Interplay of a unit-speed constraint and time-delay in the flocking model with internal variables

1.   Introduction

Emergent behaviors of interacting multi-agent systems are often observed in our daily life. For example, aggregation of bacteria ^[34], flocking of migrating birds ^[17], schooling of moving fish ^[18,33], synchronization of fireflies and pacemaker cells ^[6,19,38], etc. For a brief introduction to this subject, we also refer to the articles and books ^{[1,4,20,29,30,32,36,37]}. In this work, we are interested in self-propelled flocking dynamics in which all agents move at a common velocity with limited surroundings and simple rules. After the seminal work done by Vicsek ^[35], many mathematical models describing flocking behavior have been widely studied in the mathematics community. Among them, the Cucker–Smale (CS) model ^[17] is one of the most successful models that represents flocking which has been studied from various perspectives, to name a few, local interactions ^[27], kinetic descriptions ^[7,24], hydrodynamic descriptions ^[21,26], stochastic descriptions ^[8], time-delay ^[10,13,14], etc.

In ^[23], Ha and Ruggeri proposed a CS-type flocking model with internal variable, which was called a temperature variable in their context. They considered the standard balance laws (mass, momentum, energy conservation) for the finitely many mixtures of spatially homogeneous ideal gases, together with specific constitutive equations suggested in ^[31] consistent with the entropy principle in thermodynamics. To fix the idea, let $x_i, v_i, T_i$ be the position, velocity, and temperature of the $i$-th flocking agent, respectively. Then, the thermodynamic Cucker-Smale model (TCS in short) is given by the following second-order ordinary differential equations (ODEs) for position–velocity–temperature variables $\{(x_i, v_i, T_i)\}_{i = 1}^N$:

where $\kappa_1$ and $\kappa_2$ are nonnegative scale parameters of the velocity and temperature communication weights, respectively. As in the CS model, the authors in ^[23] assumed that $\phi, \zeta:[0, \infty)\rightarrow [0, \infty)$ are nonnegative, locally Lipschitz, and monotonically decreasing, so that (1.1) is locally well-posed and the interaction becomes weaker as the distance between the two agents increases. However, it might be questionable whether the fact that

is natural in terms of a flocking model. Therefore, when we think of the TCS-type models as the flocking models, we would like to suggest another interpretation for the temperature variable $T_i$. For instance, we may interpret $1/T_i$ as a time-varying measure of how attractive each bird is relative to other birds in the population. Then, each bird decides which direction to accelerate in proportion to the velocities and attractiveness of others, even for birds with the same velocity, and it becomes a matter of finding the initial conditions of all birds' velocities and attractiveness to converge to the same value. However, out of respect for the first authors' expressions, we decided to continue to refer to it as the TCS model throughout the paper.

Meanwhile, recent experiments on starling flocks ^[5,9] indicate that the speed fluctuations of birds are very small during their flights, demonstrating the need for us to study constant speed flocking models as ^[35]. Since the velocities of CS and TCS models converge to the same value under well-prepared initial conditions, the CS model and the TCS model can be said to have similar asymptotic behavior to their constant speed counterparts. Recently, ^[2] studied a new TCS-type flocking model in which the speed of each particle is constant. The main idea of ^[2] to create the new model was the same idea that led to the unit speed CS model ^[12] from the CS model: replace the right-hand side of (1.1b) by its orthogonal projection onto $v_i$, i.e.,

Note that because it is an orthogonal projection to $v_i$, the $v_i$ term disappears from the original TCS model, and the only interaction that appears to be asymmetric remains. As a result, the TCS model with a unit-speed constraint was given as

where the assumptions for $\kappa_1$, $\kappa_2$, $\phi$, and $\zeta$ are the same as for the system (1.1), and $\mathbb{S}^{d-1}$ is the $(d-1)$-dimensional unit sphere embedded in $\mathbb{R}^d$, i.e.,

In addition to ^[2] and ^[12], there have been several mathematical studies on CS-type models with constant speed constraint. For example, ^[15] provided a sufficient condition to exhibit a bi-cluster flocking of unit speed CS ensemble, and ^[22] found a critical coupling strength to exhibit an asymptotic flocking for the same model. In particular, ^[11] conducted a study on a modified unit speed CS model which also considers the constant time-delay effects. Such time-delay effects are in fact prevalent in both physical and biological systems due to the fundamental constraint that information transmission cannot occur instantaneously during mutual communication processes ^[16,25]. This limitation on the speed of communication propagation has found significant application within the mathematical biology literature.

In this paper, we study the unit speed TCS model with nonconstant time-delays. More precisely, we define $\tau_{ji}(t)$ as the time-delay for the $i$-th agent to detect a signal from the $j$-th agent at time $t\geq 0$, where we assume $\tau_{ii}(t) = 0$ for all $i\in [N]$ to avoid a self-processing time-delay. Furthermore, we assume $\tau_{ji}(t): [0, \infty)\to [0, \infty)$ nonnegative, continuous, and uniformly bounded by a constant $\tau$ for each pair $(i, j)\in [N]^2$, i.e.,

Taking account these time-delay effects to Eq (1.2), we propose the following unit speed TCS model with time-delay (in short TCSUT):

For this flocking model, our main concern in this paper is to find sufficient conditions on the communication weights and initial data that guarantee the asymptotic flocking to occur. To do this, we first define the asymptotic flocking phenomenon rigorously.

Definition 1.1. (Asymptotic flocking) Let $Z = \{(x_i, v_i, T_i)\}_{i = 1}^N$ be a solution to the system (1.3). The configuration $Z$ exhibits asymptotic flocking if

Note that the nonconstant time-delay in (1.3) means that each $i$-th bird receives information about the other birds at different points in time to determine its velocity and temperature. Therefore, if we want to analyze the diameters

to show the asymptotic flocking in Definition 1.1, there are some additional technical difficulties compared to the constant time-delay models. Since it may not be enough to simply use the diameters in (1.4) to fully control the right-hand side of (1.3), we need to measure the upper bound of the error that each velocity and temperature can have due to the time-delay effect. For simplicity, we set

and for every $Z\in \{X, V, T\}$, we define the delayed diameter $D_Z^{\tau}$ and perturbation $\Delta_Z^\tau$ as

which satisfy

Then, the right-hand side of (1.3) can be controlled by using Eq (1.5), since

and the main result of this paper is to find sufficient conditions for $D_X$ to be uniformly bounded and $D_V, D_T$ to converge to zero.

To this end, we will consider two different approaches: constructing a differential inequality for diameters $(D_X, D_V, D_T)$ and constructing a differential inequality for delayed diameters $(D_X^\tau, D_V^\tau, D_T^\tau)$. The analysis of diameters is similar to the method used in ^[3] and other related works, but the analysis of delayed diameter is, to the best of the authors' knowledge, the first attempt on studies of flocking models with time-delays.

The rest of this paper is organized as follows. In Section 2, we provide some basic properties and previous flocking estimate for the unit speed TCS model (1.2) without time-delay. In Section 3, we present several preparatory lemmas which are crucially used to guarantee the global well-posedness of the system (1.3) and also used to construct differential inequalities for the delayed diameters in Section 4. In Section 4, we derive a system of differential inequalities for the delayed-diameters $(D_X^\tau, D_V^\tau, D_T^\tau)$ and present the first main result. In Section 5, we present a system of differential inequalities for the diameters $(D_X, D_V, D_T)$ and the second main result. We also present several numerical experiments in Section 6 to demonstrate our theoretical results. Finally, Section 7 is devoted to summarizing the main results of this paper and some discussion on the remaining issues to be investigated in a future work. In Appendix A and B, we provide the detailed proof of Lemma 3.1 and Lemma 4.1, respectively.

