Multi-agent system for target tracking on a sphere and its asymptotic behavior

Sun-Ho Choi Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin 17104, Republic of Korea [email protected] , Dohyun Kwon Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr., Madison, WI 53706, USA [email protected] and Hyowon Seo Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin 17104, Republic of Korea [email protected]

Abstract.

We propose a second-order multi-agent system for target tracking on a sphere. The model contains a centripetal force, a bonding force, a velocity alignment operator to the target, and cooperative control between flocking agents. We propose an appropriate regularized rotation operator instead of Rodrigues’ rotation operator to derive the velocity alignment operator for target tracking. By the regularized rotation operator, we can decompose the phase of agents into translational and structural parts. By analyzing the translational part of this reference frame decomposition, we can obtain rendezvous results to the given target. If the multi-agent system can obtain the target’s position, velocity, and acceleration vectors, then the complete rendezvous occurs. Even in the absence of the target’s acceleration information, if the coefficients are sufficiently large enough, then the practical rendezvous occurs.

1. Introduction

Target tracking refers to designing a dynamical system that agents follow given maneuvering target agents using the information of the targets, such as position, velocity, and acceleration. The target tracking problem is applied in various fields, such as mobile sensor networks, virtual reality, and surveillance systems using unmanned aerial vehicles (UAVs) [18, 22, 24]. Most of the relevant literature focuses on the uncertainty of target motions. From a technical point of view, we can divide the models for this field into measurement models, target motion models, and filtering models. The measurement model deals with target information in a sensor coordinate containing additive noise such as image sensors and radar sensor networks [2, 3, 20]. The target motion model is a coupled dynamical system for target tracking. The filtering model is based on the particle filter method and stochastic frameworks estimating the target state such as nonlinear filtering [13, 15] and adaptive filtering [14, 16].

Depending on the structure of the system, we also divide the models for target tracking into two types of systems: single integrator model and double integrator model. For the single integrator model, one can control the velocity of the agents directly. For example, in [10], the authors proposed a tracking algorithm for a slowly moving target using the target’s position and bearing angle. Many researchers assume agents can obtain only the target’s position and bearing angle for targets maneuvering underwater. From the engineering point of view, it is a reasonable assumption. For the double integrator model, one can control the acceleration of agents. After Olfati-Saber’s seminal work [11], researches for the dynamic tracking system using the double integrator model have been extensively conducted. For this kind of model, the tracking agents can have the position and velocity information of the target. Moreover, to avoid collisions between agents or make a formation flight of the agents, a flocking algorithm and cooperative control are frequently used.

The domain or manifolds of agents are also one of the main topics in this field [1, 18] such as the surveillance system for the restricted area or target tracking system on the whole planet. Our goal is to provide a robust navigational feedback system for the target tracking problem on a sphere. Let $\gamma$ -agent be a given target governed by the following system:

\displaystyle\begin{aligned} \dot{q}_{\gamma}&=p_{\gamma},\\ \dot{p}_{\gamma}&=-\frac{\|p_{\gamma}\|^{2}}{\|q_{\gamma}\|^{2}}q_{\gamma}+U_{\gamma}(t),\end{aligned}

(1.1)

where $q_{\gamma}\in\mathbb{S}^{2}$ , $p_{\gamma}\in T_{q_{\gamma}}\mathbb{S}^{2}$ , and $U_{\gamma}$ are the position, velocity, and control law of the target agent ( $\gamma$ -agent) on sphere, respectively. To conserve the modulus of $q_{\gamma}(t)\in\mathbb{S}^{2}$ , we additionally assume that the following condition holds for all $t\geq 0$ .

q_{\gamma}(t)\perp U_{\gamma}(t).

Therefore, the control law $U_{\gamma}(t)$ has the following form: for some $u_{\gamma}(t)\in\mathbb{R}^{3}$ ,

U_{\gamma}(t)=\|q_{\gamma}(t)\|^{2}u_{\gamma}(t)-\langle u_{\gamma}(t),q_{\gamma}(t)\rangle q_{\gamma}(t).

For simplicity, we assume that $u_{\gamma}(t)$ is continuous.

For a given $\gamma$ -agent, we propose a novel multi-agent system for the target tracking on a spherical space:

\displaystyle\begin{aligned} \dot{q}_{i}(t)&=p_{i}(t),\\ \dot{p}_{i}(t)&=-\frac{\|p_{i}\|^{2}}{\|q_{i}\|^{2}}q_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})\\ &\quad+c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i},\end{aligned}

(1.2)

where $q_{i}\in\mathbb{S}^{2}$ and $p_{i}\in T_{q_{i}}\mathbb{S}^{2}$ are the position and velocity of the $i$ th agent, respectively. The first term on the right-hand side of the second equation is the centripetal force term to conserve the modulus of $q_{i}$ . The second term

\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})

is the cooperative control term between agents where the inter-particle force parameter is given by

\displaystyle\sigma_{ij}=\sigma(\|x_{i}-x_{j}\|^{2}).

The next two terms, $c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})$ and $c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})$ , are the bonding force and a velocity alignment term between the target and the $i$ th agent, respectively, where $c_{q}>0$ and $c_{p}>0$ are target tracking coefficients for the position and velocity, respectively. The last term $U_{i}$ is an extra control law based on the target’s information, which will be determined later in (1.5) and (1.6) for each purpose.

Throughout this paper, we assume the initial data satisfies the following admissible conditions on $\mathbb{S}^{2}$ :

\displaystyle\|q_{i}(0)\|=1,\quad\langle p_{i}(0),q_{i}(0)\rangle=0,\quad\mbox{ for all $i\in\{1,\ldots,N\}$}.

(1.3)

Definition 1.1.

For a given target $(q_{\gamma}(t),p_{\gamma}(t))$ , let $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ be the solution to (1.2). We define the two kinds of rendezvouses.

(1)

An asymptotic complete rendezvous occurs between the agents and the given target, if

$\lim_{t\to\infty}\max_{1\leq i\leq N}\|q_{i}(t)-q_{\gamma}(t)\|=0.$
(2)

An asymptotic practical rendezvous occurs between the agents and the given target, if

$\lim_{c_{q},c_{p}\to\infty}\lim_{t\to\infty}\max_{1\leq i\leq N}\|q_{i}(t)-q_{\gamma}(t)\|=0.$

In what follows, we will show that our model contains many robust properties, including the complete rendezvous. Even in the absence of the target acceleration information, the practical rendezvous occurs when the coefficients are large enough. In particular, we obtain a sharp estimate of the distance between the target and agents. There are many other papers on the dynamics on $\mathbb{S}^{2}$ as well as $\mathbb{R}^{n}$ , but our asymptotic analysis including exponential convergence and practical rendezvous is new on the target tracking problem, to the best of our knowledge.

The derivation of our model is motivated by the decomposition property of flocking dynamics on a flat space. On a flat space, from momentum conservation, the dynamics is represented by the composition of frame reference dynamics and local alignment dynamics as in [11]. In contrast to previous results in $\mathbb{R}^{n}$ , it is hard to expect such a decomposition for the flocking model on $\mathbb{S}^{2}$ . See Sections 2 and 3 for details. In particular, in our previous papers [6, 7, 8], we used Rodrigues’ rotation operator ${R_{\cdot\rightarrow\cdot}}$ to derive a flocking system on a sphere since Rodrigues’ rotation operator ${R_{\cdot\rightarrow\cdot}}$ is the most natural flocking operator. However, its composition is complex so that it is difficult to analyze. Moreover, it contains an unavoidable singularity at antipodal points due to its geometric characteristics. From this singularity, even though agents are located on $\mathbb{S}^{2}$ , the vanishing point on the communication rate is necessary [6]. Due to this difficulty, the target tracking problem on $\mathbb{S}^{2}$ has not been well understood.

We remove the singular term from the natural rotation operator ${R_{\cdot\rightarrow\cdot}}$ to obtain a rotation operator in two dimensions:

\displaystyle{P}_{z_{1}\rightarrow z_{2}}:=\langle z_{1},z_{2}\rangle I+z_{2}z_{1}^{T}-z_{1}z_{2}^{T},\quad\hbox{ for $z_{1}$ and $z_{2}$ in a unit sphere. }

(1.4)

See also Appendix A for the motivation of the non-singularity rotation operator $P$ and its properties. We will prove that its dynamics consists of the composition of the rigid motion part on $\mathbb{S}^{2}$ and the local alignment part. Using this property, we derive an $\mathbb{S}^{2}$ -version of the reference frame decomposition in Proposition 3.2 and provide a sufficient condition to obtain a target tracking estimate between multiple agents $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ and the given target $(q_{\gamma}(t),p_{\gamma}(t))$ . Moreover, by the regularity of the operator ${P}$ , we can obtain the following global existence result.

Theorem 1.

Assume that for a continuous function $u_{\gamma}$ , a given target $(q_{\gamma}(t),p_{\gamma}(t))$ satisfies (1.1). If the initial data $\{(q_{i}(0),p_{i}(0))\}_{i=1}^{N}$ satisfies (1.3) and $U_{i}$ is Lipschitz continuous with respect to $\{(q_{i},p_{i})\}_{i=1}^{N}$ with $\langle U_{i},q_{i}\rangle=0$ , then there exists a unique global-in-time solution $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ to system (1.2) and $\{q_{i}(t)\}_{i=1}^{N}$ are located on $\mathbb{S}^{2}$ for all time $t>0$ .

As in $\mathbb{R}^{d}$ , we notice that the velocity alignment operator between the target and the agents plays an important role in target tracking. In particular, the bonding force between the target and the agents, $c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})$ , alone is not enough to track a target on $\mathbb{S}^{2}$ . The velocity alignment operator $c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})$ is crucial for the target tracking algorithm. See the simulations in Section 5. In the next two theorems, we present a quantitative analysis of the velocity alignment operator with two different $U_{i}$ ’s;

\displaystyle U_{i}=2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i}

(1.5)

\displaystyle U_{i}=0,

(1.6)

where $w_{\gamma}$ is the angular velocity of the target given by

\displaystyle w_{\gamma}=q_{\gamma}\times p_{\gamma}.

(1.7)

From Theorem 2, if the agents can obtain the exact target information containing acceleration, then the agents can accurately track the target, and the position differences between the target and the agents decay exponentially fast.

Theorem 2.

Let $(q_{\gamma}(t),p_{\gamma}(t))$ be a given target satisfying (1.1) with a continuous target control $u_{\gamma}$ and $\{q_{i}(t),p_{i}(t)\}_{i=1}^{N}$ be the solution to (1.2) satisfying (1.3). We assume that $\sigma_{ij}=\sigma$ is a positive constant and

U_{i}=2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i},

where $w_{\gamma}$ is the angular velocity defined in (1.7).

If $c_{q}>\sigma>0$ or

	$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\\|p_{i}(0)-w_{\gamma}(0)\times q_{i}(0)\\|^{2}$
	$\displaystyle\quad+\frac{\sigma}{2N^{2}}\sum_{i,j=1}^{N}\\|q_{i}(0)-q_{j}(0)\\|^{2}+\frac{c_{q}}{N}\sum_{i=1}^{N}\\|q_{\gamma}(0)-q_{i}(0)\\|^{2}<\sigma\left(1+\frac{c_{q}}{\sigma}\right)^{2},$

then the asymptotic complete rendezvous occurs and its convergence rate is exponential, i.e., there are positive constants $\mathcal{C}$ , $\mathcal{D}$ such that

\|q_{i}(t)-q_{\gamma}(t)\|,~{}\|p_{i}(t)-p_{\gamma}(t)\|\leq\mathcal{C}e^{-\mathcal{D}t}.

