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In this note we continue our study of unidirectional solutions to hydrodynamic Euler alignment systems with strongly singular communication kernels ϕ (x): = |x|(n+α) for α ∊ (0, 2). Here, we consider the critical case α = 1 and establish a couple of global existence results of smooth solutions, together with a description of their long time dynamics. The first one is obtained via Schauder-type estimates under a null initial entropy condition and the other is a small data result. We extend the notion of weak solution and prove the existence of global Leray--Hopf solutions for α ∊ [1, 2). Furthermore, we give anisotropic Onsager-type criteria for the validity of the natural energy law of the system. Finally, we provide quantitative estimates that show how far the density of the limiting flock is from a uniform distribution, depending solely on the size of the initial entropy. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Steinerberger [Proc. Amer. Math. Soc., 147 (2019), pp. 4733—4744] to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to describe collective behavior of agents (such as birds, fish, or robots) in mathematical biology. We consider a periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data. [ABSTRACT FROM AUTHOR]

We derive an explicit form for the Cucker--Smale (CS) model on the special orthogonal group SOp3q by identifying closed form expressions for geometric quantities such as covariant derivative and parallel transport in exponential coordinates. We study the emergent dynamics of the model by using a Lyapunov functional approach and LaSalle's invariance principle. Specifically, we show that velocity alignment emerges from some admissible class of initial data, under suitable assumptions on the communication weight function. We characterize the \omega -limit set of the dynamical system and identify a dichotomy in the asymptotic behavior of solutions. Several numerical examples are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

We introduce and analyze a model for the dynamics of flocking and steering of a finite number of agents. In this model, each agent's acceleration consists of flocking and steering components. The flocking component is a generalization of many of the existing models and allows for the incorporation of many real world features such as acceleration bounds, partial masking effects, and orientation bias. The steering component is also integral to capture real world phenomena. We provide rigorous sufficient conditions under which the agents flock and steer together. We also provide a formal singular perturbation study of the situation where flocking happens much faster than steering. We end our work by providing some numerical simulations to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We study the pattern formation problem of the Cucker--Smale (CS) flocking model. A distributed CS type flocking model for formation control is introduced based on our previous collision-avoiding model. We present two sufficient frameworks guaranteeing the emergence of flocking and formation achievement in terms of the initial state and system parameters. In the first framework, we show a threshold phenomenon between unconditional and conditional flocking, embodying all three rules of Reynolds: velocity alignment, cohesion, and collision avoidance. Furthermore, with an additional condition on the initial state and system parameters, we show that the agents also converge to a desired spatial configuration in the second framework. Finally, we perform some numerical simulations to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We present a simple proof of asymptotic consensus in the discrete Hegselmann-Krause model and flocking in the discrete Cucker-Smale model with normalization and variable delay. This proof utilizes the convexity of the normalized communication weights and a Gronwall-Halanay-type inequality. The main advantage of our method, compared to previous approaches to the delay Hegselmann-Krause model, is that it does not require any restriction on the maximal time delay, or the initial data, or decay rate of the influence function. From this point of view the result is optimal. For the Cucker-Smale model it provides an analogous result in the regime of unconditional flocking with sufficiently slowly decaying communication rate, but still without any restriction on the length of the maximal time delay. Moreover, we demonstrate that the method can be easily extended to the mean-field limits of both the Hegselmann-Krause and Cucker-Smale systems, using appropriate stability results on the measure-valued solutions. [ABSTRACT FROM AUTHOR]

We introduce a new class of models for emergent dynamics. It is based on a new communication protocol which incorporates two main features: short-range kernels which restrict the communication to local metric balls, and anisotropic communication kernels, adapted to the local density in these balls, which form topological neighborhoods. We prove flocking behavior-the emergence of global alignment for regular, nonvacuous solutions of the n-dimensional models based on short-range topological communication. Moreover, global regularity (and hence unconditional flocking) of the one-dimensional model is proved via an application of a De Giorgi-type method. To handle the nonsymmetric singular kernels that arise with our topological communication, we develop a new analysis for local fractional elliptic operators (interesting in its own right) encountered in the construction of our class of models. [ABSTRACT FROM AUTHOR]

We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and nonlocal diffusion and isentropic pressure. The local viscosity is allowed to degenerate as the density approaches vacuum. The method provides a tool to propagate initial regularity of classical solutions provided no vacuum has formed and serves as an alternative to the classical energy method. We obtain a series of global well-posedness results for state laws in previously uncovered cases, including p(ρ) = cpρ. As an application we prove global well-posedness of collective behavior models with pressure arising from an agent-based Cucker-Smale system. [ABSTRACT FROM AUTHOR]

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 neighbors. Such a design represents a multidimensional extension of existing control laws for 1D platoon formation control. For the proposed controller, we study consensus emergence, collisionavoidance, 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. [ABSTRACT FROM AUTHOR]

We study a hydrodynamic Cucker-Smale-type model with time delay in communication and information processing, in which agents interact with each other through normalized communication weights. The model consists of a pressureless Euler system with time-delayed nonlocal alignment forces. We resort to its Lagrangian formulation and prove the existence of its global-in-time classical solutions. Moreover, we derive a sufficient condition for the asymptotic flocking behavior of the solutions. Finally, we show the presence of a critical phenomenon for the Eulerian system posed in the spatially one-/two-dimensional setting. [ABSTRACT FROM AUTHOR]