Notation: Throughout this paper, we employ the following notation for simplicity.

2.   Preliminaries

In this section, we briefly review several basic properties and previous results of the TCS model with a unit speed constraint obtained in ^[2]. Readers with sufficient background knowledge in this subject may skip this section.

2.1.   The unit speed TCS model

In this subsection, we introduce the unit speed TCS model in the absence of time-delay. Recall that the unit speed TCS model was given as

Even if we consider $T_i$ as an internal variable other than temperature, the 1st and 2nd laws of thermodynamics still hold for the original TCS model, since it is derived from the standard balance laws for spatially homogeneous gas mixtures. Similarly, in the model with the constant speed condition added, the law of energy conservation and entropy principle hold for $T_i$'s, which are as follows.

Proposition 2.1. Let $(X, V, T)$ be a solution to the system (2.1). Then, the following assertions hold:

1. (Energy conservation) The total internal energy $\sum_{i = 1}^{N}T_i$ is conserved.

2. (Entropy principle) The total entropy $\mathcal{S}: = \sum_{i = 1}^{N}\ln T_i$ is monotonically increasing.

However, these laws can be considered only as long as each $T_i$ exists as a positive real number, since the solution of (2.1) blows up if any $T_i$ approaches zero. Therefore, in order to see how the solution of the equation behaves for $t\to\infty$, it must be ensured that all $T_i$'s cannot be smaller than some positive real number, which can be seen in the following proposition.

Proposition 2.2. Let $\{(x_i^0, v_i^0, T_i^0)\}_{i = 1}^N\subset \mathbb{R}^d\times \mathbb{S}^{d-1}\times (0, \infty)$ be an arbitrary initial configuration. Then, the equation (2.1) admits a unique global solution on $t > 0$ and

What Proposition 2.2 means is that we can always be guaranteed that the extreme values of the temperatures will approach each other, even though they may not actually converge to the same value. Therefore, a natural question to ask is whether we can expect a similar behavior for velocity vectors, and the answer to this question can be found in the following proposition.

Proposition 2.3. Let $(X, V, T)$ be a solution to (2.1) subject to initial data $\{(x_i^0, v_i^0, T_i^0)\}_{i = 1}^N$ satisfying

Then, the function

is monotonically increasing in $t$.

2.2.   Previous results

In this subsection, we review previous flocking estimates for (2.1) obtained from ^[2] and discuss whether they can be improved. We begin with a system of differential inequalities in terms of $D_X, D_V$ and $D_T$.

Proposition 2.4. (Differential inequalities for diameters) Let $(X, V, T)$ be a solution to the system (2.1) subject to initial data $\{(x_i^0, v_i^0, T_i^0) \}_{i = 1}^N$ satisfying $A(0) > 0$. Then, we have

In particular, the function

is monotonically decreasing with respect to $t$, where

Since $\mathcal{L}(D_X, D_V)$ is monotonically decreasing, $\Phi(D_X)$ has a uniform in time upper bound determined by the initial data. Therefore, if this upper bound is smaller than $\Phi(\infty)\in [0, \infty]$, we know that the diameter $D_X$ is uniformly bounded in time. Then, Proposition 2.4 determines the exponential decay rate of the diameters $D_V$ and $D_T$.

Proposition 2.5. (Asymptotic flocking) Let $(X, V, T)$ be a solution to the system (2.1) subject to initial data $\{(x_i^0, v_i^0, T_i^0) \}_{i = 1}^N$ satisfying

Then, we have

where $D_X^\infty$ is the unique positive number satisfying

Remark 2.1. We make a few remarks about the optimality of Proposition 2.5.

(1) In ^[2], the author provided two sufficient frameworks for the asymptotic flocking of (2.1). One is the result stated in Proposition 2.5, which uses the Lyapunov functional $\mathcal{L}_+$, and there was another result using 'the bootstrapping argument' according to their context. However, with more precise calculations we can show that the result is weaker under stronger conditions than Proposition 2.5.

(2) In fact, according to the proof of Proposition 2.4, one can obtain

Then, the function

is monotonically decreasing with respect to $t$, and we have

whenever the initial data satisfies

In Section 3, we will show that the generalizations of Proposition 2.2 and Proposition 2.3 hold in the time-delayed model (1.3). These will be crucially used to prove the well-posedness of (1.3) and to find some flocking estimates corresponding to Proposition 2.5.

3.   Basic properties of the time-delayed model

In this section, we present the global well-posedness and some basic properties of the system (1.3). To fix the idea, consider a following time-delayed ODE:

Then, $x(t-\tau(t))$ does not play a crucial role to the well-posedness of the solution. Instead, it is more appropriate to view it as an independently given function that does not depend on the value of $x(t)$. Therefore, the local well-posedness of the ODE can be obtained when $f(y, z)$ is Lipschitz in $y$, continuous in $z$, and $t\mapsto x(t-\tau(t))$ is continuous. Because of the above reasons, it is clear that the global well-posedness of the system (1.3) can be obtained by the invariance of $\|v_i\|$ and the boundedness of $1/T_i$ for each $i\in [N]$, and the invariance of each $\|v_i\|$ is given by the relation

Then, due to the uniqueness of the solution and the unit-speed constraint for initial data, we can rewrite the system (1.3) as the following simplified form:

In ^[10], the authors proved the monotonicity of the maximum and minimum temperatures for the TCS model with time-delay, which does not have the invariance of speeds as Eq (3.1). Although not rigorously proved in previous works, their basic idea was to use the following lemma to the minimum and maximum temperatures. We here present a rigorous proof for the completeness of this paper and for the use of this lemma in future work.

Lemma 3.1. Let $f\in C(\mathbb{R}; \mathbb{R})$, and define $F = \mathcal{T}[f]:\mathbb{R}\to \mathbb{R}$ as

Then, the following assertions hold:

1. If $f(t_0) > F(t_0)$ for some $t_0\in\mathbb{R}$, there exists $\delta > 0$ such that

2. $F$ is monotonically increasing if $D_+f(t_*)\geq 0$ for all $t_*\in \mathbb{R}$ satisfying $F(t_*) = f(t_*)$.

3. If $f\in \text{Lip}_L(\mathbb{R}, \mathbb{R})$ for some $L > 0$, then $F\in \text{Lip}_L(\mathbb{R}, \mathbb{R})$.

Proof. Since the proof is lengthy and technical, we leave it to Appendix A. □

Remark 3.1. To preempt some possible misconceptions about the relation between $f$ and $\mathcal{T}[f]$, we present the following examples.