Remark 1.1.

(1)

If the above sufficient condition in Theorem 2 does not hold, then we can find a steady-state solution. This means that the sufficient condition is almost optimal to lead the convergence result in Theorem 2. See Section 5.
(2)

The author in [11] does not deal with the estimate of the distance between the target and agents. Our model is inspired by [11], but the target tracking estimate and practical rendezvous are novel.
(3)

The derivation of $U_{i}$ in the above theorem is technical, but from the frame decomposition in Proposition 3.2, it is a very natural choice to obtain the complete rendezvous.

The former one in (1.5) corresponds to the case with the target acceleration, while it is unknown in the latter case (1.6). These choices with the different amounts of the target information induce the different accuracies of the target tracking. Since the target information obtained by the agents through observation is usually incomplete, there have been many studies to overcome this incompleteness. For example, many researchers proposed target tracking systems including restricted target information [10, 19], communication-induced delays [12, 17], and additive noise from measurement [9, 23]. The result in Theorem 3 below means that the large coefficients of the system allow the agents to get close enough to the target as needed without acceleration information of the target. In other words, the practical rendezvous occurs.

Theorem 3.

For $(q_{\gamma}(t),p_{\gamma}(t))$ satisfying (1.1) with a continuous target control $u_{\gamma}$ , let $\{q_{i}(t),p_{i}(t)\}_{i=1}^{N}$ be the solution to (1.2) subject to the initial data satisfying (1.3) and

U_{i}=0.

Assume that $\sigma_{ij}=\sigma$ is a positive constant and the angular velocity of the target and its time derivative are bounded

\|w_{\gamma}\|,\|\dot{w}_{\gamma}\|<C_{\gamma}.

If $\|p_{i}(0)-p_{\gamma}(0)\|\neq 2$ for all $i\in\{1,\ldots,N\}$ , then the asymptotic practical rendezvous occurs and

\|q_{i}(t)-q_{\gamma}(t)\|\leq\mathcal{C}e^{-\frac{\mathcal{D}}{4}t}+\frac{\mathcal{C}}{\mathcal{D}},

where $\mathcal{C}$ is a positive constant depending on the initial data, $\sigma$ , and $C_{\gamma}$ . The constant $\mathcal{D}$ is given by

\displaystyle\mathcal{D}:=\left\{\begin{array}[]{ll}\displaystyle~{}c_{p}-\sqrt{-4c_{q}+c_{p}^{2}},&\quad\mbox{if}\quad c_{p}^{2}\geq-4c_{q},\\ \displaystyle~{}c_{p},&\quad\mbox{if}\quad c_{p}^{2}<-4c_{q}.\end{array}\right.

There are technical issues in the proofs of Theorems 2 and 3. We can obtain the complete rendezvous result in Theorem 2 through Lasalle’s invariance principle with an energy functional. However, Lasalle’s invariance principle does not give a convergence rate. An appropriate Lyapunov functional will be used to obtain the exponential convergence result. In particular, in this case, we derive a closed differential inequality by using six functionals including information on the distance between the target and agents and the distance between agents. The practical rendezvous in Theorem 3 has a more subtle issue. It is necessary to control the distance between the target and agents through the size of the coefficients. However, it is impossible if the coefficients appear in the nonlinear higher-order terms except for the linear terms. If we use a standard functional, the coefficients necessarily occur in the nonlinear terms due to the geometrical characteristics of $\mathbb{S}^{2}$ . This problem will be solved by using new functionals inspired by hyperbolic geometry.

The rest of the paper is organized as follows. In Section 2, we present the global-in-time existence and uniqueness of the solution to (1.2) and target tracking results for $\mathbb{R}^{3}$ . Section 3 is devoted to a reference frame decomposition for the main system. From this decomposition, the solution to the main system is represented by the composition of operators for the translational part and the structural part. Next, we reduce the system for the structural part to a linearized system in Section 4. Using this, we prove the complete and practical rendezvouses of Theorems 2 and 3 in Section 5. In Section 6, we verify our analytic results using numerical simulations. Section 7 is devoted to the summary of our results.

2. Preliminary: Global well-posedness and Motivations

2.1. The global existence and uniqueness

In this section, we provide the proof of Theorem 1: there is a unique global-in-time solution to (1.2) and this solution is located on the sphere when the initial data satisfies the admissible conditions in (1.3).

For the local existence and uniqueness, we use the same argument in [6, 7]. For given $C^{1}$ functions $q_{\gamma}$ , $p_{\gamma}$ , and $w_{\gamma}=q_{\gamma}\times p_{\gamma}$ , we consider the following system of ODEs:

\displaystyle\begin{aligned} \dot{q}_{i}(t)&=p_{i}(t),\\ \dot{p}_{i}(t)&=-\frac{\|p_{i}\|^{2}}{\|q_{i}\|^{2}}q_{i}+\sum_{j=1}^{N}\frac{\sigma(\|x_{i}-x_{j}\|^{2})}{N}(\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})\\ &\quad+c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})+c_{p}(\langle q_{\gamma},q_{i}\rangle p_{\gamma}-\langle q_{i},p_{\gamma}\rangle q_{\gamma}-p_{i})+U_{i}.\end{aligned}

(2.1)

Here, we will choose $U_{i}=2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i}$ for the complete rendezvous and $U_{i}=0$ for the practical rendezvous.

We assume that the initial data $\{(q_{i}(0),p_{i}(0))\}_{i=1}^{N}$ satisfies the admissible condition in (1.3). Then the right-hand side of (2.1) is Lipschitz continuous with respect to $\{(q_{i},p_{i})\}_{i=1}^{N}$ in a small neighborhood of $\{(q_{i}(0),p_{i}(0))\}_{i=1}^{N}$ in $\mathbb{R}^{6N}$ . By the Picard-Lindelöf Theorem, there is the maximum time interval $[0,T_{M})$ in which a solution of (2.1) exists and it is unique.

We next follow the same argument in [6, 7]. On the maximum time interval $[0,T_{M})$ , we take the inner product between the second equation of (2.1) and $x_{i}$ to obtain that

\displaystyle\langle\dot{p}_{i},q_{i}\rangle

\displaystyle=-\|p_{i}\|^{2}-c_{p}\langle p_{i},q_{i}\rangle.

(2.2)

By (2.2) and the first equation of (2.1), we obtain that

	$\displaystyle\frac{d}{dt}\sum_{i=1}^{N}\|\langle p_{i},q_{i}\rangle\|^{2}$	$\displaystyle=2\sum_{i=1}^{N}(\langle\dot{p}_{i},q_{i}\rangle+\langle p_{i},\dot{q}_{i}\rangle)\langle p_{i},q_{i}\rangle$
		$\displaystyle=2\sum_{i=1}^{N}(\langle\dot{p}_{i},q_{i}\rangle+\\|p_{i}\\|^{2})~{}\langle p_{i},q_{i}\rangle$
		$\displaystyle=-2c_{p}\sum_{i=1}^{N}\|\langle p_{i},q_{i}\rangle\|^{2}.$

Note that the initial data satisfies $\sum_{i=1}^{N}|\langle v_{i}(0),x_{i}(0)\rangle|^{2}=0$ . Therefore, the Gronwall inequality implies that

\displaystyle\sum_{i=1}^{N}|\langle v_{i}(t),x_{i}(t)\rangle|\equiv 0,\quad\mbox{for}~{}t>0,

and this implies that

\displaystyle\langle v_{i}(t),x_{i}(t)\rangle\equiv 0.

We take the inner product between $\dot{q}_{i}$ and $q_{i}$ . By the first equation of (2.1),

\displaystyle\frac{d}{dt}\|q_{i}\|^{2}=2\langle\dot{q}_{i},q_{i}\rangle=2\langle p_{i},q_{i}\rangle=0.

Since initial conditions satisfy $\|x_{i}(0)\|=1$ and $\langle v_{i}(0),x_{i}(0)\rangle=0$ for all $i\in\{1,\ldots,N\}$ , we have

\displaystyle\|x_{i}(t)\|\equiv 1,\quad\mbox{for}~{}t>0,~{}i\in\{1,\ldots N\}.

In conclusion, we can apply the extensibility of solutions in [21, Corollary 2.2] to obtain that

T_{M}=\infty.

Moreover, we can easily check that $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ is the unique solution to (1.2) by a standard argument. Therefore, we can obtain the following proposition.

Proposition 2.1.

Let $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ be a solution to (1.2) with (1.3). Then for all $i\in\{1,\ldots,N\}$ and $t>0$ ,

\displaystyle\langle q_{i}(t),p_{i}(t)\rangle=0\quad\hbox{ and }\quad\|q_{i}(t)\|=1.

2.2. Target tracking problem in $\mathbb{R}^{3}$

In this section, we estimate the distance between the target and agents for the following model in $\mathbb{R}^{3}$ :

	$\displaystyle\dot{q}_{i}$	$\displaystyle=p_{i},$
	$\displaystyle\dot{p}_{i}$	$\displaystyle=\sum_{j=1}^{N}\frac{\psi_{ij}}{N}(p_{j}-p_{i})+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(q_{j}-q_{i})+c_{q}(q_{\gamma}-q_{i})+c_{p}(p_{\gamma}-p_{i})+u_{i},$

where $q_{i}\in\mathbb{R}^{3}$ and $p_{i}\in\mathbb{R}^{3}$ are the position and velocity of the $i$ th agent, respectively. Here, $q_{\gamma},p_{\gamma}$ , and $u_{\gamma}$ are the position, velocity, and acceleration of a given target ( $\gamma$ -agent) satisfying

	$\displaystyle\dot{q}_{\gamma}$	$\displaystyle=p_{\gamma},$
	$\displaystyle\dot{p}_{\gamma}$	$\displaystyle=u_{\gamma}.$

A new input parameter $u_{i}$ will be determined later. Depending on the information of the target, we choose two different $u_{i}$ ’s and analyze the corresponding asymptotic behaviors. The argument is straightforward, and thus the reader familiar with target tracking problems in $\mathbb{R}^{3}$ may skip this section.

If $u_{i}=0$ , then the above model corresponds to the one in Olfati-Saber’s seminal paper [11]. As studied in [11], the system of equations can be decomposed as two second-order systems for the structural dynamics and translational dynamics. For simplicity, we assume that $\psi_{ij}=0$ and $\sigma_{ij}=\sigma_{ji}$ for all indices $i$ and $j$ in $\{1,\ldots,N\}$ . We note that the effect of the flocking term $\sum_{j=1}^{N}\frac{\psi_{ij}}{N}(p_{j}-p_{i})$ is negligible, when $\max_{1\leq i,j\leq N}\|p_{j}-p_{i}\|\ll 1$ . See the numerical simulations in Figures 6 and 7.

Let

q_{c}=\frac{1}{N}\sum_{i=1}^{N}q_{i},\quad p_{c}=\frac{1}{N}\sum_{i=1}^{N}p_{i},

and

\displaystyle x_{i}=q_{i}-q_{c},\quad v_{i}=p_{i}-p_{c}.

(2.3)

Then, the above dynamics can be decomposed into the translational dynamics (2.4) and the structural dynamics (2.5):

\displaystyle\begin{aligned} \dot{q}_{c}&=p_{c},\\ \dot{p}_{c}&=c_{q}(q_{\gamma}-q_{c})+c_{p}(p_{\gamma}-p_{c})+u_{i},\end{aligned}

(2.4)

and

\displaystyle\begin{aligned} \dot{x}_{i}&=x_{i},\\ \dot{v}_{i}&=\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(x_{j}-x_{i})-c_{q}x_{i}-c_{p}v_{i}.\end{aligned}

(2.5)

The structural dynamics part in (2.5) has been analyzed in [11].

