# Invariance of velocity angles and flocking in the Inertial Spin model

Ioannis Markou

###### Abstract

We study the invariance of velocity angles and flocking properties of the
Inertial Spin model introduced by Cavagna et al. [J. Stat. Phys., 158, (2015),
601–627]. We present a novel approach, based on a second order nonlinear
Gronwall inequality for the velocity diameter, to identify invariant regions
and study the flocking behavior of the model. This is an approach inspired by
the work of Choi-Ha-Yun in the study of synchronization for kuramoto
oscillators with finite inertia [Physica D, 240, (2011), 32–44]. We give
sufficient conditions in terms of the parameters of the model and the initial
data, so that invariant regions and asymptotic alignment of velocities is
possible for coupling interactions that satisfy general assumptions.

Keywords: Inertial Spin model, Vicsek model, unit speed constraint.

2010 MR Subject Classification: 82C22, 34D05, 34C15.

##  1 Introduction to the Model

Collective behavior of self-propelled individual agents is observed everywhere
in nature, e.g. in swarming insects, herding of land animals, synchronization
in chemical oscillations etc. Many individual-based models (IBM models) have
been proposed to study otherwise complex phenomena such as synchronization,
flocking and swarming of birds, turning in schools of fish etc. In the most
successful models we should include the Kuramoto model for synchronization
[32, 42], the now classical three-zone model of Reynolds [37], the Cucker-
Smale model for flocking [18, 19], the Vicsek model for flocking [40], models
that study turning in schools of fish [21, 22], other orientation based
flocking models e.g. [20, 27] etc. The research in the above otherwise simple
models has followed so many different paths, that it would be futile to
mention the entire list. For instance, we may direct the reader to following
literature e.g. in Vicsek type models [7, 17, 24, 38], delayed Cucker-Smale
systems [10, 23, 29, 30], Cucker-Smale with asymmetric weights [36],
synchronization with inertial effects [12, 14, 16], or the survey articles [3,
4, 41] etc.

In this paper we study emergence behavior in a relatively new model called the
Inertial Spin (IS) model that was introduced to describe the mechanism in
which collective turns occur in groups of birds. The IS model is a model for
the collective dynamics of particles that takes into account, not only
position and velocity of particles, but also the inertial angular momentum
(spin) of each agent. The model was introduced in [1, 6] and was later studied
in [5, 26, 39]. Its introduction and justification came to treat one major
flaw of the classical Vicsek model [40]. Indeed, the classical Vicsek model
describes the evolution of self-propelled particles in 2-d having a constant
speed, where the direction of the velocity is updated by computing the average
of neighboring directions. One of the drawbacks of the Vicsek model is the
inadequacy to describe how collective turns occur. In experiments that have
been performed in real flocks it has been observed that when a bird starts to
turn, then the turning information is propagated along the group and the whole
group performs a collective turn. This has been addressed with the
introduction of an extra (spin) variable that accounts for the turning
information. In the new system each particle is described by vectors
$x_{i},v_{i},s_{i}\in\mathbb{R}^{3}$ that stand for the position, velocity and
spin of the $i^{\prime}$ th particle.

We describe the Inertial Spin model by a system of ODEs for the positions,
velocities, and spins, i.e.

$\displaystyle\left\\{\begin{array}[]{ll}\dot{x}_{i}(t)&=v_{i}(t),\quad
i=1\ldots,N,\quad t>0,\\\ \\\ \dot{v}_{i}(t)&=\frac{1}{\chi}s_{i}(t)\times
v_{i}(t),\\\ \\\
\dot{s}_{i}(t)&=v_{i}(t)\times\left[\frac{k}{N}\sum\limits_{j=1}^{N}\psi_{ij}(v_{j}(t)-v_{i}(t))-\gamma\dot{v}_{i}(t)\right],\end{array}\right.$
(6)

with prescribed initial conditions that satisfy:

$(x_{i},v_{i},s_{i})\Big{|}_{t=0}=(x_{i0},v_{i0},s_{i0}),\quad s_{i0}\cdot
v_{i0}=1,\,\text{and normalized initial speed}\quad|v_{i0}|=1,\quad 1\leq
i\leq N.$ (7)

Here the constant $\chi>0$ is the moment of inertia, $\gamma>0$ is the
friction (damping) parameter, and $k>0$ is the coupling strength of
interactions. The matrix $(\psi_{ij})_{i,j}$ is the matrix of communication
weights between particles $i$ and $j$ which measures the strength of the
social interaction between the two agents. Since all the vectors are in
$\mathbb{R}^{3}$ we denote by $\cdot$ and $\times$ the usual inner (dot) and
outer (cross) product between two vectors. Similarly, by $|\cdot|$ we denote
the induced euclidian distance in $\mathbb{R}^{3}$. We also consider the
vectors of positions, velocities and spins defined by

$x:=(x_{1},\ldots,x_{N})\in\mathbb{R}^{3N},\qquad
v:=(v_{1},\ldots,v_{N})\in\mathbb{R}^{3N},\qquad
s:=(s_{1},\ldots,s_{N})\in\mathbb{R}^{3N}.$ (8)

The derivation of the IS system is beyond the scope of this paper and can be
found in e.g. [6, 26].

The final component of system (6) that we address is the communication matrix
$(\psi_{ij})_{ij}$. The $\psi_{ij}$ element of the communication matrix
describes the social interaction between the $i$ and $j$ agents. The minimum
of assumptions that we make for $\psi_{ij}$ is that the $(\psi_{ij})_{ij}$
matrix is nonnegative, symmetric i.e. $\psi_{ij}=\psi_{ji}$ for all pairs
$i,j$, and its elements are bounded by some $\psi_{M}>0$. There are two main
frameworks: The first one is a general framework that considers variable in
time weights, with only basic assumption a lower bound on the $\psi_{ij}$
values, i.e that

$\psi_{m}\leq\psi_{ij}(t)\leq\psi_{M},\qquad\text{for
some}\quad\psi_{M}>\psi_{m}>0,\quad\forall i,j\qquad\forall t\geq 0.$ (9)

This assumption is considerably more general than the case of constant
communication weights, and is very useful in systems with variable social
interactions. Another realistic framework is the case where
$\psi_{ij}:=\psi(|x_{i}-x_{j}|)$ is a function of the metric distance between
the $i,j$ particles. The function $\psi:[0,\infty)\to\mathbb{R}_{+}$ in this
framework is bounded, decreasing, and Lipschitz continuous,

$\displaystyle 0<\psi(r)\leq\psi(0),\qquad[\psi]_{Lip}:=\sup\limits_{x\neq
y}\frac{|\psi(x)-\psi(y)|}{|x-y|}<\infty,$ (10)
$\displaystyle(\psi(r_{2})-\psi(r_{1}))(r_{2}-r_{1})\leq 0,\quad\forall
r,r_{1},r_{2}>0.$

A classical example in this category is the communication rate in the
classical Cucker-Smale model [18, 19], where $\psi(r)$ takes the form
$\psi(r)=(1+r^{2})^{-\beta/2}$ for some $\beta>0$. In this article we will
work mainly with the first framework, where communication weights are allowed
to vary with time but have a strong positive bound from below.

The rest of this article is structured as follows. In Section 2 we give some
of the most important properties of the IS model, and then we present the most
important existing result on its aggregate behavior. After that, we present
the main results of this paper. In the short Section 3 that follows, we show
how the IS model is connected to other systems of collective behavior. In
Section 4, we derive the basic differential inequality for the velocity
diameter. In Section 5, we give important lemmas that allow the treatment of
the inequalities we derived in Section 4. Finally, in Section 6 we prove the
main results.

##  2 Preliminaries and Main results

###  2.1 Preliminaries and Definitions

The Inertial Spin model enjoys some important properties (e.g. conservations,
invariances etc) which we try to include in this section, starting with the
following Proposition from [26] (see Propositions 2.1 & 2.2).

######  Proposition 1.

[26] Assume that $(x(t),v(t),s(t))$ is the solution to system (6), with
initial conditions $(x_{0},v_{0},s_{0})$ that satisfy (7). Then, the following
assertions hold:
(i) The speed of each particle is conserved,

$\frac{d}{dt}|v_{i}(t)|=0,\quad t>0,\quad i=1,\ldots,N.$

(ii) The inner product of the spin and velocity is conserved,

$\frac{d}{dt}(s_{i}(t)\cdot v_{i}(t))=0,\quad t>0,\quad i=1,\ldots,N.$

(iii) The average spin defined by
$s_{c}(t):=\frac{1}{N}\sum\limits_{i=1}^{N}s_{i}(t)$ dissipates with rate
$\frac{\gamma}{\chi}$, i.e.

$s_{c}(t)=s_{c}(0)e^{-\frac{\gamma}{\chi}t}.$

(iv) System (6) has rotational symmetry, i.e. for an orthogonal $3\times 3$
matrix $O\in O(3)$ and the new dynamical variables
$\widetilde{x}_{i},\widetilde{v}_{i},\widetilde{s}_{i}$ defined by

$\widetilde{x}_{i}:=Ox_{i},\qquad\widetilde{v}_{i}:=Ov_{i},\qquad\widetilde{s}_{i}:=Os_{i},$

then $(\widetilde{x},\widetilde{v},\widetilde{s})$ is also a solution of (6).

Simple geometric identities offer some further insights on the system (6). For
a start, the identity $v_{i}\times(s_{i}\times v_{i})=(v_{i}\cdot
v_{i})s_{i}-(s_{i}\cdot v_{i})v_{i}=s_{i}$ means that we can rewrite the spin
equation $(1)_{3}$ as

$\dot{s}_{i}(t)=\frac{k}{N}\sum\limits_{j=1}^{N}\psi_{ij}v_{i}(t)\times
v_{j}(t)-\frac{\gamma}{\chi}s_{i}(t).$ (11)

We may derive a second order equation for $\ddot{v}_{i}$ using the system (6).
Indeed, differentiating $(1)_{2}$ and then using $(1)_{3}$ we get

$\displaystyle\chi\ddot{v}_{i}$ $\displaystyle=\dot{s}_{i}\times
v_{i}+s_{i}\times\dot{v}_{i}$
$\displaystyle=\left[v_{i}\times\left(\frac{k}{N}\sum\limits_{j=1}^{N}\psi_{ij}v_{j}-\gamma\dot{v}_{i}\right)\right]\times
v_{i}+\frac{1}{\chi}s_{i}\times(s_{i}\times v_{i})$
$\displaystyle=\frac{k}{N}\sum\limits_{j=1}^{N}\psi_{ij}(v_{i}\times
v_{j})\times v_{i}-\gamma(v_{i}\times\dot{v}_{i})\times
v_{i}+\frac{1}{\chi}s_{i}\times(s_{i}\times v_{i}).$ (12)

We now use the following identities in (12) (all of which follow from the
triple product formula $a\times(b\times c)=b(a\cdot c)-c(a\cdot b)$ for
$a,b,c\in\mathbb{R}^{3}$)

$\displaystyle(v_{i}\times v_{j})\times v_{i}$
$\displaystyle=v_{j}-(v_{i}\cdot
v_{j})v_{i},\qquad(v_{i}\times\dot{v}_{i})\times v_{i}=(v_{i}\cdot
v_{i})\dot{v}_{i}-(v_{i}\cdot\dot{v}_{i})v_{i}=\dot{v}_{i},$ $\displaystyle
s_{i}\times(s_{i}\times v_{i})$ $\displaystyle=(s_{i}\cdot
v_{i})s_{i}-(s_{i}\cdot
s_{i})v_{i}=-|s_{i}|^{2}v_{i}=-\chi^{2}|\dot{v}_{i}|^{2}v_{i},$

and we end up with the following second order equation for velocities

$\chi\ddot{v}_{i}+\gamma\dot{v}_{i}+\chi|\dot{v}_{i}|^{2}v_{i}=\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}-(v_{i}\cdot
v_{k})v_{i}).$ (13)

The formulation of (13) allows us to have a first glimpse of the mechanism
that drives the velocity alignment in certain configurations. Indeed, in this
formulation the RHS can be written as

$\frac{k}{N}\sum\limits_{j=1}^{N}\psi_{ij}\Gamma(v_{i},v_{j}),\qquad\text{where}\quad\Gamma(v_{i},v_{j})=v_{j}-(v_{i}\cdot
v_{j})v_{i}$

is the orthogonal projection of vector $v_{j}$ on $v_{i}$. The term
$\Gamma(v_{i},v_{j})$ is a nonlinear and non-symmetric coupling term that
plays a role on preserving the speed of the agents due to the property
$\Gamma(v_{i},v_{j})\cdot v_{i}=0$. This equation reduces to the Cucker-Smale
model with constant speed when we take $\chi\to 0$, i.e.

