endpoint estimates for one dimensional oscillatory integral operators abstract the one dimensional oscillatory integral operator associated to a real analytic phase s is given by t f x e i s x y x y f y d y in their fundamental work phong and stein established sharp l 2 estimates for t the goal of this paper is to extend their results to all endpoints in particular we obtain a complete characterization for the mapping properties for t on l p r more precisely we show that t f p f p holds for some 0 if and only if 1 p 1 p lies in the reduced newton polygon of s

meromorphy of local zeta functions in smooth model cases abstract it is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane but in the case of general c smooth functions the meromorphic extension problem is not obvious indeed it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles in order to understand the situation of the meromorphic extension in the smooth case we investigate a simple but essentially important case in which the respective function is expressed as u x y x a y b flat function where u 0 0 0 and a b are nonnegative integers after classifying flat functions into four types we precisely investigate the meromorphic extension of local zeta functions in each case our results show new interesting phenomena in one of these cases actually when a b local zeta functions can be meromorphically extended to the half plane re s 1 a and their poles on the half plane are contained in the set k b k n with k b a

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis we study the boundedness problem for maximal operators mathcal m associated with averages along smooth hypersurfaces s of finite type in 3 dimensional euclidean space for p 2 we prove that if no affine tangent plane to s passes through the origin and s is analytic then the associated maximal operator is bounded on l p left mathbb r 3 right if and only if p h s where h s denotes the so called height of the surface s defined in terms of certain newton diagrams for non analytic s we obtain the same statement with the exception of the exponent p h s our notion of height h s is closely related to a n varchenko s notion of height h for functions such that s can be locally represented as the graph of after a rotation of coordinates

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

inverse problems for lorentzian manifolds and non linear hyperbolic equations we study two inverse problems on a globally hyperbolic lorentzian manifold m g the problems are 1 passive observations in spacetime consider observations in an open set v subset m the light observation set corresponding to a point source at q in m is the intersection of v and the light cone emanating from the point q let w subset m be an unknown open relatively compact set we show that under natural causality conditions the family of light observation sets corresponding to point sources at points q in w determine uniquely the conformal type of w 2 active measurements in spacetime we develop a new method for inverse problems for non linear hyperbolic equations that utilizes the non linearity as a tool this enables us to solve inverse problems for non linear equations for which the corresponding problems for linear equations are still unsolved to illustrate this method we solve an inverse problem for semilinear wave equations with quadratic non linearities we assume that we are given the neighborhood v of the time like path mu and the source to solution operator that maps the source supported on v to the restriction of the solution of the wave equation to v when m is 4 dimensional we show that these data determine the topological differentiable and conformal structures of the spacetime in the maximal set where waves can propagate from mu and return back to mu

technical communique on the degree of polynomial parameter dependent lyapunov functions for robust stability of single parameter dependent lti systems a counter example to barmish s conjecture in this paper we consider the robust hurwitz stability analysis problems of a single parameter dependent matrix a q a email protected 1 over email protected 1 1 where a 0 a email protected r n x n with a 0 being hurwitz stable in particular we are interested in the degree n of the polynomial parameter dependent lyapunov matrix ppdlm of the form p q email protected i 0 email protected ip i that ensures the robust hurwitz stability of a q via p q 0 p q a q a t q p q 3 in general

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

resolution of singularities in two dimensions and the stability of integrals abstract for a small disk d centered at the origin in r 2 a smooth real valued function s x y on d and a positive epsilon we consider the measure of the points in d where s x y as well as oscillatory integral analogues specifically we consider the effect of perturbing s x y on these quantities besides being of intrinsic interest these questions are important in the analysis of fourier transforms of surface supported measures complex and higher dimensional analogues of these questions are also connected to various issues in algebraic and complex geometry for real analytic s x y this question has been investigated for example by karpushkin using versal deformation theory and by phong stein sturm who developed a method often referred to as the method of algebraic estimates in this paper we show how the use of resolution of singularities algorithms in two dimensions along with some one dimensional van der corput type lemmas provides another method for dealing with such questions as a result we prove new estimates and theorems for these and related quantities furthermore since these algorithms apply to all smooth functions the theorems will hold for all smooth functions as opposed to the earlier real analytic results

