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As a preliminary overview, this work provides first a broad tutorial on the fluidization of discrete event dynamic models, an efficient technique for dealing with the classical state explosion problem. Even if named as continuous or fluid, the relaxed models obtained are frequently hybrid in a technical sense. Thus, there is plenty of room for using discrete, hybrid and continuous model techniques for logical verification, performance evaluation and control studies. Moreover, the possibilities for transferring concepts and techniques from one modeling paradigm to others are very significant, so there is much space for synergy. As a central modeling paradigm for parallel and synchronized discrete event systems, Petri nets (PNs) are then considered in much more detail. In this sense, this paper is somewhat complementary to David and Alla (2010). Our presentation of fluid views or approximations of PNs has sometimes a flavor of a survey, but also introduces some new ideas or techniques. Among the aspects that distinguish the adopted approach are: the focus on the relationships between discrete and continuous PN models, both for untimed, i.e., fully non-deterministic abstractions, and timed versions; the use of structure theory of (discrete) PNs, algebraic and graph based concepts and results; and the bridge to Automatic Control Theory. After discussing observability and controllability issues, the most technical part in this work, the paper concludes with some remarks and possible directions for future research.

The purpose of this study is to investigate image segmcntation from the viewpoint of image data regularized clustering. From this viewpiint. segmentation into a fixed but arbitrary number N of regions is stated as the simultaneous minimizatiiin o f N -1 energy functionals. each involving a single rcgion and its complement. The resulting Eulcr-Lagrange curve evolution equations yield a partition at convergencc provided the Curves are initialized so as to define an arbitrary partition of the image domain. The m e t h d is implemented via level sets. and results are shown on synthetic and natural vectorial images. . INTRODUCTIONImage segmentation is a fundamental problem in digital image processing and computer vision. The introduction of active contours and Icvel-sets brought forth a new class of tractable algorithms which have succeeded in segmenting difficult images. Active contour methods map regions to the interior o f simple closed planar curves which cvolve to segment the image. Thc two-region segmentation problem i s rather straightforward to statc. However. two-region algorithms are dilticult to generalize to an arbitrary, albeit fixed. number of regions. The difficulty comes mainly from the Fact that while a simple closed curve unambiguously defines a partition of the image domain. thc interior and thc exterior o f the curve, two or more curves a n intersect. causing regions they define to overlap and, therefore, ambiguity in segmentation, The classic study of Zhu and Yuille [ I ] has firmly established the capability of curve evolution methods in image segmentation. I n their method. each curve segment between two regions of an initial partition ofthe image domain i s made to evolve according to a region competition strategy that preserves a partition o f the image domain at all times. The method dms not accommodate the levelset framework. and thus lacks the numerical stability and topology independence that level-sets afford. Also. a good initialization seems critical to a successful completion of thc algorithm. An alternative approach is to use several curves. the interior of each corresponding to a region of segmentation [21 131 141 [51. In [21. Yezzi er al. use what they call a fully global functional to maximally separate the characteristics of the different regions of segmentation. This method yields a partition at convergence. I t s This work wm supponed in pm by the National Sciencc and Engineering Research Council of Canada undcrgnnt DGP 0004234.extension to more than two regions, however. is quite complex because it calls for the maximization of the volume of a polyhedmn with as many vertices as regions. Another difficulty is that segmentntion in N regions requires an N-I-dimensional image function. In 131 Chan and Wse introduce what they refer to as multi-phase active contours. The method seeks a scgmentation into up to a power of 2 number of regions. There i s no clrnr indication on the actual number o f regions the method yields since this depends not just on the image but also on the weight ...

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The aim of this paper is to analyze a class of consensus algorithms with finite-time or fixed-time convergence for dynamic networks formed by agents with first-order dynamics. In particular, in the analyzed class a single evaluation of a nonlinear function of the consensus error is performed per each node. The classical assumption of switching among connected graphs is dropped here, allowing to represent failures and intermittent communications between agents. Thus, conditions to guarantee finite and fixed-time convergence, even while switching among disconnected graphs, are provided. Moreover, the algorithms of the considered class are shown to be computationally simpler than previously proposed finite-time consensus algorithms for dynamic networks, which is an important feature in scenarios with computationally limited nodes and energy efficiency requirements such as in sensor networks. The performance of the considered consensus algorithms is illustrated through simulations, comparing it to existing approaches for dynamic networks with finite-time and fixed-time convergence. It is shown that the settling time of the considered algorithms grows slower when the number of nodes increases than with other consensus algorithms for dynamic networks.Inspired by the ability of certain social insects to self-organize and mutually cooperate by relying only on neighborto-neighbor communication, there has been an increasing interest during the last decade in the distributed algorithms that control agent networks by local interactions. In particular, the consensus algorithm [1-3] allows a network of agents to agree on a common value for its internal state in a distributed fashion (see e.g. [4-9]), by using only communication among neighbors. Several works have been published proposing consensus algorithms in which the agents are first-order integrator systems, second-order integrator systems or high-order linear systems. Some works consider static communication topologies while others consider dynamic topologies, modeling intermittent communications, movement of the agents and the switching between different transmission/reception power levels.Regarding first-order agents, it is known that if the graph topology is strongly connected then consensus can be achieved by the standard protocol (the input of an agent is a linear combination of the errors between the agent's state

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