$\dot{x}_{i}=v_{i},\qquad\dot{v}_{i}=\frac{\bar{k}}{N}\sum\limits_{j=1}^{N}\psi_{ij}\Gamma(v_{i},v_{j}),\qquad\text{where}\quad\bar{k}=\frac{k}{\gamma}.$
(14)

This version of CS model with constant speed has been studied in [8, 9, 28].
The idea of taking a coupling term $\Gamma(v_{i},v_{j})$ which takes the
orthogonal projection of $v_{j}$ on $v_{i}$ instead of $v_{j}-v_{i}$ goes back
to a model of quantum synchronization due to Lohe [34]. We should also note
that other instances of C-S dynamics with a non linear coupling
$\Gamma(v_{i},v_{j})$ have been studied in the literature. For instance, in
[25, 35] the authors studied a velocity coupling of the type
$\Gamma(v_{i},v_{j})=(v_{j}-v_{i})|v_{j}-v_{i}|^{2(\gamma-1)}$ for a constant
$\gamma\in\left(\frac{1}{2},\frac{3}{2}\right)$, and gave conditions for
flocking as well as collision avoidance.

For a start we define the diameters of positions, velocities, and spins in the
usual way

$D(x(t)):=\max\limits_{1\leq i,j\leq N}|x_{i}(t)-x_{j}(t)|,\quad
D(v(t)):=\max\limits_{1\leq i,j\leq N}|v_{i}(t)-v_{j}(t)|,\quad
D(s(t)):=\max\limits_{1\leq i,j\leq N}|s_{i}(t)-s_{j}(t)|.$

Our point of view is to study the evolution of velocity diameters by means of
studying the maximal angles between velocities since these two are
geometrically connected. For this, if we assume that all velocities $v_{i}$
lie in a cone with opening (apex) angle $\pi/2$, then we can study the
evolution of angle between $v_{i}$ and $v_{j}$ by studying the inner product
of these two velocities. In this situation, by taking the minimum of the inner
product of all velocity pairs, we can control the velocity diameter of the
system.

Thus, given the considerations above, we introduce the geometric factor of
motion $\pazocal{A}(v)$ just as in [8], which was defined as

$\pazocal{A}(v(t)):=\min\limits_{i\neq j}v_{i}(t)\cdot v_{j}(t).$ (15)

When all the velocities lie inside a cone with angle in $\pi/2$, then
$\pazocal{A}(v)>0$ and we can have some control of the velocity diameter. As a
result, the geometric factor plays a crucial role in identifying invariance
velocity regions. Furthermore, the pair of velocities that is the minimizer in
the factor $\pazocal{A}(v)$ is the same pair that maximizes the velocity
diameter with the exact relationship between these two quantities being
$\pazocal{A}(v(t))=1-\frac{1}{2}D(v(t))^{2}$ when $\pazocal{A}(v(t))>0$. We
now proceed with some definitions:

######  Definition 1.

Let $(x(t),v(t),s(t))$ be a smooth solution to system (6), with initial data
$(x_{0},v_{0},s_{0})$ that satisfy conditions (7). The following definitions
hold:

  * •

(Spin vanishing) The solution exhibits asymptotic vanishing of spins if and
only if

$D(s(t))\to 0\quad\text{as}\quad t\to\infty,$

which due to average spin dissipation implies $s_{i}(t)\to 0$ for all $1\leq
i\leq N$.

  * •

(Orientation invariance) We say that the solution exhibits orientational
invariance if and only if all velocity angles remain inside a cone with slant
angle less than $\pi/2$ for all times. In other words, if there exists some
$0<\beta<1$ such that $\pazocal{A}(v(t))>\beta$ for all $t\geq 0$.

  * •

(Velocity alignment) The solution exhibits velocity alignment if and only if
$D(v(t))\to 0$ as $t\to\infty$.

  * •

(Asymptotic flocking) We say that the solution exhibits asymptotic flocking if
and only if we have velocity alignment and group coherence i.e.

$D(v(t))\to 0\quad\text{as}\quad t\to\infty,\qquad\exists\quad
0<D^{\infty}<\infty\quad\text{s.t.}\quad\sup\limits_{t\geq
0}D(x(t))<D^{\infty}.$

If this holds true for all initial data $(x_{0},v_{0},s_{0})$ we speak of
unconditional flocking, otherwise if flocking is possible only for a specific
subset of initial conditions, then we speak of conditional flocking.

In this article we want to address the following question:

  * •

Q1 : _Under what assumptions in terms of the parameters $\chi,\gamma,k$ of the
problem, the communication matrix $\psi_{ij}$, and in terms of the initial
conditions, we can establish invariant regions? Also, what are the assumptions
we need to show velocity alignment or even emergence of flocks?_

A more specific question we ask is the following:

  * •

Q2 : _Does the smallness of initial diameters of the state variables suffice
for the proof of orientation invariance for the velocities and/or flocking, or
do we need further assumptions on the parameters $k,\chi,\gamma$ in the
model?_

Notation: For brevity we might have to suppress the dependence of any state
variable from time $t$, especially when we are treating longer formulas e.g.
we may write $\dot{s}_{i}$ or $D(v)^{2}$ instead of $\dot{s}_{i}(t)$ and
$D(v(t))^{2}$ etc.

###  2.2 State of the art and main Theorems

We should start with the state of art results in the study of asymptotics for
the IS model. So far, the only known result is the one presented in [26] for
weights that are multiplicative, i.e. for weights that satisfy
$\psi_{ij}=p_{i}p_{j}$ for any $1\leq i,j\leq N$, and a given set
$\\{p_{i}\\}_{1\leq i\leq N}$ of positive elements $p_{i}>0$. This result was
given in two important steps. In the first step, it is shown that for constant
weights (which is a more general assumption than that of multiplicative
weights) we can deduce asymptotic spin vanishing as is shown in the following:

######  Proposition 2.

[26] Suppose that parameters $k,\gamma,\chi$ are all positive. If
$(x(t),v(t),s(t))$ is a solution to the IS model with communication weights
that are nonnegative constant and symmetric, then asymptotic spin vanishing
occurs. In particular, we have that

$\lim\limits_{t\to\infty}|s_{i}(t)|=0\quad\text{and}\quad\lim\limits_{t\to\infty}|\dot{s}_{i}(t)|=0.$

###### Proof.

A sketch of the main steps in the proof of Proposition 2 is possible in one
paragraph. First, we introduce the following functionals

$\pazocal{E}(t):=\frac{1}{N^{2}}\sum\limits_{i,j}\psi_{ij}|v_{i}-v_{j}|^{2},\qquad\pazocal{S}(t):=\frac{1}{N}\sum\limits_{i}|s_{i}|^{2}.$

We can show with a direct computation that
$\frac{\chi}{2}\pazocal{E}(t)+\frac{1}{k}\pazocal{S}(t)$ is dissipative for
symmetric, and constant communication weights by showing that

$\frac{d}{dt}\left(\frac{\chi}{2}\pazocal{E}(t)+\frac{1}{k}\pazocal{S}(t)\right)+\frac{2\gamma}{\chi
k}\pazocal{S}(t)=0,\qquad t>0.$

This in turn implies the a priori energy estimate

$\frac{\chi}{2}\pazocal{E}(t)+\frac{1}{k}\pazocal{S}(t)+\frac{2\gamma}{\chi
k}\int_{0}^{t}\pazocal{S}(s)\,ds\leq\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0),$

or equivalently

$\int_{0}^{\infty}\pazocal{S}(t)\,dt\leq\frac{\chi
k}{2\gamma}\left(\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0)\right).$
(16)

The next step is to show that $\dot{\pazocal{S}}(t)$ is uniformly bounded, and
as a result $\pazocal{S}(t)$ is uniformly continuous. Then, with the help of
Barbalat’s lemma [2] the integrability of $\pazocal{S}(t)$ together with its
uniform continuity implies that $\pazocal{S}(t)\to 0$ as $t\to\infty$.
Similarly, it can be shown that $\dot{\pazocal{S}}(t)$ is also uniformly
continuous and since $\pazocal{S}(t)\to 0$ it also follows that
$\dot{\pazocal{S}}(t)\to 0$. ∎

We now impose the additional assumption of multiplicativity in the
communication weights. Then, with the help of Proposition 2 and assumptions on
the smallness of initial data $\pazocal{E}(0),\pazocal{S}(0)$, it was proven
in [26] that the IS system reaches a 2-flock or 1-flock formation
asymptotically in time. For this it helps to introduce the following averaging
quantities

$p_{c}:=\frac{1}{N}\sum\limits_{i=1}^{N}p_{i}\qquad\bar{v}_{c}:=\frac{1}{N}\sum\limits_{i=1}^{N}p_{i}v_{i}.$

The following main result was proven in [26] (see Theorem 4.1):

######  Proposition 3.

[26] In addition to the assumptions made in Proposition 2 we assume that the
communication weights are also multiplicative, i.e. that
$\psi_{ij}=p_{i}p_{j}$ where $p_{i}>0$ for any $1\leq i,j\leq N$. Then the
following statements hold :
(i) If the initial data $(x_{0},v_{0},s_{0})$ satisfies the condition

$\pazocal{E}(0)+\frac{2}{\chi k}\pazocal{S}(0)<2p_{c}^{2},$

then each velocity $v_{i}$ converges to either $\bar{v}_{c}/|\bar{v}_{c}|$ or
$-\bar{v}_{c}/|\bar{v}_{c}|$.
(ii) If the initial data $(x_{0},v_{0},s_{0})$ satisfies the more restrictive
condition

$\pazocal{E}(0)+\frac{2}{\chi k}\pazocal{S}(0)<\min\limits_{1\leq i\leq
N}\left\\{\frac{8p_{i}(Np_{c}-p_{i})}{N^{2}}\right\\},$

then we have velocity alignment and all velocities converge to
$\bar{v}_{c}/|\bar{v}_{c}|$.

