We consider stationary consensus protocols for networks of dynamic agents with fixed topologies. At each time instant, each agent knows only its and its neighbors' state, but must reach consensus on a group decision value that is function of all the agents' initial state. We show that the agents can reach consensus if the value of such a function is time-invariant when computed over the agents' state trajectories. We use this basic result to introduce a non-linear protocol design rule allowing consensus on a quite general set of values. Such a set includes, e.g., any generalized mean of order p of the agents' initial states. As a second contribution we show that our protocol design is the solution of individual optimizations performed by the agents. This notion suggests a game theoretic interpretation of consensus problems as mechanism design problems. Under this perspective a supervisor entails the agents to reach a consensus by imposing individual objectives. We prove that such objectives can be chosen so that rational agents have a unique optimal protocol, and asymptotically reach consensus on a desired group decision value. We use a Lyapunov approach to prove that the asymptotical consensus can be reached when the communication links between nearby agents define a time-invariant undirected network. Finally we perform a simulation study concerning the vertical alignment maneuver of a team of unmanned air vehicles

We classify the contributions of DEA literature assessing Decision Making Units (DMUs) whose internal structure is known. Starting from an elementary framework, we define the main research areas as shared flow, multilevel and network models, depending on the assumptions they are subject to. For each model category, the principal mathematical formulations are introduced along with their main variants, extensions and applications. We also discuss the results of aggregating efficiency measures and of considering DMUs as submitted to a central authority that imposes constraints or targets on them. A common feature among the several models is that the efficiency evaluation of the DMU depends on the efficiency values of its subunits thereby increasing the discrimination power of DEA methodology with respect to the black box approach.Keywords Efficiency evaluation · Data envelopment analysis · Networks · Hierarchy · Multi-stage production processes Data Envelopment Analysis (DEA) has been a standard tool for evaluating the relative efficiencies of Decision Making Units (DMUs) since the paper of Charnes et al. (1978) based on the seminal work of Farrell (1957). Some underlying assumptions are common to classical DEA models. The efficiency of a DMU is defined as the weighted ratio of the outputs (products or outcomes) yielded by the DMU over the inputs (resources used or consumed). All DMUs considered are homogeneous, i.e., they all have the same types of inputs and outputs, and are independent, i.e., no constraint binds input and output levels of a DMU with the inputs and outputs of other DMUs. Furthermore, DMUs are seen as black boxes, i.e., their internal structures are not considered. As a consequence, generally, there is no clear L. Castelli ( ) · W. Ukovich

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We consider stationary consensus protocols for networks of dynamic agents. The measure of the neighbors' states is affected by unknown but bounded disturbances. Here the main contribution is the formulation and solution of what we call the $\epsilon$-consensus problem, where the states are required to converge in a target set of radius $\epsilon$ asymptotically or in finite time. We introduce as a solution a dead-zone policy that we denote as the lazy rule

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Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that it may visit. We prove the existence of a mean field game equilibrium. A mean field game equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field game equilibrium.Keywords Tourist flow optimal control · mean field games · switching variables · dynamics on networks Mathematics Subject Classification (2010) 91A13 · 49L20 · 90B20 · 91A80

In this paper we consider multi-inventory systems in presence of uncertain demand. We assume that i) demand is unknown but bounded in an assigned compact set and ii) the control inputs (controlled flows) are subject to assigned constraints. Given a long-term average demand, we select a nominal flow that feeds such a demand. In this context, we are interested in a control strategy that meets at each time all possible current demands and achieves the nominal flow in the average. We provide necessary and sufficient conditions for such a strategy to exist and we characterize the set of achievable flows. Such conditions are based on linear programming and thus they are constructive. In the special case of a static flow (i.e. a system with 0-capacity buffers) we show that the strategy must be affine. The dynamic problem can be solved by a linear-saturated control strategy (inspired by the previous one). We provide numerical analysis and illustrating examples.