controllability of linear single input positive discrete time systems linear positive systems are utilized in various scientific areas such as economic modelling behavioural science stochastic processes etc a recent study has focused on some sufficient conditions for the controllability of these systems a full characterization is provided of the controllability for discrete time single input positive systems necessary and sufficient conditions are derived using a digraph theoretic approach the main results show that strong constraints are to be imposed on the digraph associated with the pair a b to ensure the controllability of the system

subharmonic oscillations of a semilinear wave equation soit u tt u xx f t x u 0 u t 0 0 u t 1 ou f est continue sur r 0 r et de periode 2 en t soit f t x u 0 u f t x s ds si f est convexe en u f t x u u f t x u u 2 0 u uniformement en t alors pour chaque k n il existe une solution 2k periodique u k l de 1 telle que u k l k

nonnegative realization of autonomous systems in the behavioral approach nonnegative linear systems which have traditionally been investigated within the state space framework have been recently introduced and analyzed by means of the behavioral approach in a couple of recent papers j w nieuwenhuis linear algebra appl 281 1998 pp 43 58 m e valcher linear algebra appl 319 2000 pp 147 162 several general definitions and results about nonnegative behaviors as well as a complete analysis of nonnegativity property for autonomous behaviors have been presented in this contribution by focusing our interest again on autonomous behaviors we explore the nonnegative realization problem by deriving an extended set of necessary and sufficient geometric conditions for an autonomous behavior to be nonnegative realizable in the scalar case in particular necessary and sufficient conditions for nonnegative realizability which refer to the set of zeros of any polynomial involved in the kernel description of the behavior are provided finally a comparison between the nonnegative realizability property here investigated and k realizability addressed in h maeda and s kodama ieee trans ircuits systems i fund theory appl cas 281 1981 pp 39 47 is carried on

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

global optimization of polynomials using gradient tentacles and sums of squares we consider the problem of computing the global infimum of a real polynomial f on mathbb r n every global minimizer of f lies on its gradient variety i e the algebraic subset of mathbb r n where the gradient of f vanishes if f attains a minimum on mathbb r n it is therefore equivalent to look for the greatest lower bound of f on its gradient variety nie demmel and sturmfels proved recently a theorem about the existence of sums of squares certificates for such lower bounds based on these certificates they find arbitrarily tight relaxations of the original problem that can be formulated as semidefinite programs and thus be solved efficiently we deal here with the more general case when f is bounded from below but does not necessarily attain a minimum in this case the method of nie demmel and sturmfels might yield completely wrong results in order to overcome this problem we replace the gradient variety by larger semialgebraic subsets of mathbb r n which we call gradient tentacles it now gets substantially harder to prove the existence of the necessary sums of squares certificates

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

a new proof and an extension of a theorem of p rabinowitz concerning nonlinear wave equations abstract in 1978 p rabinowitz proved a theorem concerned with the existence of a nontrivial periodic solution of a nonlinear wave equation with a continuous increasing superlinear nonlinear term in this paper we present a new and simpler proof of this theorem and relax an assumption on the nonlinear term which is a discontinuous nondecreasing function author

a note on discrete time positive realizations abstract a discrete time positive linear system has the property that any nonnegative initial state and any nonnegative input produces a nonnegative trajectory in the state space and output for all time such systems are r n reachable if every state of the nonnegative orthant r n is reachable from origin with a nonnegative input function in a finite time interval a simple necessary and sufficient condition for r n realizability i e for the existence of a nonnegative realization that is also r n reachable is stated the realization procedure is straightforward

an example of how positivity may force realizations of large dimension abstract a standard result of linear system theory states that an siso proper rational transfer function of degree n always has a realization of dimension n in some applications one is interested in having a realization with nonnegative entries and it is known that when the dominant poles display a specific pattern forcing nonnegativity leads to a system which is not jointly reachable and observable in this paper we show that the minimal dimension of a positive realization may be large even in the case of a single dominant pole more precisely we provide a family of transfer functions each of which is of degree n 3 such that for any integer n 3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than n

spectral properties of rational matrix functions with nonnegative realizations abstract if a b c is an entrywise nonnegative realization of a rational matrix function w i e w c a 1 b for a vanishing at infinity then r w inf r 0 w has no poles with r is a pole of w and r a spectral radius of a is an eigenvalue of a we prove that if the realization is minimal nonnegative then 1 1 r w r a 2 2 order of the pole r w of w order of the pole r a of a 1 we characterize the order of these poles in the spirit of rothblum s index theorem namely as the length of the longest chains of singular vertices in the reduced graph of a with a suitable new access relation which incorporates b and c into the familiar access relation of a