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

new strategy for robust stability analysis of discrete time uncertain systems in this paper a new robust stability analysis approach is developed for uncertain discrete time linear time invariant systems with polytopic or affine parameter dependent uncertainty models the proposed approach is based on a combination of a branch and bound like strategy with linear matrix inequality lmi based analysis formulations two sufficient conditions are considered one for the robust stability of the uncertain system and other one for the contrary situation if both sufficient conditions fail to characterize the polytope then it is iteratively subdivided into subpolytopes until some one proves to be unstable or all ones are verified to be robustly stable the polytope subdivision is implemented by means of a specially developed simplex subdivision algorithm exhaustive numerical tests prove the efficiency of the proposed approach when compared with the most recent lmi based formulations

a dual formulation of mixed spl mu and on the losslessness of d g scaling this paper studies the mixed structured singular value spl mu and the well known d g scaling upper bound spl nu a dual characterization of spl mu and spl nu is derived which intimately links the two values using the duals it is shown that spl nu is guaranteed to be lossless i e equal to spl mu if and only if 2 m sub r m sub e m sub c spl les 3 where m sub r m sub c and m sub c are the numbers of repeated real scalar blocks repeated complex scalar blocks and full complex blocks respectively the losslessness result further leads to a variation of the well known kalman yakubovich popov lemma and lyapunov inequalities

lmi characterization of structural and robust stability the discrete time case abstract this paper extends to the discrete time case some robust stability conditions recently obtained for continuous time systems those conditions are expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable as in the continuous time case parameter dependent lyapunov functions can be constructed and consequently the new approach can yield much sharper and less conservative results than the simultaneous stability approach in particular well known stability problems namely d stability and robust stability in the presence of diagonally structured uncertainty can be more efficiently addressed numerical examples are included to illustrate the advantages of the new stability conditions

on the growth and stability of real analytic functions this paper addresses the issue of whether integrals of real analytic functions remain finite under small deformations an approach based on uniform estimates for certain classes of one dimensional integrals is introduced it is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of karpushkin as well as produce new results in higher dimensions in dimension three the new stability results are sharp as shown by the well known example of varchenko

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties this paper studies the static output feedback sof control problem of continuous time linear systems with polytopic uncertainties novel lmi conditions with a line search over a scalar variable for designing robust sof controllers are proposed where the uncertain output matrix of the considered system is allowed to be not of full row rank in particular it is shown that the new method can give less or at least the same conservative results than those methods by inserting a matrix equality constraint between system output matrix and lyapunov matrix furthermore the result is extended to the case of h control numerical examples are given to illustrate the effectiveness of the proposed method

brief paper comparisons among robust stability criteria for linear systems with affine parameter uncertainties this paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems where the system state matrices considered are affinely dependent on the uncertain parameters at first a robust stability criterion for the class of systems to be affinely quadratically stable aqs is derived based on the vertex separator approach where affine parameter dependent lyapunov functions are exploited to prove stability then comparison results between the robust stability criterion and the existing tests for aqs are given in terms of degree of conservatism a numerical example is given to illustrate the results

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

real analytic desingularization and subanalytic sets an elementary approach we give a proof of a theorem on desingularization of real analytic functions which is a weaker version of h hironaka s result but has the advantage of being completely self contained and elementary and not involving any machinery from algebraic geometry we show that the basic facts about subanalytic sets can be proved from this result

maximal averages over hypersurfaces and the newton polyhedron abstract using some resolution of singularities and oscillatory integral methods in conjunction with appropriate damping and interpolation techniques l p boundedness theorems for p 2 are obtained for maximal averages over hypersurfaces in r n for n 2 these estimates are sharp in various situations including the convex hypersurfaces of finite line type considered by several authors as a corollary we also give a generalization of the result of sogge and stein that for some finite p the maximal operator corresponding to a hypersurface whose gaussian curvature does not vanish to infinite order is bounded on l p for some finite p analogous estimates are proven for fourier transforms of surface measures and these are sharp for the same hypersurfaces as the maximal operators

