In this paper, we present a distributed control strategy, enabling agents to converge onto and travel along a consensually selected curve among a class of closed planar curves. Individual agents identify the number of neighbors within a finite circular sensing range and obtain information from their neighbors through local communication. The information is then processed to update the control parameters and force the swarm to converge onto and circulate along the aforementioned planar curve. The proposed mathematical framework is based on stochastic differential equations driven by white Gaussian noise (diffusion processes). Using this framework, there is maximum probability that the swarm dynamics will be driven toward the consensual closed planar curve. In the simplest configuration where a circular consensual curve is obtained, we are able to derive an analytical expression that relates the radius of the circular formation to the agent’s interaction range. Such an intimate relation is also illustrated numerically for more general curves. The agent-based control strategy is then translated into a distributed Braitenberg-inspired one. The proposed robotic control strategy is then validated by numerical simulations and by implementation on an actual robotic swarm. It can be used in applications that involve large numbers of locally interacting agents, such as traffic control, deployment of communication networks in hostile environments, or environmental monitoring.

The collective behaviour of stochastic multi-agents swarms driven by Gaussian and non-Gaussian environments is analytically discussed in a mean-field approach. We first exogenously implement long range mutual interactions rules with strengths that are weighted by the real-time distance separating each agent with the swarm barycentre. Depending on the form of this barycentric modulation, a transition between two drastically different collective behaviours can be unveiled. A behavioural bifurcation threshold due to the tradeoff between the desynchronisation effects of the stochastic environment and the synchronising interactions is analytically calculated. For strong enough interactions, the emergence of a soliton propagating wave is established. Alternatively, weaker interactions cannot overcome the environmental noise and evanescent diffusive waves result. In a second and complementary approach, we show that the emergent solitons can alternatively be interpreted as being the optimal equilibrium of mean-field games (MFG) models with ad-hoc running cost functions which are here exactly determined. These MFG's soliton equilibria are therefore endogenously generated. Hence for the classes of models here proposed, an explicit correspondence between exogenous and endogenous interaction rules leading to similar collective effects is explicitly constructed. For non-Gaussian environments our results offer a new class of exactly solvable mean-field games dynamics.

This paper is based on the premise that economic growth is driven by an interplay between innovation and imitation in an economy composed of interacting firms operating in a stochastic environment. A novel approach to modeling imitation is presented based on range-dependent processes that describe how firms consider proximity when imitating peers who are found in a given neighborhood in terms of productivity. Using a particularly tractable approach, we are able to analyze how drastically different economic growth scenarios emerge from different imitation strategies. These emerging scenarios range from diffusive growth where the variance of productivity grows indefinitely, to balanced growth described by a traveling wave with fixed variance. The latter scenario is sustained only when imitation strength among firms exceeds a critical bifurcation threshold.

The mean-field dynamics of a collection of stochastic agents evolving under local and nonlocal interactions in one dimension is studied via analytically solvable models. The nonlocal interactions between agents result from $(a)$ a finite extension of the agents interaction range and $(b)$ a barycentric modulation of the interaction strength. Our modeling framework is based on a discrete two-velocity Boltzmann dynamics which can be analytically discussed. Depending on the span and the modulation of the interaction range, we analytically observe a transition from a purely diffusive regime without definite pattern to a flocking evolution represented by a solitary wave traveling with constant velocity.

We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, requires that agents have infinite interaction ranges. Using the probabilistic size of the swarm’s support, we analytically estimate the critical interaction range below that flocked swarms cannot survive. In the second part of the paper, we consider the interactions between two flocked swarms of Brownian agents with finite interaction ranges. Both swarms travel with different barycentric velocities, and agents from both swarms indifferently interact with each other. For appropriate initial configurations, both swarms eventually collide (i.e., all agents interact). Depending on the values of the control parameters, one of the following patterns emerges after collision: (i) Both swarms remain essentially flocked, or (ii) the swarms become ultimately quasi-free and recover their nominal barycentric speeds. We derive a set of analytical flocking conditions based on the generalized rank-based Brownian motion. An extensive set of numerical simulations corroborates our analytical findings.