1. Convergence of $F = \mathcal{T}[f]$ does not imply the convergence of $f$. For example, if $f(t) = \cos\left(\frac{2\pi t}{\tau}\right)$, the function $F = \mathcal{T}[f]$ is a constant function $F\equiv -1$. In fact, if $F$ is monotonically increasing, we have

2. There may not be a 'first point' that does not satisfy the condition in Lemma 3.1 (2). For example, if $f(t) = -\max\left\{0, t\right\}^2$, the function $f$ is monotonically decreasing. Then, $\mathcal{T}[f]\equiv f$ and $f'(t) = -2\max\left\{0, t\right\}\geq 0$ for only $t\leq 0$, and therefore

Using Lemma 3.1, one can prove monotonic increase in minimum temperature and monotonic decrease in maximum temperature, similar to the original models without delay ^[2,23]. More precisely, we can prove that $T_m^\tau$ is monotonically increasing and $T_M^\tau$ is monotonically decreasing when the initial values of both functions are positive real numbers.

Lemma 3.2. (Monotonicity of $T_m^\tau$ and $T_M^\tau$) Let $(X, V, T)$ be a solution to Eq (3.1), where the initial data

is continuous for every $i\in[N]$. Then, $T_m^\tau$ and $-T_M^\tau$ are monotonically increasing, i.e.,

Proof. Let us denote by

Then, the constraint for the initial data can be written as $F(0) > 0$, and we want to prove $F(t)\geq F(0)$ for every $t \geq 0$. If $\varepsilon > 0$ and $t_0$ is the first time satisfying $F(t) = F(0)-\varepsilon > 0$, then whenever $F(t_*) = f(t_*)$ in $t_*\in [0, t_0]$, we can consider the maximal index set $\mathcal{I}\subset [N]$ such that

Then, since each $T_k(t)$ is differentiable in $t$, we have

and by taking the minimum in $k$, one can obtain

Therefore, the Dini derivative $D_+f$ at $t = t_*$ satisfies

where we used $T_k^{\tau_{kj}}(t_*)\geq F(t_*) > F(0)-\varepsilon > 0$ in the last inequality. By using Lemma 3.1, the function $F$ is therefore monotonically increasing in $[0, t_0]$, which contradicts to $F(t_0) = F(0)-\varepsilon < F(0)$. Therefore, we have

In addition, by letting each $s\geq 0$ be a new starting time, this also shows that $F$ is monotonically increasing in $[0, \infty)$:

We can also obtain the monotonic decreasing property of $T_M^\tau$ by choosing $f = -T_M$ and $F = \mathcal{T}[f] = -T_M^\tau$. □

What we want to emphasize in this proof is that we assumed the first point $t_0$ where the value of $F$ is less than or equal to $F(0)-\varepsilon$ and then applied Lemma 3.1 on the interval $[0, t_0]$ to prove its contradiction. This is because, as we pointed out in Remark 3.1. (2), there may not be the first point at which the condition of Lemma 3.1(2) does not hold.

Then, by applying the classical Cauchy-Lipschitz theory, the global existence of the solution to (3.1) can be guaranteed from the invariance of the speed and the existence of positive lower bound for temperatures. However, the use of Lemma 3.1 goes beyond simply showing the monotonicity of extreme temperatures. To be specific, every open hemisphere $U\subset \mathbb{S}^{d-1}$ contains all $v_1(t), \ldots, v_N(t)$ uniformly in $t\geq 0$, whenever $U$ contains all initial velocities.

Lemma 3.3. Let $(X, V, T)$ be a solution to Eq (3.1), where the initial data

is continuous for every $i\in[N]$. If initial data satisfies

for some $u\in\mathbb{S}^{d-1}$, the function

is monotonically increasing.

Proof. We can employ a similar argument to Lemma 3.2. In this case, we set

If $\varepsilon > 0$ and $t_0$ is the first time satisfying $F(t) = F(0)-\varepsilon > 0$, then whenever $F(t_*) = f(t_*)$ in $t_*\in [0, t_0]$, we can consider the maximal index set $\mathcal{I}\subset [N]$ such that

Then, the Dini derivative $D_+f$ at $t = t_*$ satisfies

where we used $\langle u, v_k^{\tau_{kj}}(t_*)\rangle\geq F(t_*)\geq 0$ in the second inequality. By using Lemma 3.1, the function $F$ is monotonically increasing in $[0, t_0]$, which contradicts to $F(t_0) = F(0)-\varepsilon < F(0)$. Therefore, we have $F(t)\geq F(0)$ for all $t\geq 0$, and by letting each $s\geq 0$ be a new starting time, this also shows that $F$ is monotonically increasing in $[0, \infty)$. □

What should be noted here is that Lemma 3.3 alone cannot prove the desired flocking phenomenon, since the unit vector $u\in\mathbb{S}^{d-1}$ satisfying (3.2) is not unique if it exists. Instead, it is better to use an indicator that can show that individual velocities are getting closer to each other. The most intuitive way to achieve this is to assume that all initial velocities can be used as the vector '$u$' in Eq (3.2), so that all velocities approach each other. Under such initial conditions, the lemma below implies that the inner product $\langle v_i(t_1), v_j(t_2)\rangle$ is strictly positive for all $i, j\in [N]$ and $t_1, t_2\geq -\tau$.

Lemma 3.4. Let $(X, V, T)$ be a solution to (3.1), where the initial data

is continuous for every $i\in[N]$. If the initial data satisfies

then the function

is monotonically increasing in both $t_1$ and $t_2$.

Proof. Since $A^{\tau, \tau}(t_1, t_2) = A^{\tau, \tau}(t_2, t_1)$ for all $t_1, t_2\geq 0,$ we only need to show the monotonic increasing property in one variable, which is sufficient to verify

by taking each positive number as a new starting time as in Lemma 3.2 and Lemma 3.3. For every $s_1\in [-\tau, 0]$ and $i\in [N]$, we apply Lemma 3.3 to $u = v_i(s_1)$ and obtain

is monotonically increasing. Therefore, we have

which is the desired result. □

Before we close this section, we provide several estimates which will play crucial roles in Section 4. Recall that we defined the diameter and perturbation functions affected by the time-delay as follows: for $Z = (z_1, \ldots, z_N)\in \{X, V, T\}$,

Consequently, one can easily obtain $\Delta_Z^\tau\leq D_Z^\tau$ by definition, and Lemma 3.4 yields

provided that Eq (3.3) holds, i.e., $A^{\tau, \tau}(0, 0) > 0$. In addition, we can estimate perturbation functions $\Delta_V^\tau(t)$ and $\Delta_T^\tau(t)$ by using the integration of $\|\dot{z}_i\|$ in $[t-\tau, t]$.

Lemma 3.5. Let $(X, V, T)$ be a solution to (3.1), where the initial data

is Lipschitz continuous for every $i\in[N]$. Then, for every $t \geq 0$, we have

Proof. For given $Z\in \{V, T\}$ and $t\geq 0$, suppose we have

Then, by using the triangle inequality, one has

Below, we estimate the upper bound of $\|\dot{z}_i(u)\|$ in $u\geq 0$ one by one. For the velocity derivative, we have

where we used

in the first and the last inequality, respectively. Similarly, for the temperature derivative, we apply Lemma 3.2 to Eq (3.1) to obtain

where we used

in the last inequality. □

Remark 3.2. Since $\dot{z}_i^0(s)$ is not given in the differential equation (3.1), we need to use the Lipschitz constant of the initial data $\{z_i^0\}_{i = 1}^N$ to evaluate $\Delta_Z^\tau(t)$ in $t\in [0, \tau)$. In addition, the integration of $\|\dot{v}_i(u)\|$ in $u\in [s, t]$ is greater than or equal to the 'geodesic distance' between $v_i(s)$ and $v_i(t)$, since velocities are moving on the unit sphere $\mathbb{S}^{d-1}$. Therefore, it is possible to modify Lemma 3.5 to

Finally, the last thing we need to check is that the diameter functions $D_Z^\tau$ are absolutely continuous, so it satisfies the fundamental theorem of calculus. If we verify this, we can use the system of differential inequalities for $D_X^\tau, D_V^\tau$, and $D_T^\tau$, which we will prove in the next section, to find sufficient framework that guarantees the desired flocking phenomenon.