newton polygons and local integrability of negative powers of smooth functions in the plane let f x y be any smooth real valued function with f 0 0 0 for a sufficiently small neighborhood u of the origin we study the number sup u f x f it is known that sometimes this number can be expressed in a natural way using the newton polygon of f we provide necessary and sufficient conditions for this newton polygon characterization to hold the behavior of the integral at the supremal e is also analyzed

lmi tests for positive definite polynomials slack variable approach the considered problem is assessing non negativity of a function s values when indeterminates are in domains constrained by scalar polynomial inequalities the tested functions are multi indeterminates polynomial matrices which are required to be positive semi definite new tests based on linear matrix inequalities are provided in a slack variables type approach the results are compared to those obtained via the sum of squares approach they are proved to be equivalent in case of unbounded domains and less conservative if polytopic type bounds are known

discrete symmetries of differential equations the determination of the continuous symmetries of differential equations follows a well known algorithm and is reduced to solution of a set of linear equations this is based on considering infinitesimal generators of the symmetries so that the method does not extend to discrete symmetries in this paper we present a method to determine discrete symmetries in a certain class by means of a linear system although this is considerably more difficult to solve than the one connected with continuous symmetries we also consider the inverse and simpler problem of determining the most general equation admitting a given discrete symmetry in the last part we consider a number of examples dealing in particular with symmetries of relevance to physics

stability of polytopes of matrices via affine parameter dependent lyapunov functions asymptotically exact lmi conditions abstract this paper investigates necessary and sufficient conditions for the existence of an affine parameter dependent lyapunov function assuring the hurwitz or schur stability of a polytope of matrices a systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given at each step a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter dependent lyapunov function necessity is asymptotically attained through a relaxation based on a generalization of polya s theorem to the case of matrix valued functions numerical experiments illustrate the results

symmetry group of partial differential equations and of differential difference equations the toda lattice versus the korteweg de vries equation the authors correlate the symmetry group of the continuous transformations of the toda lattice to that of the korteweg de vries equation they show how by taking into account the continuous limit of the toda lattice the four parameter symmetry group of the toda lattice is contained in that of the kdv equation by an inverse process discretization of the symmetry group of the kdv equation they find a discrete element of the symmetry group of the toda lattice which gives by symmetry reduction its soliton solution

oscillatory integral decay sublevel set growth and the newton polyhedron extending some resolution of singularities methods greenblatt in j funct anal 255 8 1957 1994 2008 of the author a generalization of a well known theorem of varchenko funct anal appl 18 3 175 196 1976 relating decay of oscillatory integrals to the newton polyhedron is proven they are derived from analogous results for sublevel integrals proven here varchenko s theorem requires a certain nondegeneracy condition on the faces of the newton polyhedron on the phase in this paper it is shown that the estimates of varchenko s theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold thus in problems where one wants to switch coordinates to a coordinate system where varchenko s estimates are valid one has greater flexibility some additional estimates are also proven for more degenerate situations including some too degenerate for the newton polyhedron to give the optimal decay in the sense of varchenko

a multi dimensional resolution of singularities with applications to analysis we formulate a resolution of singularities algorithm for analyzing the zero sets of real analytic functions in dimensions geq 3 rather than using the celebrated result of hironaka the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature as an application we define a new notion of the height of real analytic functions compute the critical integrability index and obtain sharp growth rate of sublevel sets this also leads to a characterization of the oscillation index of scalar oscillatory integrals with real analytic phases in all dimensions

