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The goal of this paper is to introduce a new theoretical framework for Optimal Transport (OT), using the terminology and techniques of Fully Probabilistic Design (FPD). Optimal Transport is the canonical method for comparing probability measures and has been successfully applied in a wide range of areas (computer vision Rubner et al. [2004], computer graphics Solomon et al. [2015], natural language processing Kusner et al. [2015], etc.). However, we argue that the current OT framework suffers from two shortcomings: first, it is hard to induce generic constraints and probabilistic knowledge in the OT problem; second, the current formalism does not address the question of uncertainty in the marginals, lacking therefore the mechanisms to design robust solutions. By viewing the OT problem as the optimal design of a probability density function with marginal constraints, we prove that OT is an instance of the more generic FPD framework. In this new setting, we can furnish the OT framework with the necessary mechanisms for processing probabilistic constraints and deriving uncertainty quantifiers, hence establishing a new extended framework, called FPD-OT. Our main contribution in this paper is to establish the connection between OT and FPD, providing new theoretical insights for both. This will lay the foundations for the application of FPD-OT in a subsequent work, notably in processing more sophisticated knowledge constraints, as well as in designing robust solutions in the case of uncertain marginals., Keywords: Optimal Transport, Fully Probabilistic Design, Convex optimization

The AIMD algorithm, which underpins the Transmission Control Protocol (TCP) for transporting data packets in communication networks, is perhaps the most successful control algorithm ever deployed. Recently, its use has been extended beyond communication networks, and successful applications of the AIMD algorithm have been reported in transportation, energy, and mathematical biology. A very recent development in the use of AIMD is its application in solving large-scale optimization and distributed control problems without the need for inter-agent communication. In this context, an interesting problem arises when multiple AIMD networks that are coupled in some sense (usually through a nonlinearity). The purpose of this note is to prove that such systems in certain settings inherit the ergodic properties of individual AIMD networks. This result has important consequences for the convergence of the aforementioned optimization algorithms. The arguments in the paper also correct conceptual and technical errors in [1].

In many sharing-economy applications, as well as in conventional economy applications, one wishes to regulate the behaviour of an ensemble of agents with guarantees on both the regulation of the ensemble in aggregate and the revenue or quality of service associated with each agent. Previous work [Automatica, Volume 108, 108483, arXiv:1807.03256 ] has developed guarantees of unique ergodicity when there are linear filters. Here, we extend the guarantees to systems including non-linear elements, such as non-linear filters.

In power systems, one wishes to regulate the aggregate demand of an ensemble of distributed energy resources (DERs), such as controllable loads and battery energy storage systems. We suggest a notion of predictability and fairness, which suggests that the long-term averages of prices or incentives offered should be independent of the initial states of the operators of the DER, the aggregator, and the power grid. We show that this notion cannot be guaranteed with many traditional controllers used by the load aggregator, including the usual proportional-integral (PI) controller. We show that even considering the non-linearity of the alternating-current model, this notion of predictability and fairness can be guaranteed for incrementally input-to-state stable (iISS) controllers, under mild assumptions.

There has been much recent interest in two-sided markets and dynamics thereof. In a rather a general discrete-time feedback model, which we show conditions that assure that for each agent, there exists the limit of a long-run average allocation of a resource to the agent, which is independent of any initial conditions. We call this property the unique ergodicity. Our model encompasses two-sided markets and more complicated interconnections of workers and customers, such as in a supply chain. It allows for non-linearity of the response functions of market participants. Finally, it allows for uncertainty in the response of market participants by considering a set of the possible responses to either price or other signals and a measure to sample from these.

This paper describes the use of Distributed Ledger Technologies as a mean to enforce social contracts and to orchestrate the behaviour of agents in a smart city environment. Specifically, we present a scheme to price personalised risk in sharing economy applications. We provide proofs for the convergence of the proposed stochastic system and we validate our approach through the use of extensive Monte Carlo simulations.

This paper studies the boundary output feedback stabilization of general 1-D reaction-diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition while the system output is selected as a either Dirichlet or Neumann boundary trace. The control strategy takes the form of a finite-dimensional observer-based controller with feedback and observer gains that are computed in order to dominate the state delayed term. For any arbitrarily given value of the state delay, we show the exponential stability of the resulting closed-loop system provided the order of the observer is selected large enough.