We focus on the translational dynamics part in (2.4) for two different cases of $u_{i}$ . We first suppose that all of the position $p_{\gamma}$ , velocity $q_{\gamma}$ , and acceleration $u_{\gamma}$ of the target are given. In this case, it is natural to choose $u_{i}:=u_{\gamma}$ . Let

q_{d}=q_{c}-q_{\gamma},\quad p_{d}=p_{c}-p_{\gamma}.

Then the translational dynamics in (2.4) can be rewritten as

	$\displaystyle\dot{q}_{d}$	$\displaystyle=p_{d},$
	$\displaystyle\dot{p}_{d}$	$\displaystyle=-c_{q}q_{d}-c_{p}p_{d}.$

This is a simple linear system of ODEs and it has the following solution;

	$\displaystyle q_{d}(t)$	$\displaystyle=\frac{1}{2\sqrt{c_{p}^{2}-4c_{q}}}\bigg{[}-c_{p}q_{d}(0)e^{\frac{1}{2}t\left(-\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}+q_{d}(0)\sqrt{c_{p}^{2}-4c_{q}}e^{\frac{1}{2}t\left(-\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}$
		$\displaystyle\quad\qquad\qquad\qquad\qquad+c_{p}q_{d}(0)e^{\frac{1}{2}t\left(\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}+q_{d}(0)\sqrt{c_{p}^{2}-4c_{q}}e^{\frac{1}{2}t\left(\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}$
		$\displaystyle\quad\qquad\qquad\qquad\qquad\qquad-2p_{d}(0)e^{\frac{1}{2}t\left(-\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}+2p_{d}(0)e^{\frac{1}{2}t\left(\sqrt{c_{p}^{2}-4c_{q}}-c_{p}\right)}\bigg{]}.$

Therefore, we can easily check that $q_{d}$ and $p_{d}$ converge to zero exponentially. This means that the complete rendezvous with an exponential decay rate occurs for any positive $c_{q}$ and $c_{p}$ .

If we only know the position and velocity of the target, we cannot expect a complete rendezvous. On the other hand, we can control the maximum position difference between the target and agents if the tracking coefficients for the target are sufficiently large. We refer to [4, 5] for related issues.

For $u_{i}=0$ , the translational dynamics is given by

	$\displaystyle\dot{q}_{d}$	$\displaystyle=p_{d},$
	$\displaystyle\dot{p}_{d}$	$\displaystyle=-c_{q}q_{d}-c_{p}p_{d}-u_{\gamma}.$

As we mentioned above, we cannot expect the complete rendezvous for this case. Alternatively, to obtain the practical rendezvous estimate, we additionally assume that the acceleration of the target is bounded:

\displaystyle\limsup\|u_{\gamma}\|\leq C_{\gamma},

(2.6)

for some $C_{\gamma}>0$ . Then we define auxiliary variables as follows.

X^{1}_{d}=\langle q_{d},q_{d}\rangle,\quad X^{2}_{d}=\langle q_{d},p_{d}\rangle,\quad X^{3}_{d}=\langle p_{d},p_{d}\rangle.

By the system of the translational dynamics, we can obtain

	$\displaystyle\dot{X}^{1}_{d}$	$\displaystyle=2X^{2}_{d},$
	$\displaystyle\dot{X}^{2}_{d}$	$\displaystyle=X^{3}_{d}-c_{q}X^{1}_{d}-c_{p}X^{2}_{d}-\langle q_{d},u_{\gamma}\rangle,$
	$\displaystyle\dot{X}^{3}_{d}$	$\displaystyle=-2c_{q}X^{2}_{d}-2c_{p}X^{3}_{d}-2\langle p_{d},u_{\gamma}\rangle.$

We rewrite the above system of equations as the following inhomogeneous linear system of ODEs:

\displaystyle\dot{X}_{d}=A_{d}X_{d}+F_{d},

where $X_{d}=(X^{1}_{d},X^{2}_{d},X^{3}_{d})^{T}$ and $F_{d}=(0,-\langle q_{d},u_{\gamma}\rangle,-2\langle p_{d},u_{\gamma}\rangle)^{T}$ , and the coefficient matrix is given by

\displaystyle M_{d}=\begin{bmatrix}0&2&0\\ -c_{q}&-c_{p}&1\\ 0&-2c_{q}&-2c_{p}\end{bmatrix}.

Note that $M_{d}$ has the following eigenvalues.

\left\{-c_{p},~{}-c_{p}-\sqrt{c_{p}^{2}-4c_{q}},~{}-c_{p}+\sqrt{c_{p}^{2}-4c_{q}}\right\}.

Let $D_{d}<0$ be the greatest real part in the above eigenvalues and let

-\mu_{d}=D_{d}.

Then, we have

	$\displaystyle\frac{d}{dt}\\|X_{d}\\|^{2}$	$\displaystyle=2\langle X_{d},M_{d}X_{d}\rangle+2\langle X_{d},F_{d}\rangle$
		$\displaystyle\leq-2\mu_{d}\\|X_{d}\\|^{2}+2\\|X_{d}\\|\\|F_{d}\\|,$

this implies that

\displaystyle\frac{d}{dt}\|X_{d}\|\leq-\mu_{d}\|X_{d}\|+\|F_{d}\|.

From elementary calculations, it follows that for any $\epsilon>0$ ,

	$\displaystyle\\|F_{d}\\|$	$\displaystyle\leq\\|q_{d}\\|\\|u_{\gamma}\\|+2\\|p_{d}\\|\\|u_{\gamma}\\|$
		$\displaystyle\leq\frac{\epsilon\\|q_{d}\\|^{2}}{2}+\frac{1}{2\epsilon}\\|u_{\gamma}\\|^{2}+\frac{\epsilon\\|p_{d}\\|^{2}}{2}+\frac{2}{\epsilon}\\|u_{\gamma}\\|^{2}$
		$\displaystyle\leq\epsilon\\|X_{d}\\|+\frac{5}{2\epsilon}\\|u_{\gamma}\\|^{2}.$

We choose $\displaystyle\epsilon=\mu_{d}/2$ and use the Gronwall inequality and (2.6) to obtain that

	$\displaystyle\\|X_{d}\\|$	$\displaystyle\leq e^{-(\mu_{d}-\epsilon)t}\\|X_{d}(0)\\|+\frac{5}{2\epsilon}e^{(\mu_{d}-\epsilon)t}\int_{0}^{t}\\|u_{\gamma}(s)\\|^{2}e^{(\mu_{d}-\epsilon)s}ds$
		$\displaystyle\leq e^{-(\mu_{d}-\epsilon)t}\\|X_{d}(0)\\|+C_{\gamma}^{2}\frac{5}{2\epsilon}e^{-(\mu_{d}-\epsilon)t}\frac{e^{(\mu_{d}-\epsilon)t}-1}{\mu_{d}-\epsilon}.$

This implies that

\limsup\|X_{d}\|\leq\frac{10C_{\gamma}^{2}}{\mu_{d}^{2}}.

Thus, if we choose a sufficiently large tracking coefficients $c_{q},c_{p}>0$ , then we obtain that

\limsup_{t\to\infty}\|q_{i}(t)-q_{\gamma}(t)\|,~{}\limsup_{t\to\infty}\|p_{i}(t)-p_{\gamma}(t)\|\ll 1.

3. Generalized rotation operator on sphere and reference frame decomposition

In this section, we decompose our model (1.2) on $\mathbb{S}^{2}$ into structural dynamics and translational dynamics. Due to the complexity of (1.2), the decomposition of agents’ positions into a sum of two vectors as the model in $\mathbb{R}^{3}$ is not suitable for our case. Instead, we observe that a rigid body motion on $\mathbb{S}^{2}$ can be used as a reference frame. Choosing an appropriate rigid body motion, our model can be represented as the composition of a rigid body motion and local alignment dynamics. The rigid body motion can be derived based on the angular velocity tensor $W_{\gamma}(t)$ of the $\gamma$ -agent and a generalized rotation operator $S_{\gamma}$ along the given target described below. Recall the given $\gamma$ -agent trajectory on $\mathbb{S}^{2}$ :

\dot{q}_{\gamma}=p_{\gamma},

where $q_{\gamma}\in\mathbb{S}^{2}$ and $p_{\gamma}\in T_{x}\mathbb{S}^{2}$ are the position and velocity of the given $\gamma$ -agent, respectively.

Let

w_{\gamma}=q_{\gamma}\times p_{\gamma}.

By elementary calculation, we have $q_{\gamma}\times w_{\gamma}=-p_{\gamma}$ and

\displaystyle\dot{q}_{\gamma}=w_{\gamma}\times q_{\gamma}.

For the angular velocity vector $w_{\gamma}=(w_{\gamma}^{1},w_{\gamma}^{2},w_{\gamma}^{3})^{T}$ , we define the angular velocity tensor $W_{\gamma}(t)$ of the $\gamma$ -agent by

\displaystyle W_{\gamma}^{t}=\begin{bmatrix}0&-w_{\gamma}^{3}(t)&w_{\gamma}^{2}(t)\\ w_{\gamma}^{3}(t)&0&-w_{\gamma}^{1}(t)\\ -w_{\gamma}^{2}(t)&w_{\gamma}^{1}(t)&0\end{bmatrix}.

From the above notation, the equation for the $\gamma$ -agent is written by

\displaystyle\dot{q}_{\gamma}=p_{\gamma}=W_{\gamma}^{t}q_{\gamma}.

Now, we consider the following system of ODEs:

\displaystyle\dot{x}(t)=W_{\gamma}^{t}x(t).

(3.1)

We can define the corresponding solution operator $S_{\gamma}(x_{0},t)=S_{\gamma}^{t}x_{0}:\mathbb{S}^{2}\times[0,\infty)\mapsto\mathbb{S}^{2}$ such that

\displaystyle S_{\gamma}^{t}x_{0}=x(t;x_{0}),

(3.2)

where $x(t;x_{0})$ is the solution to (3.1) subject to

\displaystyle x(0;x_{0})=x_{0}\in\mathbb{S}^{2}.

(3.3)

One can easily check that $S_{\gamma}^{t}$ is a rigid body motion on $\mathbb{S}^{2}$ .

Lemma 3.1.

Let $x_{\gamma}(t)\in\mathbb{S}^{2}$ be the position of a $\gamma$ -agent which is a $C^{2}$ function with respect to $t\geq 0$ . For the given $\gamma$ -agent, the solution operator $S_{\gamma}^{t}$ defined above is represented by a matrix and the matrix product. Moreover, for any $x,y\in\mathbb{R}^{3}$ ,

\|x\|^{2}=\|S_{\gamma}^{t}x\|^{2},\quad\langle x,y\rangle=\langle S_{\gamma}^{t}x,S_{\gamma}^{t}y\rangle.

Proof.

Let $x_{\gamma}(t)$ be a given $C^{2}$ function with $\|x_{\gamma}(t)\|=1$ . We define the solution operator $S_{\gamma}^{t}$ by (3.1)-(3.3). Take any two vectors $x_{1}^{0}$ and $x_{2}^{0}$ on $\mathbb{S}^{2}$ . Let(3.2)

x_{1}(t)=S_{\gamma}^{t}x_{1}^{0},\quad x_{2}(t)=S_{\gamma}^{t}x_{2}^{0}.

Equivalently,

\dot{x}_{1}(t)=W_{\gamma}^{t}x_{1}(t),\quad\dot{x}_{2}(t)=W_{\gamma}^{t}x_{2}(t),

subject to

x_{1}(0)=x_{1}^{0},\quad x_{2}(0)=x_{2}^{0}.