###### Proof.

The proof is presented in [26]. ∎

There are some remarks regarding Proposition 3 that we have to address before
we go on with the presentation of the results of this work :

######  Remark 1.

It is important to note that Proposition 3 applies only to a very restrictive
class of weights that adhere to the multiplicativity property. As it was
pointed in [26], this property appears to be relevant in fragmentation-
coagulation processes of polymers, but it is still a very restrictive
framework to work with in the general problem. In fact, even the class of
constant communication weights is not wide enough to use in many modelling
examples e.g. in cases where the interaction weights depend on the metric
distance between agents, or when the network topology changes in time etc.

######  Remark 2.

The convergence that is proven in Proposition 3 does not possess any explicit
rate. Hence it is impossible to deduce coherence in group formation from this
result alone, i.e. we don’t have sufficient information to show flocking in
the traditional way that was defined earlier. In many physically realistic
cases we would like to have a way to show that when alignment in velocities
happens, the group of particles remains bounded uniformly in time, or at
least, we have the vanishing of the velocity diameter in some explicit rate.

######  Remark 3.

The IS model has a natural tendency to induce orientation alignment in the
particle group when the velocities and spins are not that far apart. This
property of the IS system becomes evident when we write the system as a second
order equation for the velocities of agents. Unfortunately, the functional
approach that is the cornerstone of Proposition 3 does not make any use of the
structure that is “build in” the IS system. In this paper, our approach is to
make full use of this structure.

Having these considerations in mind, we are in position to state the main two
results of this work. First, we state a theorem that works for positive,
constant and symmetric communication weights (not necessarily multiplicative).
This theorem states that in such a case we have invariant regions for the
velocity vectors and we can achieve velocity alignment in these regions but
with no explicit rate.

######  Theorem 1.

Let $(x(t),v(t),s(t))$ be a global smooth solution to (6), with a symmetric
and constant communication matrix $\\{\psi_{ij}\\}_{i,j=1}^{N}$, and initial
data $(x_{0},v_{0},s_{0})$ that satisfy conditions (7). Suppose also that
$\psi_{m}=\min\limits_{1\leq i,j\leq N}\psi_{ij}$, and
$\psi_{M}=\max\limits_{1\leq i,j\leq N}\psi_{ij}$. Finally, we assume some
$\delta_{0}\in(0,1)$ s.t. the following two conditions hold:

$\pazocal{A}(v_{0})>\delta_{0}\qquad\text{and}\qquad\pazocal{C}_{0}<1-\delta_{0},$

where the exact form of
$\pazocal{C}_{0}:=\pazocal{C}(D(v_{0}),\dot{D}(v_{0}),D(s_{0}),s_{0},\gamma,k,\chi)$
is given by the following formulas :
(i) If $\gamma>\sqrt{8k\chi\psi_{m}\delta_{0}}:$

$\pazocal{C}_{0}:=\frac{1}{2}D(v_{0})^{2}+\frac{\chi}{\gamma\sqrt{1-8k\chi\psi_{m}\delta_{0}/\gamma^{2}}}\left(D(v_{0})|\dot{D}(v_{0})|+\frac{1}{2}\nu_{1}D(v_{0})^{2}\right)+\frac{2kN}{\gamma\sqrt{2k\chi\psi_{m}\delta_{0}}}\left(\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0)\right)$

(ii) If $\gamma\leq\sqrt{8k\chi\psi_{m}\delta_{0}}:$

$\displaystyle\pazocal{C}_{0}:=\left\\{\begin{array}[]{ll}&\frac{1}{2}D(v_{0})^{2}\left(1+\frac{2\chi}{\gamma}\right)+\frac{4kN}{\gamma^{2}}\left(\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0)\right),\qquad\text{for}\quad\dot{D}(v_{0})\geq
0\\\ \\\
&\frac{1}{2}D(v_{0})^{2}+\frac{4kN}{\gamma^{2}}\left(\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0)\right),\qquad\text{for}\quad\dot{D}(v_{0})<0.\end{array}\right.$

Then the particle system has the property $\pazocal{A}(v(t))>\delta_{0}$ for
all $t\geq 0$, and the velocity diameters vanish asymptotically and $D(v)\to
0$ as $t\to\infty$.

The second result that we prove is a stronger result that can be achieved in
the “Small Inertia” regime, and gives exponentially fast alignment of
velocities under certain assumptions on the initial configurations.

######  Theorem 2.

Let $(x(t),v(t),s(t))$ be a global smooth solution to the IS system (6), with
initial data $(x_{0},v_{0},s_{0})$ that satisfy (7). We have also assumed that
the communication matrix in (6) satisfies
$\psi_{m}\leq\psi_{ij}(t)\leq\psi_{M}$ for all $1\leq i,j\leq N$ and $t\geq
0$, and that $\pazocal{A}(v_{0})>\delta_{0}$ for some $\delta_{0}\in(0,1)$.
Finally, we assume that the initial configurations and parameters satisfy the
conditions

$(H1)\quad\gamma>\sqrt{\frac{6k\chi\psi_{M}}{\delta_{0}\psi_{m}}},\qquad\qquad(H2)\quad\pazocal{C}_{0}<1-\delta_{0},$

where the exact form of $\pazocal{C}_{0}$ is given by

$\pazocal{C}_{0}:=\frac{1}{2}D(v_{0})^{2}+\frac{\chi
D(v_{0})|\dot{D}(v_{0})|}{\gamma}+\frac{2}{\gamma^{2}}\left(D(s_{0})^{2}+2\max\limits_{1\leq
i\leq N}|s_{i0}|^{2}\right).$

Then the particle flow exhibits orientation invariance with
$\pazocal{A}(v(t))>\delta_{0}$ for $t\geq 0$. Furthermore, we have the
estimates

$\displaystyle
D(v(t))\leq\left\\{\begin{array}[]{ll}O(e^{-\frac{\mu_{*}}{2}t}),\qquad\text{for}\quad\mu_{*}<\frac{\gamma}{2\chi},\\\
\\\
O(\sqrt{t}e^{-\frac{\gamma}{4\chi}t}),\qquad\text{for}\quad\mu_{*}\geq\frac{\gamma}{2\chi},\end{array}\right.$
(20)

where $\mu_{*}=\sup\pazocal{D}$ for the non-empty set

$\pazocal{D}=\left\\{0<\mu<\frac{\gamma}{\chi}\quad:\quad\mu^{3}-\frac{2\gamma}{\chi}\mu^{2}+\frac{2k\chi\psi_{m}\delta_{0}+\gamma^{2}}{\chi^{2}}\mu\leq\frac{2k}{\chi}\left(\frac{\gamma}{\chi}\psi_{m}\delta_{0}-\frac{6k}{\gamma}\psi_{M}\right)\right\\}.$
(21)

######  Remark 4.

The way we have defined $\pazocal{C}_{0}$ in both theorems ensures that the
$\pazocal{C}_{0}<1-\delta_{0}$ assumption is stronger than
$\pazocal{A}(v_{0})>0$, and hence $\pazocal{A}(v_{0})>\delta_{0}$ could have
been omitted. We have kept it because it signifies the importance of the
geometric factor in the analysis, and it is also helpful, since we work with a
continuity argument on $\pazocal{A}(v(t))$ to show that
$\pazocal{A}(v(t))>\delta_{0}$ for all $t\geq 0$ in both proofs.

##  3 Some special cases of IS models

In this brief Section we discuss some results for two simplified instances of
the IS model: Namely, the IS model on 2-d plane where it reduces to the
inertial Kuramoto model, and the overdamped regime $\chi\to 0$, where the IS
reduces to the Cucker-Smale with constant speed (for weights that are
functions of the metric distance of particles).

###  3.1 Inertial Spin on the plane and connection to the inertial Kuramoto
model

If we assume a motion restricted on a plane then the velocity vector $v_{i}$
can be expressed in polar form as

$v_{i}=(\cos{\theta_{i}},\sin{\theta_{i}}),$

for $\theta_{i}\in[0,2\pi)$. We Differentiate the velocity vector to get the
first and second derivatives

$\dot{v}_{i}=(-\sin{\theta_{i}},\cos{\theta_{i}})\dot{\theta}_{i}\qquad\text{and}\qquad\ddot{v}_{i}=(-\cos{\theta_{i}},-\sin{\theta_{i}})|\dot{\theta}_{i}|^{2}+(-\sin{\theta_{i}},\cos{\theta_{i}})\ddot{\theta}_{i}.$

Substituting the derivatives of $v_{i}$ into equation (13), then using the
identity
$\chi|\dot{v}_{i}|^{2}v_{i}=\chi(\cos{\theta_{i}},\sin{\theta_{i}})|\dot{\theta}_{i}|^{2}$
and the fact that

$\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}-(v_{i}\cdot
v_{k})v_{i})=(-\sin{\theta_{i}},\cos{\theta})\sum\limits_{k=1}^{N}\psi_{ik}\sin{(\theta_{k}-\theta_{i})},$

we get the equation for the evolution of $\theta_{i}$, i.e.

$\chi\ddot{\theta_{i}}+\gamma\dot{\theta_{i}}=\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}\sin{(\theta_{k}-\theta_{i})},\quad
1\leq i\leq N.$ (22)

This is the Kuramoto model equation when inertial effects are involved, for a
network topology where $\psi_{ij}:=a_{ij}$ is the coupling strength between
oscillators $i$ and $j$, and with zero natural frequencies $\Omega_{i}=0$.
This model is coupled with the set of initial conditions

$\theta_{i}(0)=\theta_{i0},\quad\dot{\theta}_{i}(0)=\dot{\theta}_{i0},\qquad
1\leq i\leq N.$ (23)

The inertial Kuramoto model has been studied extensively in many works e.g.
[11, 12, 13, 14, 15, 16, 31]. In particular, we may consider the more general
model

$m_{i}\ddot{\theta}_{i}+\gamma_{i}\dot{\theta}_{i}=\Omega_{i}+k\sum\limits_{j=1}^{N}a_{ij}\sin(\theta_{j}-\theta_{i}),\quad
t\geq 0,\quad i=1,\ldots,N,$ (24)

with prescribed initial data given by (23). Here $m_{i},\gamma_{i}$ are the
inertia and damping parameters of the i’th oscillator respectively, $k$ is the
coupling strength, and the $\Omega_{i}$’s are the natural frequencies of the
oscillators which are independent to one another and each one follows a common
distribution $g(\Omega)$. The unknown function $\theta_{i}(t)$ corresponds to
the phase of the i’th oscillator, and its derivative
$\omega_{i}(t):=\dot{\theta}_{i}(t)$ to its frequency. We use the shorthand
vector notation
$(\theta(t),\omega(t)):=(\theta_{1},\ldots,\theta_{N},\omega_{1},\ldots,\omega_{N})$
and the initial data are represented by the vector $(\theta_{0},\omega_{0})$.
We also define the diameters for the phases and frequencies by

$D_{\theta}(t):=\max\limits_{1\leq i\leq N}|\theta_{j}-\theta_{i}|,\qquad
D_{\omega}(t):=\max\limits_{1\leq i\leq N}|\omega_{j}-\omega_{i}|.$

We say that the system exhibits complete phase synchronization if
$D_{\theta}(t)\to 0$, and complete frequency synchronization if
$D_{\omega}(t)\to 0$ as $t\to\infty$.

