a non trivial solution to a riemann hilbert problem g with and satisfying the corona conditions in c and c respectively however determining such a solution and verifying that the corona conditions are satisfied are in general difficult problems in this paper on one hand we establish conditions on which are equivalent to the corona conditions but easier to verify if g 1 are analytic and bounded in a strip this happens in particular with almost periodic symbols on the other hand we identify new classes of symbols g for which a non trivial solution to g can be explicitly determined and the corona conditions can be verified by the above mentioned approach thus obtaining invertibility criteria for the associated toeplitz operators

spectral properties of truncated toeplitz operators by equivalence after extension abstract we study truncated toeplitz operators in model spaces k p for 1 p with essentially bounded symbols in a class including the algebra c r h as well as sums of analytic and anti analytic functions satisfying a separation condition using their equivalence after extension to toeplitz operators with 2 2 matrix symbols we establish fredholmness and invertibility criteria for truncated toeplitz operators with separated symbols and in particular we identify a class of operators for which semi fredholmness is equivalent to invertibility for symbols in c r h we extend to all p 1 the spectral mapping theorem for the essential spectrum stronger results are obtained in the case of operators with rational symbols or if the underlying model space is finite dimensional

elastic fields of interacting elliptic inhomogeneities abstract this paper presents a method for the calculation of two dimensional elastic fields in a solid containing any number of inhomogeneities under arbitrary far field loadings the method called pseudo dislocations method is illustrated for the solution of interacting elliptic inhomogeneities it reduces the interacting inhomogeneities problem to a set of linear algebraic equations numerical results are presented for a variety of elliptic inhomogeneity arrangements including the special cases of elliptic holes cracks and circular inhomogeneities all these complicated problems can be solved with high accuracy and efficiency

toeplitz operators of finite interval type and the table method abstract we solve a riemann hilbert problem with almost periodic coefficient g associated to a toeplitz operator t g in a class which is closely connected to finite interval convolution equations based on a generalization of the so called table method the explicit determination of solutions to that problem allows one to establish necessary and sufficient conditions for the invertibility of the corresponding toeplitz operator and to determine an appropriate factorization of g providing explicit formulas for the inverse of t g some unexpected properties of the fourier spectrum of the solutions are revealed which are not apparent through other approaches to the same problem

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

estimating the domain of attraction via union of continuous families of lyapunov estimates this paper proposes a new approach to estimate the domain of attraction of equilibrium points of polynomial systems the idea consists of estimating the domain of attraction via the union of a continuous family of lyapunov estimates rather than via one lyapunov estimate only as done in existing methods this family is obtained through a convex lmi optimization by deriving a stability condition which takes simultaneously into account all considered lyapunov functions moreover inner approximations of the union of this family via a set with simple shape are also derived

on algebraic proofs of stability for homogeneous vector fields we prove that if a homogeneous continuously differentiable vector field is asymptotically stable then it admits a lyapunov function which is the ratio of two polynomials i e a rational function we further show that when the vector field is polynomial the lyapunov inequalities on both the rational function and its derivative have sum of squares certificates and hence such a lyapunov function can always be found by semidefinite programming this generalizes the classical fact that an asymptotically stable linear system admits a quadratic lyapunov function which satisfies a certain linear matrix inequality in addition to homogeneous vector fields the result can be useful for showing local asymptotic stability of nonhomogeneous systems by proving asymptotic stability of their lowest order homogeneous component this paper also includes some negative results we show that in absence of homogeneity globally asymptotically stable polynomial vector fields may fail to admit a global rational lyapunov function and in presence of homogeneity the degree of the numerator of a rational lyapunov function may need to be arbitrarily high even for vector fields of fixed degree and dimension on the other hand we also give a family of homogeneous polynomial vector fields that admit a low degree rational lyapunov function but necessitate polynomial lyapunov functions of arbitrarily high degree this shows the potential benefits of working with rational lyapunov functions particularly as the ones whose existence we guarantee have structured denominators and are not more expensive to search for than polynomial ones

a hybrid micro macro bem formulation for micro crack clusters in elastic components local analysis schemes capable of detailed representations of the micro features of a problem are integrated with a macro scale bem technique capable of handling complex finite geometries and realistic boundary conditions the micro scale effects are introduced into the macro scale bem analysis through an augmented fundamental solution obtained from an integral equation representation of the micro scale features the proposed hybrid micro macro bem formulation allows decomposition of the complete problem into two sub problems one residing entirely at the micro level and the other at the macro level this allows for investigations of the effects of the micro structural attributes while retaining the macro scale geometric features and actual boundary conditions for the component or structure under consideration as a first attempt elastic fracture mechanics problems with interacting cracks at close spacings are considered the numerical results obtained from the hybrid bem analysis establish the accuracy and effectiveness of the proposed micro macro computational scheme for this class of problems the proposed micro macro bem formulation can easily be extended to investigate the effects of other micro features e g interfaces short or continuous fibre reinforcements voids and inclusions in the context of linear elasticity on macroscopic failure modes observed in structural components