realization of continuous time positive linear systems positive linear systems are used in biomathematics economics and other research areas for discrete time positive linear systems part of the realization problem has been solved in this paper the solution of the corresponding problem for continuous time positive linear systems will be presented which can be deduced from that of the discrete time case by a transformation sufficient and necessary conditions for the existence of a positive realization are presented to solve the problem of minimality the solution of the factorization of positive matrices is needed

primes in several classes of the positive matrices the classification of primes in the semi ring of the positive matrices is of interest to control and system theory a few examples of primes in the positive matrices are known in this paper results on the classification of primes in the positive matrices in the doubly stochastic matrices and in the doubly stochastic circulants are presented

brief stabilization of positive systems with first integrals positive systems possessing first integrals are considered these systems frequently occur in applications this paper is devoted to two stabilization problems the first is concerned with the design of feedbacks to stabilize a given level set secondly it is shown that the same feedback allows to globally stabilize an equilibrium point if it is asymptotically stable with respect to initial conditions in its level set two examples are provided and the results are compared with those in the literature

large amplitude time periodic solutions of a semilinear wave equation abstract this paper studies the existence of periodic solutions for a family of semilinear wave equations where the restoring force is independent of time monotone and grows at a more rapid rate than linear near infinity with appropriate technical assumptions it is shown that there is an unbounded sequence of such free vibrations i e there are solutions of arbitrarily large amplitude if the restoring force is independent of x the monotonicity assumption can be omitted

eigenvalue regions for positive systems positive systems are systems in which the state variables take only nonnegative values for all times in this paper we derive eigenvalue regions for discrete and continuous time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix

nonnegative realizations of matrix transfer functions abstract various aspects of the nonnegative finite dimensional realizability of time invariant discrete linear systems are considered a new proof of the basic result of the nonnegative realizability of a primitive scalar valued transfer function with nonnegative impulse response function is given an algorithm for establishing whether a scalar valued transfer function with nonnegative impulse response has a nonnegative realization is presented the main result characterizes the nonnegative realizability of a scalar valued transfer function with the help of primitive transfer functions and is extended to the general case of matrix valued transfer functions then conditions for the existence of some special nonnegative realizations of transfer functions are presented e g where the middle main matrix is irreducible strictly positive or primitive

necessary conditions for positive realizability of continuous time linear systems abstract this paper deals with the positive realization problem the problem is to find from a given transfer function a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative necessary conditions are investigated and a new one is given together with some related results

geometrical conditions for the reachability and realizability of positive periodic discrete systems reachability and realizability of discrete time linear systems in which the state variable and the input are constrained to lie in the positive orthant r are studied more precisely a characterization of positive reachability is given in terms of certain geometrical conditions on the reachability cones further it is shown that the existence of certain polyhedral cones is equivalent to positive periodic realizations the connection between the above two kinds of cones is established elsevier science inc 1997

the karpelevic region revisited abstract we consider the karpelevic region n c consisting of all eigenvalues of all stochastic matrices of order n we provide an alternative characterisation of n that sharpens the original description given by karpelevic in particular for each 0 2 we identify the point on the boundary of n with argument we further prove that if n n with n 2 and t n then t is a subdominant eigenvalue of some stochastic matrix of order n

brief identification of positive linear systems with poisson output transformation positive systems are systems in which the input state output variables are always positive since they represent quantities we propose an identification procedure for a class of positive linear systems

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

positive realizations of linear systems abstract this paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization a b c t this problem arises in the context of positive linear systems i e systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative using a unitary framework for continuous time and discrete time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis moreover the complete solution in the case of a transfer function of mcmillan degree n 3 is obtained

a counterexample on dynamical systems described by linear difference inequalities abstract in this note we will start a preliminary investigation of discrete time systems described by a finite number of linear inequalities as such it can be seen as an extension of willems 1991 behavioral approach of linear systems