Lemma 3.6. Let $(X, V, T)$ be a solution to (3.1), where the initial data

is Lipschitz continuous for every $i\in[N]$. Then, $D_X^\tau, D_V^\tau$, and $D_T^\tau$ are globally Lipschitz in $t\geq 0$.

Proof. By using the uniform boundedness of $\frac{dz_i}{dt}$ obtained from Lemma 3.2, one can see that

is globally Lipschitz. Since the minimum of finitely many Lipschitz continuous functions is Lipschitz by the relation

we have the global Lipschitz continuity of the function

for every fixed $w$. In addition, by using Lemma 3.1(3), one can also obtain the Lipschitz continuity of

Now, let us denote a nonempty, compact set $C_Z(t)$ by

Then, the function

is globally Lipschitz, and the function

is also globally Lipschitz since $z_j(s_2)$ is Lipschitz in $s_2$ and the Lipschitz constant of the function $w\mapsto \max_{z\in C_Z(t)}\|z-w\|$ is not greater than 1. Therefore, all delayed diameter functions are globally Lipschitz, i.e.,

is globally Lipschitz in $t$. □

4.   Asymptotic behavior of the delayed diameters

In this section, we present a system of differential inequalities on delayed diameters $D_X^\tau, D_V^\tau$, and $D_T^\tau$ to deduce suitable sufficient frameworks for the asymptotic flocking of the system (3.1). Unlike the commonly used diameter $D_Z(t) = \max_{i, j\in [N]}\|z_i(t)-z_j(t)\|$, each delayed diameter does not specify at which point in time the distance between two vectors are evaluated. Therefore, in order to estimate the derivative of the delayed diameter, we need to know the behavior of the function given by a maximum value of a differentiable function over a certain range.

Lemma 4.1. Let $f:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ be a continuous function, and define

If there is a continuous function $\lambda:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ and $t_0\in\mathbb{R}$ such that

we have

Proof. Basically, the main idea to prove this lemma is finding a uniform upper bound of the Dini derivative of $h\mapsto f(t_1+h, t_2+h)$ for all points $(t_1, t_2)$ which maximizes $f$ on $\mathcal{S}(t)$. However, since we are considering the values of $f$ on an infinite set $\mathcal{S}(t)$ and analyzing the temporal evolution of their maximum, we want the Dini derivative to be uniformly bounded by a reasonable value in the neighborhood of every maximum point in $\mathcal{S}(t)$. This is why we required a continuous upper bound $\lambda$, and the technical difficulties of the proof can be resolved with some simple preparation and with the help of the Berge maximum theorem (see ^[28] for details). We leave the detailed proof to Appendix B. □

Now, we present the differential inequalities of the delayed diameters. For every $Z\in \{X, V\}$, $D_Z^\tau(t)^2$ can be represented as the maximum value of

Once we find a continuous upper bound $\lambda$ of the Dini derivative of the function $f_Z$ in the $(1, 1)$ direction, we can use the maximum of $\lambda$ over the maximizing set of $f_Z$ as the upper bound of the Dini derivative of $(D_Z^\tau)^2$. Therefore, by expressing the value of the upper bound $\lambda$ in terms of the value of $f_Z$, we can obtain a differential inequality for the delayed diameters.

Lemma 4.2. Let $(X, V, T)$ be a solution to (3.1), where the initial data

are Lipschitz continuous functions satisfying $\frac{dx_i^0}{dt} = v_i^0$ for all $i\in [N]$ and

In addition, we define a continuous function $S_X^\tau$ as

Then, the following differential inequalities hold:

1. $\frac{dD_X^\tau(t)}{dt}\leq D_V^\tau(t), \quad \mathit{\mbox{a.e.}}\quad t > 0$.

2. $\frac{dD_V^\tau(t)}{dt}\leq -\frac{\kappa_1\phi(S_X^\tau(t))\cos\delta}{T_M^\tau(0)}D_V^\tau(t)+\frac{3\kappa_1\phi(0)}{T_m^\tau(0)}\max_{s\in[t-\tau, t]}\Delta_V^\tau(s), \quad \mathit{\mbox{a.e.}}\quad t > \tau$.

3. $\frac{dD_T^\tau(t)}{dt}\leq -\frac{\kappa_2\zeta(S_X^\tau(t))}{T_M^\tau(0)^2}D_T^\tau(t)+\frac{3\kappa_2\zeta(0)}{T_m^\tau(0)^2}\max_{s\in[t-\tau, t]}\Delta_T^\tau(s), \quad \mathit{\mbox{a.e.}}\quad t > \tau$.

Proof. (1) For every $i, j\in N$ and $t_1, t_2\geq 0$, define

Then, for the continuous function $f_X: = \max\limits_{i, j\in [N]}f_{ij}$,

In particular, for every $(t_1, t_2)\in\mathcal{S}(t)$, we have

Therefore, by using Lemma 4.1, one can obtain

which implies

whenever $D_X^\tau$ is differentiable at $t$. Finally, from the Lipschitz continuity of $D_X^\tau$ in Lemma 3.6, we have the desired result.

(2) For the velocity diameter $D_V^\tau$, we set

Then, we have

where each $\mathcal{I}_{ijk}(t_1, t_2)$ satisfies

In particular, if $f_{ij}(t_1, t_2) = f_V(t_1, t_2)$ for some $i, j\in [N]$ and $t_1, t_2\geq 0$,

where we used $\langle v_i(t_1), v_j(t_2)\rangle \geq A^{\tau, \tau}(t_1, t_2) > 0$ and

in the last inequality. In addition, by using the inequality

we have

Therefore, by using Lemma 4.1, one can obtain

which implies the following inequalities whenever $D_V^\tau$ is differentiable at $t$:

Similar to the case of $D_X^\tau$, we reach the desired result due to the Lipschitz continuity of Lemma 3.6.

(3) In this case, we define

Then, whenever $f_{ij}(t_1, t_2) = f_T(t_1, t_2)$ and $t_1, t_2\leq t$, we get

By using the relations

the above inequality reduces to

Therefore, by using Lemma 4.1, one can show that

which implies

whenever $D_T^\tau$ is differentiable at $t$. □

Note that the differential inequality in Lemma 4.2 holds for (almost every) $t > \tau$. This is because we need to estimate the derivative of $f_Z$ at all points on $\mathcal{S}(t)$. Since the ODE (3.1) only gives the derivative of $\dot{z}_i(s)$ for $s > 0$, we require $t-\tau > 0$ in Lemma 4.2.

The next lemma allows us to present the upper bound of the objective function when the differential inequalities as Lemma 3.5 and Lemma 4.2 are given.

