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This paper considers the problem of optimizing the trajectory of an Unmanned Aerial Vehicle (UAV) Base Station (BS). A map is considered, characterized by a traffic intensity of users to be served. The UAV BS must travel from a given initial location at an initial time to a final position within a given duration and serves the traffic on its way. The problem consists in finding the optimal trajectory that minimizes a certain cost depending on the velocity and on the amount of served traffic. The problem is formulated using the framework of Lagrangian mechanics. When the traffic intensity is quadratic (single-phase), we derive closed-form formulas for the optimal trajectory. When the traffic intensity is bi-phase, necessary conditions of optimality are provided and an Alternating Optimization Algorithm is proposed, that returns a trajectory satisfying these conditions. The Algorithm is initialized with a Model Predictive Control (MPC) online algorithm. Numerical results show how the trajectory is improved with respect to the MPC solution. 

The envisioned Internet of Things (IoT) will involve a massive deployment of objects connected through wireless cells. While commercial solutions are already available, the fundamental limits of such networks in terms of node density, achievable rates or reliability are not known. To address this question, this paper uses a large scale Multiple Access Channel (MAC) to model IoT nodes randomly distributed over the coverage area of a unique base station. The traffic is represented by an information rate spatial density ρ(x). This model, referred to as the Spatial Continuum Multiple Access Channel, is defined as the asymptotic limit of a sequence of discrete MACs. The access capacity region of this channel is defined as the set of achievable information rate spatial densities achievable with vanishing transmission errors and under a sum-power constraint. Simulation results validate the model and show that this fundamental limit theoretically achievable when all nodes transmit simultaneously over an infinite time, may be reached even with a relatively small number of simultaneous transmitters (typically around 20 nodes) which gives credibility to the model. The results also highlight the potential interest of non-orthogonal transmissions for IoT uplink transmissions when compared to an ideal time sharing strategy.