Then we have

\dot{x}_{1}(t)-\dot{x}_{2}(t)=W_{\gamma}^{t}(x_{1}(t)-x_{2}(t)).

This implies that

\displaystyle\frac{1}{2}\frac{d}{dt}\|x_{1}(t)-x_{2}(t)\|^{2}

\displaystyle=\langle x_{1}(t)-x_{2}(t),W_{\gamma}^{t}(x_{1}(t)-x_{2}(t))\rangle.

We note that $W_{\gamma}$ is a skew symmetric matrix and this implies that

	$\displaystyle\langle x_{1}(t)-x_{2}(t),W_{\gamma}^{t}(x_{1}(t)-x_{2}(t))\rangle$	$\displaystyle=\langle W_{\gamma}^{T}(t)(x_{1}(t)-x_{2}(t)),x_{1}(t)-x_{2}(t)\rangle$
		$\displaystyle=-\langle W_{\gamma}(t)(x_{1}(t)-x_{2}(t)),x_{1}(t)-x_{2}(t)\rangle$
		$\displaystyle=-\langle x_{1}(t)-x_{2}(t),W_{\gamma}(t)(x_{1}(t)-x_{2}(t))\rangle.$

Therefore, we can obtain that

\langle x_{1}(t)-x_{2}(t),W_{\gamma}^{t}(x_{1}(t)-x_{2}(t))\rangle=0

and

\displaystyle\frac{d}{dt}\|x_{1}(t)-x_{2}(t)\|^{2}=0.

Since we choose $x_{1}^{0}$ and $x_{2}^{0}$ arbitrary, $S_{\gamma}^{t}:\mathbb{S}^{2}\mapsto\mathbb{S}^{2}$ is a rigid body motion of $\mathbb{S}^{2}$ . This implies that $S_{\gamma}^{t}$ is represented by a matrix and the matrix product. Moreover, the following holds.

\|x\|^{2}=\|S_{\gamma}^{t}x\|^{2},\quad\langle x,y\rangle=\langle S_{\gamma}^{t}x,S_{\gamma}^{t}y\rangle,

for any $x,y\in\mathbb{R}^{3}$ . ∎

In $\mathbb{R}^{3}$ , the agent’s position can be decomposed into a sum of two vectors as described in (2.3)-(2.5). Similarly, the agent’s position on $\mathbb{S}^{2}$ is expressed as the composition of the translational operator $S_{\gamma}^{t}$ and the structural vector $x_{i}$ :

\displaystyle q_{i}(t)=S_{\gamma}^{t}x_{i}(t).

(3.4)

Notice that $x_{\gamma}(t):=q_{\gamma}(0)$ is a time-independent fixed point on $\mathbb{S}^{2}$ and satisfies

\displaystyle q_{\gamma}(t)=S_{\gamma}^{t}x_{\gamma}(t).

(3.5)

In the proposition below, we derive a second-order system of $x_{i}$ in the moving frame.

Proposition 3.2.

Let $(q_{\gamma}(t),p_{\gamma}(t))$ be a given $\gamma$ -agent satisfying

\dot{q}_{\gamma}=p_{\gamma},

where $q_{\gamma}\in\mathbb{S}^{2}$ and $p_{\gamma}\in T_{x}\mathbb{S}^{2}$ . Let $S_{\gamma}^{t}$ be the solution operator defined by (3.1)-(3.3). If (3.4) and (3.5) hold, then the followings are equivalent.

(1)

$\{(x_{i}(t),v_{i}(t))\}_{i=1}^{N}$ satisfies the following structural system of ODEs:

\displaystyle\begin{aligned} \dot{x}_{i}&=v_{i},\\ \dot{v}_{i}&=-\frac{\|v_{i}\|^{2}}{\|x_{i}\|^{2}}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|x_{i}\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})\\ &\qquad\qquad\qquad\qquad+c_{q}(\|x_{i}\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i},\end{aligned}

(3.6)

subject to initial data $x_{i}(0)\in\mathbb{S}^{2}$ , $v_{i}(0)\in T_{x_{i}(0)}\mathbb{S}^{2}$ for all $i\in\{1,\ldots,N\}$ .

(2)

$\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ is the solution to main system (1.2) subject to (1.3) with

$\displaystyle U_{i}=2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i}+S_{\gamma}^{t}A_{i}.$ (3.7)

Proof.

For any $x_{0}\in\mathbb{S}^{2}$ , we consider $x(t)=S_{\gamma}^{t}x_{0}$ . Then

\displaystyle\dot{S}_{\gamma}^{t}x_{0}=\frac{d}{dt}(S_{\gamma}^{t}x_{0})=\dot{x}(t)=W_{\gamma}^{t}x(t)=W_{\gamma}^{t}S_{\gamma}^{t}x_{0}.

(3.8)

Since $x_{0}$ is arbitrary and $S_{\gamma}^{t}$ is a $3\times 3$ matrix by Lemma 3.1, we have

\displaystyle\dot{S}_{\gamma}^{t}=W_{\gamma}^{t}S_{\gamma}^{t}.

(3.9)

We note that for any $x\in\mathbb{R}^{3}$ ,

\displaystyle W_{\gamma}^{t}x=w_{\gamma}\times x.

(3.10)

We first prove that if $\{(x_{i}(t),v_{i}(t))\}_{i=1}^{N}$ satisfies (3.6), then $\{(q_{i}(t),\dot{q}_{i}(t))\}_{i=1}^{N}$ is the solution to the main system with (3.7), where $q_{i}(t)=S_{\gamma}^{t}x_{i}(t)$ . By the definition,

\frac{d}{dt}q_{i}=\dot{S}_{\gamma}^{t}x_{i}+S_{\gamma}^{t}\dot{x}_{i}.

Motivated by the above, we naturally define the corresponding velocity as follows.

\displaystyle p_{i}=\dot{S}_{\gamma}^{t}x_{i}+S_{\gamma}^{t}\dot{x}_{i}.

(3.11)

Thus, we have

\displaystyle\frac{d}{dt}p_{i}=\ddot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}+S_{\gamma}^{t}\ddot{x}_{i}.

By (3.6) and Lemma 3.1,

\displaystyle\begin{aligned} S_{\gamma}^{t}\ddot{x}_{i}=&-\frac{\|v_{i}\|^{2}}{\|x_{i}\|^{2}}S_{\gamma}^{t}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}\left[\|x_{i}\|^{2}S_{\gamma}^{t}x_{j}-\langle x_{i},x_{j}\rangle S_{\gamma}^{t}x_{i}\right]\\ &\quad+c_{q}\left[\|x_{i}\|^{2}S_{\gamma}^{t}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle S_{\gamma}^{t}x_{i}\right]-c_{p}S_{\gamma}^{t}v_{i}+S_{\gamma}^{t}A_{i}.\end{aligned}

(3.12)

From the property of $S_{\gamma}^{t}$ in Lemma 3.1, it follows that

\|x_{i}\|^{2}=\|S_{\gamma}^{t}x_{i}\|^{2},\quad\langle x_{i},x_{j}\rangle=\langle S_{\gamma}^{t}x_{i},S_{\gamma}^{t}x_{j}\rangle.

As [6, 7, 8], we can easily prove that

\displaystyle x_{i}(t)\in\mathbb{S}^{2},\quad v_{i}(t)\in T_{x_{i}(t)}\mathbb{S}^{2},\quad\mbox{ for all}\quad t\geq 0,~{}i\in\{1,\ldots,N\}.

(3.13)

By this modulus conservation and (3.12),

\displaystyle\begin{aligned} S_{\gamma}^{t}\ddot{x}_{i}=&-\|v_{i}\|^{2}q_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}\left[\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i}\right]\\ &\quad+c_{q}\left[\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i}\right]+c_{p}\left[W_{\gamma}^{t}q_{i}-p_{i}\right]+S_{\gamma}^{t}A_{i}.\end{aligned}

(3.14)

Here, we used (3.9) and (3.11) to obtain

\displaystyle-S_{\gamma}^{t}v_{i}=W_{\gamma}^{t}q_{i}-p_{i}.

(3.15)

By (3.8), (3.9), (3.15) and the definition of $q_{i}$ and $p_{i}$ ,

\displaystyle\begin{aligned} \ddot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}&=\dot{W}_{\gamma}^{t}S_{\gamma}^{t}x_{i}+W_{\gamma}^{t}\dot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}\\ &=\dot{W}_{\gamma}^{t}q_{i}+W_{\gamma}^{t}W_{\gamma}^{t}q_{i}+2W_{\gamma}^{t}S_{\gamma}^{t}\dot{x}_{i}\\ &=\dot{W}_{\gamma}^{t}q_{i}+W_{\gamma}^{t}W_{\gamma}^{t}q_{i}+2W_{\gamma}^{t}(p_{i}-W_{\gamma}^{t})q_{i})\\ &=\dot{W}_{\gamma}^{t}q_{i}-W_{\gamma}^{t}W_{\gamma}^{t}q_{i}+2W_{\gamma}^{t}p_{i}.\end{aligned}

(3.16)

Clearly, by the skew symmetric property of $W_{\gamma}$ ,

	$\displaystyle\\|p_{i}\\|^{2}$	$\displaystyle=\\|W_{\gamma}^{t}S_{\gamma}^{t}x_{i}\\|^{2}+2\langle W_{\gamma}^{t}S_{\gamma}^{t}x_{i},S_{\gamma}^{t}\dot{x}_{i}\rangle+\\|S_{\gamma}^{t}\dot{x}_{i}\\|^{2}$
		$\displaystyle=\\|W_{\gamma}^{t}q_{i}\\|^{2}+2\langle W_{\gamma}^{t}q_{i},p_{i}-W_{\gamma}q_{i}\rangle+\\|v_{i}\\|^{2}$
		$\displaystyle=\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle+\\|v_{i}\\|^{2}.$

This implies that

\displaystyle-\|v_{i}\|^{2}=-\|p_{i}\|^{2}+\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle.

(3.17)

By (3.16) and (3.17), we have

\displaystyle\begin{aligned} \ddot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}-\|v_{i}\|^{2}q_{i}&=-\|p_{i}\|^{2}q_{i}+\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle q_{i}-W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\\ &\quad-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle q_{i}+2W_{\gamma}^{t}p_{i}+\dot{W}_{\gamma}^{t}q_{i}.\end{aligned}

(3.18)

Thus, by (3.14) and (3.18),

\displaystyle\begin{aligned} \dot{p}&=\ddot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}+S_{\gamma}^{t}\ddot{x}_{i}\\ &=-\|p_{i}\|^{2}q_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})+c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})\\ &\quad+c_{p}(W_{\gamma}^{t}q_{i}-p_{i})+\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle q_{i}-W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\\ &\quad-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle q_{i}+2W_{\gamma}^{t}p_{i}+\dot{W}_{\gamma}^{t}q_{i}+S_{\gamma}^{t}A_{i}.\end{aligned}

(3.19)

We note that for any $x\in\mathbb{R}^{3}$ ,

\displaystyle W_{\gamma}^{t}x=w_{\gamma}\times x.

(3.20)

From (3.19)-(3.20) and the modulus conservation property of $S_{\gamma}^{t}$ with $x_{i}(t)\in\mathbb{S}^{2}$ , it follows that

	$\displaystyle\dot{p}$	$\displaystyle=-\frac{\\|p_{i}\\|^{2}}{\\|q_{i}\\|^{2}}q_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\\|q_{i}\\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})+c_{q}(\\|q_{i}\\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})$
		$\displaystyle\quad+c_{p}(w_{\gamma}\times q_{i}-p_{i})+2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}\times q_{i}+S_{\gamma}^{t}A_{i}.$

Now, if we choose $A_{i}$ such as

2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i}+S_{\gamma}^{t}A_{i}=0,

then our model corresponds to $u_{i}=0$ case in the flat space case, and if we choose $A_{i}=0$ then our model corresponds to $u_{i}=u_{\gamma}$ case in the flat space case. From the uniqueness of the solution to the main system, we obtain the desired result.