Lemma 4.3. Let $y:[0, \infty)\to[0, \infty)$ be a Lipschitz continuous function and $f:[0, \infty)\to\mathbb{R}$ be a continuous function. Assume that $y$ and $f$ satisfy

for some constants $a, d > 0$ and $\tau\geq 0$. If $\tau$ is sufficiently small to satisfy

and the following assertions hold.

(1) There are two constants $b, c > 0$ such that

(2) Whenever $b, c > 0$ satisfy (4.2), we have

Proof. Since $L$ is continuous and

one can find a positive number $c\in(0, a)$ satisfying $L(c)\leq \frac{1}{d}$, and one can also choose a sufficiently large $b$ to satisfy (4.2). Now, assume there exists the first time $t_*\in (2\tau, \infty)$ such that

Then, from the definition of $t_*$, we have

and we substitute (4.5) to (4.1) to obtain

Consequently, the Grönwall inequality (4.6) yields

where we used Eq (4.2) in the last inequality. On the other hand, by using Eq (4.7) to Eq (4.1b) at $t = t_*$, one can also obtain

which leads to a contradiction in Eq (4.4). Therefore, we have

for all $t\geq 0$, which implies the desired result for Eq (4.3).

□

Remark 4.1. For the case when $\tau = 0$, the function $L$ becomes a constant function $0$. Then, we can choose any number $c$ from $(0, a)$ to satisfy $L(c)\leq \frac{1}{d}$, and every positive $b$ satisfying $b\geq(a-c)y(0)$ also satisfies the condition (4.2). Therefore, we have

which implies

Since inequality (4.1) for $\tau = 0$ is

we can say that Lemma 4.3 gives the optimal upper bound for $\tau = 0$.

Now, we introduce the first sufficient framework for the emergence of flocking.

● $(\mathcal{F}1)$: $A^{\tau, \tau}(0, 0) = \cos\delta > 0, \quad \delta\in (0, \frac{\pi}{2})$.

● $(\mathcal{F}2)$: There exists a constant $D_X^{\tau, \infty} > 0$ satisfying

● $(\mathcal{F}3)$: The time-delay bound $\tau\geq 0$ is sufficiently small to satisfy

for some $c_1\in(0, a_1], c_2\in(0, a_2]$, where the constants $a_1, a_2, d_1, d_2$, and $\beta$ are given by

Now, we are ready to provide the first main result on the asymptotic flocking of time-delayed unit speed TCS model (3.1), by showing that $D_X^\tau$ is uniformly bounded and $D_V^\tau, D_T^\tau$ converge to zero}.

Theorem 4.1. Let $(X, V, T)$ be a solution to (3.1), where the initial data

are Lipschitz continuous functions satisfying $\frac{dx_i^0}{dt} = v_i^0$ and $(\mathcal{F}1)$–$(\mathcal{F}3)$. Then, we estimate

where the constant $\gamma$ is given by

Therefore, the solution $(X, V, T)$ exhibits the asymptotic flocking.

Proof. (Step 1) First, we find the range of $t$ such that $D_V^\tau$ and $D_T^\tau$ satisfy the inequalities playing the role of $y(t)$ in Lemma 4.3. Assume there exists a minimum $t^* < \infty$ among all $t$ satisfying $S_X^\tau(t)\geq D_X^{\tau, \infty}$. Then, by using Lemma 3.4, we have

which implies $t^* > \tau$ from the condition (4.8). We then use the monotone decreasing property of $\phi, \zeta$ and Lemma 3.5 and Lemma 4.2 to obtain

for the delayed diameter $D_V^\tau$, and similarly,

for the delayed diameter $D_T^\tau$.

(Step 2) Now, we will show that $t^*$ cannot be a finite number so that Eqs (4.9) and (4.10) hold for all $t\in (\tau, \infty)$. For every positive number $b_1, b_2$ satisfying

one can verify that (4.2) holds for $(b_1, c_1)$ and $(b_2, c_2)$. More precisely, by using Eqs (4.8) and (4.11) and Lemma 3.4, we show that

In addition, we use the monotonic increasing property of $A^{\tau, \tau}(t, t)$ (see Lemma 3.4) to verify

which implies

Similarly, by using Eqs (4.8) and (4.11) and Lemma 3.2, we have

and we use the monotonic decreasing property of $D_T^\tau(t) = T_M^\tau(t)-T_m^\tau(t)$ (see Lemma 3.2) to derive

which implies

Therefore, we apply Lemma 4.3 to (4.9) and (4.10) for all $b_1, b_2$ satisfying (4.11) to obtain

In particular, we employ the result of Lemma 4.2 to (4.12) to get

However, from direct calculation, (4.8) and (4.13) yield

which contradicts to $S_X^\tau(t^*) = D_X^{\tau, \infty}$ obtained from the existence of $t^* < \infty$. Therefore, we have $S_X^\tau(t) < D_X^{\tau, \infty}$ for all $t > \tau$, and we apply Lemma 4.3 to (4.9) and (4.10) for all $b_1, b_2$ satisfying (4.11) to reach the desired results. □

Remark 4.2. Theorem 4.1 is also applicable when time-delay $\tau$ is set to $0$. If $\tau = 0$, the framework $(\mathcal{F}1)$–$(\mathcal{F}3)$ becomes

which coincides with the condition (2.2). In addition, the condition (4.8) reduces to

Therefore, Theorem 4.1 exactly coincides with Proposition 2.5 for $\tau = 0$.

5.   Asymptotic behavior of the diameters

In this section, we provide a system of differential inequalities on diameters $D_X, D_V$, and $D_T$. In this case, the Lipschitz continuity of the diameters can be easily obtained from the Lipschitz continuity of $X, V, T$. The goal of this section is to find a sufficient framework without the condition $(\mathcal{F}1)$, so that the initial data might allow $\langle v_i(t_1), v_j(t_2)\rangle < 0$ for some $i, j\in [N]$ and $t_1, t_2\in [-\tau, 0]$. More precisely, we will replace $(\mathcal{F}1)$ to the weaker condition that

In this case, since we cannot apply the result of Lemma 3.4, the proof of Theorem 4.1 cannot be applied as is, and we must consider the possibility that $A^{\tau, \tau}$ may oscillate at the beginning rather than increasing monotonically. The overall flow of this section is that we can still control the diameter $D_V$ under the condition that $\tau$ is sufficiently small.

Lemma 5.1. Let $(X, V, T)$ be a solution to Eq (3.1) subject to the continuous initial data $\{(x_i^0, v_i^0, T_i^0)\}_{i = 1}^N$ satisfying $\frac{dx_i^0}{dt} = v_i^0$ for all $i\in [N]$. Then, the following differential inequalities hold:

1. Whenever $D_X$ is differentiable at time $t > -\tau$, we have

2. Whenever $D_V$ is differentiable at time $t > 0$ and $D_V(t)\leq \sqrt{2}$, we have

3. Whenever $D_T$ is differentiable at time $t > 0$, we have

Proof. We can prove this lemma by using a similar argument to Lemma 4.2.

(1) For every $i, j\in [N]$ and $t > -\tau,$ we have

Therefore, one can obtain

which implies

whenever $D_X$ is differentiable at time $t$.

(2) Let $\mathcal{J}(t)$ be the set of all index pairs $(i, j)$ which maximize the difference between two velocities at time $t$, i.e.,

Then, for every $(i, j)\in \mathcal{J}(t)$, we can split the derivative of $\|v_i-v_j\|^2$ at time $t$ into four parts as follows:

Below, we estimate these four parts separately into two parts.