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

stability analysis of polynomially dependent systems by eigenvalue perturbation in this technical note we present a stability analysis approach for polynomially dependent one parameter systems the approach which appears to be conceptually appealing and computationally efficient and is referred to as an eigenvalue perturbation approach seeks to characterize the analytical and asymptotic properties of eigenvalues of matrix valued functions or operators the essential problem dwells on the asymptotic behavior of the critical eigenvalues on the imaginary axis that is on how the imaginary eigenvalues may vary with respect to the varying parameter this behavior determines whether the imaginary eigenvalues cross from one half plane into another and hence plays a critical role in determining the stability of such systems our results reveal that the eigenvalue asymptotic behavior can be characterized by solving a simple generalized eigenvalue problem leading to numerically efficient stability conditions

convergence of two dimensional weighted integrals a two dimensional weighted integral in r 2 is proposed as a tool for analyzing higher dimensional unweighted integrals and a necessary and sufficient condition for the finiteness of the weighted integral is obtained

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

an elementary coordinate dependent local resolution of singularities and applications abstract an elementary classical analysis resolution of singularities method is developed extensively using explicit coordinate systems the algorithm is designed to be applicable to subjects such as oscillatory integrals and critical integrability exponents as one might expect the trade off for such an elementary method is a weaker theorem than hironaka s work h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or its subsequent simplications and extensions such as e bierstone p milman canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant invent math 128 2 1997 207 302 s encinas o villamayor good points and constructive resolution of singularities acta math 181 1 1998 109 158 j kollar resolution of singularities seattle lectures preprint a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196 nonetheless the methods of this paper can be used to prove a variety of theorems of interest in analysis as illustration two consequences are given first and most notably a general theorem regarding the existence of critical integrability exponents are established secondly a quick proof of a well known inequality of lojasiewicz s lojasiewicz ensembles semi analytiques inst hautes etudes sci bures sur yvette 1964 is given the arguments here are substantially different from the general algorithms such as h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero i ann of math 2 79 1964 109 203 h hironaka resolution of singularities of an algebraic variety over a field of characteristic zero ii ann of math 2 79 1964 205 326 or the elementary arguments of e bierstone p milman semianalytic and subanalytic sets inst hautes etudes sci publ math 67 1988 5 42 and h sussman real analytic desingularization and subanalytic sets an elementary approach trans amer math soc 317 2 1990 417 461 the methods here have as antecedents the earlier work of the author m greenblatt a direct resolution of singularities for functions of two variables with applications to analysis j anal math 92 2004 233 257 phong and stein d h phong e m stein the newton polyhedron and oscillatory integral operators acta math 179 1997 107 152 and varchenko a n varchenko newton polyhedra and estimates of oscillatory integrals funct anal appl 18 3 1976 175 196

on archimedean zeta functions and newton polyhedra abstract let f be a polynomial function over the complex numbers and let be a smooth function over c with compact support when f is non degenerate with respect to its newton polyhedron we give an explicit list of candidate poles for the complex local zeta function attached to f and the provided list is given just in terms of the normal vectors to the supporting hyperplanes of the newton polyhedron attached to f more precisely our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant and necessary in the study of local zeta functions through resolution of singularities our results refine the corresponding results of varchenko and generalize the results of denef and sargos in the real case to the complex setting

on a conjecture about the norm of lyapunov mappings abstract the norm properties of lyapunov mappings and their restrictions on symmetric and skew symmetric subspaces are investigated for non negative non positive and tridiagonal matrices this paper gives an affirmative answer to an open conjecture which says that the norm of a lyapunov mapping over the space of real square matrices equals that of its restricted map over the subspace of real symmetric matrices in addition matrix transformation and norm relationships on certain matrices arising from the lyapunov mapping and its restrictions are provided

convergent lmi relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter dependent lyapunov functions abstract this paper investigates the problems of checking robust stability and evaluating robust 2 performance of uncertain continuous time linear systems with time invariant parameters lying in polytopic domains the novelty is the ability to check robust stability by constructing a particular parameter dependent lyapunov function which is a polynomial function of the uncertain system matrices as opposed to a general polynomial function of the uncertain parameter the degree of the polynomial is tied to a certain integer the existence of such lyapunov function can be proved by solving parameter dependent linear matrix inequalities lmis which are guaranteed to be solvable for a sufficiently large yet finite value of whenever the system is robustly stable extensions to guaranteed 2 cost computation are also provided numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