We next prove that if $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ is the solution to the main system with (3.7), then $\{(x_{i}(t),\dot{x}_{i}(t))\}_{i=1}^{N}$ satisfies (3.6), where $x_{i}(t)=S_{\gamma}^{-1}(t)q_{i}(t)$ . By the first equation of (1.2), we have

\displaystyle p_{i}=\dot{q}_{i}=\dot{S}_{\gamma}^{t}x_{i}+S_{\gamma}^{t}\dot{x}_{i}=W_{\gamma}^{t}S_{\gamma}^{t}x_{i}+S_{\gamma}^{t}\dot{x}_{i}.

(3.21)

This implies that

\displaystyle\begin{aligned} \ddot{q}_{i}&=\ddot{S}_{\gamma}^{t}x_{i}+2\dot{S}_{\gamma}^{t}\dot{x}_{i}+S_{\gamma}^{t}\ddot{x}_{i}\\ &=\dot{W}_{\gamma}^{t}S_{\gamma}^{t}x_{i}+W_{\gamma}^{t}W_{\gamma}^{t}S_{\gamma}^{t}x_{i}+2W_{\gamma}^{t}S_{\gamma}^{t}\dot{x}_{i}+S_{\gamma}^{t}\ddot{x}_{i}.\end{aligned}

(3.22)

By (3.21), we have

\displaystyle\begin{aligned} \|p_{i}\|^{2}&=\|W_{\gamma}^{t}S_{\gamma}^{t}x_{i}\|^{2}+2\langle W_{\gamma}^{t}S_{\gamma}^{t}x_{i},S_{\gamma}^{t}\dot{x}_{i}\rangle+\|S_{\gamma}^{t}\dot{x}_{i}\|^{2}\\ &=\|W_{\gamma}^{t}q_{i}\|^{2}+2\langle W_{\gamma}^{t}q_{i},p_{i}-W_{\gamma}q_{i}\rangle+\|v_{i}\|^{2}\\ &=\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle+\|v_{i}\|^{2}.\end{aligned}

(3.23)

The second equation in (1.2) and $q_{i}(t)\in\mathbb{S}^{2}$ imply that

\displaystyle\begin{aligned} \ddot{q}_{i}&=-\|p_{i}\|^{2}q_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|q_{i}\|^{2}q_{j}-\langle q_{i},q_{j}\rangle q_{i})+c_{q}(\|q_{i}\|^{2}q_{\gamma}-\langle q_{i},q_{\gamma}\rangle q_{i})\\ &\quad+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i},\\ &=-\|p_{i}\|^{2}S_{\gamma}^{t}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|S_{\gamma}^{t}x_{i}\|^{2}S_{\gamma}^{t}x_{j}-\langle S_{\gamma}^{t}x_{i},S_{\gamma}^{t}x_{j}\rangle S_{\gamma}^{t}x_{i})\\ &\quad+c_{q}(\|S_{\gamma}^{t}x_{i}\|^{2}S_{\gamma}^{t}x_{\gamma}-\langle S_{\gamma}^{t}x_{i},S_{\gamma}^{t}x_{\gamma}\rangle S_{\gamma}^{t}x_{i})+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i}.\end{aligned}

From the property of $S_{\gamma}^{t}$ in Lemma 3.1, it follows that

\displaystyle\begin{aligned} \ddot{q}_{i}&=-\|p_{i}\|^{2}S_{\gamma}^{t}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|x_{i}\|^{2}S_{\gamma}^{t}x_{j}-\langle x_{i},x_{j}\rangle S_{\gamma}^{t}x_{i})\\ &\quad+c_{q}\left(\|x_{i}\|^{2}S_{\gamma}^{t}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle S_{\gamma}^{t}x_{i}\right)+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i}.\end{aligned}

(3.24)

By (3.22)–(3.24), (3.23)

\displaystyle\begin{aligned} S_{\gamma}^{t}\ddot{x}_{i}&=-\left[\dot{W}_{\gamma}^{t}S_{\gamma}^{t}x_{i}+W_{\gamma}^{t}W_{\gamma}^{t}S_{\gamma}^{t}x_{i}+2W_{\gamma}^{t}S_{\gamma}^{t}\dot{x}_{i}\right]\\ &\quad-\|p_{i}\|^{2}S_{\gamma}^{t}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|x_{i}\|^{2}S_{\gamma}^{t}x_{j}-\langle x_{i},x_{j}\rangle S_{\gamma}^{t}x_{i})\\ &\quad+c_{q}\left(\|x_{i}\|^{2}S_{\gamma}^{t}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle S_{\gamma}^{t}x_{i}\right)+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i}\\ &=-\left[\dot{W}_{\gamma}^{t}S_{\gamma}^{t}x_{i}+W_{\gamma}^{t}W_{\gamma}^{t}S_{\gamma}^{t}x_{i}+2W_{\gamma}^{t}S_{\gamma}^{t}\dot{x}_{i}\right]\\ &\quad-\left(\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle+\|v_{i}\|^{2}\right)S_{\gamma}^{t}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|x_{i}\|^{2}S_{\gamma}^{t}x_{j}-\langle x_{i},x_{j}\rangle S_{\gamma}^{t}x_{i})\\ &\quad+c_{q}\Big{(}\|x_{i}\|^{2}S_{\gamma}^{t}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle S_{\gamma}^{t}x_{i}\Big{)}+c_{p}({P}_{q_{\gamma}\rightarrow q_{i}}(p_{\gamma})-p_{i})+U_{i}.\end{aligned}

Note that

	$\displaystyle-\left[\dot{W}_{\gamma}^{t}S_{\gamma}^{t}x_{i}+W_{\gamma}^{t}W_{\gamma}^{t}S_{\gamma}^{t}x_{i}+2W_{\gamma}^{t}S_{\gamma}^{t}\dot{x}_{i}\right]-\Big{(}\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle\Big{)}S_{\gamma}^{t}x_{i}$
	$\displaystyle\qquad\qquad=-\left[\dot{W}_{\gamma}^{t}q_{i}+W_{\gamma}^{t}W_{\gamma}^{t}q_{i}+2W_{\gamma}^{t}(p_{i}-W_{\gamma}q_{i})\right]-\Big{(}\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle\Big{)}q_{i}$
	$\displaystyle\qquad\qquad=-\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle q_{i}+W_{\gamma}^{t}W_{\gamma}^{t}q_{i}+2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle q_{i}-2W_{\gamma}^{t}p_{i}-\dot{W}_{\gamma}^{t}q_{i}$
	$\displaystyle\qquad\qquad=-2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})-\dot{w}_{\gamma}\times q_{i}.$

Therefore, by the property of $S_{\gamma}$ and the above two equalities, we obtain that $\{(x_{i}(t),v_{i}(t))\}_{i=1}^{N}$ satisfies (3.6) with (3.7).

∎

4. Reduction to a linearized system with a negative definite coefficient matrix

In this section, we derive a linearized system from the structural system in (3.6). We define auxiliary variables motivated by the flat case in Section 2 and we extract leading order terms using $\|q_{i}(t)\|=1$ and $\langle q_{i}(t),p_{i}(t)\rangle=0$ for all $t\geq 0$ and $i\in\{1,\ldots,N\}$ . In the system with respect to auxiliary variables, leading order terms form an inhomogeneous linear system of ODEs with a negative definite coefficient matrix.

We consider the following system of ODEs with $\sigma_{ij}=\sigma>0$ and $c_{q},c_{p}>0$ .

\displaystyle\begin{aligned} \dot{x}_{i}&=v_{i},\\ \dot{v}_{i}&=-\frac{\|v_{i}\|^{2}}{\|x_{i}\|^{2}}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\|x_{i}\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})\\ &\qquad\qquad\qquad\qquad\qquad+c_{q}(\|x_{i}\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}.\end{aligned}

(4.1)

For consistency, we additionally assume that for all $t\geq 0$ ,

\langle A_{i}(t),x_{i}(t)\rangle=0,\quad\mbox{for all}~{}i\in\{1,\ldots,N\},

and the initial data satisfies

\|x_{i}(0)\|=1\quad\mbox{and}\quad\langle v_{i}(0),x_{i}(0)\rangle=0,\quad\mbox{for all}~{}i\in\{1,\ldots,N\}.

We now define the auxiliary variables as follows.

\displaystyle X_{\gamma}^{1}=\frac{1}{N}\sum_{i=1}^{N}\|x_{i}-x_{\gamma}\|^{2},\quad X_{\gamma}^{2}=\frac{1}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},v_{i}\rangle,\quad X_{\gamma}^{3}=\frac{1}{N}\sum_{i=1}^{N}\langle v_{i},v_{i}\rangle,

and

	$\displaystyle X^{1}=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle x_{i}-x_{k},x_{i}-x_{k}\rangle,\quad X^{2}=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle v_{i}-v_{k},x_{i}-x_{k}\rangle,$
	$\displaystyle X^{3}=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle v_{i}-v_{k},v_{i}-v_{k}\rangle.$

We also define the corresponding inhomogeneous terms as follows.

	$\displaystyle F_{\gamma}^{1}$	$\displaystyle=0,$
	$\displaystyle F_{\gamma}^{2}$	$\displaystyle=-\frac{1}{N}\sum_{i=1}^{N}\frac{\\|v_{i}\\|^{2}}{2}\\|x_{i}-x_{\gamma}\\|^{2}+\frac{\sigma}{4N^{2}}\sum_{i,j=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{\gamma}\\|^{2}$
		$\displaystyle\quad+\frac{c_{q}}{4N}\sum_{i=1}^{N}\\|x_{i}-x_{\gamma}\\|^{4}+\frac{1}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},A_{i}\rangle,$
	$\displaystyle F_{\gamma}^{3}$	$\displaystyle=\frac{2}{N}\sum_{i=1}^{N}\langle v_{i},A_{i}\rangle,$

and

	$\displaystyle F^{1}$	$\displaystyle=0,$
	$\displaystyle F^{2}$	$\displaystyle=-\frac{1}{N^{2}}\sum_{i,k=1}^{N}\frac{\\|v_{i}\\|^{2}+\\|v_{k}\\|^{2}}{2}\\|x_{i}-x_{k}\\|^{2}+\frac{\sigma}{2N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\quad+\frac{c_{q}}{2N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},x_{i}-x_{k}\rangle,$
	$\displaystyle F^{3}$	$\displaystyle=\frac{2}{N^{2}}\sum_{i,k=1}^{N}\left(\\|v_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\\|v_{k}\\|^{2}\langle x_{x},v_{i}\rangle\right)+\frac{2\sigma}{N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\langle x_{i},v_{k}\rangle$
		$\displaystyle\quad+\frac{c_{q}}{N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\frac{2}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},v_{i}-v_{k}\rangle.$

Let

\displaystyle X=(X_{\gamma}^{1},X_{\gamma}^{2},X_{\gamma}^{3},X^{1},X^{2},X^{3})^{T},\quad F=(F_{\gamma}^{1},F_{\gamma}^{2},F_{\gamma}^{3},F^{1},F^{2},F^{3})^{T}.

(4.2)

Proposition 4.1.

For the auxiliary variable $X$ and the inhomogeneous term $F$ , the following holds.

