regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

the extended euler lagrange condition for nonconvex variational problems this paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a differential inclusion with a possibly nonconvex right side they take the form of an euler lagrange inclusion involving convexification in only one coordinate supplemented by the transversality and weierstrass conditions it is also shown that for time invariant free time problems the adjoint arc can be chosen so that the hamiltonian function is constant along the minimizing state arc the methods used here based on simple finite dimensional nonsmooth calculus clarke decoupling and a rudimentary version of the maximum principle offer an alternative and somewhat simpler derivation of such results to those used by ioffe and rockafellar in concurrent research

the existence of positive periodic solutions of a class of lotka volterra type impulsive systems with infinitely distributed delay in this paper the existence of positive periodic solutions of a class of periodic lotka volterra type impulsive systems with distributed delays is studied by using the continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are obtained which improve and generalize some existing results

inverse numerical range and determinantal representation abstract the numerical range w a of an n n matrix a is the collection of quadratic forms z a over unit sphere 1 the inverse numerical range problem aims to find a unit vector which corresponds to a given point z of the numerical range w a given a matrix a the helton vinnikov theorem produces a symmetric matrix s so that a and s have the same numerical range w a w s in this paper we investigate the inverse numerical range problem for the boundary points and points on the boundary generating curve of the numerical range in place of the construction of unit vectors satisfying z a we express the kernel vector function of the linear pencil x s y s z i n as a function on the abel jacobi variety of the associated elliptic curve of a the kernel function plays a key role for the inverse numerical problem we perform this process when s is a generic 3 3 symmetric matrix

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

the hamiltonian inclusion for nonconvex velocity sets since clarke s 1973 proof of the hamiltonian inclusion for optimal control problems with convex velocity sets there has been speculation and more recently speculation relating to a stronger partially convexified version of the hamiltonian inclusion as to whether these necessary conditions are valid in the absence of the convexity hypothesis the issue was in part resolved by clarke himself when in 2005 he showed that l infty local minimizers satisfy the hamiltonian inclusion in this paper it is shown by counterexample that the hamiltonian inclusion and so also the stronger partially convexified hamiltonian inclusion are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a w 1 1 local minimizer not an l infty local minimizer the counterexample demonstrates that the need to consider l infty local minimizers not w 1 1 local minimizers in the proof of the hamiltonian inclusion for nonconvex velocity sets is fundamental not j

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

real zero polynomials and a horn s problem abstract a horn s problem concerns finding two self adjoint matrices so that a b and a b have prescribed spectrum in this paper we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem in addition we outline an algorithm to construct a pair a b

on nonlinear two point boundary value problems for impulsive differential algebraic problems in this paper we investigate boundary value problems for first order impulsive differential algebraic problems with causal operators note that a corresponding boundary condition is given by a nonlinear function using a monotone iterative method we formulate general sufficient conditions under which such problems have solutions extremal or a unique an example shows that corresponding assumptions are satisfied the results are new

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

containment problems for polytopes and spectrahedra we study the computational question of whether a given polytope or spectrahedron s a as given by the positive semidefiniteness region of a linear matrix pencil a x is contained in another s b first we classify the computational complexity extending results on the polytope polytope case by gritzmann and klee to the polytope spectrahedron case for various restricted containment problems np hardness is shown we then study in detail semidefinite conditions to certify containment building upon work by ben tal and nemirovski and helton klep and mccullough in particular we discuss variations of a sufficient semidefinite condition to certify containment of a spectrahedron in a spectrahedron it is shown that these sufficient conditions even provide exact semidefinite characterizations for containment in several important cases including containment of a spectrahedron in a polyhedron moreover in the case of bounded s a the criteria will always succeed in certifying containment of some scale

periodic boundary value problem for first order impulsive delay differential equations in this paper by means of the method of upper and lower solutions and the monotone iterative technique the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive delay differential equations is considered

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

remarks on the lax conjecture for hyperbolic polynomials abstract a conjecture of lax p lax differential equations difference equations and matrix theory commun pure appl math 11 1958 175 194 says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix the conjecture has recently been proved by lewis parillo ramana based on previous work of dubrovin and helton vinnikov in this note we prove related results for polynomials in several space variables which have rotational symmetries

construction of determinantal representation of trigonometric polynomials abstract for a pair of n n hermitian matrices h and k a real ternary homogeneous polynomial defined by f t x y det ti n xh yk is hyperbolic with respect to 1 0 0 the fiedler conjecture or lax conjecture is recently affirmed namely for any real ternary hyperbolic polynomial f t x y there exist real symmetric matrices s 1 and s 2 such that f t x y det ti n xs 1 ys 2 in this paper we give a constructive proof of the existence of symmetric matrices for the ternary forms associated with trigonometric polynomials

