stability analysis and control of rigid body systems with impacts and friction many critical tasks in robotics such as locomotion or manipulation involve collisions between a rigid body and the environment or between multiple bodies methods based on sums of squares sos for numerical computation of lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems and can additionally be used to automatically synthesize stabilizing feedback controllers here we present a method for applying sums of squares verification to rigid bodies with coulomb friction undergoing discontinuous inelastic impact events the proposed algorithm explicitly generates lyapunov certificates for stability positive invariance and safety over admissible non penetrating states and contact forces we leverage the complementarity formulation of contact which naturally generates the semialgebraic constraints that define this admissible region the approach is demonstrated on multiple robotics examples including simple models of a walking robot a perching aircraft and control design of a balancing robot

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

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

robust global optimization with polynomials we consider the optimization problems maxz minx k p z x and minx k maxz p z x where the criterion p is a polynomial linear in the variables z the set can be described by lmis and k is a basic closed semi algebraic set the first problem is a robust analogue of the generic sdp problem maxz p z whereas the second problem is a robust analogue of the generic problem minx k p x of minimizing a polynomial over a semi algebraic set we show that the optimal values of both robust optimization problems can be approximated as closely as desired by solving a hierarchy of sdp relaxations we also relate and compare the sdp relaxations associated with the max min and the min max robust optimization problems

existence results for semilinear differential equations with nonlocal and impulsive conditions abstract this paper is concerned with the existence for impulsive semilinear differential equations with nonlocal conditions using the techniques of approximate solutions and fixed point existence results are obtained for mild solutions when the impulsive functions are only continuous and the nonlocal item is lipschitz in the space of piecewise continuous functions is not lipschitz and not compact is continuous in the space of bochner integrable functions respectively

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

global stability analysis of fluid flows using sum of squares this paper introduces a new method for proving global stability of uid ows through the construction of lyapunov functionals for nite dimensional approximations of uid systems we show how one can exploit recently developed optimization methods based on sum of squares decomposition to construct a polynomial lyapunov function we then show how these methods can be extended to innite dimensional navier stokes systems using robust optimization techniques crucially this extension requires only the solution of innite dimension al linear eigenvalue problems and nite dimensional sum of squares optimization problems we further show that subject to minor technical constraints a general polynomial lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower bound on the maximum reynolds number for which a ow is globally stable if the ow does remain globally stable for reynolds numbers at least slightly beyond the energy stability limit such polynomial functions can be searched for eciently using the sos technique we propose

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

a procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

discriminant of symmetric matrices as a sum of squares and the orthogonal group it is proved that the discriminant of n n real symmetric matrices can be written as a sum of squares where the number of summands equals the dimension of the space of n variable spherical harmonics of degree n the representation theory of the orthogonal group is applied to express the discriminant of 3 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares it is proved that the discriminant of 4 4 real symmetric matrices can be written as a sum of seven squares these improve results of kummer from 1843 and borchardt from 1846 2010 wiley periodicals inc

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

asymptotic analysis of linearly elastic shells iii justification of koiter s shell equations we consider as in part i a family of linearly elastic shells of thickness 2 all having the same middle surfaces r3 where r2 is a bounded and connected open set with a lipschitz continuous boundary and l3 r3 the shells are clamped on a portion of their lateral face whose middle line is 0 where 0 is any portion of withlength 0 0 we make an essential geometrical assumption on the middle surfaces and on the set 0 which states that the space of inextensional displacements begin gathered v f omega eta eta i in h 1 omega times h 1 omega times h 2 omega hfill eta i partial v eta 3 0 on gamma 0 gamma alpha beta eta 0 in omega hfill end gathered where gamma alpha beta are the components of the linearized change is metric tensor ofs contains non zero functions this assumption is satisfied in particular ifs is a portion of cylinder and 0 is contained in a generatrix ofs

design of continuous twisting algorithm abstract for the double integrator with matched lipschitz disturbances we propose a continuous homogeneous controller providing finite time stability of the origin the disturbance is compensated exactly in finite time using a discontinuous function through an integral action since the controller is dynamic the closed loop is a third order system that achieves a third order sliding mode in the steady state the stability and robustness properties of the controller are proven using a smooth and homogeneous strict lyapunov function lf in a first stage the gains of the controller and the lf are designed using a method based on polya s theorem in a second stage the controller s gains are adjusted through a sum of squares representation of the lf

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

nonlocal cauchy problems governed by compact operator families abstract we study the existence of mild solutions for the nonlocal cauchy problem u t au t f t u t t t 0 t 0 t u t 0 g u u 0 and give new criterions which extend some existing results in this area

