Control of agent swarms in random environments

The collective dynamic behavior of large groups of interacting autonomous agents (swarms) have inspired much research in both fundamental and engineering sciences. It is now widely acknowledged that the intrinsic nonlinearities due to mutual interactions can generate highly collective spatio-temporal patterns. Moreover, the resulting self-organized behavior cannot be simply guessed by solely investigating the elementary dynamic rules of single individuals. With a view to apply swarm collective behaviors to engineering, it is mandatory to thoroughly understand and master the mechanism of emergence to ultimately address the basic question: What individual dynamics and what type of interactions generate a given stable collective spatio-temporal behavior ? The present doctoral work is a contribution to the general common effort devoted to give an engineering operational answer to this simple and yet still highly challenging question. Swarms modeling is based on the dynamic properties of multi-agents systems (MAS). Methodological approaches for studying MAS are i) mathematics, ii) numerical simulation and iii) experimental validation on physical systems. While in this work we strive to construct and analytically solve new classes of mathematical MAS models, we also make a very special effort to develop new MAS modeling platforms for which one is simultaneously able to offer exact analytical results, corroborate these via simulation and finally implement the resulting control mechanism on swarms of actual robots. In full generality, MAS are formed by mutually interacting autonomous agents evolving in random environments. The presence of noise sources will indeed be unavoidable in any actual implementation. This drives us to consider coupled sets of stochastic nonlinear differential equations as being the natural mathematical modeling framework. We first focus on the simplest situations involving homogeneous swarms. Here, for large homogeneous swarms, the mean-field approach (borrowed from statistical physics) can be used to analytically characterize the resulting spatio-temporal patterns from the individual agent dynamics. In this context, we propose a new modeling platform (the so-called mixed-canonical dynamics) for which we are able to fully bridge the gap between pure mathematics and actual robotic implementation. In a second approach, we then consider heterogeneous swarms realized either when one agent behaves either as a leader or a shill (i.e as an infiltrated agent), or when two different sub-swarms compose the whole MAS. Analytical results are generally very hard to find for heterogeneous swarms, since the mean-field approach cannot be used. In this context, we use recent results in rank-based Brownian motions to approach some heterogeneous MAS models. In particular, we are able to analytically study i) a case of soft control of the swarm by a shill agent, and ii) the mutual interactions between two different societies (i.e., sub-groups) of homogeneous agents. Finally, the same mathematical framework enables us to consider a class of MAS where agents mutually interact via their environment (stigmergic interactions). Here, we can once again simultaneously present analytical results, numerical simulations and to ultimately implement the controller on a swarm of robotic boats.