\dot{X}=MX+F,

where the coefficient matrix $M$ is given by

\displaystyle M=\begin{bmatrix}0&2&0&0&0&0\\ -c_{q}&-c_{p}&1&-\sigma/2&0&0\\ 0&-2c_{q}&-2c_{p}&0&\sigma&0\\ 0&0&0&0&2&0\\ 0&0&0&-(c_{q}+\sigma)&-c_{p}&1\\ 0&0&0&0&-2(c_{q}+\sigma)&-2c_{p}\\ \end{bmatrix}.

Proof.

Clearly,

\displaystyle\frac{d}{dt}X_{\gamma}^{1}=2X_{\gamma}^{2}.

For $X_{\gamma}^{2}$ , we have

	$\displaystyle\frac{d}{dt}X_{\gamma}^{2}$	$\displaystyle=X_{\gamma}^{3}+\frac{1}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},\dot{v}_{i}\rangle$
		$\displaystyle=X_{\gamma}^{3}+\frac{1}{N}\sum_{i=1}^{N}\bigg{\langle}x_{i}-x_{\gamma},~{}-\\|v_{i}\\|^{2}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+c_{q}(\\|x_{i}\\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}\bigg{\rangle}$
		$\displaystyle=X_{\gamma}^{3}-\frac{1}{N}\sum_{i=1}^{N}\frac{\\|v_{i}\\|^{2}}{2}\\|x_{i}-x_{\gamma}\\|^{2}+\frac{\sigma}{N^{2}}\sum_{i,j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle$
		$\displaystyle\quad+\frac{c_{q}}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i}\rangle-\frac{c_{p}}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},v_{i}\rangle+\frac{1}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},A_{i}\rangle.$

Note that by $x_{i}\in\mathbb{S}^{2}$ and changing the indices,

\displaystyle\begin{aligned} \sum_{i,j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle&=-\sum_{i,j=1}^{N}\langle x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle\\ &=-\sum_{i,j=1}^{N}\langle x_{\gamma},x_{i}-\langle x_{i},x_{j}\rangle x_{i}\rangle\\ &=-\sum_{i,j=1}^{N}\frac{\|x_{i}-x_{j}\|^{2}}{2}\langle x_{\gamma},x_{i}\rangle\\ &=-\frac{1}{2}\sum_{i,j=1}^{N}\|x_{i}-x_{j}\|^{2}+\frac{1}{4}\sum_{i,j=1}^{N}\|x_{i}-x_{j}\|^{2}\|x_{i}-x_{\gamma}\|^{2}.\end{aligned}

(4.3)

By (4.3), we have

	$\displaystyle\frac{d}{dt}X_{\gamma}^{2}$	$\displaystyle=X_{\gamma}^{3}-\frac{1}{N}\sum_{i=1}^{N}\frac{\\|v_{i}\\|^{2}}{2}\\|x_{i}-x_{\gamma}\\|^{2}-\frac{\sigma}{2}X^{1}+\frac{\sigma}{4N^{2}}\sum_{i,j=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{\gamma}\\|^{2}$
		$\displaystyle\quad-\frac{c_{q}}{N}\sum_{i=1}^{N}\\|x_{i}-x_{\gamma}\\|^{2}+\frac{c_{q}}{4N}\sum_{i=1}^{N}\\|x_{i}-x_{\gamma}\\|^{4}-c_{p}X_{\gamma}^{2}+\frac{1}{N}\sum_{i=1}^{N}\langle x_{i}-x_{\gamma},A_{i}\rangle$
		$\displaystyle=-c_{q}X_{\gamma}^{1}-c_{p}X_{\gamma}^{2}+X_{\gamma}^{3}-\frac{\sigma}{2}X^{1}+F_{\gamma}^{2}.$

Similarly, we have

	$\displaystyle\frac{1}{2}\frac{d}{dt}X_{\gamma}^{3}$	$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\langle v_{i},\dot{v}_{i}\rangle$
		$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\bigg{\langle}v_{i},~{}-\\|v_{i}\\|^{2}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})$
		$\displaystyle\qquad\qquad\qquad\qquad+c_{q}(\\|x_{i}\\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}\bigg{\rangle}$
		$\displaystyle=\frac{\sigma}{N^{2}}\sum_{i,j=1}^{N}\langle v_{i},x_{j}\rangle-\frac{1}{N}\sum_{i=1}^{N}c_{q}\langle v_{i},x_{i}-x_{\gamma}\rangle-\frac{1}{N}\sum_{i=1}^{N}c_{p}\langle v_{i},v_{i}\rangle+\frac{1}{N}\sum_{i=1}^{N}\langle v_{i},A_{i}\rangle.$

Thus, we have

\displaystyle\frac{d}{dt}X_{\gamma}^{3}=-2c_{q}X^{2}_{\gamma}-2c_{p}X^{3}_{\gamma}-\sigma X^{2}+F_{\gamma}^{3}.

(4.4)

For $X^{1}$ ,

\frac{d}{dt}X^{1}=2X^{2}.

Similar to the previous cases, we use the second equation in (4.1) to obtain

	$\displaystyle\frac{d}{dt}X^{2}$	$\displaystyle=X^{3}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle\dot{v}_{i}-\dot{v}_{k},x_{i}-x_{k}\rangle$
		$\displaystyle=X^{3}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\bigg{\langle}-\\|v_{i}\\|^{2}x_{i}+\\|v_{k}\\|^{2}x_{k}+\sum_{j=1}^{N}\frac{\sigma}{N}[-\langle x_{i},x_{j}\rangle x_{i}+\langle x_{k},x_{j}\rangle x_{k}]$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad+c_{q}\left[-\langle x_{i},x_{\gamma}\rangle x_{i}+\langle x_{k},~{}x_{\gamma}\rangle x_{k}\right]-c_{p}v_{i}+c_{p}v_{k},x_{i}-x_{k}\bigg{\rangle}$
		$\displaystyle\qquad+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},x_{i}-x_{k}\rangle.$

By $x_{i}\in\mathbb{S}^{2}$ , we have

	$\displaystyle\frac{d}{dt}X^{2}$	$\displaystyle=X^{3}-\frac{1}{N^{2}}\sum_{i,k=1}^{N}\frac{\\|v_{i}\\|^{2}+\\|v_{k}\\|^{2}}{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-\sigma X^{1}+\frac{\sigma}{4N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{\sigma}{4N^{3}}\sum_{i,j,k=1}^{N}\\|x_{k}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-c_{q}X^{1}+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{k}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-c_{p}X^{2}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},x_{i}-x_{k}\rangle.$

Changing the indices implies that

\frac{d}{dt}X^{2}=-\sigma X^{1}-c_{q}X^{1}-c_{p}X^{2}+X^{3}+F^{2}.

Finally, for $X^{3}$ , we obtain

	$\displaystyle\frac{1}{2}\frac{d}{dt}X^{3}$	$\displaystyle=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\bigg{\langle}-\\|v_{i}\\|^{2}x_{i}+\\|v_{k}\\|^{2}x_{k}+\sum_{j=1}^{N}\frac{\sigma}{N}(-\langle x_{i},x_{j}\rangle x_{i}+\langle x_{k},x_{j}\rangle x_{k})$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad+c_{q}\left[-\langle x_{i},x_{\gamma}\rangle x_{i}+\langle x_{k},x_{\gamma}\rangle x_{k}\right]-c_{p}v_{i}+c_{p}v_{k},~{}v_{i}-v_{k}\bigg{\rangle}$
		$\displaystyle\qquad+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},v_{i}-v_{k}\rangle$
		$\displaystyle=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\left(\\|v_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\\|v_{k}\\|^{2}\langle x_{x},v_{i}\rangle\right)$
		$\displaystyle\quad-\sigma X^{2}+\sum_{i,j,k=1}^{N}\frac{\sigma}{2N^{3}}\\|x_{i}-x_{j}\\|^{2}\langle x_{i},v_{k}\rangle+\sum_{i,j,k=1}^{N}\frac{\sigma}{2N^{3}}\\|x_{k}-x_{j}\\|^{2}\langle x_{k},v_{i}\rangle$
		$\displaystyle\quad-c_{q}X^{2}+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{k}\\|^{2}\langle x_{k},v_{i}\rangle-c_{p}X^{3}$
		$\displaystyle\quad+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},v_{i}-v_{k}\rangle.$

Thus, we conclude that

\frac{d}{dt}X^{3}=-2\sigma X^{2}-2c_{q}X^{2}-2c_{p}X^{3}+F^{3}.

∎

Note that the eigenvalues of the $6\times 6$ coefficient matrix $M$ have the only negative real part. The above result will be used for the complete rendezvous case.

Remark 4.2.

In [8], we use $l^{\infty}$ -framework to obtain a uniform decay estimate which is independent of $N$ . However, due to $X^{2}$ term on the right-hand side of (4.4), we cannot use this $l^{\infty}$ -framework. We obtain only the convergence result depending on $N$ by using the $6\times 6$ system with $l^{2}$ -framework.

For the practical rendezvous result, we use a different framework, weighted $l^{\infty}$ -framework. To obtain $l^{\infty}$ -estimate, we define the following functionals:

\displaystyle X_{i}^{1}=\frac{4\|x_{i}-x_{\gamma}\|^{2}}{4-\|x_{i}-x_{\gamma}\|^{2}},\quad X_{i}^{2}=\frac{16\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}},\quad X_{i}^{3}=\frac{16\langle v_{i},v_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}},

(4.5)

and

\displaystyle\begin{aligned} F_{i}^{1}&=0,\\ F_{i}^{2}&=-\frac{\|v_{i}\|^{2}}{2}\frac{16\|x_{i}-x_{\gamma}\|^{2}}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}+\frac{16\sigma}{N\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle\\ &\quad+\frac{16\langle x_{i}-x_{\gamma},A_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{3}}\\ F_{i}^{3}&=\frac{32\sigma}{N\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle v_{i},x_{j}\rangle+\frac{32\langle v_{i},A_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}+\frac{64\langle v_{i},v_{i}\rangle\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{3}}.\end{aligned}

(4.6)

We note that due to the geometric structure of $\mathbb{S}^{2}$ , the quartic terms with the coefficient $c_{q}$ in $F_{\gamma}^{2}$ and $F^{2}$ appear. Thus, the standard functional $X(t)$ in the previous argument and Section 2 does not work for this practical rendezvous case. For the complete rendezvous case, we will use the energy functional method and Lasalle’s invariance principle to control the quartic terms. However, for the practical rendezvous case, we cannot use the same methodology since the system is not autonomous. Thus, if an extra term with the coefficient $c_{q}$ appears in $F$ , then it is hard to obtain the desired result. Alternatively, using the functionals in (4.5), we can remove the quartic term with the coefficient $c_{q}$ as in (4.6).

By the same argument in Proposition 4.3, we have

\displaystyle\frac{d}{dt}X_{i}^{1}=2X_{i}^{2}.

Using the second equation for the structural system, we obtain the following for $X_{\gamma}^{2}$ .