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

stability analysis for a class of partial differential equations via semidefinite programming this paper studies scalar integral inequalities in one dimensional bounded domains with polynomial integrands we propose conditions to verify the integral inequalities in terms of differential matrix inequalities these conditions allow for the verification of the inequalities in subspaces defined by boundary values of the dependent variables the results are applied to solve integral inequalities arising from the lyapunov stability analysis of partial differential equations examples illustrate the results

the energy operator and new scaling relations for the incompressible navier stokes equations the incompressible navier stokes equations have been widely studied due to their vast ability to model hydrodynamic phenomena the nonlinear character of these equations produces a hierarchical system of dynamical scales that offer challenging barriers to computation simulation and scale elimination modeling here the incompressible navier stokes equations are decomposed and studied under the context of global eigensystems that have a direct relation to the dynamical character of the scales of motion in particular the self adjoint linear reynolds orr operator is shown to correspond to the energy operator of this nonlinear system of equations and has the distinguishing property of having an eigensystem that orders the time derivative of the integral kinetic energy for the nonlinear navier stokes operator the eigensystem of the reynolds orr operator is shown to yield a dynamical range of scales that can be characterized by an energy containing range an inertial range and a dissipation range it i

existence of nondensely defined evolution equations with nonlocal conditions abstract this paper is concerned with the existence for nondensely defined evolution equations with nonlocal conditions using the techniques of fixed point theory and approximate solutions existence results are obtained for integral solutions when the nonlocal item is lipschitz continuous or continuous respectively examples are also given to illustrate our results

on the locking phenomenon for a class of elliptic problems in this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework at first the statements of the problem and its galerkin approximations are defined and an asymptotic analysis is performed then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter finally we study an example in computation of arches working in linear elasticity conditions we build one finite element scheme giving a locking behaviour and another one which does not

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

global existence for semilinear evolution equations with nonlocal conditions abstract in this paper we study the global existence of solutions for semilinear evolution equations with nonlocal conditions via a fixed point analysis approach using the leray schauder alternative we derive conditions under which a solution exists globally

existence results and asymptotic behavior for nonlocal abstract cauchy problems the purpose of this paper is to study the existence and asymptotic behavior of solutions for cauchy problems with nonlocal initial datum generated by accretive operators in banach spaces

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

on integral solutions of some nonlocal fractional differential equations with nondense domain abstract in this paper we prove the existence and uniqueness of integral solution for some nondensely defined fractional semilinear differential equation with nonlocal conditions the results are obtained by means of fixed point methods we also give some examples of such problems

sparse sos relaxations for minimizing functions that are summations of small polynomials this paper discusses how to find the global minimum of functions that are summations of small polynomials small means involving a small number of variables some sparse sum of squares sos techniques are proposed we compare their computational complexity and lower bounds with prior sos relaxations under certain conditions we also discuss how to extract the global minimizers from these sparse relaxations the proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems numerical experiments are presented which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems lastly we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization

existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions abstract in this work we establish several results about the existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions we assume that the linear part generates a strongly continuous semigroup on a general banach space

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

semi infinite programming using high degree polynomial interpolants and semidefinite programming in a common formulation of semi infinite programs the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval if this function is sufficiently closely approximable by a polynomial or a rational function then the semi infinite program can be reformulated as an equivalent semidefinite program which in turn can be solved with interior point methods very efficiently to high accuracy on the other hand solving this semidefinite program is challenging if the polynomials involved are of high degree due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum of squares representation of nonnegative polynomials are badly scaled we combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties using sum of squares interpolants specifically it is shown that the c

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

on solutions of neutral nonlocal evolution equations with nondense domain in this paper with sadovskii s and banach s fixed point theorems applied we establish some results on the existence of integral solutions strong solutions and strict solutions for a class of nondensely defined neutral evolution equations with nonlocal conditions also an example is given in the end to show the applications of the obtained results

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 and performance verification of optimization based controllers this paper presents a method to verify closed loop properties of optimization based controllers for deterministic and stochastic constrained polynomial discrete time dynamical systems the closed loop properties amenable to the proposed technique include global and local stability performance with respect to a given cost function both in a deterministic and stochastic setting and the mathcal l 2 gain the method applies to a wide range of practical control problems for instance a dynamical controller e g a pid plus input saturation model predictive control with state estimation inexact model and soft constraints or a general optimization based controller where the underlying problem is solved with a fixed number of iterations of a first order method are all amenable to the proposed approach the approach is based on the observation that the control input generated by an optimization based controller satisfies the associated karush kuhn tucker kkt conditions which provided all data is polynomial are a system of polynomial equalities and inequalities the closed loop properties can then be analyzed using sum of squares sos programming