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

nonhomogeneous boundary value problem for first order impulsive differential equations with delay the authors employ the method of upper and lower solution coupled with the monotone iterative technique to obtain results of existence and uniqueness for a nonhomogeneous boundary value problem of impulsive differential equations with delay

applications of stable polynomials to mixed determinants johnson s conjectures unimodality and symmetrized fischer products for n x n matrices a and b define eta a b sigma s det a s det b s where the summation is over all subsets of 1 n s is the complement of s and a s is the principal submatrix

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

a maximum principle for generalized control systems on definit un systeme de commande generalise comme une collection de fonctions f t x definies pour t x 0 1 r n dans r n et satisfaisant les hypotheses de caratheodory dans t x on etablit un theoreme donnant une condition necessaire pour la trajectoire d un tel systeme d appartenance a la frontiere de l ensemble atteignable on applique ce theoreme a des cas particuliers

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

boundary value problems for first order functional differential equations with impulsive integral conditions in this paper by using the method of lower and upper solutions coupled with the monotone iterative technique we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition some comparison results are also formulated

the tracial moment problem and trace optimization of polynomials the main topic addressed in this paper is trace optimization of polynomials in noncommuting nc variables given an nc polynomial f what is the smallest trace f underline a can attain for a tuple of matrices underline a a relaxation using semidefinite programming sdp based on sums of hermitian squares and commutators is proposed while this relaxation is not always exact it gives effectively computable bounds on the optima to test for exactness the solution of the dual sdp is investigated if it satisfies a certain condition called flatness then the relaxation is exact in this case it is shown how to extract global trace optimizers with a procedure based on two ingredients the first is the solution to the truncated tracial moment problem and the other crucial component is the numerical implementation of the artin wedderburn theorem for matrix algebras due to murota kanno kojima kojima and maehara trace optimization of nc polynomials is a nontrivial extension of polynomial optimization in commuting variables on one side and eigenvalue optimization of nc polynomials on the other side two topics with many applications the most prominent being to linear systems engineering and quantum physics the optimization problems discussed here facilitate new possibilities for applications e g in operator algebras and statistical physics

necessary conditions for multiobjective optimal control problems with free end time necessary conditions of optimality are derived for multiobjective optimal control problems with free end time in which the dynamics constraint is modeled as a nonconvex differential inclusion the obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems

proof of the bmv conjecture we prove the bmv bessis moussa villani 1 conjecture which states that the function t mapsto mathop rm tr exp a tb t geqslant 0 is the laplace transform of a positive measure on 0 if a and b are n times n hermitian matrices and b is positive semidefinite a semi explicit representation for this measure is given

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

boundary value problems for first order impulsive delay differential equations with a parameter in this paper by means of the method of upper and lower solutions and monotone iterative technique the existence of maximal and minimal solutions of the boundary value problems for first order impulsive delay differential equations is established

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

on the occurrence of intermediate local minimizers that are not strong local minimizers abstract recently it has been shown by mordukhovich that certain necessary conditions of optimality in control theory are satisfied at intermediate local minimizers in addition to strong local minimizers it is of interest to know then whether the class of intermediate local minimizers differs from that of strong local minimizers we give examples illustrating that the two concepts of minimizers are indeed distinct and also distinguish intermediate minimizers of different rank we also extend the class of necessary conditions which are known to be satisfied at an intermediate local minimizer

unitary similarity of the determinantal representation of unitary bordering matrices abstract let f x y z be a hyperbolic ternary form of degree n the helton vinnikov theorem asserts that there exists an n n complex symmetric matrix s such that f s x y z f x y z where the determinantal hyperbolic ternary form of an n n matrix b is defined by f b x y z det x b y b z i n let a be an n n unitary bordering matrix it is known that a is unitarily similar to a symmetric matrix in this paper we investigate the unitary similarity between the unitary bordering matrix a and the helton vinnikov symmetric matrix s admitted the determinantal representation of the ternary form f a x y z satisfying f s x y z f a x y z for n 3 4

equivalent subgradient versions of hamiltonian and euler llagrange equations in variational analysis much effort in recent years has gone into generalizing the classical hamiltonian and euler lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints these generalizations in which nonsmoothness abounds and gradients are systematically replaced by subgradients have succeeded in furnishing necessary conditions for optimality that reduce to the classical ones in the classical setting but important issues have remained unsettled especially concerning the exact relationship of the subgradient versions of the hamiltonian equations versus those of the euler lagrange equations here it is shown that new tighter subgradient versions of these equations are actually equivalent to each other the theory of epi convergence of convex functions provides the technical basis for this development

variational problems with nonconvex noncoercive highly discontinuous integrands characterization and existence of minimizers we consider the functional f v int a b f t v prime t dt in cal h p v in w 1 p v a 0 v b d p in 1 infty under only the assumption that the integrand is cal l otimes cal b n measurable we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the euler lagrange inclusion as an application we provide necessary and sufficient conditions for the existence of the minimum expressed in terms of a limitation on the width of the slope d

on the validity of the euler lagrange equation the purpose of the present paper is to establish the validity of the euler lagrange equation for the solution x to the classical problem of the calculus of variations

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when