the corona theorem and the canonical factorization of triangular ap matrix functions effective criteria and explicit formulas abstract using results of the companion paper the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost periodic matrix functions of the form formula the important role in the proof of necessity in these criteria is played by a result on the stability of canonical ap factorizability obtained in the present paper by finding corona solutions for a pair of binomial almost periodic functions it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomial f

an effective inclusion model for effective moduli of heterogeneous materials with ellipsoidal inhomogeneities abstract in the present study an effective inclusion model for effective elastic moduli of heterogeneous materials is proposed to analyze the problem of an infinite matrix containing an ellipsoidal rve it is assumed that the strain energy changes of the infinite matrix due to embedding the rve and its effective inclusion into the matrix are identical a system of equations for effective moduli is then formulated using the new energy balance equation that interrelate effective moduli to a problem of an infinite matrix containing n inhomogeneities in an ellipsoidal sub region a generalized non interacting solution derived based on the present formulation coincides with the estimates of the hashin shtrikman type obtained by ponte casta n eda and willis j mech phys solids 43 1995 1919 the effect of the shapes of rves on the approximate solution is also discussed in detail as further application of the present formulation the numerical models for the effective moduli of solids with cracks and heterogeneous materials with spherical inhomogeneities are proposed which account for the interactions among many cracks or spherical inhomogeneities the numerical results are then compared with the existing micromechanics models and experimental data

convergent relaxations of polynomial matrix inequalities and static output feedback using a moment interpretation of recent results on sum of squares decompositions of nonnegative polynomial matrices we propose a hierarchy of convex linear matrix inequality lmi relaxations to solve nonconvex polynomial matrix inequality pmi optimization problems including bilinear matrix inequality bmi problems this hierarchy of lmi relaxations generates a monotone sequence of lower bounds that converges to the global optimum results from the theory of moments are used to detect whether the global optimum is reached at a given lmi relaxation and if so to extract global minimizers that satisfy the pmi the approach is successfully applied to pmis arising from static output feedback design problems

flows in fibers let h a be the algebra of all bounded analytic functions on the open unit disc a and let 9j h a be the maximal ideal space of h a using a flow we represent a reasonable portion of a fiber in 9j h a this indicates a relation between the corona theorem and the individual ergodic theorem as an application we answer a question of forelli 3 by showing that there exists a minimal flow on which the induced uniform algebra is not a dirichlet algebra the proof rests on the fact that the closure of a nonhomeomorphic part in 9 h a may contain a homeomorphic copy of 9 h a taking suitable factors we may construct a lot of minimal flows which are not strictly ergodic

constructive almost periodic factorization of some triangular matrix functions abstract a factorability criterion is obtained constructively and the respective factorization obtained explicitly for 2 2 triangular almost periodic matrix functions of the form e 0 f e here f c 1 e c 0 c 1 e e x e i x c j are non zero constants and 0 max with being irrational note that the factorization problem even for triangular matrix functions as above with an arbitrary trinomial f is open the result obtained is yet another step towards its solution

a piecewise approximation approach to nonlinear systems stability and region of attraction this paper discusses piecewise bilinear models and recurrent fuzzy systems as particular classes of dynamic fuzzy systems which can be dealt with in the same framework due to their structural similarity as universal approximators they are capable of representing any continuous nonlinear system dynamics with arbitrary accuracy by means of a rule base first basic definitions of both systems are revisited here and unifying canonical forms are given then a stability criterion is derived for the system classes which is based on approximating piecewise quadratic lyapunov functions and formulated in terms of linear matrix inequalities the main contribution is the demonstration of the stability analysis and estimation of region of attraction by means of these models outperforming methods reported in the literature

integral equation methods and numerical solutions of crack and inclusion problems in planar elastostatics we present algorithms for crack and inclusion problems in planar linear elastostatics the algorithms are based on new integral equations for the pure crack problem the integral equations are of fredholm s second kind our algorithms show great stability and allow for solutions to problems more complex than have previously been possible our results are orders of magnitudes more accurate than those of previous investigators which rely on integral equations of fredholm s first kind