The following result regarding the case of identical natural frequencies, and
homogeneous inertia and friction, was the first result of its kind for (24)
that was proven in [14]. Instead of using the formulation of the theorem from
[14] (where the aggregate result is presented in two Theorems 4.1 & 4.2), we
choose the more synoptic formulation in [12]:

######  Proposition 4.

[12] Suppose that the parameters $m_{i},\gamma_{i},\Omega_{i},k$ and initial
data satisfy the following relations:

$\displaystyle m_{i}=m,\quad\gamma_{i}=1,\quad
a_{ij}=\frac{1}{N},\quad\Omega_{i}=\Omega,\quad m,k>0$ $\displaystyle
0<\pazocal{C}_{1}(0)<\pi\qquad\text{and}\qquad
mk\in\left(0,\frac{1}{4}\right)\cup\left(\frac{\pazocal{C}_{1}(0)}{4\sin{\pazocal{C}_{1}(0)}},\infty\right),$

where
$\pazocal{C}_{1}(m,\theta(t),\dot{\theta}(t)):=\max\\{D_{\theta}(t),\,D_{\theta}(t)+m\dot{D}_{\theta}(t)\\}$.
Then, for any solution $(\theta(t),\omega(t))$ to (24), there exists some
positive constant $\Lambda_{1}>0$ s.t.

$D_{\theta}(t)+D_{\omega}(t)\leq O(1)e^{-\Lambda_{1}t}\quad\text{for}\quad
t\to\infty.$

######  Remark 5.

The observant reader might have already noticed the similarity between the
above result and Theorem 2 of this article. It is important for instance to
see that the initial configurations in Proposition 4 (as manifested by the
assumption $0<\pazocal{C}_{1}(m,\theta_{0},\omega_{0})<\pi$) are chosen so
that all initial phases are contained on the semi-circle. This implies a
certain orientation that is assumed in initial conditions, and it is a common
assumption in both Proposition 4 and Theorem 2 (in Theorem 2 this assumption
is $\pazocal{C}_{0}<1-\delta_{0}$). On the other hand, there is also a major
difference between the two results. Although Theorem 2 makes the assumption of
a “small inertia” regime with the condition
$\gamma>\sqrt{\frac{6k\chi\psi_{M}}{\delta_{0}\psi_{m}}}$, Proposition 4
applies in both small and large inertia regions, as manifested by
$mk\in\left(0,\frac{1}{4}\right)\cup\left(\frac{\pazocal{C}_{1}(0)}{4\sin{\pazocal{C}_{1}(0)}},\infty\right)$.
This shouldn’t come as a surprise, as the 2-d setting of the IS system allows
for a much sharper inequality for the phase diameter, than the one we derive
in Section 4 for the velocity diameter of the IS in 3-d. The problem of
showing invariant regions for the velocities in the large inertia regime, when
the communication matrix is not constant, remains open.

###  3.2 Formal zero inertial limit $\chi\to 0$

In the overdamped regime, when the inertia becomes negligible compared to the
spin dissipation and $\psi_{ij}=\psi(|x_{i}-x_{j}|)$, the IS model reduces to
the CS model with constant speed. By taking the formal limit $\chi\to 0$
(derivation could be made rigorous!) we get system (14), which we rewrite as

$\dot{x}_{i}=v_{i},\qquad\dot{v}_{i}=\frac{\bar{k}}{N}\sum\limits_{j=1}^{N}\psi_{ij}(v_{j}-(v_{i}\cdot
v_{j})v_{i}),\qquad\text{with}\quad\bar{k}=\frac{k}{\gamma}.$ (25)

This model has been studied in [8] and later in [9, 28]. In general, it is
easy to construct initial configurations for a simple three particle system
where a global flock is not achieved (see Appendix. A in [8]). Furthermore,
despite the simplicity of this system, the asymptotic behavior is not yet
fully understood for all initial configurations. That being said, for the
restricted class of initial conditions s.t. $\pazocal{A}(v_{0})>0$ it is
possible to employ a Lyapunov functional approach to prove asymptotic
flocking. Indeed, if we consider initial data with $\pazocal{A}(v_{0})>0$ then
we can prove the following system of dissipating differential inequalities
(SDDI) holds

$\Big{|}\frac{d}{dt}D(x)\Big{|}\leq
D(v),\qquad\frac{d}{dt}D(v)\leq-\bar{k}\psi(D(x))D(v)\pazocal{A}(v_{0}),$ (26)

which directly implies that $\pazocal{A}(v(t))\geq\pazocal{A}(v_{0})$ for all
$t\geq 0$. From this point, the standard treatment is to introduce the
Lyapunov functional

$\pazocal{H}_{\pm}(t):=D(v)\pm\bar{k}\pazocal{A}(v_{0})\int_{0}^{D(x)}\psi(s)\,ds,$
(27)

and show with the help of (26) that this functional dissipates, i.e.
$\pazocal{H}_{\pm}(t)\leq\pazocal{H}_{\pm}(0)$ for all $t\geq 0$. Then the
following flocking estimate for (25) was proven.

######  Proposition 5.

[8] Suppose that the communication weight $\psi(\cdot)$ and initial
configurations $(x_{0},v_{0})$ satisfy the following conditions:

$\pazocal{A}(v_{0})>0,\qquad
0<D(v_{0})<\bar{k}\pazocal{A}(v_{0})\min{\left\\{\int_{D(x_{0})}^{\infty}\psi(s)\,ds,\int_{0}^{D(x_{0})}\psi(s)\,ds\right\\}}.$

Then, there exists a unique solution $(x(t),v(t))$ to system (14) satisfying
the asymptotic flocking conditions:

$\sup\limits_{t\geq 0}D(x)<D^{\infty}\qquad\text{and}\qquad\,D(v)\leq
D(v_{0})e^{-\bar{k}\pazocal{A}(v_{0})\psi(D^{\infty})t},\quad t\geq 0,$

where $D^{\infty}$ is defined implicitly by
$D(v_{0}):=\bar{k}\pazocal{A}(v_{0})\int_{D(x_{0})}^{D^{\infty}}\psi(s)\,ds$.

We should also note here that invariant regions of velocities and the flocking
property have also been studied in a “cone-vision” variant of the C-S model in
[33]. In the “cone-vision” model the agents update their velocities by only
considering other agents that belong in a certain vision-cone of their
velocity vector. In general, such a model can give birth to multi-cluster
formations, but conditions for the emergence of a unique flock in the 2-d and
3-d models were presented in [33].

##  4 A second order differential inequality for the velocity diameter

We now prove the main inequality for velocity diameters. In our proof we
assume the more general communication $\psi:=\psi(|x_{j}-x_{i}|)$ where
$\psi(\cdot)$ is a decreasing weight function s.t. $0<\psi(\cdot)<\psi_{M}$.
We make no assumption of a lower bound for $\psi(\cdot)\geq\psi_{m}>0$ in
this.

######  Proposition 6.

Let $(x,v,s)$ be a solution to (6) with initial data $(x_{0},v_{0},s_{0})$
that satisfy (7). We also assume that this solution has all flight directions
(orientations of velocities) chosen s.t. $\pazocal{A}(v(t))>0$ for all $t\geq
0$. Then we have

$\displaystyle\frac{d^{2}}{dt^{2}}D(v)^{2}+\frac{\gamma}{\chi}\frac{d}{dt}D(v)^{2}+\frac{2k}{\chi}\psi(D(x))D(v)^{2}\pazocal{A}(v)\leq\frac{4}{\chi^{2}}\left(D(s)^{2}+\max\limits_{1\leq
i\leq N}|s_{i}|^{2}D(v)^{2}\right).$ (28)

Moreover, the inequality for $D(v)$ is

$\displaystyle\frac{d^{2}}{dt^{2}}D(v)^{2}+\frac{\gamma}{\chi}\frac{d}{dt}D(v)^{2}+\frac{2k}{\chi}\psi(D(x))D(v)^{2}\pazocal{A}(v)\leq\pazocal{C}_{1}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}D(v(s))^{2}+\pazocal{C}_{2}e^{-\frac{\gamma}{\chi}t},$
(29)

where

$\pazocal{C}_{1}=\frac{4k^{2}\psi_{M}}{\gamma\chi}(1+D(v)^{2})<\frac{12k^{2}\psi_{M}}{\gamma\chi}\quad\text{and}\quad\pazocal{C}_{2}=\frac{4}{\chi^{2}}\left(D(s_{0})^{2}+2\max\limits_{1\leq
i\leq N}|s_{i}(0)|^{2}\right)$

###### Proof.

We write down eq. (13) for two particles $i,j$ and take the inner product with
$v_{j}$ and $v_{i}$ respectively. The idea is that by obtaining an equation
for the inner product $v_{i}\cdot v_{j}$ we can then derive an equation for
$|v_{i}-v_{j}|^{2}$.

$\chi\ddot{v}_{i}\cdot v_{j}+\gamma\dot{v}_{i}\cdot
v_{j}+\chi|\dot{v}_{i}|^{2}v_{i}\cdot
v_{j}=\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot v_{j}-(v_{i}\cdot
v_{k})v_{i}\cdot v_{j}),$ $\chi\ddot{v}_{j}\cdot v_{i}+\gamma\dot{v}_{j}\cdot
v_{i}+\chi|\dot{v}_{j}|^{2}v_{i}\cdot
v_{j}=\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot v_{i}-(v_{j}\cdot
v_{k})v_{j}\cdot v_{i}).$

Now if we sum both equations we get

$\displaystyle\chi(\ddot{v}_{i}\cdot v_{j}+\ddot{v}_{i}\cdot
v_{j})+\gamma\frac{d}{dt}(v_{i}\cdot
v_{j})+\chi(|\dot{v}_{i}|^{2}+|\dot{v}_{j}|^{2})v_{i}\cdot v_{j}=$
$\displaystyle\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot
v_{j})+\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot v_{i}-(v_{k}\cdot
v_{j})v_{i}\cdot v_{j}).$ (30)

Note that we have $\frac{d^{2}}{dt^{2}}(v_{i}\cdot v_{j})=\ddot{v}_{i}\cdot
v_{j}+2\dot{v}_{i}\cdot\dot{v}_{j}+\ddot{v}_{j}\cdot v_{i}$, so eq. (30)
yields

$\displaystyle\chi\frac{d^{2}}{dt^{2}}(v_{i}\cdot
v_{j})+\gamma\frac{d}{dt}(v_{i}\cdot
v_{j})-2\chi\dot{v}_{i}\cdot\dot{v}_{j}+\chi(|\dot{v}_{i}|^{2}+|\dot{v}_{j}|^{2})v_{i}\cdot
v_{j}=$ $\displaystyle\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot
v_{j})+\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot v_{i}-(v_{k}\cdot
v_{j})v_{i}\cdot v_{j})$ (31)

We now use the identity $v_{i}\cdot v_{j}=1-\frac{1}{2}|v_{i}-v_{j}|^{2}$ in
(31) to get

$\displaystyle\frac{\chi}{2}\frac{d^{2}}{dt^{2}}|v_{i}-v_{j}|^{2}+\frac{\gamma}{2}\frac{d}{dt}|v_{i}-v_{j}|^{2}=-2\chi\dot{v}_{i}\cdot\dot{v}_{j}+\chi(|\dot{v}_{i}|^{2}+|\dot{v}_{j}|^{2})v_{i}\cdot
v_{j}$ $\displaystyle-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot
v_{j})-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot v_{i}-(v_{k}\cdot
v_{j})v_{i}\cdot v_{j}).$ (32)