$\diamond$ (Estimate of $\mathcal{L}_{11}+\mathcal{L}_{12}$): In this case, one can verify the following inequalities at time $t$:

where we used $\langle v_i, v_j\rangle = 1-\frac{1}{2}D_V^2\geq 0$ in the last inequality. This implies that both $\mathcal{L}_{11}$ and $\mathcal{L}_{12}$ are nonpositive numbers, and therefore

where we used the monotonic decreasing property of $\phi$ and

in the first inequality. Then, we use $\langle v_i+v_j, v_k\rangle\geq 2\langle v_i, v_j\rangle$ and

in Eq (5.1) to obtain

$\diamond$ (Estimate of $\mathcal{L}_{13}+\mathcal{L}_{14}$): From direct calculation, we have

where we used

in the last inequality.

Therefore, we combine Eqs (5.3) and (5.4) to obtain

which implies

whenever $D_V$ is differentiable and $D_V\leq \sqrt{2}$.

(3) Let $M(t), m(t)$ be the set of all indices such that

Then, for every $i\in M(t)$ and $j\in m(t)$, we can split the derivative of $T_i-T_j$ at time $t$ into two parts as follows:

Then, we estimate these two parts separately.

$\diamond$ (Estimate of $\mathcal{L}_{21}$): From a direct calculation, it follows that

where we used $T_j(t)\leq T_k(t)\leq T_i(t)$ for all $i\in M(t)$ and $j\in m(t)$ in the first inequality.

$\diamond$ (Estimate of $\mathcal{L}_{22}$): In this case, we use Lemma 3.2 to obtain

Therefore, we combine these two estimates to obtain

which implies the desired result. □

Remark 5.1. One thing to keep in mind during the proof of Lemma 5.1 is that the acceleration may not be zero even if there is a point at which the velocity diameter $D_V$ is zero, due to the time-delay effect. Therefore, even if we find $i, j$ with $D_V = \|v_i-v_j\|$, we cannot exclude the possibility that $i$ and $j$ are equal. Another thing to keep in mind is that the condition $D_V(t)\leq \sqrt{2}$ is necessary to obtain a meaningful inequality for $D_V$ because of the technical reason, otherwise the values

can be negative when $\langle v_i, v_k\rangle = \langle v_j, v_k\rangle = \langle v_i, v_j\rangle < 0$.

Similar to what we have done in Section 4, we prepare a lemma to present the upper bound of $(D_X, D_V, D_T)$ and $(\Delta_V, \Delta_T)$ by using Lemma 3.5 and Lemma 5.1.

Lemma 5.2. Let $y:[0, \infty)\to [0, \infty)$ be a Lipschitz continuous function and $f:[0, \infty)\to\mathbb{R}$ be a continuous function. Assume that $y$ and $f$ satisfy

for some constants $\chi, \theta, L > 0$ and $\tau\geq 0$. If $\tau$ is sufficiently small to satisfy

the following assertions hold.

(1) There are two constants $\xi > 0$ and $\mu \in (0, \chi)$ such that

(2) Whenever $\xi, \mu > 0$ satisfy Eq (5.7), we have

Proof. From Eqs (5.5) and (5.6), we can choose a sufficiently large $\xi$ satisfying

Then, the condition (5.7) holds for sufficiently small positive number $\mu\in (0, \chi)$. Now, assume there exists a minimum $t_*\in (0, \infty)$ among all $t$ satisfying

Then, we have

and we substitute (5.10) to (5.5) to obtain

Consequently, this Grönwall's inequality yields

where we used Eq (5.7) in the last inequality. On the other hand, by using Eqs (5.10) and (5.11) to (5.5b) at $t = t_*$, we have

where we used Eq (5.7) in the last inequality. However, this leads a contradiction to Eq (5.9), and therefore

which implies the desired inequality (5.8). □

Unlike in Section 4, we need to guarantee $D_V(t)\leq \sqrt{2}$ to apply Lemma 5.2 to Lemma 5.1. From Eqs (5.7) and (5.8), one can find a time-invariant upper bound for $y(t)$. Since the function

has a maximum at $t = t_0$ satisfying

we have

Therefore, we have the following time-invariant upper bound for $y(t)$:

Now, by using Lemma 5.1 and 5.2, we can construct another sufficient framework leading to the asymptotic flocking. In this case, we assume $D_X(0)$ and $D_V(0)$ sufficiently small, instead of assuming the smallness of $D_X^\tau(0)$ and $D_V^\tau(0)$.

● $(\mathcal{F}1)'$: $D_V(0) = 2\sin\frac{\delta}{2}, \quad \delta\in(0, \frac{\pi}{2})$.

● $(\mathcal{F}2)'$: There exists a constant $D_X^\infty > 0$ satisfying

● $(\mathcal{F}3)'$: The time-delay bound $\tau > 0$ is sufficiently small to satisfy

for some $\mu_1\in (0, \chi_1)$ and $\mu_2\in (0, \chi_2)$, where the constants $\chi_1, \chi_2, \theta_1, \theta_2, L_1, L_2$ are defined as

Note that from the equality conditions in (5.12), $\mu_1$ tends to $\chi_1$ during $\tau\to +0$. Therefore, $(\mathcal{F}3)'$ is indeed a condition that holds for sufficiently small $\tau$.

Finally, we are ready to provide the second main result, which also shows that $D_X$ is uniformly bounded and $D_V, D_T$ converge to zero. Although the proof might not be intuitive and its implications are hard to grasp at first glance, we wrote it down in detail for the completeness.

Theorem 5.1. Let $(X, V, T)$ be a solution to Eq (3.1), where the initial data

are Lipschitz continuous functions satisfying $\frac{dx_i^0}{dt} = v_i^0$ and $(\mathcal{F}1)'$–$(\mathcal{F}3)'$. Then, the following inequalities hold for all $t > 0$:

Therefore, the solution $(X, V, T)$ exhibits the asymptotic flocking.

Proof. (Step 1) Assume there exists a minimum $t^* < \infty$ among all $t$ satisfying

Then, we will show that, in fact, only the first condition among the three can be satisfied, i.e., $D_X(t^*) = D_X^\infty$. First, one can obtain $t^* > 0$ by using $(\mathcal{F}1)', (\mathcal{F}2)'$, and the continuity of $D_V$, and we use the monotonic decreasing property of $\phi, \zeta$, Lemma 3.5, and Lemma 5.1 to obtain

for the diameter $D_V$, and similarly,

for the diameter $D_T$. Now, for the two constants $\xi_1, \xi_2$ defined as

one can verify that (5.7) holds for $(L_1, \chi_1, \mu_1, \xi_1)$ and $(L_2, \chi_2, \mu_2, \xi_2)$. More precisely, the first condition of (5.7) is immediately obtained from (5.12b) and (5.12c), and the third condition is obtained from

which is a consequence of Lemma 3.5 for $t = 0$. The second condition of (5.7) can also be verified by the relation

As a consequence, we apply Lemma 5.2 to Eqs (5.14) and (5.15) to get the following inequalities for $t\in (0, t^*)$:

Then, we use Eq (5.12d) to obtain

and therefore, the only possible case to satisfy (5.13) is the first case, i.e., $D_X(t^*) = D_X^\infty$.