	$\displaystyle\frac{d}{dt}X_{i}^{2}$	$\displaystyle=X_{i}^{3}+\frac{16\langle x_{i}-x_{\gamma},\dot{v}_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=X_{i}^{3}+16\bigg{\langle}x_{i}-x_{\gamma},~{}-\\|v_{i}\\|^{2}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+c_{q}(\\|x_{i}\\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}\bigg{\rangle}/\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}$
		$\displaystyle\quad+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=X_{i}^{3}-\frac{\\|v_{i}\\|^{2}}{2}\frac{16\\|x_{i}-x_{\gamma}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{16\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle$
		$\displaystyle\quad+\frac{16c_{q}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\langle x_{i}-x_{\gamma},x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i}\rangle-\frac{16c_{p}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\langle x_{i}-x_{\gamma},v_{i}\rangle$
		$\displaystyle\quad+\frac{16\langle x_{i}-x_{\gamma},A_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}.$

Note that

	$\displaystyle\langle x_{i}-x_{\gamma},x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i}\rangle$	$\displaystyle=\langle x_{i}-x_{\gamma},x_{\gamma}\rangle-\langle x_{i},x_{\gamma}\rangle\langle x_{i}-x_{\gamma},x_{i}\rangle$
		$\displaystyle=-\\|x_{i}-x_{\gamma}\\|^{2}-\langle x_{i}-x_{\gamma},x_{\gamma}\rangle\langle x_{i}-x_{\gamma},x_{i}\rangle$
		$\displaystyle=-\\|x_{i}-x_{\gamma}\\|^{2}+\frac{\\|x_{i}-x_{\gamma}\\|^{4}}{4}.$

This implies that

\displaystyle\begin{aligned} \frac{d}{dt}X_{i}^{2}&=X_{i}^{3}-\frac{\|v_{i}\|^{2}}{2}\frac{16\|x_{i}-x_{\gamma}\|^{2}}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}+\frac{16\sigma}{N\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle\\ &\quad-c_{q}X_{i}^{1}-c_{p}X_{i}^{2}+\frac{16\langle x_{i}-x_{\gamma},A_{i}\rangle}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\|x_{i}-x_{\gamma}\|^{2}\right)^{3}}.\end{aligned}

For $X_{i}^{3}$ , we have

	$\displaystyle\frac{d}{dt}X_{\gamma}^{3}$	$\displaystyle=\frac{32\langle v_{i},\dot{v}_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle v_{i},v_{i}\rangle\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=32\bigg{\langle}v_{i},~{}-\\|v_{i}\\|^{2}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})$
		$\displaystyle\qquad\qquad+c_{q}(\\|x_{i}\\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}\bigg{\rangle}/\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}+\frac{64\langle v_{i},v_{i}\rangle\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=\frac{32\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle v_{i},x_{j}\rangle-32c_{q}\frac{\langle v_{i},x_{i}-x_{\gamma}\rangle}{{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}}-32c_{p}\frac{\langle v_{i},v_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}$
		$\displaystyle\quad+\frac{32\langle v_{i},A_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle v_{i},v_{i}\rangle\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=\frac{32\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle v_{i},x_{j}\rangle-2c_{q}X_{i}^{2}-2c_{p}X_{i}^{3}+\frac{32\langle v_{i},A_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle v_{i},v_{i}\rangle\langle x_{i}-x_{\gamma},v_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}.$

In conclusion, we have

	$\displaystyle\frac{d}{dt}X_{i}^{1}$	$\displaystyle=2X_{i}^{2}+F_{i}^{1},$
	$\displaystyle\frac{d}{dt}X_{i}^{2}$	$\displaystyle=-c_{q}X^{1}_{i}-c_{p}X_{i}^{2}+X_{i}^{3}+F_{i}^{2}$
	$\displaystyle\frac{d}{dt}X_{i}^{3}$	$\displaystyle=-2c_{q}X^{2}_{i}-2c_{p}X^{3}_{i}+F_{i}^{3}.$

Therefore, we have proved the following proposition.

Proposition 4.3.

Let

X_{i}=(X_{i}^{1},X_{i}^{2},X_{i}^{3})^{T},\quad F_{i}=(F_{i}^{1},F_{i}^{2},F_{i}^{3})^{T},

where $X_{i}^{k}$ , $F_{i}^{k}$ , $k=1,2,3$ are functionals defined in (4.5) and (4.6).

Then the following holds.

\dot{X}_{i}=M_{\infty}X_{i}+F_{i},

where the coefficient matrix $M_{\infty}$ is given by

\displaystyle M_{\infty}=\begin{bmatrix}0&2&0\\ -c_{q}&-c_{p}&1\\ 0&-2c_{q}&-2c_{p}\end{bmatrix}.

5. Asymptotic analysis on the target tracking models: complete and practical rendezvouses

In this section, we provide the proofs of Theorems 2 and 3 in Section 1. Let $(q_{\gamma},p_{\gamma})$ be the phase of the target. We assume that the target satisfies (1.1) for some continuous $u_{\gamma}(t)\in\mathbb{R}^{3}$ . For the given target $(q_{\gamma}(t),p_{\gamma}(t))$ , let $\{(q_{i}(t),p_{i}(t))\}_{i=1}^{N}$ be the solution to (1.2). By the argument in Section 4, we have the following equivalent system for $x_{i}(t)=S_{\gamma}^{-1}(t)p_{i}(t)$ .

\displaystyle\begin{aligned} \dot{x}_{i}&=v_{i},\\ \dot{v}_{i}&=-\frac{\|v_{i}\|^{2}}{\|x_{i}\|^{2}}x_{i}+\sum_{j=1}^{N}\frac{\sigma_{ij}}{N}(\|x_{i}\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})+c_{q}(\|x_{i}\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i},\end{aligned}

(5.1)

where $S_{\gamma}^{t}$ is the solution operator defined by (3.1)-(3.3). For the angular velocity $w_{\gamma}=q_{\gamma}\times p_{\gamma}$ , $A_{i}$ is the extra control law given by

A_{i}=S_{\gamma}^{-1}(t)U_{i}-2\langle w_{\gamma},q_{i}\rangle S_{\gamma}^{-1}(t)[q_{i}\times p_{i}]-S_{\gamma}^{-1}(t)[\dot{w}_{\gamma}(t)\times q_{i}].

5.1. Complete rendezvouses

We assume that $\sigma_{ij}=\sigma>0$ and $A_{i}=0$ , i.e.,

U_{i}=2\langle w_{\gamma},q_{i}\rangle q_{i}\times p_{i}-\dot{w}_{\gamma}(t)\times q_{i}.

We first use an energy functional method to obtain the convergence result in Theorem 2 without convergence rate. We now define an energy functional $\mathcal{E}=\mathcal{E}(\{(x_{i},v_{i})\}_{i=1}^{N})$ as follows.

\mathcal{E}=\mathcal{E}_{k}+\mathcal{E}_{c},

where $\mathcal{E}_{k}$ is the kinetic energy given by

\mathcal{E}_{k}=\frac{1}{2N}\sum_{i=1}^{N}\|v_{i}\|^{2},

and $\mathcal{E}_{c}$ is the configuration energy given by

\mathcal{E}_{c}=\frac{\sigma}{4N^{2}}\sum_{i,j=1}^{N}\|x_{i}-x_{j}\|^{2}+\frac{c_{q}}{2N}\sum_{i=1}^{N}\|x_{\gamma}-x_{i}\|^{2}.

This energy functional has a dissipation property. To obtain this, we take the inner product between $v_{i}$ and $\dot{v}_{i}$ to obtain

	$\displaystyle\frac{1}{2}\frac{d}{dt}\\|v_{i}\\|^{2}$	$\displaystyle=-\frac{\\|v_{i}\\|^{2}}{\\|x_{i}\\|^{2}}\langle x_{i},v_{i}\rangle+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}\langle x_{j},v_{i}\rangle-\langle x_{i},x_{j}\rangle\langle x_{i},v_{i}\rangle)$
		$\displaystyle\quad\quad+c_{q}(\\|x_{i}\\|^{2}\langle x_{\gamma},v_{i}\rangle-\langle x_{i},x_{\gamma}\rangle\langle x_{i},v_{i}\rangle)-c_{p}\langle v_{i},v_{i}\rangle.$

Using the orthogonality $\langle x_{i},v_{i}\rangle=0$ and $\|x_{i}\|=1$ in (3.13), we have

\displaystyle\frac{1}{2}\frac{d}{dt}\|v_{i}\|^{2}

\displaystyle=\sum_{j=1}^{N}\frac{\sigma}{N}\langle x_{j},v_{i}\rangle+c_{q}\langle x_{\gamma},v_{i}\rangle-c_{p}\|v_{i}\|^{2}.

Therefore,

\frac{d}{dt}\mathcal{E}_{k}=\sum_{i,j=1}^{N}\frac{\sigma}{N^{2}}\langle x_{j},v_{i}\rangle+\frac{c_{q}}{N}\sum_{i=1}^{N}\langle x_{\gamma},v_{i}\rangle-\frac{c_{p}}{N}\sum_{i=1}^{N}\|v_{i}\|^{2}.

Similarly,

	$\displaystyle\frac{d}{dt}\mathcal{E}_{c}$	$\displaystyle=\frac{\sigma}{2N^{2}}\sum_{i,j=1}^{N}\langle x_{i}-x_{j},v_{i}-v_{j}\rangle-\frac{c_{q}}{N}\sum_{i=1}^{N}\langle x_{\gamma},v_{i}\rangle$
		$\displaystyle=-\frac{\sigma}{2N^{2}}\sum_{i,j=1}^{N}(\langle x_{i},v_{j}\rangle+\langle x_{j},v_{i}\rangle)-\frac{c_{q}}{N}\sum_{i=1}^{N}\langle x_{\gamma},v_{i}\rangle$
		$\displaystyle=-\frac{\sigma}{N^{2}}\sum_{i,j=1}^{N}\langle x_{j},v_{i}\rangle-\frac{c_{q}}{N}\sum_{i=1}^{N}\langle x_{\gamma},v_{i}\rangle.$

Therefore, we have

\frac{d}{dt}(\mathcal{E}_{k}+\mathcal{E}_{c})=-\frac{c_{q}}{N}\sum_{i=1}^{N}\|v_{i}\|^{2}=-2c_{q}\mathcal{E}_{k}.

We notice that (5.1) is autonomous, since $x_{\gamma}$ is a constant vector. Moreover, the energy functional $\mathcal{E}$ is zero if and only if

v_{i}=0~{}\mbox{for all}~{}i\in\{1,\ldots,N\}.

We can easily prove that the union of the following two sets is the maximum invariant set of $\mathcal{E}$ .

\left\{\{(x_{i},v_{i})\}_{i=1}^{N}:v_{i}=0,\quad x_{i}=x_{\gamma}~{}\mbox{for all}~{}i\in\{1,\ldots,N\}\right\}

and

\bigg{\{}\{(x_{i},v_{i})\}_{i=1}^{N}:v_{i}=0,\quad\frac{\sigma}{N}\sum_{j=1}^{N}x_{j}+c_{q}x_{\gamma}=0~{}\mbox{for all}~{}i\in\{1,\ldots,N\}\bigg{\}}.

If we assume that $c_{q}>\sigma$ or $\displaystyle\mathcal{E}(0)<\frac{\sigma}{2}\left(1+\frac{c_{q}}{\sigma}\right)^{2}$ , then $\displaystyle\frac{\sigma}{N}\sum_{j=1}^{N}x_{j}+c_{q}x_{\gamma}\neq 0$ . Thus, by Lasalle’s invariance principle,

\|v_{i}(t)\|\to 0\quad\mbox{and}\quad x_{i}(t)\to x_{\gamma}

as $t\to\infty$ . Therefore, we have proved the following proposition.

Proposition 5.1.

If $c_{q}>\sigma$ or $\displaystyle\mathcal{E}(0)<\frac{\sigma}{2}\left(1+\frac{c_{q}}{\sigma}\right)^{2}$ , then

v_{i}(t)\to 0

and

x_{i}(t)\to x_{\gamma}(t)

as $t\to\infty$ for any initial data satisfying $x_{i}(0)\neq-x_{\gamma}(0)$ for all $i\in\{1,\ldots,N\}$ .