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

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

about nonnegative realizations we will state sufficient conditions such that a discrete time time invariant finite dimensional system with a nonnegative impulse response has a nonnegative finite dimensional realization the conditions are in terms of the minimal polynomial

a new look at nonnegativity on closed sets and polynomial optimization we first show that a continuous function f is nonnegative on a closed set k subseteq r n if and only if countably many moment matrices of some signed measure d nu fd mu with support equal to k are all positive semidefinite if k is compact mu is an arbitrary finite borel measure with support equal to k in particular we obtain a convergent explicit hierarchy of semidefinite outer approximations with it no lifting of the cone of nonnegative polynomials of degree at most d wen used in polynomial optimization on certain simple closed sets k like e g the whole space r n the positive orthant a box a simplex or the vertices of the hypercube it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable this convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

proofs of conjectures on the karpelevich arcs in the region of eigenvalues of stochastic matrices abstract the subset of the complex plane that consists of all eigenvalues of all stochastic matrices of a fixed order was completely determined by karpelevich 1951 the boundary of this region consists of so called karpelevich arcs johnson and paparella 2017 6 made several conjectures on properties of these arcs in this paper we prove two of their conjectures specifically we prove that the karpelevich arcs are regular differentiable curves and establish that some powers of certain karpelevich arcs correspond to some other karpelevich arcs

realization of positive linear systems abstract positive linear systems are frequently used as mathematical models in research areas like biology and economics the problem of classifying all minimal realizations of these systems is treated in this paper extensive use is made of the theory of polyhedral cones sufficient and necessary conditions for the existence of a positive realization are given but the problem of minimality leads to the factorization problem for positive matrices ideas and results are given to come towards a solution of this factorization problem

cyclic polygons roots of polynomials with decreasing nonnegative coefficients and eigenvalues of stochastic matrices abstract a polygon in the complex plane is called cyclic if it is the convex hull of 1 2 for some complex number we determine the regions where should lie in order that this polygon be n sided as an application of this result we determine the region in the complex plane in which the roots of the equation n 1 a 1 n 2 a n 1 0 lie when the coefficients are subject to the inequalities 1 a 1 a 2 a n 1 0 as another application we show that a conjecture of dmitriev and dynkin concerning eigenvalues of stochastic matrices is false for 6 by 6 matrices we also simplify the statement of karpelevich s theorem which describes the region m n where the eigenvalues of n by n stochastic matrices lie

a geometrical representation of the spectra of four dimensional nonnegative matrices abstract a geometrical representation of the set of four complex numbers which are the spectrum of 4 dimensional entrywise nonnegative real matrices is provided the characterization is based on the result for the nonnegative inverse eigenvalue problem niep from the coefficients of the characteristic polynomial given in 16

periodic solutions of a dirac equation with concave and convex nonlinearities abstract this paper is concerned with multiplicity of periodic solutions of a dirac equation with concave and convex nonlinearities we establish a variational setting and obtain two sequences of periodic solutions with large and small energies respectively for the problem by variational method

a new statement about the theorem determining the region of eigenvalues of stochastic matrices abstract let mn denote the set of points in the complex plane that are eigenvalues of n dimensional stochastic matrices the set mn is completely determined by the karpelevich theorem the statement of which however is lengthy and intricate the paper shortens the presentation of the theorem

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

bounds on the size of minimal nonnegative realizations for discrete time lti systems nonnegative linear time invariant lti state space models are characterized by the property that their state variables and outputs remain nonnegative whenever the initial state and the inputs are nonnegative necessary and sufficient conditions for the existence of a nonnegative state space realization of a given discrete time dt impulse response were obtained recently in this letter we look at minimal nonnegative realizations of dt impulse responses and develop bounds on the number of state variables that are necessary

new approximations for the cone of copositive matrices and its dual we provide convergent hierarchies for the convex cone mathcal c of copositive matrices and its dual mathcal c the cone of completely positive matrices in both cases the corresponding hierarchy consists of nested spectrahedra and provide outer resp inner approximations for mathcal c resp for its dual mathcal c thus complementing previous inner resp outer approximations for mathcal c for mathcal c in particular both inner and outer approximations have a very simple interpretation finally extension to mathcal k copositivity and mathcal k complete positivity for a closed convex cone mathcal k is straightforward