We prove unconditional and conditional flocking results for a Cucker--Smale flocking model enhanced with switching topologies. These topologies can be arbitrary; in particular, they are not necessarily symmetric. Our proofs hold for both discrete and continuous time. In both cases, the critical exponent discriminating between unconditional and conditional flocking is shown to be at most 1 2(k 1), where k is the number of agents in the population. [ABSTRACT FROM AUTHOR]

The Cucker-Smale (C-S) flocking model can be applied to design several mobile autonomous agent systems such as unmanned aerial vehicles and robot systems. In these applications, the problem of collision is usually a nonnegligible issue that should be addressed. In the literature, extra bonding/repulsive forces are added to solve the problem of collision. However, all the known results are limited to the symmetric coupling topology. In this paper, we address this problem under a group-hierarchical multileadership consisting of symmetric coupling topologies and the hierarchical leadership. We obtain the following three main results. First, we show that the augmented C-S model achieves asymptotic flocking provided, roughly, that the ratio between the rate of bounding force and communication rate is sufficiently large. Second, we prove that, for a given augmented C-S model, there exists a set of initial states, which can be explicitly described, such that their corresponding solutions maintain some minimum distance between them. Finally, we show that, for a given set of initial states, we can control the system so that its corresponding solution is collision free by suitably choosing bounding forces, which are independent of given initial states. This in turn suggests that the corresponding system can be collision free by having tolerance for error in their initial states. [ABSTRACT FROM AUTHOR]

A generalization of the Cucker-Smale model for collective animal behavior is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modeling: (i) stochasticity (imperfections) of individual behavior and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized Cucker-Smale model are derived by using a suitable Lyapunov functional. As a by-product, a new result regarding the asymptotic behavior of delayed geometric Brownian motion is obtained. In the second part of the paper, results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behavior of the system--namely, that the introduction of an intermediate time delay may facilitate flocking. [ABSTRACT FROM AUTHOR]

The Cucker-Smale flocking model is one of the many aggregation and flocking models describing a collective, self-driven motion of self-propelled particles with some predefined tendency (e.g., to form a flock, to move in a specific direction). We are interested in the discrete Cucker- Smale flocking model with a singular kernel Ψ(s) = s-α with 0 < α. The Cucker-Smale model with a singular kernel possesses particularly interesting and difficult-to-analyze dynamics. Even the most basic question of well-posedness for the particle system remains open. In this paper we prove that in case of α < 1/2 the velocity component of a certain type of weak solutions is absolutely continuous. This result enables us to obtain existence and uniqueness of global solutions. [ABSTRACT FROM AUTHOR]

We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or "agents," with the tendency to adjust to their "environmental averages." This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents, and what types of "rules of engagement" influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish, or cells, rendezvous of mobile agents, and, in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological, and physical sciences assume that the intensity of alignment increases as agents get closer, reflecting a common tendency to align with those who think or act alike. Moreover, "similarity breeds connection" reflects our intuition that increasing the intensity of alignment as the difference of positions decreases is more likely to lead to a consensus. We argue here that the converse is true: when the dynamics is driven by local interactions, it is more likely to approach a consensus when the interactions among agents increase as a function of their difference in position. Heterophily, the tendency to bond more with those who are different rather than with those who are similar, plays a decisive role in the process of clustering. We point out that the number of clusters in heterophilious dynamics decreases as the heterophily dependence among agents increases. In particular, sufficiently strong heterophilious interactions enhance consensus. [ABSTRACT FROM AUTHOR]

Microscopic models of flocking and swarming take into account large numbers of interacting individuals. Numerical resolution of large flocks implies huge computational costs. Typically for interacting individuals we have a cost of (N²). We tackle the problem numerically by considering approximated binary interaction dynamics described by kinetic equations and simulating such equations by suitable stochastic methods. This approach permits us to compute approximate solutions as functions of a small scaling parameter e at a reduced complexity of O(N)operations. Several numerical results show the efficiency of the algorithms proposed. [ABSTRACT FROM AUTHOR]

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under consideration include the kinetic Cucker--Smale equation [Cucker and Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862, Cucker and Smale, Japan. J. Math., 2 (2007), pp. 197-227] with possibly nonsymmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [J. Statist. Phys., 141 (2011), pp. 923-947]. The main tools employed in the analysis are the velocity-averaging lemma and the Schauder fixed-point theorem along with various integral bounds. [ABSTRACT FROM AUTHOR]

Consider a flock of birds that fly interacting between them. This flock is a hierarchical system in which each bird, at each time step, adjusts its own velocity according to its past velocity and a linear combination of the relative velocities of its superiors in the hierarchy. This linear combination has nonnegative random coefficients, including the case in which each of the birds can fail to see any of its superiors with certain probability. For this model with random interactions we prove that the flocking results, obtained for similar deterministic models, hold true. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale [IEEE Trans. Automat. Control, 52 (2007), pp. 852-862], which describes the collective behavior of an ensemble of organisms, animals, or devices. This kinetic version introduced in [S.-Y. Ha and E. Tadmor, Kinet. Relat. Models, 1 (2008), pp. 415-435] is here obtained starting from a Boltzmann-type equation. The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. A continuous analogue of the theorems of [F. Cucker and S. Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852-862] is shown to hold for the solutions on the kinetic model. More precisely, the solutions will concentrate exponentially fast in velocity to the mean velocity of the initial condition, while in space they will converge towards a translational flocking solution. [ABSTRACT FROM AUTHOR]