sum of squares of polynomials approach to nonlinear stability of fluid flows an example of application with the goal of providing the first example of application of a recently proposed method thus demonstrating its ability to give results in principle global stability of a version of the rotating couette flow is examined the flow depends on the reynolds number and a parameter characterizing the magnitude of the coriolis force by converting the original navier stokes equations to a finite dimensional uncertain dynamical system using a partial galerkin expansion high degree polynomial lyapunov functionals were found by sum of squares of polynomials optimization it is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterizing the coriolis force outside this range a lower bound for the global stability limit was obtained which is still better than the energy stability limit in the course of the study several results meaningful in the context of the method used were also obtained overall the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows to the best of our knowledge it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a reynolds number greater than that which could be achieved with the energy stability approach

nonlocal cauchy problems for neutral functional differential and integrodifferential inclusions in banach spaces in this paper we investigate the existence of mild solutions on a compact interval to initial value problems for neutral functional differential and integrodifferential inclusions in banach spaces with nonlocal conditions the results are obtained by using a fixed point theorem for condensing maps due to martelli

existence and uniqueness of mild solutions to impulsive fractional differential equations this paper is concerned with the existence and uniqueness of a mild solution to impulsive fractional semilinear differential equations the results are obtained by means of fixed point methods we also give an example of such problems

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

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 existence and asymptotic behaviour of solutions of an evolution equation and an application to the feynman kac theorem abstract in this paper we study the existence of mild solutions of a semilinear evolution equation on an unbounded interval the rapidity of the growth of those solutions is characterized moreover we investigate the local and global attractivity of solutions of an equation in question and we describe their asymptotic behaviour our investigations are conducted in a real separable banach space

existence of classical solutions to nonautonomous nonlocal parabolic problems abstract we obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditions u t a t u t f t u t t s t u s g u u 0 in a banach space x where t s 0 f g are given x valued functions and a t is a sectorial operator not necessarily densely defined in x for each t 0 t both banach s contraction principle and schauder s fixed point theorem as well as the theory of evolution families and interpolation spaces are employed in our approach a concrete example is shown to illustrate the existence theorem

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

stability analysis of linear partial differential equations with generalized energy functions we present a method for the stability analysis of a large class of linear partial differential equations pdes in one spatial dimension we rely on lyapunov analysis to establish the exponential stability of the systems under consideration the proposed test for the verification of the underlying lyapunov inequalities relies on the existence of solutions of a system of coupled differential equations we illustrate the application of this method using a pde actuated by a backstepping computed feedback law furthermore for the case of pdes defined by polynomial data we formulate a numerical methodology in the form of a convex optimization problem which can be solved algorithmically we show the effectiveness of the proposed numerical methodology using examples of different types of pdes

an η approximation method in nonlinear vector optimization abstract in this paper a new approach for a solution of a nonlinear multiobjective programming problem is introduced an equivalent approximated vector optimization problem is constructed by a modification of the objective and the constraint functions in the original multiobjective programming problem the connection between weak efficient points in the original multiobjective programming problem and its equivalent approximated vector optimization problem is proved in this way optimality conditions for nonlinear constrained multiobjective programming problems having invex and or generalized invex objective and constraint functions with respect to the same functions are obtained

semidefinite approximations for global unconstrained polynomial optimization we consider the problem of minimizing a polynomial function on mathbb r n known to be hard even for degree 4 polynomials therefore approximation algorithms are of interest lasserre siam j optim 11 2001 pp 796 817 and parrilo math program 96 2003 pp 293 320 have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations we propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming the method is applied to several examples

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

impulsive problems for semilinear differential equations with nonlocal conditions abstract this paper is concerned with the existence of impulsive semilinear differential equations with nonlocal conditions using the technique of fixed point theory existence results are obtained for mild solutions when the nonlocal item is lipschitz is not lipschitz and not compact respectively

necessary conditions to avoid numerical locking in singular perturbations application to arch equations abstract we study the problem of numerical computation of the solution of equations posed in infinite dimensional vector spaces for example partial differential equations we focus specially on the case of linear equations depending on a real parameter t 0 1 the solution is supposed to exist and to be unique for any given t 0 1 but for t 0 there are several solutions the kernel of the operator defining the equation degenerates for t 0 it is not reduced to 0 several specific equations of this type have been investigated in the literature the authors usually try to build approximation schemes which converge to the solution of the equation uniformly with respect to the parameter when it tends to zero in this paper we study first in a general setting the existence of any approximation scheme which might converge uniformly with respect to t 0 1 this is done using the notion of nth which of a subset of a banach space then the result is applied to the equation governing the linear static equilibrium of an arch the small parameter is the thickness of the arch the result says that the best possible convergence is uniform in t furthermore one finite element method which is very commonly used in industrial problems degenerates when t becomes small but another finite element method which has been proposed more recently does converge uniformly with respect to t our result says that it is optimal