Note also that since
$-2\chi\dot{v}_{i}\cdot\dot{v}_{j}+\chi(|\dot{v}_{i}|^{2}+|\dot{v}_{j}|^{2})v_{i}\cdot
v_{j}\leq\chi|\dot{v}_{i}-\dot{v}_{j}|^{2}$ inequality (32) yields

$\displaystyle\frac{\chi}{2}\frac{d^{2}}{dt^{2}}|v_{i}-v_{j}|^{2}+\frac{\gamma}{2}\frac{d}{dt}|v_{i}-v_{j}|^{2}\leq\pazocal{I}_{1}+\pazocal{I}_{2},$
(33)

where $\pazocal{I}_{1}:=\chi|\dot{v}_{i}-\dot{v}_{j}|^{2}$ and

$\displaystyle\pazocal{I}_{2}:=\pazocal{I}_{21}+\pazocal{I}_{22}:=-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot
v_{j})-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot v_{i}-(v_{k}\cdot
v_{j})v_{i}\cdot v_{j}).$

To control the term $\pazocal{I}_{1}$ we first write $\dot{v}_{i}-\dot{v}_{j}$
as

$\dot{v}_{i}-\dot{v}_{j}=\frac{1}{\chi}(s_{i}-s_{j})\times
v_{i}+\frac{1}{\chi}s_{j}\times(v_{i}-v_{j}),$

by virtue of (6). We therefore have that

$\pazocal{I}_{1}\leq\frac{2}{\chi}|s_{i}-s_{j}|^{2}+\frac{2}{\chi}|s_{j}|^{2}|v_{i}-v_{j}|^{2},$

which yields

$\pazocal{I}_{1}\leq\frac{2}{\chi}D(s)^{2}+\frac{2}{\chi}\max\limits_{1\leq
i\leq N}|s_{i}|^{2}D(v)^{2}.$ (34)

We proceed with the treatment of $\pazocal{I}_{2}$. In this treatment we make
the assumption that the $i,j$ indices are chosen so that $v_{i}$ and $v_{j}$
are the extremal velocities that minimize $v_{i}\cdot v_{j}$, i.e $v_{j}\cdot
v_{k}\geq v_{i}\cdot v_{j}$ for all $1\leq k\leq N$. Indeed, with this choice
for the $i,j$ pair we have that

$\displaystyle\pazocal{I}_{21}$
$\displaystyle=-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{ik}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot v_{j})$
$\displaystyle\leq-\frac{k}{N}\min\limits_{i,j}\psi_{ij}\sum\limits_{k=1}^{N}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot v_{j})$
$\displaystyle\leq-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot v_{j}).$

Note that we used the fact that $v_{k}\cdot v_{j}-(v_{k}\cdot v_{i})v_{i}\cdot
v_{j}\geq 0$ (since $v_{k}\cdot v_{j}\geq v_{i}\cdot v_{j}$ and $|v_{k}\cdot
v_{i}|\leq 1$). In similar manner we show for $\pazocal{I}_{22}$ that

$\displaystyle\pazocal{I}_{22}$
$\displaystyle=-\frac{k}{N}\sum\limits_{k=1}^{N}\psi_{jk}(v_{k}\cdot
v_{i}-(v_{k}\cdot v_{j})v_{i}\cdot v_{j})$
$\displaystyle\leq-\frac{k}{N}\min\limits_{i,j}\psi_{ij}\sum\limits_{k=1}^{N}(v_{k}\cdot
v_{i}-(v_{k}\cdot v_{j})v_{i}\cdot v_{j})$
$\displaystyle\leq-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(v_{k}\cdot
v_{i}-(v_{k}\cdot v_{j})v_{i}\cdot v_{j}).$

If we add the two expressions after we re-arrange terms we have

$\displaystyle\pazocal{I}_{2}$
$\displaystyle\leq-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(v_{k}\cdot
v_{j}-(v_{k}\cdot v_{j})v_{i}\cdot
v_{j})-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(v_{k}\cdot v_{i}-(v_{k}\cdot
v_{i})v_{i}\cdot v_{j})$
$\displaystyle\leq-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(1-v_{i}\cdot
v_{j})v_{k}\cdot v_{j}-\frac{k}{N}\psi(D(x))\sum\limits_{k=1}^{N}(1-v_{i}\cdot
v_{j})v_{k}\cdot v_{i}$
$\displaystyle\leq-\frac{k}{2N}\psi(D(x))D(v)^{2}\sum\limits_{k=1}^{N}v_{k}\cdot
v_{j}-\frac{k}{2N}\psi(D(x))D(v)^{2}\sum\limits_{k=1}^{N}v_{k}\cdot
v_{i}\leq-k\psi(D(x))D(v)^{2}v_{i}\cdot v_{j}.$

This yields the following bound for $\pazocal{I}_{2}$

$\pazocal{I}_{2}\leq-k\psi(D(x))D(v)^{2}\pazocal{A}(v).$ (35)

Inserting the estimates (34)-(35) in (33), and keeping in mind that we have
chosen $i,j$ s.t. $D(v)=|v_{i}-v_{j}|$, we get the first inequality (28).

Finally, we turn to the term $\pazocal{I}_{1}$ which controls the RHS of (28).
To control $|s_{i}-s_{j}|^{2}$ we use the spin equation (11) for
$s_{i}-s_{j}$, and after we multiply by $s_{i}-s_{j}$ we get

$\frac{d}{dt}|s_{i}-s_{j}|^{2}+\frac{2\gamma}{\chi}|s_{i}-s_{j}|^{2}=\frac{2k}{N}\sum\limits_{k}\left((\psi_{ik}v_{i}-\psi_{jk}v_{j})\times
v_{k}\right)\cdot(s_{i}-s_{j}).$

We note that when $0\leq\psi(\cdot)\leq\psi_{M}$ then
$|\psi_{ik}v_{i}-\psi_{jk}v_{j}|\leq\psi_{M}|v_{i}-v_{j}|$. Now using the
Young inequality we get

$\displaystyle\frac{d}{dt}|s_{i}-s_{j}|^{2}+\frac{2\gamma}{\chi}|s_{i}-s_{j}|^{2}\leq\frac{k^{2}\chi\psi_{M}}{\gamma}|v_{i}-v_{j}|^{2}+\frac{\gamma}{\chi}|s_{i}-s_{j}|^{2}.$

With the help of Duhamel’s principle we get

$\displaystyle|s_{i}-s_{j}|^{2}\leq\frac{k^{2}\chi\psi_{M}}{\gamma}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}|v_{i}(s)-v_{j}(s)|^{2}+e^{-\frac{\gamma}{\chi}t}|s_{i0}-s_{j0}|^{2}.$

Thus, picking $i,j$ s.t. $D(s)=|s_{i}-s_{j}|$ we have

$D(s)^{2}\leq\frac{k^{2}\chi\psi_{M}}{\gamma}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}D(v(s))^{2}+e^{-\frac{\gamma}{\chi}t}D(s_{0})^{2}.$

A similar bound can be obtained for $|s_{i}|^{2}$, i.e.

$\frac{d}{dt}|s_{i}|^{2}+\frac{\gamma}{\chi}|s_{i}|^{2}\leq\frac{k^{2}\chi\psi_{M}}{\gamma
N}\sum\limits_{k=1}^{N}|v_{i}\times v_{k}|^{2}.$

Now using the fact that $|v_{i}\times
v_{j}|^{2}=|v_{i}-v_{j}|^{2}-\frac{1}{4}|v_{i}-v_{j}|^{4}$, which implies
$|v_{i}\times v_{j}|^{2}\leq|v_{i}-v_{j}|^{2}$, we get the estimate

$|s_{i}|^{2}\leq
e^{-\frac{\gamma}{\chi}t}|s_{i0}|^{2}+\frac{k^{2}\chi\psi_{M}}{\gamma}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}D(v(s))^{2}.$

In this case, collecting the terms for $\pazocal{I}_{1}$ we get

$\displaystyle\pazocal{I}_{1}$
$\displaystyle\leq\frac{2k^{2}\psi_{M}}{\gamma}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}D(v(s))^{2}+\frac{2}{\chi}e^{-\frac{\gamma}{\chi}t}D(s_{0})^{2}$
$\displaystyle+\frac{2k^{2}\psi_{M}}{\gamma}D(v)^{2}\int_{0}^{t}ds\,e^{-\frac{\gamma}{\chi}(t-s)}D(v(s))^{2}+\frac{2}{\chi}e^{-\frac{\gamma}{\chi}t}D(v)^{2}\max\limits_{1\leq
i\leq N}|s_{i0}|^{2}.$

∎

######  Remark 6.

In the derivation of inequalities (28) – (29) we have used the fact that the
flight orientations are all contained in an invariant region
$\pazocal{A}(v(t))>0\,\,\text{for}\,\,t\geq 0$. The purpose of this work is to
show that if $\pazocal{A}(v_{0})>\delta_{0}$ (for any choice
$\delta_{0}\in(0,1)$), then under certain assumptions on the initial data and
parameters of the model, and with the help of a continuity argument on (28) –
(29), we have that $\pazocal{A}(v(t))>\delta_{0}$ for all $t\geq 0$.

######  Remark 7.

In addition to identifying invariant regions with the help of Proposition 6 we
want to know whether convergence to a common velocity can be derived from (28)
– (29). Some of the obvious difficulties in treating (28) – (29) are: (i) the
nonlinearity in the term $\psi(D(x))D(v)^{2}\pazocal{A}(v)$, (ii) the non-
homogeneity in the RHS, and (iii) the fact that they are both second order
differential inequalities. All of the above imply the necessity to work with
new types of Gronwall inequalities. In the Section that follows, we derive
such inequalities.

##  5 Second order Gronwall inequalities.

We begin with a Lemma that is a generalization of a result presented in [14].
In [14], a second order Gronwall inequality for the differential inequality
$a\ddot{y}+b\dot{y}+cy+d\leq 0$ with $a>0$, $b,c,d\in\mathbb{R}$, was proven.
Here we show a similar result, but with a continuous function $g(t)$ in the
place of the constant $d$.

######  Lemma 1.

Let $y=y(t)$ be a nonnegative $C^{2}$-function satisfying the second order
differential inequality

$a\ddot{y}+b\dot{y}+cy\leq g(t),\quad y(0)=y_{0},\quad\dot{y}(0)=y_{1},\quad
t>0,$ (36)

where $a>0$, $b,c$ constants, and $g(t)$ is a positive continuous function.
Then, we have the estimates for

(i) $b^{2}-4ac>0:$

$\displaystyle y(t)\leq
e^{-\nu_{1}t}y_{0}+a\frac{e^{-\nu_{2}t}-e^{-\nu_{1}t}}{\sqrt{b^{2}-4ac}}\left(y_{1}+\nu_{1}y_{0}\right)+\frac{1}{a}\int_{0}^{t}ds\,e^{-\nu_{1}(t-s)}\int_{0}^{s}ds^{\prime}e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime}),$
(37)

where $\nu_{1}=\frac{b+\sqrt{b^{2}-4ac}}{2a}$ and
$\nu_{2}=\frac{b-\sqrt{b^{2}-4ac}}{2a}$.

