the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

weak dual residuations applied to tropical linear equations an extension to an algorithm of r a cuninghame green and k zimmermann for solving equations with residuated functions is presented this extension relies on the concept of weak residuation and in the so called strong property it is shown that a contextualization of this method to tropical linear equations which will be denoted as primal method in contrast with the dual method another algorithm described in literature generates a non decreasing sequence which converges to the smallest solution in a special semimodule it is also shown the connections of this method with previously published works

just in time control of timed event graphs update of reference input presence of uncontrollable input a linear system theory has been developed for the class of discrete event systems subject to synchronization this paper presents the just in time control of such systems when reference input is updated and or in the presence of uncontrollable input s the proposed controls are the solutions to an optimization problem under equality constraint

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

optimal control synthesis of timed event graphs with interval model specifications the purpose of this technical note is the optimal control synthesis of a timed event graph when the state and control trajectories should follow the specifications defined by an interval model the problem is reformulated in the fixed point form and the spectral theory gives the conditions of existence of a solution

on feedback invariance properties for systems over a principal ideal domain simple necessary and sufficient conditions for the solvability of many control problems for linear systems over a field are based on the equivalence between the a b invariance property and the a bf invariance or feedback invariance property for systems over a ring this equivalence is no longer true and many results of the geometric control theory cannot be extended we present new algorithmically checkable characterizations of the a bf invariance property for systems defined over a principal ideal domain and a new solvability condition for the disturbance rejection problem

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

homological algebra modulo a regular sequence with special attention to codimension two abstract let m be a finite module over a ring r obtained from a commutative ring q by factoring out an ideal generated by a regular sequence the homological properties m over r and over q are intimately related their links are analyzed here from the point of view of differential graded homological algebra over a koszul complex that resolves r over q one outcome of this approach is a transparent derivation of some central results of the theory another is a new insight into codimension two phenomena yielding an explicit finitistic construction of the generally infinite minimal r free resolution of m it leads to theorems on the structure and classification of finite modules over codimension two local complete intersections that are exact counterparts of eisenbud s results for modules over hypersurfaces

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

max plus a b invariant spaces and control of timed discrete event systems the concept of a b invariant subspace or controlled invariant of a linear dynamical system is extended to linear systems over the max plus semiring although this extension presents several difficulties which are similar to those encountered in the same kind of extension to linear dynamical systems over rings it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings sufficient conditions are given for computing the maximal a b invariant subspace contained in a given space and the existence of linear state feedbacks is discussed an application to the study of transportation networks which evolve according to a timetable is considered

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

a solution to the problem of a b invariance for series the aim of this article is to study a standard problem of control theory the a b invariance problem which amounts to computing a maximal element x subject to conditions of the form ax 6 x b and x 6 k we give a solution to the problem in the framework of formal series over particular complete idempotent semirings over nite idempotent semirings we show that under the assumption that b and k are recognizable series the maximal solution exists and is also recognizable we obtain a similar result for the in nite tropical semiring with additional hypothesis that the series a is a language but the notion of recognizable series has to be extended to the weaker notion of pseudo recognizable series c 2002 elsevier science b v all rights reserved

an algorithm to describe the solution set of any tropical linear system a x b x abstract an algorithm to give an explicit description of all the solutions to any tropical linear system a x b x is presented the given system is converted into a finite rather small number p of pairs s t of classical linear systems a system s of equations and a system t of inequalities the notion introduced here that makes p small is called compatibility the particular feature of both s and t is that each item equation or inequality is bivariate i e it involves exactly two variables one variable with coefficient 1 and the other one with 1 s is solved by gaussian elimination we explain how to solve t by a method similar to gaussian elimination to achieve this we introduce the notion of sub special matrix the procedure applied to t is therefore called sub specialization

the derived category of a locally complete intersection ring abstract in this paper we answer a question of dwyer greenlees and iyengar by proving a local ring r is a complete intersection if and only if every complex of r modules with finitely generated homology is proxy small moreover we establish that a commutative noetherian ring r is locally a complete intersection if and only if every complex of r modules with finitely generated homology is virtually small

observer design for max linear systems this technical note deals with the state estimation for max plus linear systems this estimation is carried out following the ideas of the observer method for classical linear systems the system matrices are assumed to be known and the observation of the input and of the output is used to compute the estimated state the observer design is based on the residuation theory which is suitable to deal with linear mapping inversion in idempotent semiring

the equation a x b y over max for the two sided homogeneous linear equation system a x b y over max with no infinite rows or columns in a or b an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists if the finite elements of a b are all integers convergence is in a finite number of steps for which a precise bound can be calculated if moreover one of a b has only finite elements the algorithm is thus pseudopolynomial in complexity

duality and interval analysis over idempotent semirings abstract in this paper semirings with an idempotent addition are considered these algebraic structures are endowed with a partial order this allows to consider residuated maps to solve systems of inequalities a x b see 3 the purpose of this paper is to consider a dual product denoted and the dual residuation of matrices in order to solve the following inequality a x x b x sufficient conditions ensuring the existence of a non linear projector in the solution set are proposed the results are extended to semirings of intervals such as they were introduced in 25

observer based controllers for max plus linear systems max plus algebra is a suitable algebraic setting to model discrete event systems involving synchronization and delay phenomena which are often found in transportation networks communications systems and manufacturing systems one way of controlling this kind of systems consists in choosing the dates of input events in order to achieve the desired performances e g to obtain output events in order to respect given dates this kind of control is optimal according to a just in time criterion if the input event dates are delayed as much as possible while ensuring the output events to occur before a desired reference date this paper presents the observer based controller for max plus linear systems where only estimations of system states are available for the controller as in the classical sense this is a state feedback control problem which is solved in two steps first an observer computes an estimation of the state by using the input and the output measurements then this estimated state is used to compute the state feedback control action as a main result it is shown that the optimal solution of this observer based control problem leads to a greater control input than the one obtained with the output feedback strategy a high throughput screening example in drug discovery illustrates this main result by showing that the scheduling obtained from the observer based controller is better than the scheduling obtained from the output feedback controller

