Local well-posedness of the generalized Cucker-Smale model

In this paper, we study the local well-posedness of two types of generalized Cucker-Smale (in short C-S) flocking models. We consider two different communication weights, singular and regular ones, with nonlinear coupling velocities $v|v|^{\beta-2}$ for $\beta > \frac{3-d}{2}$. For the singular communication weight, we choose $\psi^1(x) = 1/|x|^{\alpha}$ with $\alpha \in (0,d-1)$ and $\beta \geq 2$ in dimension $d > 1$. For the regular case, we select $\psi^2(x) \geq 0$ belonging to (L_{loc}^\infty \cap \mbox{Lip}_{loc})(\mathbb{R}^d) and $\beta \in (\frac{3-d}{2},2)$. We also remark the various dynamics of C-S particle system for these communication weights when $\beta \in (0,3)$

Flocking with Obstacle Avoidance

In this paper, we provide a dynamic graph theoretical framework for flocking in presence of multiple obstacles. In particular, we give formal definitions of nets and flocks as spatially induced graphs. We provide models of nets and flocks and discuss the realization/embedding issues related to structural nets and flocks. This allows task representation and execution for a network of agents called alpha-agents. We also consider flocking in the presence of multiple obstacles. This task is achieved by introducing two other types of agents called beta-agents and gamma-agents. This framework enables us to address split/rejoin and squeezing maneuvers for nets/flocks of dynamic agents that communicate with each other. The problems arising from switching topology of these networks of mobile agents make the analysis and design of the decision-making protocols for such networks rather challenging. We provide simulation results that demonstrate the effectiveness of our theoretical and computational tools

Singular Cucker-Smale Dynamics

The existing state of the art for singular models of flocking is overviewed, starting from microscopic model of Cucker and Smale with singular communication weight, through its mesoscopic mean-filed limit, up to the corresponding macroscopic regime. For the microscopic Cucker-Smale (CS) model, the collision-avoidance phenomenon is discussed, also in the presence of bonding forces and the decentralized control. For the kinetic mean-field model, the existence of global-in-time measure-valued solutions, with a special emphasis on a weak atomic uniqueness of solutions is sketched. Ultimately, for the macroscopic singular model, the summary of the existence results for the Euler-type alignment system is provided, including existence of strong solutions on one-dimensional torus, and the extension of this result to higher dimensions upon restriction on the smallness of initial data. Additionally, the pressureless Navier-Stokes-type system corresponding to particular choice of alignment kernel is presented, and compared - analytically and numerically - to the porous medium equation

Determining interaction rules in animal swarms

In this paper we introduce a method for determining local interaction rules in animal swarms. The method is based on the assumption that the behavior of individuals in a swarm can be treated as a set of mechanistic rules. The principal idea behind the technique is to vary parameters that define a set of hypothetical interactions to minimize the deviation between the forces estimated from observed animal trajectories and the forces resulting from the assumed rule set. We demonstrate the method by reconstructing the interaction rules from the trajectories produced by a computer simulation.Comment: v3: text revisions to make the article more comprehensibl