(ii) $b^{2}-4ac\leq 0:$

$\displaystyle y(t)\leq
e^{-\frac{b}{2a}t}\Bigg{[}y_{0}+\left(\frac{b}{2a}y_{0}+y_{1}\right)t\Bigg{]}+\frac{1}{a}\int_{0}^{t}ds\int_{0}^{s}ds^{\prime}e^{-\frac{b}{2a}(t-s^{\prime})}g(s^{\prime}).$
(38)

Moreover, if $g(t)$ vanishes asymptotically as $t\to\infty$, i.e. $g(t)\to 0$
as $t\to\infty$. Then, it follows that $y(t)\to 0$ as $t\to\infty$.

###### Proof.

The proof follows the ideas in the proof of Lemma 3.1 pg. 34 in [14] with
modifications.

(i) For $b^{2}-4ac>0$ we consider the functional

$\displaystyle
L(t):=\dot{y}+Ay-\frac{1}{a}\int_{0}^{t}\,ds\,e^{-\frac{h}{a}(t-s)}g(s),$

where $A=\frac{b-h}{a}$ and $h$ any of the two positive solutions of
polynomial $h^{2}-bh+ac=0$. For this Lyapunov functional we find easily that
$\dot{L}(t)\leq-\frac{h}{a}L(t)$. Then, we have

$L(t)\leq e^{-\frac{h}{a}t}L(0)=e^{-\frac{h}{a}t}(y_{1}+Ay_{0}),$

which in turn yields

$\dot{y}+Ay\leq\frac{1}{a}\int_{0}^{t}ds\,e^{-\frac{h}{a}(t-s)}g(s)+e^{-\frac{h}{a}t}(y_{1}+Ay_{0}).$

Now solving in $y(t)$ by integrating factor and taking
$h=\frac{b-\sqrt{b^{2}-4ac}}{2}$ as the solution that optimizes the exponents
we derive the bound in (37).

(ii) For $b^{2}-4ac\leq 0$ we make the substitution
$Y(t):=e^{\frac{b}{2a}t}y(t)$ and derive the following equation for $Y(t)$,
i.e.

$\ddot{Y}(t)-\frac{b^{2}-4ac}{4a}Y(t)\leq\frac{1}{a}e^{\frac{b}{2a}t}g(t).$

Now since $y(t)$ is nonegative we have
$\ddot{Y}(t)\leq\frac{1}{a}e^{\frac{b}{2a}t}g(t)$ which yields after two
integrations

$Y(t)\leq
y_{0}+\left(\frac{b}{2a}y_{0}+y_{1}\right)t+\frac{1}{a}\int_{0}^{t}ds\,\int_{0}^{s}ds^{\prime}e^{\frac{b}{2a}s^{\prime}}g(s^{\prime}).$

The desired inequality (38) for $y(t)$ follows trivially after we multiply
both sides by $e^{-\frac{b}{2a}t}$. We leave the last part of our proof, i.e.
that when the continuous $g(t)\to 0$, as $t\to\infty$, then it follows that
$y(t)\to 0$ for both (37)-(38), in the following autonomous lemma. ∎

######  Lemma 2.

Assume that we have a continuous function $g(t)$ on $[0,\infty)$ s.t. $g(t)\to
0$ as $t\to\infty$. Then the following double integrals (for
$a,b,\nu_{1},\nu_{2}>0$ and $\nu_{1}>\nu_{2}$)

$\pazocal{T}_{1}(t):=e^{-\nu_{1}t}\int_{0}^{t}ds\,e^{(\nu_{1}-\nu_{2})s}\int_{0}^{s}ds^{\prime}\,e^{\nu_{2}s^{\prime}}g(s^{\prime}),\qquad\pazocal{T}_{2}(t):=e^{-\frac{b}{2a}t}\int_{0}^{t}ds\int_{0}^{s}ds^{\prime}\,e^{\frac{b}{2a}s^{\prime}}g(s^{\prime}),$

also vanish as $t\to\infty$.

###### Proof.

We fix some $\epsilon>0$, and due to the continuity of $g(t)$ we know there
exists some $T_{\epsilon}$ s.t. $g(t)<\epsilon$ for $t\geq T_{\epsilon}$. We
can break the double integral $\pazocal{T}_{1}(t)$ into 3 terms as follows

$\displaystyle\pazocal{T}_{1}(t)$
$\displaystyle=e^{-\nu_{1}t}\int_{0}^{t}ds\ldots\int_{0}^{s}ds^{\prime}\ldots=\pazocal{T}_{11}(t)+\pazocal{T}_{12}(t)+\pazocal{T}_{13}(t)$
$\displaystyle=e^{-\nu_{1}t}\int_{0}^{T_{\epsilon}}ds\ldots\int_{0}^{s}ds^{\prime}\ldots+e^{-\nu_{1}t}\int_{T_{\epsilon}}^{t}ds\ldots\int_{0}^{T_{\epsilon}}ds^{\prime}\ldots+e^{-\nu_{1}t}\int_{T_{\epsilon}}^{t}ds\ldots\int_{T_{\epsilon}}^{s}ds^{\prime}\ldots.$

We now compute and integral term separately using the bounds
$g(t)\leq\|g\|_{\infty}:=\max\limits_{t\geq 0}g(t)$ on the interval
$[0,T_{\epsilon}]$, and $g(t)<\epsilon$ on $[T_{\epsilon},\infty)$, i.e.

$\displaystyle\pazocal{T}_{11}(t)$ $\displaystyle\leq
e^{-\nu_{1}t}\frac{\|g\|_{\infty}}{\nu_{2}}\int_{0}^{T_{\epsilon}}ds\,e^{(\nu_{1}-\nu_{2})s}(e^{\nu_{2}s}-1)\leq
e^{-\nu_{1}t}\frac{\|g\|_{\infty}}{\nu_{2}}\left(\frac{1}{\nu_{1}}(e^{\nu_{1}T_{\epsilon}}-1)-\frac{1}{\nu_{1}-\nu_{2}}\left(e^{(\nu_{1}-\nu_{2})T_{\epsilon}}-1\right)\right),$
$\displaystyle\pazocal{T}_{12}(t)$ $\displaystyle\leq
e^{-\nu_{1}t}\frac{\|g\|_{\infty}}{\nu_{2}}\int_{T_{\epsilon}}^{t}ds\,e^{(\nu_{1}-\nu_{2})s}(e^{\nu_{2}T_{\epsilon}}-1)\leq
e^{-\nu_{1}t}\frac{\|g\|_{\infty}}{\nu_{2}(\nu_{1}-\nu_{2})}\left(e^{(\nu_{1}-\nu_{2})T_{\epsilon}}-1\right)\left(e^{\nu_{2}T_{\epsilon}}-1\right),$
$\displaystyle\pazocal{T}_{13}(t)$ $\displaystyle\leq
e^{-\nu_{1}t}\frac{\epsilon}{\nu_{2}}\int_{T_{\epsilon}}^{t}ds\,e^{(\nu_{1}-\nu_{2})s}(e^{\nu_{2}s}-e^{\nu_{2}T_{\epsilon}})$
$\displaystyle\leq
e^{-\nu_{1}t}\frac{\epsilon}{\nu_{2}}\left(\frac{1}{\nu_{1}}\left(e^{\nu_{1}t}-e^{\nu_{1}T_{\epsilon}}\right)-\frac{1}{\nu_{1}-\nu_{2}}\left(e^{(\nu_{1}-\nu_{2})t}-e^{(\nu_{1}-\nu_{2})T_{\epsilon}}\right)\right)$
$\displaystyle\leq\frac{\epsilon}{\nu_{2}}\left(\frac{1}{\nu_{1}}\left(1-e^{-\nu_{1}(t-T_{\epsilon})}\right)-\frac{1}{\nu_{1}-\nu_{2}}\left(e^{-\nu_{2}t}-e^{-\nu_{1}(t-T_{\epsilon})}e^{-\nu_{2}T_{\epsilon}}\right)\right).$

It becomes clear from the
$\pazocal{T}_{11}(t),\pazocal{T}_{12}(t),\pazocal{T}_{13}(t)$ estimates that
we can choose
$T_{\epsilon}^{\prime}=T_{\epsilon}^{\prime}(\epsilon,T_{\epsilon},\|g\|_{\infty})>0$
s.t $\pazocal{T}_{11}(t),\pazocal{T}_{12}(t)<\epsilon$ and also
$\pazocal{T}_{13}(t)<\frac{2}{\nu_{1}\nu_{2}}\epsilon$ for
$t>T_{\epsilon}^{\prime}$, i.e.
$\pazocal{T}_{1}<\left(2+\frac{2}{\nu_{1}\nu_{2}}\right)\epsilon$ for
$t>T_{\epsilon}^{\prime}$. The proof for the term $\pazocal{T}_{2}(t)$ is
similar and therefore omitted. ∎

It is useful to derive uniform (in time) bounds based on the above estimates,
which also involve the integrability of $g(t)$ in time, i.e. a uniform bound
with respect to the integral $\int_{0}^{\infty}g(t)\,dt<\infty$. We show the
uniform bounds in the following corollary.

######  Corollary 1.

Let $y=y(t)$ be a nonnegative $C^{2}$-function satisfying the second order
differential inequality (36), and $g(t)$ is an integrable function. Then we
have the following uniform bounds for $y(t)$:
(i) $b^{2}-4ac>0$ :

$\displaystyle y(t)\leq
y_{0}+\frac{a}{\sqrt{b^{2}-4ac}}(|y_{1}|+\nu_{1}y_{0})+\frac{1}{2\sqrt{ac}}\int_{0}^{\infty}ds\,g(s)$

(ii) $b^{2}-4ac\leq 0$ :

if $y_{1}\geq 0$ :

$\displaystyle y(t)\leq
e^{-\frac{y_{0}}{\frac{b}{2a}y_{0}+y_{1}}}\left(1+\frac{2a}{b}\right)y_{0}+\frac{1}{b}\int_{0}^{\infty}ds\,g(s),$

if $y_{1}<0$ :

$\displaystyle y(t)\leq y_{0}+\frac{1}{b}\int_{0}^{\infty}ds\,g(s).$

###### Proof.

It is an elementary exercise in calculus to give a uniform bound for the non
integral terms that appear in (37) and (38) in Lemma 1. The most important
terms to bound are the integral terms

$\int_{0}^{t}ds\,e^{-\nu_{1}(t-s)}\int_{0}^{s}ds^{\prime}e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime})\quad\text{and}\quad\int_{0}^{t}ds\int_{0}^{s}ds^{\prime}e^{-\frac{b}{2a}(t-s^{\prime})}g(s^{\prime}).$

We have the following Hölder inequality for the first integral

$\displaystyle\int_{0}^{t}ds\,e^{-\nu_{1}(t-s)}\int_{0}^{s}ds^{\prime}e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime})\leq\left(\int_{0}^{t}ds\,e^{-2\nu_{1}(t-s)}\right)^{1/2}\left(\int_{0}^{t}ds\,\Big{|}\int_{0}^{s}ds^{\prime}\,e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime})\Big{|}^{2}\right)^{1/2}.$

The first term computed by direct calculation is
$\sqrt{\frac{1}{2\nu_{1}}(1-e^{-2\nu_{1}t})}\leq\sqrt{\frac{1}{2\nu_{1}}}$.
The second term is bounded by the integral
$\left(\int_{0}^{\infty}ds\,\Big{|}\int_{0}^{s}ds^{\prime}\,e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime})\Big{|}^{2}\right)^{1/2}$
which is non other than the $L^{2}$ norm of the convolution of the function
$f(t)=e^{-\nu_{2}t}$ with $g(t)$, i.e. $\|f\ast g\|_{2}$. Now according to
Young’s inequality for convolutions we have $\|f\ast
g\|_{r}\leq\|f\|_{p}\|g\|_{q}$ for $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$ and
$1\leq p,q,r\leq\infty$, and if we take $p=2$, $q=1$, and $r=2$ then we have
that the last term is bounded by $\frac{1}{\sqrt{2\nu_{2}}}\|g\|_{1}$ so we
have

$\displaystyle\int_{0}^{t}ds\,e^{-\nu_{1}(t-s)}\int_{0}^{s}ds^{\prime}e^{-\nu_{2}(s-s^{\prime})}g(s^{\prime})\leq\frac{1}{2\sqrt{\nu_{1}\nu_{2}}}\|g\|_{1}=\frac{1}{2}\sqrt{\frac{a}{c}}\|g\|_{1}.$

In similar fashion we can show that

$\displaystyle\int_{0}^{t}ds\int_{0}^{s}ds^{\prime}e^{-\frac{b}{2a}(t-s^{\prime})}g(s^{\prime})\leq\frac{a}{b}\|g\|_{1}.$

∎

We now proceed to prove a different Lemma for a Gronwall inequality which is
more tailored to the structure of (29), in the way that Lemma 1 was more
specific to the structure of (28). This is a second order differential
inequality that includes a convolution term and an exponential decay term in
the RHS.