a simple technique for finding effective elastic constants of cracked solids for arbitrary crack orientation statistics abstract the effective elastic properties of cracked solids are anisotropic if the cracks have preferred orientations in this paper a simple scheme for evaluating the elastic stiffness tensor for an arbitrary orientation distribution of cracks at finite crack densities is presented the approach is based on a tensorial transformation of the effective elastic constants for isotropic orientation statistics through the use of a second order crack density tensor

the numerical calculation of two dimensional effective moduli for microcracked solids abstract there exist several micromechanics models for the determination of the effective moduli of microcracked solids and crack density is the only parameter in these models that characterizes the effect of microcracking a numerical hybrid bem method in conjunction with a unit cell model is proposed in the present paper to evaluate these micromechanics models a unit cell which can be considered as a representative block in the solid contains randomly distributed microcracks the unit cell is then assumed to be periodic in the solid so as to account for interactions between cracks inside and outside the cell there are stochastic variations of the estimated moduli for different microcrack distributions two groups of microcracks with the same crack density one with a low number of large cracks and the other with a large number of small cracks show the same range of stochastic variations and the same mean of effective moduli for random distributions of microcracks the effective moduli based on this numerical method for randomly distributed cracks and parallel cracks are compared with those from various micromechanics models while the differential method provides the closest estimation to the mean of the numerical results at low crack density the generalized self consistent method is much more accurate at relatively high crack density

generalised absolute stability and sum of squares this paper introduces a general framework for analysing nonlinear systems using absolute stability theory and sum of squares programming the technique decomposes a vector field into a system with a polynomial vector field in feedback with a nonlinear memoryless term which is contained in a generalised sector inequality with polynomial bounds this decomposition can be used to model uncertainty in the nonlinearity or to bound difficult to analyse terms by simpler polynomial functions such as time varying non polynomial or higher order nonlinearities conditions for stability and regions of attraction are found using polynomial and lur e type lyapunov functions which generalise those used for the derivation of the multivariable circle and popov criteria in classical absolute stability the technique extends both absolute stability theory and the applicability of sum of squares programming the usefulness of the technique is demonstrated with illustrative examples

local stability analysis of continuous time takagi sugeno fuzzy systems a fuzzy lyapunov function approach this paper proposes a strategy to estimate invariant subsets of the domain of attraction da for asymptotically stable zero equilibrium points of continuous time takagi sugeno t s fuzzy systems specifically by using lyapunov s stability theory and the linear matrix inequality lmi technique sufficient conditions for proving the local stability are provided in terms of single parameter minimization problems subject to lmi constraints or eigenvalue problems which are solvable via convex optimizations the fuzzy lyapunov functions flfs expressed by the so called multi dimensional fuzzy summations are employed to characterize invariant subsets of the da as sublevel sets of the flfs to compute a larger inner estimate of the da an iterative lmi algorithm is also developed finally illustrative examples show the efficacy of the approach

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

estimating the domain of attraction for uncertain polynomial systems estimating the domain of attraction da of equilibrium points is a problem of fundamental importance in systems engineering several approaches have been proposed for the case of known polynomial systems allowing one to find the largest estimate of the da leda for a given lyapunov function lf however the problem of estimating the robust da rda that is the da guaranteed for all possible uncertainties in an uncertain system it is still an unsolved problem in this paper lmi methods are proposed for estimating the rda in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope specifically the issue of computing the robust leda rleda that is the intersection of all ledas is considered for common and parameter dependent lfs providing constant and parameter dependent lower bounds the computation of approximations with simple shape of the rleda in the case of parameter dependent lfs is also discussed

conditional expectations and the corona problem of ergodic hardy spaces abstract using the regular corona theorem for the upper half plane we solve the coronatype problem for generalized analytic functions associated with flows

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

on the computation of set induced control lyapunov functions for continuous time systems the paper presents a computation oriented method for characterizing and obtaining local control lyapunov functions induced by particular star shaped nonconvex sets for continuous time nonlinear systems with bounded inputs for a given set the necessary and sufficient conditions for the induced function to be a nonconvex local control lyapunov function are provided the related convex problems for computing the exact region in which the function is decreasing and the optimal control input are presented the results are applied to the brockett integrator

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

spectral factorization of bi infinite multi index block toeplitz matrices in this paper we formulate a theory of lu and cholesky factorization of bi infinite block toeplitz matrices a ai j i j zd indexed by i j z d and develop two numerical methods to compute such factorizations 2002 elsevier science inc all rights reserved