completely positive semidefinite rank an n times n matrix x is called completely positive semidefinite cpsd if there exist d times d hermitian positive semidefinite matrices p i i 1 n for some d ge 1 such that x ij mathrm tr p ip j for all i j in 1 ldots n the cpsd rank of a cpsd matrix is the smallest d ge 1 for which such a representation is possible in this work we initiate the study of the cpsd rank which we motivate in two ways first the cpsd rank is a natural non commutative analogue of the completely positive rank of a completely positive matrix second we show that the cpsd rank is physically motivated as it can be used to upper and lower bound the size of a quantum system needed to generate a quantum behavior in this work we present several properties of the cpsd rank unlike the completely positive rank which is at most quadratic in the size of the matrix no general upper bound is known on the cpsd rank of a cpsd matrix in fact we show that the cpsd rank can be sub exponential in terms of the size specifically for any n ge 1 we construct a cpsd matrix of size 2n whose cpsd rank is 2 varomega sqrt n our construction is based on gram matrices of lorentz cone vectors which we show are cpsd the proof relies crucially on the connection between the cpsd rank and quantum behaviors in particular we use a known lower bound on the size of matrix representations of extremal quantum correlations which we apply to high rank extreme points of the n dimensional elliptope lastly we study cpsd graphs i e graphs g with the property that every doubly nonnegative matrix whose support is given by g is cpsd we show that a graph is cpsd if and only if it has no odd cycle of length at least 5 as a subgraph this coincides with the characterization of cp graphs

aspects of positivity in control theory this paper studies finite and infinite dimensional linear control systems of the form df dt af bu where a is the infinitesimal generator of a c 0 semigroup that preserves a cone c and where bu takes values in c since the reachable states are all in c the system is not controllable in the usual sense of concern is positive controllability which means that the entire cone c can be approximately reached it turns out that positive controllability is rather difficult to achieve but that for stable systems an important subclass of states can be reached different examples are provided

finding a partial solution to a linear system of equations in positive integers abstract we present an algorithm to determine the values of the bounded components of the x e n n such that ax b where a e m m n z and b e z m this algorithm uses gordan s duality theorem

stabilization of positive linear systems we consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant the existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane this allows to transform the problem to the stabilization problem of compartmental systems which is known and for which a solution has been proposed already we provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback a class of stabilizing feedbacks is given explicitly

excitability stability and sign of equilibria in positive linear systems abstract the relationship between sign and stability of equilibria in positive linear systems is investigated in this paper in particular it is proved that the well known result stability and irreducibility imply strict positivity can be better specified by introducing the notion of excitability in fact stability excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others

structural properties of discrete time linear positive periodic systems abstract we study reachability and controllability properties for discrete time positive periodic systems the equivalence between n periodic systems with n states and m inputs and invariant systems with n states and nm inputs preserves the positivity of such systems and in addition closely relates their reachability cones this equivalence permits us to show that the complete contrallability reachability of a positive n periodic system is equivalent to the complete contrallability reachability of n associated positive invariant systems

positive input reachability and controllability of positive systems abstract we study controllability and reachability for discrete linear control systems x k 1 ax k bu k in which the state vector x k the control vector u k the n n matrix a and the n m matrix b have nonnegative entries for an unconstrained system controllability is equivalent to reachability from the zero state in n steps furthermore a b is controllable if and only if the matrix c n b ab a 2 b a n 1 b has rank n we show that these properties are not equivalent for positive systems this has important implications for control strategy the timing of control inputs and the size and structure of the control matrix b are much more critical the zero nonzero pattern of c n plays a crucial role we illustrate these fundamental differences with models of a two species reversible chemical reaction and with two pharmacokinetic models of drug distribution in the human body

on the boolean minimal realization problem in the max plus algebra one of the open problems in the max plus algebraic system theory for discrete event systems is the minimal realization problem in this paper we present some results in connection with the minimal realization problem in the max plus algebra first we characterize the minimal system order of a max linear discrete event system we also introduce a canonical representation of the impulse response of a max linear discrete event system next we consider a simplified version of the general minimal realization problem the boolean minimal realization problem i e we consider models in which the entries of the system matrices are either equal to the max plus algebraic zero element or to the max plus algebraic identity element we give a lower bound for the minimal system order of a max plus algebraic boolean discrete event system we show that the decision problem that corresponds to the boolean realization problem i e deciding whether or not a boolean realization of a given order exists is decidable and that the boolean minimal realization problem can be solved in a number of elementary operations that is bounded from above by an exponential of the square of any upper bound of the minimal system order we also point out some open problems the most important of which is whether or not the boolean minimal realization problem can be solved in polynomial time