(Step 2) Therefore, we can apply Lemma 5.1 (2) to $t\in (0, t^*)$ to obtain

which implies that

However, (5.17) yields

which leads a contradiction to (5.12a). Thus, there is no $t^*\in (0, \infty)$ satisfying (5.13), and the inequalities in (5.16) hold for all $t > 0$. □

Remark 5.2. Similar to the case in Remark 4.2, Theorem 5.1 is also applicable when time-delay $\tau$ is set to $0$. If $\tau = 0$, the framework $(\mathcal{F}1)'-(\mathcal{F}3)'$ becomes

which coincides with the condition (2.2). In addition, the condition (5.12) reduces to

Therefore, Theorem 5.1 also exactly coincides with Proposition 2.5 for $\tau = 0$.

6.   Numerical simulation

In this section, we perform numerical simulations on (1.3) to verify the results of Theorem 4.1 and Theorem 5.1, and to explore whether there are other properties that we could not prove due to technical reasons. For the numerical implementation, we used the Euler method with time step $\Delta t = 0.05$, and we fixed $N, \phi, \zeta$ as

In addition, for the time delay $\tau_{ij}$, we set

for each $i, j\in [N]$. Thus, we compare how the dynamics of (1.3) vary with respect to $\kappa_1, \kappa_2, \tau$, and initial data.

In particular, we prepare two types of initial data for equation (1.3), which we will refer to as 'good initial' and 'bad initial' based on the velocity range. The way we set the initial conditions for $X$ and $T$ is common to both types and is as follows. For each $i\in [N]$ and $n\in [-\tau/\Delta t, 0]$, we randomly select $x_i(n)$ from a uniform distribution on $(0, 1)^3$ and $T_i(n)$ from a uniform distribution on $(1, 3.5)$. Then, the 'good initial' refers to the initial data for which

To achieve this, for each $i\in [N]$ and $n\in [-\tau/\Delta t, 0]$, we randomly select $\tilde{v}_i(n)$ from a uniform distribution on $(0, 1)^3$, just like the initial positions $x_i(n)$. Then, we normalize their norms to $1$ as follows:

If so, the discretized version of $A^{\tau, \tau}(0, 0)$, i.e.,

is strictly positive, and we can expect that (the discrete analogue of) $A^{\tau, \tau}(t_1, t_2)$ will be monotonically increasing, as proven in Lemma 3.4. On the other hand, by 'bad initial, ' we refer to the initial data for which

In other words, it refers to the initial conditions that do not satisfy the condition $(\mathcal{F}1)$ but satisfy the condition $(\mathcal{F}1)'$. We meet these requirements by randomly selecting $\tilde{v}_i(n)$ from a uniform distribution on $(-1, 1)^3$ for each $i\in [N]$ and $n\in [-\tau/\Delta t, -1]$, and also select $\tilde{v}_i(0)$ randomly from a uniform distribution on $(0, 1)^3$. Then, we normalize their norms to $1$ as in Eq (6.3) to satisfy the unit speed constraint. In this case, we can expect that flocking can occur when $\tau$ is sufficiently small, according to Theorem 5.1. However, since we cannot apply Lemma 3.4, $A^{\tau, \tau}$ may not be increasing monotonically at the beginning. Thus, one of the goals of this section is to confirm that the initial behavior of $A^{\tau, \tau}$ differs between good initial and bad initial based on the applicability of Lemma 3.4. However, when we actually ran the simulations, we found that observing the behavior of $D_V(t)$ is sufficient to clearly distinguish between the two cases. Therefore, to save computation time, we only display $D_X(t), D_V(t), T_{M}(t)$, and $T_m(t)$ over $t$ (time) in all figures of this section.

6.1.   Simulations for good initial data

In Figures 1 and 2, we fix $\tau = 1.8, \; \kappa_1 = \kappa_2 = 8$ and used the good initial data to simulate the solution of (1.3) for $t\in [0, 50]$. The first two plots in Figure 1 are the temporal evolution of $D_X(t)$ and $D_V(t)$ over $t$, and Figure 2 shows the temporal evolution of $T_M(t)$ and $T_m(t)$. These plots show that the solution exhibited asymptotic flocking under the given setting. The last two plots in Figure 1 confirm that the speeds of all particles remained at 1 throughout the simulation. In the case of $D_V$, it exhibits a monotonically decreasing behavior over time, while we can also observe an inflection point around $t = 2.4$. In fact, such inflection points were consistently observed in $D_V$ throughout the simulations, and we found that their position was always approximately at $\frac{4\tau}{3}$. We suspect that the value $\frac{4}{3}$ itself is not particularly important, but rather that the average of $\tau_{ij}$ set in Eq (6.2) is $\frac{2\tau}{3}$, and something special seems to occur around twice that time.

In Figure 3, we fix $\tau = 6, \; \kappa_1 = \kappa_2 = 8$ and used the good initial data to simulate the solution of Eq (1.3) for $t\in [0,500]$. Once again, asymptotic flocking occurred, but the rate of convergence was significantly slower compared to the previous examples. Additionally, we observed an inflection point in $D_V$ around $t = \frac{4\tau}{3} = 8$.

In Figure 4, we fix $\tau = 30, \; \kappa_1 = \kappa_2 = 8$ and used the good initial data to simulate the solution of (1.3) for $t\in [0, 10000]$. As $\tau$ increases, the time it takes for $D_V$ to saturate becomes much longer. Therefore, to verify whether flocking occurs, we had to observe the behavior of diameters for such a long time. For this reason, the inflection point is not clearly visible; however, the position of the dashed line set in this figure is also at $t = \frac{4\tau}{3} = 40$. Additionally, in this figure, we can see that $D_V$ does not converge to 0, indicating that the solution did not exhibit flocking. This means that even if the condition $(\mathcal{F}1)$ is satisfied and $\kappa_1, \kappa_2$ are fixed, flocking will no longer occur if $\tau$ becomes sufficiently large.

In Figure 5, we fix $\tau = 1.8, \; \kappa_1 = \kappa_2 = 1$ and used the good initial data to simulate the solution of (1.3) for $t\in [0, 2500]$. In this case, although $\tau$ is small as in Figures 1 and 2, the small values of $\kappa_1, \kappa_2$ result in a situation where flocking does not occur. Notably, due to the small value of $\kappa_2$, $D_T$ did not converge to 0 in this case.

6.2.   Simulations for bad initial data

In Figures 6 and 7, we fix $\tau = 1.8, \; \kappa_1 = \kappa_2 = 8$ and used the bad initial data to simulate (1.3). In other words, we are using the same $\tau$ and $\kappa_1, \kappa_2$ as in Figures 1 and 2, but with the bad initial data. The first two plots in Figure 6 illustrate the behavior of $D_X$ and $D_V$ from the given bad initial data up to time $t = 5$, while the last two plots show the behavior up to $t = 100$ for the same initial data. A notable feature observed in this figure is that, similar to Figure 1, the behavior of $D_V$ changes around the time $t = \frac{4\tau}{3} = 2.4$. However, the behavior of $D_V(t)$ for $t\in [0, 2.4]$ becomes unstable rather than just monotonically decreasing, which is a result not observed in any simulations using good initial data. Nevertheless, as demonstrated in Theorem 5.1, even with bad initial data, the asymptotic flocking can still occur if $\tau$ is sufficiently small and $\kappa_1, \kappa_2$ are sufficiently large.