$\ddot{y}+a\dot{y}+by\leq c\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds+de^{-\nu
t},\qquad t\geq 0,\quad a,b,c,d\geq 0.$ (39)

######  Lemma 3.

Let $y=y(t)$ be a nonnegative $C^{2}$-function satisfying the differential
inequality (39), with initial conditions $y(0)=y_{0}$ and $\dot{y}(0)=y_{1}$.
Assume also that we have the set

$\pazocal{D}:=\left\\{\mu>0:\mu^{3}-(a+\nu)\mu^{2}+(b+a\nu)\mu\leq
d_{*},\quad\mu<\min\\{a,\nu\\}\right\\}$ (40)

where $d_{*}=b\nu-c$. If $d_{*}>0$ then the set $\pazocal{D}$ is nonempty, and
if we define $\mu_{*}:=\sup\pazocal{D}$ we have the following estimates

$y(t)\leq
e^{-(a-\mu_{*})t}y_{0}+\frac{1}{a-2\mu_{*}}\left(e^{-\mu_{*}t}-e^{-(a-\mu_{*})t}\right)\left(y_{1}+(a-\mu_{*})y_{0}+\frac{d}{\nu-\mu_{*}}\right),\quad\text{if}\quad\mu_{*}<\frac{a}{2}$

and

$y(t)\leq
e^{-\frac{a}{2}t}y_{0}+\left(y_{1}+\frac{a}{2}y_{0}+\frac{d}{\nu-\frac{a}{2}}\right)te^{-\frac{a}{2}t},\quad\text{if}\quad\mu_{*}\geq\frac{a}{2}.$

###### Proof.

The approach that we take is to consider the functional

$L(t):=\dot{y}(t)+d_{1}y(t)+d_{2}\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds+d_{3}e^{-\nu
t},$

and try to find values for the coefficients $d_{1},d_{2},d_{3}$ and some
$\mu>0$ so that $L(t)$ decays exponentially with rate $\mu>0$ i.e.
$\dot{L}(t)\leq-\mu L(t)$. If we can find such $\mu>0$, then (keeping in mind
that $y(t)\geq 0$) we have

$\displaystyle\dot{y}(t)+d_{1}y(t)\leq e^{-\mu t}L(0)=e^{-\mu
t}(y_{1}+d_{1}y_{0}+d_{3}).$

Solving this first order inequality by using integrating factor, we get that

$y(t)\leq e^{-d_{1}t}y_{0}+\frac{1}{d_{1}-\mu}\left(e^{-\mu
t}-e^{-d_{1}t}\right)(y_{1}+d_{1}y_{0}+d_{3})\quad\text{for}\quad
d_{1}\neq\mu$

and that

$y(t)\leq
e^{-d_{1}t}y_{0}+(y_{1}+d_{1}y_{0}+d_{3})te^{-d_{1}t}\quad\text{for}\quad
d_{1}=\mu.$

Of course, if there are many different choices for $d_{1},d_{2},d_{3},\mu>0$
the ideal scenario is to pick those that maximize the decrease rates for both
$d_{1}$ and $\mu$. In order to compute $\dot{L}(t)$ we make use the fact that

$\frac{d}{dt}\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds=y(t)-\nu\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds.$

Let us now compute $\dot{L}(t)+\mu L(t)$ and set it $\leq 0$ to derive the
system of all conditions that the parameters need to satisfy so that we have
dissipation. Indeed, we have

$\displaystyle\dot{L}(t)+\mu
L(t)=\ddot{y}+(d_{1}+\mu)\dot{y}+(d_{2}+d_{1}\mu)y+d_{2}(\mu-\nu)\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds+d_{3}(\mu-\nu)e^{-\nu
t}\leq 0.$

We use inequality (39) to identify the set of conditions for which
$\dot{L}(t)+\mu L(t)\leq 0$. This yields the following system of equalities &
inequalities that need to be satisfied all at the same time, i.e.

$\displaystyle d_{1}+\mu$ $\displaystyle=a$ (41a) $\displaystyle
d_{2}+d_{1}\mu$ $\displaystyle\leq b$ (41b) $\displaystyle d_{2}(\mu-\nu)$
$\displaystyle\leq-c$ (41c) $\displaystyle d_{3}(\mu-\nu)$
$\displaystyle\leq-d$ (41d)

We now have to check the validity of conditions in system (41) and then pick
the parameters so we get the maximum decay rates. We first choose $\mu$ such
that

$0<\mu<\min\\{a,\nu\\}.$ (42)

We see that (41d) is a condition independent of (41a)-(41c), since the $d_{3}$
term is not involved in any of (41a)-(41c), i.e. we can make the following
choice for $d_{3}$ after we find admissible $\mu$ that solve the system

$d_{3}=\frac{d}{\nu-\mu}.$

We can now solve (41a) in $d_{1}$ ($d_{1}=a-\mu$) and substitute into (41b) to
get

$d_{2}\leq b-d_{1}\mu=\mu^{2}-a\mu+b.$ (43)

What is left to do, is to find values of $\mu>0$ s.t. (43) and (41c) are
simultaneously satisfied. This is true, if we can find $\mu$ that satisfies
the inequality

$\mu^{2}-a\mu+b\geq\frac{c}{\nu-\mu},$ (44)

then by (41c) we can choose any $d_{2}$ such that

$\mu^{2}-a\mu+b\geq d_{2}\geq\frac{c}{\nu-\mu}.$

The two conditions (42) and (44) are sufficient conditions for the system
(41a)-(41d) to have admissible solutions. Practically, (44) leads to the cubic
inequality contained in the definition of set $\pazocal{D}$. Notice that when
$b\nu>c$, then some $\mu>0$ that solves both (42) and (44) exists.

The last step, after we make sure that there is some $\mu>0$ so that (42) and
(44) are satisfied, is to pick the one $\mu$ that maximizes the decay rate.
For that, we define the set $\pazocal{D}$ like we did in (40) and the supremum
$\mu_{*}$ of that set. If $\mu_{*}\geq\frac{a}{2}$ then the optimal decay rate
is $\lambda=\frac{a}{2}$, otherwise, if $\mu_{*}<\frac{a}{2}$ then the optimal
rate is $\lambda=\mu_{*}$. ∎

The following uniform estimate can be shown with the help of Lemma 3

######  Corollary 2.

Let $y=y(t)$ be a nonnegative $C^{2}$-function satisfying the differential
inequality (39), with initial conditions $y(0)=y_{0}$ and $\dot{y}(0)=y_{1}$
and that condition $b\nu>c$ is satisfied. Then we have the following bound for
$y(t)$

$y(t)\leq
e^{-at}y_{0}+\frac{1}{a}(1-e^{-at})\left(y_{1}+ay_{0}+\frac{d}{\nu}\right),\qquad
t\geq 0.$

In particular, we have the uniform bound

$y(t)<y_{0}+\frac{|y_{1}|}{a}+\frac{d}{a\nu},\qquad t\geq 0.$

###### Proof.

The proof employs the estimates for $L(t)$ that we proved in Lemma 3, only now
we restrict to the case of a functional $L(t)$ with $\mu=0$ (no decay rate),
so in effect we use the functional

$L(t)=\dot{y}(t)+ay(t)+\frac{c}{\nu}\int_{0}^{t}e^{-\nu(t-s)}y(s)\,ds+\frac{d}{\nu}e^{-\nu
t}.$

∎

##  6 Proof of Main Theorems

We are now in position to prove the main theorems of this paper.

###### Proof of Theorem 1.

The proof essentially uses Proposition 6, the Gronwall lemmas proved in
Section 5, and a continuity argument.

Step A: Let $T>0$ be an arbitrary number so that
$\pazocal{A}(v(t))>\delta_{0}$ for all $t\in[0,T)$ (we know that such a $T>0$
exists by continuity of $\pazocal{A}(v)$). Then, we have that (28) holds on
that interval $[0,T)$, and subsequently we may use Lemma 1 with

$a=1,\qquad b=\frac{\gamma}{\chi},\qquad
c=\frac{2k}{\chi}\psi_{m}\delta_{0},\qquad
g(t)=\frac{4}{\chi^{2}}\left(D(s)^{2}+D(v)^{2}\max\limits_{1\leq i\leq
N}|s_{i}|^{2}\right).$

In order to control $g(t)$ in terms of $\pazocal{S}(t)$ we have

$D(s)^{2}=\max\limits_{i,j}|s_{i}-s_{j}|^{2}\leq(|s_{l}|+|s_{m}|)^{2}\leq
2(|s_{l}|^{2}+|s_{m}|^{2})\leq 2N\pazocal{S}(t),$

where $l,m$ are the indices that realize the
$\max\limits_{i,j}|s_{i}-s_{j}|^{2}$, and also

$\max\limits_{i}|s_{i}|^{2}\leq N\pazocal{S}(t),\quad\text{which both lead
to}\quad g(t)\leq\frac{16N}{\chi^{2}}\pazocal{S}(t).$

Now if we integrate $g(t)$ and use Proposition 2, we have that

$\int_{0}^{\infty}g(t)\,dt\leq\frac{16N}{\chi^{2}}\int_{0}^{\infty}\pazocal{S}(t)\,dt\leq\frac{8Nk}{\chi\gamma}\left(\frac{\chi}{2}\pazocal{E}(0)+\frac{1}{k}\pazocal{S}(0)\right).$

We have that for all $t\in[0,T)$ the estimates (37)–(38) in Lemma 1 hold.