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

existence of solutions for neutral functional differential evolution equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii s fixed point theorem we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions the results we obtained are a generalization and continuation of the recent results on this issue in the end an example is given to show the application of our results

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

local existence and stability for some partial functional differential equations with infinite delay we study the existence regularity and stability of solutions for some partial functional dierential equations with infinite delay we assume that the linear part is not necessarily densely defined and satisfies the hille yosida condition on a banach space x the nonlinear term takes its values in space larger than x namely the extrapolated favard class of the extrapolated semigroup corresponding to the linear part our approach is based on the theory of the extrapolation spaces

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

mild solutions of semilinear evolution equation on an unbounded interval and their applications the aim of this paper is to give an existence theorem for a semilinear equation of evolution on an unbounded interval in the case when the generator of a semigroup depends on time parameter the considerations of this paper are based on the concept of a measure of noncompactness in frechet space and the classical tichonov fixed point principle moreover an application of the main theorem to the second order parabolic pde is given

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

a complete characterization of the gap between convexity and sos convexity our first contribution in this paper is to prove that three natural sum of squares sos based sufficient conditions for convexity of polynomials via the definition of convexity its first order characterization and its second order characterization are equivalent these three equivalent algebraic conditions henceforth referred to as sos convexity can be checked by semidefinite programming whereas deciding convexity is np hard if we denote the set of convex and sosconvex polynomials in n variables of degree d with cn d and cn d respectively then our main contribution is to prove that cn d cn d if and only if n 1 or d 2 or n d 2 4 we also present a complete characterization for forms homogeneous polynomials except for the case n d 3 4 which is joint work with g blekherman and is to be published elsewhere our result states that the set cn d of convex forms in n variables of degree d equals the set cn d of sos convex forms if and only if n 2 or d 2 or n d 3 4 to prove these results we present in particular explicit examples of polynomials in c2 6 c2 6 and c3 4 c3 4 and forms in c3 6 c3 6 and c4 4 c4 4 and a general procedure for constructing forms in cn d 2 cn d 2 from nonnegative but not sos forms in n variables and degree d although for disparate reasons the remarkable outcome is that convex polynomials resp forms are sos convex exactly in cases where nonnegative polynomials resp forms are sums of squares as characterized by hilbert

brief paper exact determinations of the maximal output admissible set for a class of nonlinear systems this paper is concerned with obtaining necessary and sufficient conditions for fulfilling specified state and control pointwise in time constraints against a certain class of nonlinear dynamics the results are generalizations of the maximal output admissible set theory to the case of nonlinear systems main contribution of the present paper is to propose explicit algorithmic procedures to determine the maximal output admissible set another contribution is that we discuss on a finite determinability of the maximal output admissible set for nonlinear systems some relations between observability of nonlinear systems and finite characterizations of the maximal output admissible set are clarified

nonlinear differential equations with nonlocal conditions in banach spaces abstract in this paper we study the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions in banach spaces we derive conditions under which the integral solutions exist

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

analysis of autocatalytic networks in biology autocatalytic networks in particular the glycolytic pathway constitute an important part of the cell metabolism changes in the concentration of metabolites and catalyzing enzymes during the lifetime of the cell can lead to perturbations from its nominal operating condition we investigate the effects of such perturbations on stability properties e g the extent of regions of attraction of a particular family of autocatalytic network models numerical experiments demonstrate that systems that are robust with respect to perturbations in the parameter space have an easily verifiable in terms of proof complexity region of attraction properties motivated by the computational complexity of optimization based formulations we take a compositional approach and exploit a natural decomposition of the system induced by the underlying biological structure into a feedback interconnection of two input output subsystems a small subsystem with complicating nonlinearities and a large subsystem with simple dynamics this decomposition simplifies the analysis of large pathways by assembling region of attraction certificates based on the input output properties of the subsystems it enables numerical as well as analytical construction of block diagonal lyapunov functions for a large family of autocatalytic pathways

set membership estimation of fiber laser physical parameters from input output power measurements the problem of fiber laser parameters estimation from input output transient power measurements is considered in the paper the proposed procedure is based on a suitable laser model obtained from the rate and propagation equations by proper manipulation of the simplified laser equations the estimation problem is formulated in the bounded error estimation framework where a priori information on physical parameters can be properly taken into account first the so called feasible parameter set is defined then the problem of optimally picking a single model from the set is formulated in terms of central conditional estimation and solved by exploiting recent results by the authors properties of the proposed solution are discussed the proposed algorithm is tested on a set of simulated data obtained from a commercial laser simulator