The recent uptake in popularity in vehicles with zero tailpipe emissions is a welcome development in the fight against traffic induced airborne pollutants. As vehicle fleets become electrified, and tailpipe emissions become less prevalent, non-tailpipe emissions (from tires and brake disks) will become the dominant source of traffic related emissions, and will in all likelihood become a major concern for human health. This trend is likely to be exacerbated by the heavier weight of electric vehicles, their increased power, and their increased torque capabilities, when compared with traditional vehicles. While the problem of emissions from tire wear is well-known, issues around the process of tire abrasion, its impact on the environment, and modelling and mitigation measures, remain relatively unexplored. Work on this topic has proceeded in several discrete directions including: on-vehicle collection methods; vehicle tire-wear abatement algorithms and controlling the ride characteristics of a vehicle, all with a view to abating tire emissions. Additional approaches include access control mechanisms to manage aggregate tire emissions in a geofenced area with other notable work focussing on understanding the particle size distribution of tire generated PM, the degree to which particles become airborne, and the health impacts of tire emissions. While such efforts are already underway, the problem of developing models to predict the aggregate picture of a network of vehicles at the scale of a city, has yet to be considered. Our objective in this paper is to present one such model, built using ideas from Markov chains. Applications of our modelling approach are given toward the end of this note, both to illustrate the utility of the proposed method, and to illustrate its application as part of a method to collect tire dust particles., Comment: 24 pages, 11 figures

We present the observation that the process of stochastic model predictive control can be formulated in the framework of iterated function systems. The latter has a rich ergodic theory that can be applied to study the system's long-run behavior. We show how such a framework can be realized for specific problems and illustrate the required conditions for the application of relevant theoretical guarantees.

This paper is concerned with the design of a human-in-the-loop system for deployment on a smart pedelec (e-bike). From the control-theoretic perspective, the goal is not only to use the power assistance of the e-bike to reject disturbances along the route but also to manage the possibly competitive interactions between a human and the motor intervention. Managing the competitive/cooperative nature of the interactions is crucial for applications in which we wish to control physical aspects of the cycling behavior (e.g. heart rate and breathing rate). The basis of the control is a pitchfork bifurcation system, modeling the interactions, augmented using ideas from gain-scheduling. In vivo experiments have been conducted, showing the effectiveness of the proposed control strategy., Comment: 7 figures

In machine learning, training data often capture the behaviour of multiple subgroups of some underlying human population. When the amounts of training data for the subgroups are not controlled carefully, under-representation bias arises. We introduce two natural notions of subgroup fairness and instantaneous fairness to address such under-representation bias in time-series forecasting problems. In particular, we consider the subgroup-fair and instant-fair learning of a linear dynamical system (LDS) from multiple trajectories of varying lengths and the associated forecasting problems. We provide globally convergent methods for the learning problems using hierarchies of convexifications of non-commutative polynomial optimisation problems. Our empirical results on a biased data set motivated by insurance applications and the well-known COMPAS data set demonstrate both the beneficial impact of fairness considerations on statistical performance and the encouraging effects of exploiting sparsity on run time.

In several social choice problems, agents collectively make decisions over the allocation of multiple divisible and heterogeneous resources with capacity constraints to maximize utilitarian social welfare. The agents are constrained through computational or communication resources or privacy considerations. In this paper, we analyze the convergence of a recently proposed distributed solution that allocates such resources to agents with minimal communication. It is based on the randomized additive-increase and multiplicative-decrease (AIMD) algorithm. The agents are not required to exchange information with each other, but little with a central agent that keeps track of the aggregate resource allocated at a time. We formulate the time-averaged allocations over finite window size and model the system as a Markov chain with place-dependent probabilities. Furthermore, we show that the time-averaged allocations vector converges to a unique invariant measure, and also, the ergodic property holds.

Within the study of uncertain dynamical systems, iterated random functions are a key tool. There, one samples a family of functions according to a stationary distribution. Here, we introduce an extension, where one sample functions according to a time-varying distribution over the family of functions. For such iterated piecewise-stationary random functions on Polish spaces, we prove a number of results, including a bound on the tracking error.

Internet-of-Things (IoT) enables the development of sharing economy applications. In many sharing economy scenarios, agents both produce as well as consume a resource; we call them prosumers. A community of prosumers agrees to sell excess resource to another community in a prosumer market. In this chapter, we propose a control theoretic approach to regulate the number of prosumers in a prosumer community, where each prosumer has a cost function that is coupled through its time-averaged production and consumption of the resource. Furthermore, each prosumer runs its distributed algorithm and takes only binary decisions in a probabilistic way, whether to produce one unit of the resource or not and to consume one unit of the resource or not. In the proposed approach, prosumers do not explicitly exchange information with each other due to privacy reasons, but little exchange of information is required for feedback signals, broadcast by a central agency. In the proposed approach, prosumers achieve the optimal values asymptotically. Furthermore, the proposed approach is suitable to implement in an IoT context with minimal demands on infrastructure. We describe two use cases; community-based car sharing and collaborative energy storage for prosumer markets. We also present simulation results to check the efficacy of the algorithms., To appear as a chapter in "Analytics for the Sharing Economy: Mathematics, Engineering and Business Perspectives", Editors: E. Crisostomi et al., Springer, 2019 (forthcoming book)

We consider a control problem involving several agents coupled through multiple unit-demand resources. Such resources are indivisible, and each agent's consumption is modeled as a Bernoulli random variable. Controlling the number of such agents in a probabilistic manner, subject to capacity constraints, is ubiquitous in smart cities. For instance, such agents can be humans in a feedback loop---who respond to a price signal, or automated decision-support systems that strive toward system-level goals. In this paper, we consider both single feedback loop corresponding to a single resource and multiple coupled feedback loops corresponding to multiple resources consumed by the same population of agents. For example, when a network of devices allocates resources to deliver several services, these services are coupled through capacity constraints on the resources. We propose a new algorithm with fundamental guarantees of convergence and optimality, as well as present an example illustrating its performance.