Step B: We now define the set

$\pazocal{S}:=\\{T>0:\pazocal{A}(v(t))>\delta_{0},\quad\forall
t\in[0,T)\\},\qquad T_{*}:=\sup\pazocal{S}.$

We have shown in Step A due to the continuity property that the set
$\pazocal{S}$ is non-empty. We claim that $T_{*}=\infty$ so that the estimates
above hold true for all $t\geq 0$. Indeed, assume that $T_{*}<\infty$, then we
have that $\lim\limits_{t\to T_{*}^{-}}\pazocal{A}(v(t))=\delta_{0}$ or
equivalently $\lim\limits_{t\to T_{*}^{-}}D(v(t))^{2}=2-2\delta_{0}$. Now from
Corollary 1 we have that

$D(v(t))^{2}\leq 2\pazocal{C}_{0}<2-2\delta_{0},\qquad\forall t\in[0,T_{*}).$

Taking the limit $\lim\limits_{t\to T_{*}^{-}}D(v(t))^{2}$ leads to the
contradiction $2-2\delta_{0}<2-2\delta_{0}$. Therefore $T_{*}=\infty$, and the
estimates from Step A for $D(v)$ hold. Finally, the vanishing of velocity
diameters follows from Lemma 2. ∎

###### Proof of Theorem 2.

The proof follows the lines of the proof of Theorem 1.

Step A: Let $T>0$ be an arbitrary number so that
$\pazocal{A}(v(t))>\delta_{0}$ for all $t\in[0,T)$ (we know that such a $T>0$
exists by continuity of $\pazocal{A}(v)$). Then, we have that (29) holds on
that interval $[0,T)$, and subsequently we may use Lemma 3 with

$a=\frac{\gamma}{\chi},\qquad b=\frac{2k}{\chi}\psi_{m}\delta_{0},\qquad
c=\frac{12k^{2}\psi_{M}}{\gamma\chi},\qquad
d=\frac{4}{\chi^{2}}\left(D(s_{0})^{2}+2\max\limits_{1\leq i\leq
N}|s_{i0}|^{2}\right)$

where the set $\pazocal{D}$ in Lemma 3 takes the form of (21). Since $(H1)$
holds, we have that $\pazocal{D}$ is non-empty and $\mu_{*}>0$. Then for all
$t\in[0,T)$ we have the estimates

$\displaystyle
D(v)^{2}\leq\left\\{\begin{array}[]{ll}e^{-d_{1}t}D(v_{0})^{2}+\frac{1}{d_{1}-\mu_{*}}\left(e^{-\mu_{*}t}-e^{-d_{1}t}\right)\left(2D(v_{0})|\dot{D}(v_{0})|+d_{1}D(v_{0})^{2}+\frac{d}{d_{1}}\right),\quad\text{for}\quad\mu_{*}<\frac{\gamma}{2\chi},\\\
\\\
e^{-\frac{\gamma}{2\chi}t}D(v_{0})^{2}+\left(2D(v_{0})|\dot{D}(v_{0})|+\frac{\gamma}{2\chi}D(v_{0})^{2}+\frac{2d\chi}{\gamma}\right)te^{-\frac{\gamma}{2\chi}t},\quad\text{for}\quad\mu_{*}\geq\frac{\gamma}{2\chi},\end{array}\right.$

with $d_{1}=\frac{\gamma}{\chi}-\mu_{*}$.

Step B: We follow the exact same steps in Step B, as in the proof of Theorem
1. ∎

## References

  * [1] Attanasi, A., Cavagna, A., Del castello, L., Giardina, I., et al. _Information transfer and behavioural Inertia in starling flocks._ Nat. Phys. 10, 9 (2014), 615–698.
  * [2] Barbalat, I. _Systemes dequations differentielles d’oscillations non Lineaires._ Rev. Math. Pures Appl. 4, (1959), 267–270.
  * [3] Bellomo, N., Ha, S.-Y., and Outada, N. _Towards a mathematical theory of behavioral swarms._ ESAIM - Control Optim. Calc. Var. 26, 125 (2020).
  * [4] Bellomo, N., Degond, P., and Tadmor, E. _Active Particles, Volume 1: Advances in Theory, Models, and Applications in Modeling and Simulation in Science, Engineering and Technology_ Birkhauser, 2020.
  * [5] Benedetto, D., Buttà, P., and Caglioti, E. _Some aspects of the inertial spin model for flocks and related kinetic equations._ Math. Models Methods Appl. Sci. 30, 10 (2020), 1987–2022.
  * [6] Cavagna, A., Del Castello, L., Giardina, I., Grigera, T., et. al. _Flocking and Turning: a new Model for Self-organized Collective Motion._ J. Stat. Phys. 158, (2015), 601–627.
  * [7] Chaté, H., Ginelli, F., Grégoire, G., Peruani, F., and Raynaud, F. _Modeling Collective motion : variations on the Vicsek model._ Eur. Phys. J. 64, (2008), 451–456.
  * [8] Choi, S.-H., and Ha, S.-Y. _Emergence of flocking for a multi-agent system moving with constant speed._ Commun. Math. Sci. 14, 4 (2016), 953–972.
  * [9] Choi, S.-H., and Ha, S.-Y. _Interplay of the unit-speed constraint and time-delay in Cucker-Smale flocking._ J. Math. Phys. 59, 8 (2018), 082701.
  * [10] Choi, Y.-P., and Haskovec, J. _Hydrodynamic Cucker-Smale model with normalized communication weights and time delay._ SIAM J. Math. Anal. 51, 3 (2019), 2660–2685.
  * [11] Choi, Y.-P., Ha, S.-Y., Jung, S., and Slemrod, M. _Kuramoto oscillators with inertia : a fast-slow dynamical systems approach._ Quart. Appl. Math. 73, 3 (2015), 467–482.
  * [12] Choi, Y.-P., Ha, S.-Y., and Morales, J. _Emergent dynamics of the Kuramoto ensemble under the effect of inertia._ Discrete Contin. Dyn. Syst. 38, 10 (2018), 4875–4913.
  * [13] Choi, Y.-P., Ha, S.-Y., and Noh, S. _Remarks on the nonlinear stability of the Kuramoto model with inertia._ Quart. Appl. Math. 73, 2 (2015), 391–399.
  * [14] Choi, Y.-P., Ha, S.-Y., and Yun, S.-B. _Complete synchronization of Kuramoto oscillators with finite inertia._ Phys. D 240, 1 (2011), 32–44.
  * [15] Choi, Y.-P., Ha, S.-Y., and Yun, S.-B. _Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto-Daido model with inertia._ Netw. Heterog. Media 8, 4 (2013), 943–968.
  * [16] Choi, Y.-P., Li, Z., Ha, S.-Y., Xue, X., and Yun, S.-B. _Complete entrainment of Kuramoto oscillators with inertia on networks of gradient-like flow._ J. Differ. Equations 257, 7 (2014), 2591–2621.
  * [17] Cristiani, E., Menci, M., Papi, M., and Brafman, L. _An all-leader agent-based model for turning and flocking birds._ J. Math. Biol. 45, 4 (2021), 1–22.
  * [18] Cucker, F., and Smale, S. _Emergent behavior in flocks._ IEEE Trans. Automat. Control 52, 5 (2007), 852–862.
  * [19] Cucker, F., and Smale, S. _On the mathematics of emergence._ Jpn. J. Math. 2, 1 (2007), 197–227.
  * [20] Degond, P., Frouvelle, A., and Merino-Aceituno, S. _A new flocking model through body attitude coordination._ Math. Models Methods Appl. Sci. 27, 6 (2017), 1005–1049.
  * [21] Degond, P., and Motsch, S. _Continuum limit of self-driven particles with orientation interaction._ Math. Models Methods Appl. Sci. 18, 1 (2008), 1193–1215.
  * [22] Degond, P., and Motsch, S. _Large scale dynamics of the persistent turning walker model of fish behavior._ J. Stat. Phys. 131, 6 (2008), 989–1021.
  * [23] Erban, R., Haskovec, J., and Sun, Y. _A Cucker-Smale model with noise and delay._ SIAM J. Appl. Math. 76, 4 (2016), 1535–1557.
  * [24] Ginelli, F. _The Physics of the Vicsek model._ Eur. Phys. J. Spec. Top. 225, 11-12 (2016), 2099–2117.
  * [25] Ha, S.-Y., Ha, T., and Kim J.-H. _Emergent behavior of a Cucker-Smale type particle model with nonlinear velocity couplings._ IEEE Trans. Automat. Control 55, 7 (2010), 1679–1683.
  * [26] Ha, S.-Y., Kim, D., Kim, D., and Shim, W. _Flocking Dynamics of the Inertial Spin Model with Multiplicative Communication Weight._ J. Nonlinear Sci. 29, (2019), 1301–1342.
  * [27] Ha, S.-Y., Kim, D., Lee, J., and Noh, S. E. _Emergent dynamics of an orientation flocking model for multi-agent system._ Discrete Contin. Dyn. Syst. 40, 4 (2020), 2037–2060.
  * [28] Ha, S.-Y., Ko, D., and Zhang, Y. _Remarks on the critical coupling strength for the Cucker-Smale model with unit speed._ Discrete Contin. Dyn. Syst. 38, 6 (2018), 2763–2793.
  * [29] Haskovec, J., and Markou, I. _Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime._ Kinet. Relat. Models 13, 4 (2020), 795–813.
  * [30] Haskovec, J., and Markou, I. _Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays._ Math. Biosci. Eng. 17, 5 (2020), 5651–5671.
  * [31] Ha, S.-Y., and Li, Z. _Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia._ Math. Meth. Mod. Appl. Sci. 26, (2016), 357–382.
  * [32] Kuramoto, Y. _Self-entrainment of a population of coupled non-linear oscillators_ In: Araki H. (eds) International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes in Phys. 39, 1975, 420–422.
  * [33] Liu, L., Huang, L., and Wu, J. _Flocking and invariance of velocity angles._ Math. Biosci. Eng. 13, 2 (2016), 369–380.
  * [34] Lohe, M. A. _Quantum synchronization over quantum networks._ J. Phys. A: Math. Theor. 43, 465301 (2010), 20 pp.
  * [35] Markou, I. _Collision avoiding in the singular Cucker-Smale model with nonlinear velocity couplings._ Discrete Contin. Dyn. Syst. 38, 10 (2018), 5245–5260.
  * [36] Motsch, S., and Tadmor, E. _A new model for self-organized dynamics and its flocking behavior._ J. Stat. Phys. 144, 5 (2011), 923–947.
  * [37] Reynolds, C. W. _Flocks, herds and schools: A distributed behavioral model,_ In : ACM SIGGRAPH Computer Graphics 21, 4 (1987), 25–34.
  * [38] Toner, J., and Tu, Y. _Flocks, herds, and schools: A quantitative theory of flocking._ Phys. Rev. E 58, 4 (1998), 4828–4858.
  * [39] Yang, X., and Marchetti, M. C. _Hydrodynamics of Turning Flocks._ Phys. Rev. Lett. 115, 25 (2015), 5 pp.
  * [40] Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., and Shochet, O. _Novel Type of Phase Transition in a System of Self-Driven Particles._ Phys. Rev. Lett. 75, 6 (1995), 1226–1229.
  * [41] Vicsek, T., and Zafeiris, A. _Collective motion._ Phys. Rep. 517, 3-4 (2012), 71–140.
  * [42] Winfree, A. T. _Biological rhythms and the behavior of populations of coupled oscillators._ J. Theoret. Biol. 16, 1 (1967), 15–42.

(Ioannis Markou)
Institute of Applied and Computational Mathematics (IACM-FORTH)
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