Complexity, robustness, self-organization, swarms, and system thermodynamics

In this paper, we develop a thermodynamic framework for addressing consensus problems for Eulerian swarm models. Specifically, we present a distributed boundary controller architecture involving the exchange of information between uniformly distributed swarms over an nn-dimensional (not necessarily Euclidian) space that guarantees that the closed-loop system is consistent with basic thermodynamic principles. In addition, we establish the existence of a unique continuously differentiable entropy functional for all equilibrium and non-equilibrium states of our thermodynamically consistent dynamical system. Information consensus and semistability are shown using the well-known Sobolev embedding theorems and the notion of generalized (or weak) solutions. Finally, since the closed-loop system is guaranteed to satisfy basic thermodynamic principles, robustness to individual agent failures and unplanned individual agent behavior is automatically guaranteed.