We consider the Cucker-Smale flocking model with a singular communication weight ψ(s) = s −α with α > 0. We provide a critical value of the exponent α in the communication weight leading to global regularity of solutions or finite-time collision between *

We prove existence of global C 1 piecewise weak solutions for the discrete Cucker-Smale's flocking model with the communication weightWe also discuss the possibility of finite in time alignment of the velocities of the particles.

For the discrete Cucker-Smale's flocking model with a singular communication weight ψ(s) = s −α , with 0 < α < 1 2 , we prove that the velocity component of certain type of weak solutions is absolutly continuous. This result enables us to obtain existence and uniqeness of global solutions. *

The Cucker-Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency, e.g. to aggregate, flock or disperse. This paper examines the kinetic Cucker-Smale equation with a singular communication weight. Given a compactly supported measure as an initial datum we construct a global in time weak measurevalued solution in the space C weak (0, ∞; M). The solution is defined as a meanfield limit of the empirical distributions of particles, the dynamics of which is governed by the Cucker-Smale particle system. The studied communication weight is ψ(s) = |s| −α with α ∈ 0, 1 2 . This range of singularity admits the sticking of characteristics/trajectories. The second result concerns the weak-atomic uniqueness property stating that a weak solution initiated by a finite sum of atoms, i.e. Dirac deltas in the form m i δ x i ⊗ δ v i , preserves its atomic structure. Hence these coincide with unique solutions to the system of ODEs associated with the Cucker-Smale particle system.

We address the design of decentralized feedback control laws inducing consensus and prescribed spatial patterns over a singular interacting particle system of Cucker-Smale type. The control design consists of a feedback term regulating the distance between each agent and preassigned subset of neighbours. Such a design represents a multidimensional extension of existing control laws for 1d platoon formation control. For the proposed controller we study consensus emergence, collision-avoidance and formation control features in terms of energy estimates for the closed-loop system. Numerical experiments in 1, 2 and 3 dimensions assess the different features of the proposed design.