In Figures 8 and 9, we fix $\tau = 6, \; \kappa_1 = \kappa_2 = 8$ and used the bad initial data to simulate (1.3). In other words, we are using the same $\tau$ and $\kappa_1, \kappa_2$ as in Figure 3, but with bad initial data. The first two plots in Figure 6 illustrate the behavior of $D_X$ and $D_V$ from the given bad initial data up to time $t = 10$, while the last two plots show the behavior up to $t = 250$ for the same initial data. In this case, $D_V$ shows a very unstable oscillation before $t = \frac{4\tau}{3} = 8$, and it even becomes larger than $D_V(0)$ at $t = 8$. However, this effect disappears after $t = 8$ and $D_V$ begins to decrease monotonically. Nevertheless, since both $D_V$ and $D_T$ do not converge to 0 in this case, asymptotic flocking does not occur. This contrasts with the occurrence of asymptotic flocking in Figure 3, demonstrating that the condition $(\mathcal{F}1)$ also affects the occurrence of asymptotic flocking.

Finally, in Figures 10 and 11, we fix $\tau = 1.8, \; \kappa_1 = \kappa_2 = 1$ and used the bad initial data to simulate (1.3).

That is, we are using the same $\tau$ and $\kappa_1, \kappa_2$ as in Figures 5, but with the bad initial data. The first two plots in Figure 10 illustrate the behavior of $D_X$ and $D_V$ from the given bad initial data up to time $t = 5$, while the last two plots show the behavior up to $t = 150$ for the same initial data. In this case, $D_V$ shows a small oscillation before $t = \frac{4\tau}{3} = 2.4$, but decrease monotonically after $t = 2.4$. However, as with the use of good initial data in Figure 5, asymptotic flocking did not occur in this case either.

7.   Conclusion

In this paper, we have demonstrated several sufficient frameworks for the asymptotic flocking dynamics of the thermodynamic CS model with a unit-speed constraint and time-delay. To do this, we first proved the monotonic property of extreme temperatures and maximal angle between velocities, and then we provided basic estimates concerning {position–velocity–temperature} diameters and perturbation functions. Then, we derived dissipative inequalities with respect to the diameters and delayed diameters, and proposed suitable ansatz for the decay rate of the perturbation function to find the sufficient framework to exhibit asymptotic flocking of (1.3). However, there are still some interesting topics that might be studied in the future. For instance, we wonder if it is possible to find differential inequalities for $A^{\tau, \tau}$ to show the asymptotic flocking. Since we have already shown that $A^{\tau, \tau}$ is monotonically increasing, we expect that if we succeed in obtaining a differential inequality for the $A^{\tau, \tau}$ itself, we will not need to use perturbation functions $\Delta_V^\tau$ and suggest an ansatz on its exponential decay rate. We leave this issue for future work.

Authors contribution

Hyunjin Ahn: Investigation, Funding acquisition, Writing, Validation; Woojoo Shim: Investigation, Methodology, Writing, Supervision.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The work of H. Ahn was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (2022R1C12007321).

Conflict of interest

The authors declare there is no conflict of interest.

Appendix A.   Proof of Lemma 3.1

Proof. (1) Suppose we have $f(t_0) > F(t_0)$ for some $t_0\in \mathbb{R}$. Then, there exists $t_1\in[t_0-\tau, t_0)$ such that $f(t_1) = F(t_0) < f(t_0)$. Since $f$ is continuous, one can find $\delta > 0$ such that

Therefore, we have

(2) To show that $F$ is monotonically increasing, we claim the following:

Once we prove the above claim, we can show $F(t_0)\leq F(t_1)$ for all $t_0 < t_1$. To see this, consider a function

This function has a maximum point $c\in[t_0, t_1]$ as a consequence of the Weierstrass extreme value theorem. In addition, since $G(t_0) = G(t_1) = F(t_0)$, we may assume $c\in[t_0, t_1)$. Therefore, we have

which implies $F(t_0)\leq F(t_1)$.

Now, suppose we have $D^+F(t_*) < 0$ for some $t_*\in \mathbb{R}$. Then, one can obtain $f(t_*) = F(t_*)$ from the result (1), and we have

where we used $f(t_*) = F(t_*)$ in the second equality. Therefore, there exist two constants $\delta, \varepsilon > 0$ such that

From (A.1), we can employ Lemma 3.1(1) to $t_0: = t_*+h$ for each $h\in(0, \delta)$ and obtain

and since $F$ is continuous ($\because$ Berge's maximum theorem), we have

which leads to a contradiction in $D^+F(t_*) < 0$.

(3) It is sufficient to prove that

If $f(t_0) > F(t_0)$, one can find $\delta > 0$ such that

due to the continuity of $f$. Therefore, we have

where we used the Lipschitz continuity of $f$. On the other hand, if $f(t_0) = F(t_0)$, then

where we used

in the last equality. Therefore, we have

where we used the Lipschitz continuity of $f$ in the last inequality. □

Appendix B.   Proof of Lemma 4.1

Proof. From the Berge maximum theorem, the set valued map

is upper hemi-continuous with nonempty and compact values. This means that for every $t\geq 0$, if $V(\subset \mathbb{R}^2)$ contains $C(t)$, there exists a neighborhood $U$ of $t$ such that for all $t\in U$, $C(t)$ is a subset of $V$.

Now, fix arbitrary $\varepsilon > 0$, and define $g(t_1, t_2): = f(t_1, t_2)-(c+\varepsilon)\max\{t_1, t_2 \}$. Then, we have

Since $\lambda$ is continuous, there exists an open neighborhood

for each $(s, t_0)\in C(t_0), s < t_0$, such that $\lambda(s_1, s_2) < c+\frac{\varepsilon}{2}$ for all $(s_1, s_2)\in V(s, t_0)$. Similarly, for each $(t_0, s)\in C(t_0), s < t_0$, there exists an open neighborhood

such that $\lambda(s_1, s_2) < c+\frac{\varepsilon}{2}$ for all $(s_1, s_2)\in V(t_0, s)$. Finally, there exists an open set

such that $\lambda(s_1, s_2) < c+\frac{\varepsilon}{2}$ for all $(s_1, s_2)\in V(t_0, t_0)$. Then, $\{V(s_1, s_2):(s_1, s_2)\in C(t_0) \}$ is an open cover of $C(t_0)$. In addition, since $C(t_0)$ is compact, we can find its finite subcover $\{V_1, \ldots, V_n \}$. Therefore, we can find an open subset $V_0$ of $S(t_0)$ such that

Therefore, by using the Berge maximum theorem, there exists a positive number $\delta_0 < \delta$ such that $C(t)\subset V$ for all $t \in (t_0-\delta_0, t_0+\delta_0)$. This means that the Dini derivative $D^+G_{t_1, t_2}(h)$ of

is less than or equal to $-\frac{\varepsilon}{2}$ for all $(t_1, t_2)\in V_0$ and $|h| < \delta_0$, and therefore $G_{t_1, t_2}(h)$ is monotonically decreasing in $|h| < \delta_0$ for all $(t_1, t_2)\in V_0$. By using this result, we have

which implies that for every $|h| < \delta_0$,

Since $\varepsilon$ can be any positive number, we have

□