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Maxplus algebra and system theory: Where we are and where to go now
 Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Cited by 42 (18 self)
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which maxplus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to systemtheoretic and controlsynthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the maxplus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre maxplus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre maxplus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to sep ..."
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Cited by 35 (19 self)
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Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings. We illustrate the results by showing polyhedra and halfspaces over the maxplus semiring. 1.
The Minkowski Theorem for Maxplus Convex Sets
, 2006
"... We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones a ..."
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Cited by 22 (11 self)
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We establish the following maxplus analogue of Minkowski’s theorem. Any point of a compact maxplus convex subset of (R ∪ {−∞}) n can be written as the maxplus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed maxplus convex cones and closed unbounded maxplus convex sets. In particular, we show that a closed maxplus convex set can be decomposed as a maxplus sum of its recession cone and of the maxplus convex hull of its extreme points.
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
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Cited by 19 (9 self)
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Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
Maxplus convex geometry
 of Lecture Notes in Comput. Sci
, 2006
"... Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theore ..."
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Cited by 8 (6 self)
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Abstract. Maxplus analogues of linear spaces, convex sets, and polyhedra have appeared in several works. We survey their main geometrical properties, including maxplus versions of the separation theorem, existence of linear and nonlinear projectors, maxplus analogues of the MinkowskiWeyl theorem, and the characterization of the analogues of “simplicial ” cones in terms of distributive lattices. 1
MinPlus linearity and statistical mechanics
, 1996
"... We revisit some results obtained recently in minplus algebra following theideasofstatisticalmechanics. Computationofgeodesicsinagraphcanbedoneby minplus matrix products. A minplus matrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsminplusmatrix. Thegraph ..."
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Cited by 7 (1 self)
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We revisit some results obtained recently in minplus algebra following theideasofstatisticalmechanics. Computationofgeodesicsinagraphcanbedoneby minplus matrix products. A minplus matrixisseenasakindoffinitestatesmechanicalsystem. Theenergyofthis systemistheeigenvalueofitsminplusmatrix. Thegraphinterpretation oftheeigenvaluemaybeseenasakindofMariottelaw. The Cramer transformisintroducedbystatisticsonpopulationsof independentminpluslinearsystemsseenasakindofperfectgas. It transformsprobabilitycalculusinwhatwecalldecisioncalculus. Then, dynamicprogrammingequations, which are minplus linearrecurrences, may be seen as minplus Kolmogorov equations for Markov chains. An ergodic theorem for Bellman chains, analogue of Markov chains, is given. The minplus counterparts of aggregation coherency and reversibility of Markov chains are then studied. They provide new decomposition results to compute solutions of dynamic programming equations.
Duality between invariant spaces for maxplus linear discrete event systems
, 2009
"... We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the maxplus 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 coeff ..."
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Cited by 6 (5 self)
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We extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the maxplus 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.
Information and Content
 Blackwell Guide to the Philosophy of Information and Computing, Basil
, 2004
"... ..."
Annual progress report ALAPEDES
"... s of the presentations are available upon request. 3.3.2 Other conferences and workshops Appendix C contains an overview of visits by Alapedes members to other conferences, workshops and courses, that are related to Alapedes. Probably the overview is incomplete. 3.3.3 External collaboration by Al ..."
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s of the presentations are available upon request. 3.3.2 Other conferences and workshops Appendix C contains an overview of visits by Alapedes members to other conferences, workshops and courses, that are related to Alapedes. Probably the overview is incomplete. 3.3.3 External collaboration by Alapedesmembers During the Waterford convention (the first plenary meeting of Alapedes; see x 1.3.3) several researchers from the DIAS (Dublin Institute of Applied Statistics) were welcomed. They are interested in stochastic aspects of discrete event systems. Fruitfull discussions have taken place and one presentation has been given (see Appendix D). In the framework of the establishment of DIOC's (Delft Interfaculty Research Centres), an official collaboration with the Faculty of Civil Engineering has been started. The name of the joined project is: "Seamless Multimodal Mobility". The subject areas are on mathematical applications towards transportation planning. See further in x 1.1.11. L....