kahler currents and null loci we prove that the non kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a kahler manifold equals its null locus in particular this gives an analytic proof of a theorem of nakamaye and ein lazarsfeld musta nakamaye popa as an application we show that finite time non collapsing singularities of the kahler ricci flow on compact kahler manifolds always form along analytic subvarieties thus answering a question of feldman ilmanen knopf and campana we also extend the second author s results about noncollapsing degenerations of ricci flat kahler metrics on calabi yau manifolds to the nonalgebraic case

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

fixed structure lpv discrete time controller design with induced l2 norm and h2 performance abstracta new method for the design of fixed structure dynamic output feedback linear parameter varying lpv controllers for discrete time lpv systems with bounded scheduling parameter variations is presented sufficient conditions for the stability h2 and induced l2 norm performance of a given lpv system are represented through a set of linear matrix inequalities lmis these lmis are used in an iterative algorithm with monotonic convergence for lpv controller design extension to the case of uncertain scheduling parameter value is considered as well controller parameters appear directly as decision variables in the optimisation program which enables preserving a desired controller structure in addition to the low order efficiency of the proposed method is illustrated on a simulation example with an iterative convex optimisation scheme used for the improvement of the control system performance

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

sharp l estimates for one dimensional oscillatory integral operators with c phase in this paper sharp l 2 estimates are proved for general local one dimensional oscillatory integral operators with smooth phase the proof involves a coarse resolution of singularities in conjunction with a stopping time argument and extensions of the analytic methods of d h phong and e m stein

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

lie discrete symmetries of lattice equations we extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential difference equations as an example of the application of the methods we construct the discrete symmetries of the discrete painleve i equation and of the toda lattice equation

discrete point symmetries of ordinary differential equations this paper describes a method that enables the user to construct systematically the set of all discrete point symmetries of a given ordinary differential equation ode of order two or greater provided that the ode has at least a one parameter lie group of point symmetries the method is easy to use and is based upon lie s method of constructing continuous symmetries the calculations are simple and computer algebra is not usually required various examples are used to illustrate the method the paper concludes with a proof that every ode whose lie group of point symmetries is isomorphic to the unimodular group has at least four inequivalent real discrete point symmetries

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

the asymptotic behavior of degenerate oscillatory integrals in two dimensions abstract a theorem of varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real analytic phase in two dimensions his theorem expresses this order of decay in a simple geometric way in terms of its newton polygon once one is in certain coordinate systems called adapted coordinate systems in this paper we give explicit formulas that not only provide the order of decay of the leading term but also the coefficient of this term there are three rather different formulas corresponding to three different types of newton polygon analogous results for sublevel integrals are proven as are some analogues for the more general case of smooth phase the formulas require one to be in certain superadapted coordinates these are a type of adapted coordinate system which we show exists for any smooth phase

generalized kyp lemma unified frequency domain inequalities with design applications the celebrated kalman yakubovic spl caron popov kyp lemma establishes the equivalence between a frequency domain inequality fdi and a linear matrix inequality and has played one of the most fundamental roles in systems and control theory this paper first develops a necessary and sufficient condition for an s procedure to be lossless and uses the result to generalize the kyp lemma in two aspects the frequency range and the class of systems and to unify various existing versions by a single theorem in particular our result covers fdis in finite frequency intervals for both continuous discrete time settings as opposed to the standard infinite frequency range the class of systems for which fdis are considered is no longer constrained to be proper and nonproper transfer functions including polynomials can also be treated we study implications of this generalization and develop a proper interface between the basic result and various engineering applications specifically it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain phase properties in several frequency ranges the method is illustrated by numerical design examples of digital filters and proportional integral derivative controllers