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

computing the stability number of a graph via linear and semidefinite programming we study certain linear and semidefinite programming lifting approximation schemes for computing the stability number of a graph our work is based on and refines de klerk and pasechnik s approach to approximating the stability number via copositive programming siam j optim 12 2002 pp 875 892 we provide a closed form expression for the values computed by the linear programming approximations we also show that the exact value of the stability number alpha g is attained by the semidefinite approximation of order alpha g 1 as long as alpha g leq 6 our results reveal some sharp differences between the linear and the semidefinite approximations for instance the value of the linear programming approximation of any order is strictly larger than alpha g whenever alpha g 1

controllability of positive linear discrete time systems the class of positive linear discrete time systems is considered reachability sets of such systems denned on cones are studied and some of their properties established reachability null controllability and controllability criteria are obtained in forms that are easier to implement than the corresponding criteria for linear discrete time systems of general type some classes of positive linear discrete time systems are investigated and their controllability properties are established

linear conic formulations for two party correlations and values of nonlocal games in this work we study the sets of two party correlations generated from a bell scenario involving two spatially separated systems with respect to various physical models we show that the sets of classical quantum no signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones as a by product we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the navascues pironio and acin npa hierarchy and also a sufficient condition for the set of quantum correlations to be closed furthermore by our conic formulations the value of a nonlocal game over the sets of classical quantum no signaling and unrestricted correlations can be cast as a linear conic program this allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by feige and lovasz is in fact an upper bound to the quantum value of the game and moreover it is at least as strong as optimizing over the first level of the npa hierarchy lastly we show that deciding the existence of a perfect quantum resp classical strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices resp completely positive matrices by specializing the results to synchronous nonlocal games we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature

the niep for four dimensional leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial abstract the constraints on the coefficients of the characteristic polynomials of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are determined then on the basis of these constraints the regions of the complex plane consisting of those points which can serve as characteristic roots of four dimensional doubly stochastic matrices with zero trace and leslie stochastic matrices with zero trace are characterized

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs

linear quadratic regulator problem with positive controls in this paper the linear quadratic regulator problem with a positivity constraint on the admissible control set is addressed necessary and sufficient conditions for optimality are presented in terms of inner products projections on closed convex sets pontryagin s maximum principle and dynamic programming the main results are concerned with smoothness of the optimal control and the value function the maximum principle will be extended to the infinite horizon case based on these analytical methods we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem the numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other

null controllability of discrete time positive control systems abstract the discrete time linear positive and quasilinear monotone systems are considered the set of initial points is restricted to be the cone of non negative vectors the input belongs to the cone of non positive vectors the necessary and sufficient conditions of null controllability for linear systems are obtained the conditions are sufficient for quasilinear systems

controllability of multi input positive discrete time systems linear discrete time positive systems arise in various fields of science such as economic modelling behavioural science stochastic processes etc recently some authors have examined the problem of the controllability of such systems for the single input case this paper provides necessary and sufficient conditions for characterizing the controllability properties of multi input systems the results here developed show that only a narrow class of these systems enjoys the properties of reachability and controllability

minimal max realization of convex sequences we show that the minimal dimension of a linear realization over the max semiring of a convex sequence is equal to the minimal size of a decomposition of the sequence as a supremum of discrete affine maps the minimal dimensional realization of any convex realizable sequence can thus be found in linear time the result is based on a bound in terms of minors of the hankel matrix

when to call a linear system nonnegative abstract in this paper we will consider discrete time invariant linear systems that allow for an input state output representation with a finite dimensional state space and that have a finite number of inputs and outputs the basic issue in this paper is when to call these systems nonnegative an important concept in this respect is that of the most powerful unfalsified model

a matricial view of the karpelevic theorem abstract the question of the exact region in the complex plane of the possible single eigenvalues of all n by n stochastic matrices was raised by kolmogorov in 1937 and settled by karpelevic in 1951 after a partial result by dmitriev and dynkin in 1946 the karpelevic result is unwieldy but a simplification was given by okovic in 1990 and ito in 1997 the karpelevic region is determined by a set of boundary arcs each connecting consecutive roots of unity of order less than n it is shown here that each of these arcs is realized by a single somewhat simple parameterized stochastic matrix other observations are made about the nature of the arcs and several further questions are raised the doubly stochastic analog of the karpelevic region remains open but a conjecture about it is amplified