Pollution in urban centres is becoming a major societal problem. While pollution is a concern for all urban dwellers, cyclists are one of the most exposed groups due to their proximity to vehicle tailpipes. Consequently, new solutions are required to help protect citizens, especially cyclists, from the harmful effects of exhaust-gas emissions. In this context, hybrid vehicles (HVs) offer new actuation possibilities that can be exploited in this direction. More specifically, such vehicles when working together as a group, have the ability to dynamically lower the emissions in a given area, thus benefiting citizens, whilst still giving the vehicle owner the flexibility of using an Internal Combustion Engine (ICE). This paper aims to develop an algorithm, that can be deployed in such vehicles, whereby geofences (virtual geographic boundaries) are used to specify areas of low pollution around cyclists. The emissions level inside the geofence is controlled via a coin tossing algorithm to switch the HV motor into, and out of, electric mode, in a manner that is in some sense optimal. The optimality criterion is based on how polluting vehicles inside the geofence are, and the expected density of cyclists near each vehicle. The algorithm is triggered once a vehicle detects a cyclist. Implementations are presented, both in simulation, and in a real vehicle, and the system is tested using a Hardware-In-the-Loop (HIL) platform (video provided)., 9 pages, 11 figures, video available on: http://smarttransport.ucd.ie/wordpress/spotlight-projects/

In this paper we derive stability conditions for a switched system where switching occurs between linear descriptor systems of different indices. In particular, our results can be used to analyse the stability of the important case when switching between a standard system and an index one descriptor system, and systems where switching occurs between an index one and and an index two descriptor system. Examples are given to illustrate the use of our results., 10 pages, 3 figures

The effects of real-time provision of travel-time information on the behaviour of drivers are considered. The model of Marecek et al. [arXiv:1406.7639, Int. J. Control 88(10), 2015] is extended to consider uncertainty in the response of a driver to an interval provided per route. Specifically, it is suggested that one can optimise over all distributions of a random variable associated with the driver's response with the first two moments fixed, and for each route, over the sub-intervals within the minimum and maximum in a certain number of previous realisations of the travel time per the route.

We propose a distributed algorithm to solve a special distributed multi-resource allocation problem with no direct inter-agent communication. We do so by extending a recently introduced additive-increase multiplicative-decrease (AIMD) algorithm, which only uses very little communication between the system and agents. Namely, a control unit broadcasts a one-bit signal to agents whenever one of the allocated resources exceeds capacity. Agents then respond to this signal in a probabilistic manner. In the proposed algorithm, each agent is unaware of the resource allocation of other agents. We also propose a version of the AIMD algorithm for multiple binary resources (e.g., parking spaces). Binary resources are indivisible unit-demand resources, and each agent either allocated one unit of the resource or none. In empirical results, we observe that in both cases, the average allocations converge over time to optimal allocations.

In this paper, we present new results for the synchronization and consensus of networks described by Ito stochastic differential equations. From the methodological viewpoint, our results are based on the use of stochastic Lyapunov functions. This approach allowed us to consider networks where nodes dynamics can be nonlinear and non-autonomous and where noise is not just additive but rather its diffusion can be nonlinear and depend on the network state. We first present a sufficient condition on the coupling strength and topology ensuring that a network synchronizes (fulfills consensus) despite noise. Then, we show that noise can be useful, and present a result showing how to design noise so that it induces synchronization/consensus. Motivated by our current research in Smart Cities and Internet of Things, we also illustrate the effectiveness of our approach by showing how our results can be used to analyze/control the onset of synchronization in noisy networks and to study collective decision processes., Keywords: Ito differential equations, Synchronization, Complex networks

In this paper, we consider the use of stationary vehicles as tools to enhance the localisation capabilities of moving vehicles in a VANET. We examine the idea in terms of its potential benefits, technical requirements, algorithmic design and experimental evaluation. Simulation results are given to illustrate the efficacy of the technique., This version of the paper is an updated version of the initial submission, where some initial comments of reviewers have been taken into account

In this paper we propose a new engine management system for hybrid vehicles to enable energy providers and car manufacturers to provide new services. Energy forecasts are used to collaboratively orchestrate the behaviour of engine management systems of a fleet of PHEV's to absorb oncoming energy in an smart manner. Cooperative algorithms are suggested to manage the energy absorption in an optimal manner for a fleet of vehicles, and the mobility simulator SUMO is used to show simple simulations to support the efficacy of the proposed idea., Updated typos with respect to previous version