algebraic estimates stability of local zeta functions and uniform estimates for distribution functions a method of algebraic estimates is developed and used to study the stability properties of integrals of the form r b jf z j i dv under small deformations of the function f the estimates are described in terms of a stratiflcation of the space of functionsfr z jp z j jq z j g by algebraic varieties on each of which the size of the integral of r z is given by an explicit algebraic expression the method gives an independent proof of a result on stability of tian in 2 dimensions as well as a partial extension of this result to 3 dimensions in arbitrary dimensions combined with a key lemma of siu it establishes the continuity of the mapping c r b jf z c j i dv1 dvn when f z c is a holomorphic function of z c in particular the leading pole is semicontinuous in f strengthening also an earlier result of lichtin

stability analysis and gain scheduled state feedback control for continuous time systems with bounded parameter variations the problems of robust stability analysis and state feedback control based on gain scheduling for continuous time systems with time varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article with respect to previous results in the literature two main contributions of the article are i the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones ii the conditions can be extended to cope with the problem of control design for this class of system parameter dependent linear matrix inequality lmi conditions are given for the existence of a parameter dependent lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability two convex procedures based on lmis exhibiting distinct complexities are proposed to solve the problem of robust stability an extension to deal with the computation of a stabilising parameter dependent state feedback gain for this class of time varying systems is also provided as a sequence of lmis of increasing precision examples illustrate the results including comparisons with other techniques from the literature

an existence result for polynomial solutions of parameter dependent lmis we show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set admits a branch of solutions polynomial with respect to the parameters this result is useful for studying e g parametric robustness or gain scheduling issues

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations

on adapted coordinate systems the notion of an adapted coordinate system for a given real analytic function introduced by v i arnol d plays an important role for instance in the study of asymptotic expansions of oscillatory integrals in two dimensions a n varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components varchenko s proof is based on a two dimensional resolution of singularities result in this article we present a more elementary approach to these results which is based on the puiseux series expansion of roots of the given function this approach is inspired by the work of d h phong and e m stein on the newton polyhedron and oscillatory integral operators it applies to arbitrary real analytic functions and even to arbitrary smooth functions of finite type in particular we show that varchenko s conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions even in the smooth finite type setting for analytic functions a construction of adapted coordinates by means of puiseux series expansions of roots has already been carried out in work by d h phong e m stein and j a sturm on the growth and stability of real analytic function as we learned after the completion of this paper in contrast to their work however our proof more closely follows varchenko s algorithm for the construction of an adapted coordinate system which turns out to be useful for the extension to the smooth setting

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

relaxed stability conditions for continuous time t s fuzzy control systems via augmented multi indexed matrix approach this paper is concerned with the problem of developing an advanced strategy to reduce the conservatism in stability analysis and control synthesis of continuous time takagi sugeno t s fuzzy systems a novel augmented multi indexed matrix approach is proposed to implement new right hand side slack variables technique for the homogenous polynomial setting combining with the finsler lemma with homogenous matrix lagrange multipliers convergent linear matrix inequality lmi relaxations for stability analysis are proposed by using the generalization of the polya theorem for the case of positive polynomials with matrix valued coefficients a new type of state feedback controller namely the homogeneous polynomially nonquadratic control law hpnqcl is developed to conceive less conservative stabilization conditions the obtained stability and stabilization conditions are further relaxed by using the proposed right hand side slack variables technique moreover the advantages over the existing control schemes are certificated in theory three numerical examples are also provided to illustrate the effectiveness of the proposed methods

a new method for robust schur stability analysis this article is concerned with robust stability of uncertain discrete time linear systems the matrix defining the linear system system matrix is assumed to depend affinely on a set of time invariant unknown parameters lying on a known polytope robust stability is investigated by checking whether a certain integer power of the uncertain system matrix has spectral norm less than one this peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by a unique feature is that no extra variables need to be added to the problems being solved numerical experiments reveal that the value of needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle by identifying the proposed stability test with a particular choice of a parameter dependent lyapunov function extra variables can be int