on the steady state control of timed event graphs with firing date constraints two algorithms for solving a specific class of steady state control problems for timed event graphs are presented in the first asymptotic convergence to the desired set is guaranteed the second algorithm which builds on from the recent developments in the spectral theory of min max functions guarantees lyapunov stability since the distance between the actual state and the desired set never increases simulation results show the efficiency of the proposed approach in a problem of moderate complexity

polynomial optimization with real varieties we study the optimization problem of minimizing f x rm subject to h x 0 g x geq 0 with f a polynomial and h g two tuples of polynomials in x in mathbb r n lasserre s hierarchy is a sequence of sum of squares relaxations for finding the global minimum f min let k be the feasible set we prove the following results i if the real variety v mathbb r h is finite then lasserre s hierarchy has finite convergence no matter if the complex variety v mathbb c h is finite or not this solves an open question in the survey by m laurent ima vol math appl 149 springer new york 2009 pp 157 270 ii if k and v mathbb r h have the same vanishing ideal then the finite convergence of lasserre s hierarchy is independent of the choice of defining polynomials for the real variety v mathbb r h iii if only k is finite a refined version of lasserre s hierarchy using the preordering of g has finite convergence

tensor eigenvalue complementarity problems this paper studies tensor eigenvalue complementarity problems basic properties of standard and complementarity tensor eigenvalues are discussed we formulate tensor eigenvalue complementarity problems as constrained polynomial optimization when one tensor is strictly copositive the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity when no tensor is strictly copositive we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process the complementarity eigenvalues can be computed sequentially the formulated polynomial optimization can be solved by lasserre s hierarchy of semidefinite relaxations we show that it has finite convergence for generic tensors numerical experiments are presented to show the efficiency of proposed methods

the disturbance decoupling problem for systems over a ring up to now the use of geometric methods in the study of disturbance decoupling problems ddps for systems over a ring has provided only necessary conditions for the existence of solutions in this paper we study such problems considering separately the case in which only static feedback solutions are allowed and the one in which dynamic feedback solutions are admitted in the first case we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring practical procedures for testing the solvability conditions and for constructing solutions if any exist are given in the case of systems with coefficients in a principal ideal domain pid in the second case we give a complete geometric characterization of the solvability conditions for systems with coefficients in a pid

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

on the dimension of max min convex sets we introduce a notion of dimension of max min convex sets following the approach of tropical convexity we introduce a max min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope we describe the relation between this rank and the notion of strong regularity in max min algebra which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property

max plus control design for temporal constraints meeting in timed event graphs the aim of the presented work is the control of timed event graph to meet tight temporal constraints the problem of temporal constraints is formulated in terms of control of linear max plus models first the synthesis of a control law that ensures the satisfaction of a single constraint for a single input system is presented then the single input multi constraints problem is tackled and finally the method is extended to the multi inputs multi constraints problem the proposed method is illustrated on the example of a simple production process

on max plus linear dynamical system theory a class of timed discrete event systems can be modeled by using timed event graphs a class of timed petri nets that can have its firing dynamic described by using an algebra called max plus algebra for this kind of systems it may be desirable to enforce some timing constraints in steady state in this paper this problem is called a max plus regulation problem in this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems the obtained controller is a simple linear static state feedback and can be computed using efficient pseudo polynomial algorithms simulation results will illustrate the applicability of the proposed methodology

an integrated control strategy to solve the disturbance decoupling problem for max plus linear systems with applications to a high throughput screening system this paper presents the new investigations on the disturbance decoupling problem ddp for the geometric control of max plus linear systems the classical ddp concept in the geometric control theory means that the controlled outputs will not be changed by any disturbances in practical manufacturing systems solving for the ddp would require further delays on the output parts than the existing delays caused by the system breakdown the new proposed modified disturbance decoupling problem mddp in this paper ensures that the controlled output signals will not be delayed more than the existing delays caused by the disturbances in order to achieve the just in time optimal control furthermore this paper presents the integration of output feedback and open loop control strategies to solve for the mddp as well as for the ddp if these controls can only solve for the mddp but not for the ddp an evaluation principle is established to compare the distance between two output signals generated by controls solving for the mddp and ddp respectively this distance can be interpreted as the number of tokens or firings that are needed in order for the controls to solve for the ddp moreover another alternative approach is finding a new disturbance mapping in order to guarantee the solvability of the ddp by the same optimal control for the mddp the main results of this paper are illustrated by using a timed event graph model of a high throughput screening system in drug discovery

dimension reduction and feedback stabilization for max plus linear systems and applications in vlsi array processors this paper investigates the dimension reduction and feedback stabilization of max plus linear systems and applies them to control and optimize the very large scale integration vlsi array processors we introduce the weakly similar relation between max plus matrices and the pseudoequivalent relation between autonomous max plus linear systems and point out that two systems are pseudoequivalent if and only if their state matrices are weakly similar the reduced system is defined by using the pseudoequivalence whose dimension is determined by the row rank of the original state matrix we focus on obtaining a reduced system which maintains the stability and retains the steady state period an algorithm of polynomial complexity is developed to find such a reduced system the reduced system is then used to design a state feedback controller to stabilize a max plus linear system finally we use the vlsi array processors as an example to demonstrate how the presented methods work in practical applications

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included