Next we consider exponential decay estimates for $\|x_{i}-x_{\gamma}\|$ and $\|v_{i}\|$ . For notational simplicity, we define the following two functionals.

\mathcal{D}_{x}(t)=\max_{1\leq i\leq N}\|x_{i}(t)-x_{\gamma}(t)\|^{2}

and

\mathcal{D}_{v}(t)=\max_{1\leq i\leq N}\|v_{i}(t)\|^{2}.

Proposition 5.2.

Assume that $A_{i}=0$ . Then for the functional $F$ defined in (4.2), the following estimate holds

\|F\|\leq 8(\sigma+c_{q})[\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t)]X^{1}_{\gamma}.

Proof.

By elementary calculation, we have

	$\displaystyle\|F_{\gamma}^{1}\|$	$\displaystyle=0,$
	$\displaystyle\|F_{\gamma}^{2}\|$	$\displaystyle\leq\left(\frac{\mathcal{D}_{v}(t)}{2}+\frac{\sigma\mathcal{D}_{x}(t)}{4}+\frac{c_{q}\mathcal{D}_{x}(t)}{4}\right)X^{1}_{\gamma},$
	$\displaystyle\|F_{\gamma}^{3}\|$	$\displaystyle=0,$

and

	$\displaystyle F^{1}$	$\displaystyle=0,$
	$\displaystyle F^{2}$	$\displaystyle=-\frac{1}{N^{2}}\sum_{i,k=1}^{N}\frac{\\|v_{i}\\|^{2}+\\|v_{k}\\|^{2}}{2}\\|x_{i}-x_{k}\\|^{2}+\frac{\sigma}{2N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\quad+\frac{c_{q}}{2N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},x_{i}-x_{k}\rangle,$
	$\displaystyle F^{3}$	$\displaystyle=\frac{2}{N^{2}}\sum_{i,k=1}^{N}\left(\\|v_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\\|v_{k}\\|^{2}\langle x_{x},v_{i}\rangle\right)+\frac{2\sigma}{N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\langle x_{i},v_{k}\rangle$
		$\displaystyle\quad+\frac{c_{q}}{N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\langle x_{i},v_{k}\rangle+\frac{2}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},v_{i}-v_{k}\rangle.$

Note that

\displaystyle\begin{aligned} \|x_{i}-x_{k}\|^{2}&\leq\|x_{i}-x_{\gamma}+x_{\gamma}-x_{k}\|^{2}\\ &\leq 2\|x_{i}-x_{\gamma}\|^{2}+2\|x_{\gamma}-x_{k}\|^{2}\\ &\leq 4\mathcal{D}_{x}(t),\end{aligned}

(5.2)

and

\displaystyle\begin{aligned} |\langle x_{i},v_{k}\rangle|&=|\langle x_{i}-x_{k},v_{k}\rangle|\\ &\leq|\langle x_{i}-x_{\gamma},v_{k}\rangle|+|\langle x_{\gamma}-x_{k},v_{k}\rangle|\\ &\leq\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t).\end{aligned}

(5.3)

By (5.2) and (5.3), we have

\displaystyle|F^{2}|

\displaystyle\leq\frac{\mathcal{D}_{v}(t)}{N^{2}}\sum_{i,k=1}^{N}\|x_{i}-x_{k}\|^{2}+\frac{2\sigma\mathcal{D}_{x}(t)}{N^{2}}\sum_{i,k=1}^{N}\|x_{i}-x_{k}\|^{2}+\frac{c_{q}\mathcal{D}_{x}(t)}{2N^{2}}\sum_{i,k=1}^{N}\|x_{i}-x_{k}\|^{2},

and

	$\displaystyle\|F^{3}\|$	$\displaystyle\leq 4(\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t))\frac{1}{N}\sum_{i=1}^{N}\\|v_{i}\\|^{2}+\frac{2\sigma(\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t))}{N^{2}}\sum_{i,j=1}^{N}\\|x_{i}-x_{j}\\|^{2}$
		$\displaystyle\quad+\frac{c_{q}(\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t))}{N}\sum_{i=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}.$

Similarly, we have

	$\displaystyle\frac{1}{N^{2}}\sum_{i,k=1}^{N}\\|x_{i}-x_{k}\\|^{2}$	$\displaystyle=\frac{1}{N^{2}}\sum_{i,k=1}^{N}\\|x_{i}-x_{\gamma}+x_{\gamma}-x_{k}\\|^{2}$
		$\displaystyle\leq 4X_{\gamma}^{1}.$

Therefore, we obtain that

	$\displaystyle\|F^{1}\|$	$\displaystyle=0,$
	$\displaystyle\|F^{2}\|$	$\displaystyle\leq\left(4\mathcal{D}_{v}(t)+8\sigma\mathcal{D}_{x}(t)+2c_{q}\mathcal{D}_{x}(t)\right)X^{1}_{\gamma},$
	$\displaystyle\|F^{3}\|$	$\displaystyle\leq\left(8\sigma+c_{q}\right)(\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t))X^{1}_{\gamma}+4(\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t))X_{\gamma}^{3}.$

The above implies the result in this lemma.

∎

We are ready to prove Theorem 2. We first check that the coefficient matrix $M$ has the following six eigenvalues.

\left\{-c_{p},~{}-c_{p},~{}-c_{p}\pm\sqrt{-4c_{q}+c_{p}^{2}},~{}-c_{p}\pm\sqrt{-4c_{q}+c_{p}^{2}-4\sigma}\right\}.

Thus, their real parts are all negative. Let $D$ be the greatest real part of the above eigenvalues and we define

\mu:=-D>0.

Then by Proposition 5.1, for any $\epsilon>0$ , there is $t_{0}>0$ such that if $t>t_{0}$ , then

0\leq\mathcal{D}_{x}(t)+\mathcal{D}_{v}(t)<\frac{\epsilon}{4\left(1+2\sigma+2c_{q}\right)}.

From Proposition 4.1 and 5.2, it follows that

\displaystyle X(t)

\displaystyle=e^{A(t-t_{0})}X(t_{0})+\int_{t_{0}}^{t}e^{A(t-s)}F(s)ds.

This implies that

	$\displaystyle\\|X(t)\\|$	$\displaystyle\leq e^{-\mu(t-t_{0})}\\|X(t_{0})\\|+\int_{t_{0}}^{t}e^{-\mu(t-s)}\\|F(s)\\|ds$
		$\displaystyle\leq e^{-\mu(t-t_{0})}\\|X(t_{0})\\|+\epsilon\int_{t_{0}}^{t}e^{-\mu(t-s)}\\|X(s)\\|ds.$

Therefore, by the Gronwall inequality, if $t>t_{0}$ , then

\|X(t)\|\leq\|X(t_{0})\|e^{-(\mu-\epsilon)(t-t_{0})}.

5.2. Practical rendezvouses

In this part, we consider the target tracking problem without acceleration information of the target. We assume that $\sigma_{ij}=\sigma>0$ and target speed and acceleration are bounded:

\|w_{\gamma}(t)\|,\|\dot{w}_{\gamma}(t)\|<C^{w}_{\gamma},\quad t\geq 0,

where $C^{w}_{\gamma}>0$ is a positive constant. We assume that $U_{i}=0$ . We first check that the coefficient matrix $M_{\infty}$ in Proposition 4.3 has the following eigenvalues.

\left\{-c_{p},~{}-c_{p}\pm\sqrt{-4c_{q}+c_{p}^{2}}\right\}.

Thus, their real parts are all negative. Let $D_{\infty}$ be the greatest real part of the above eigenvalues and we define

\mu_{\infty}:=-D_{\infty}>0.

Let

X_{\infty}=\max_{1\leq i\leq N}\|X_{i}\|.

By Proposition 4.3, for any fixed $t>0$ , there is an index $i_{t}\in\{1,\ldots,N\}$ such that

\|X_{i_{t}}\|=X_{\infty}

and

	$\displaystyle\frac{d}{dt}X_{\infty}^{2}$	$\displaystyle=\frac{d}{dt}\\|X_{i_{t}}\\|^{2}$
		$\displaystyle=\langle X_{i_{t}},M_{\infty}X_{i_{t}}\rangle+\langle X_{i_{t}},F_{i_{t}}\rangle$
		$\displaystyle\leq-\mu_{\infty}\\|X_{i_{t}}\\|^{2}+\\|X_{i_{t}}\\|\\|F_{i_{t}}\\|$
		$\displaystyle=-\mu_{\infty}X_{\infty}^{2}+X_{\infty}\\|F_{i_{t}}\\|.$

By direct calculation,

	$\displaystyle\|F_{i}^{1}\|$	$\displaystyle=0,$
	$\displaystyle\|F_{i}^{2}\|$	$\displaystyle\leq\frac{\\|v_{i}\\|^{2}}{2}\frac{16\\|x_{i}-x_{\gamma}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{16\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\|\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle\|$
		$\displaystyle\quad+\frac{16\|\langle x_{i}-x_{\gamma},A_{i}\rangle\|}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
	$\displaystyle\|F_{i}^{3}\|$	$\displaystyle\leq\frac{32\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\|\langle v_{i},x_{j}\rangle\|+\frac{32\|\langle v_{i},A_{i}\rangle\|}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle v_{i},v_{i}\rangle\|\langle x_{i}-x_{\gamma},v_{i}\rangle\|}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}.$

We note that

	$\displaystyle\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle$	$\displaystyle=\langle x_{i}-x_{\gamma},x_{j}-x_{\gamma}\rangle+\langle x_{i}-x_{\gamma},x_{\gamma}\rangle-\langle x_{i}-x_{\gamma},\langle x_{i},x_{j}\rangle x_{i}\rangle$
		$\displaystyle=\langle x_{i}-x_{\gamma},x_{j}-x_{\gamma}\rangle-\frac{1}{2}\\|x_{i}-x_{\gamma}\\|^{2}-\frac{\langle x_{i},x_{j}\rangle}{2}\\|x_{i}-x_{\gamma}\\|^{2}.$

This implies that

|\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle|\leq 2\max_{1\leq i\leq N}\|x_{i}-x_{\gamma}\|^{2}.

Similarly,

|\langle v_{i},x_{j}\rangle|=|\langle v_{i},x_{j}-x_{i}\rangle|\leq|\langle v_{i},x_{j}-x_{\gamma}\rangle|+|\langle v_{i},x_{\gamma}-x_{i}\rangle|\leq\|v_{i}\|^{2}+\max_{1\leq i\leq N}\|x_{i}-x_{\gamma}\|^{2},

\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}\leq 4\|v_{i}\|^{2}.

Thus,

	$\displaystyle\|F_{i}^{1}\|$	$\displaystyle=0,$
	$\displaystyle\|F_{i}^{2}\|$	$\displaystyle\leq 2X_{\infty}+\frac{32\sigma\max_{1\leq i\leq N}\\|x_{i}-x_{\gamma}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{16\|\langle x_{i}-x_{\gamma},A_{i}\rangle\|}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{256\\|v_{i}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}},$
	$\displaystyle\|F_{i}^{3}\|$	$\displaystyle\leq 2\sigma X_{\infty}+\frac{32\sigma\max_{1\leq i\leq N}\\|x_{i}-x_{\gamma}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{32\|\langle v_{i},A_{i}\rangle\|}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{256\\|v_{i}\\|^{3}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}.$

By elementary calculation, we have

\displaystyle\left|\langle x_{i}-x_{\gamma},A_{i}\rangle\right|\leq\|x_{i}-x_{\gamma}\|^{2}+\frac{\|A_{i}\|^{2}}{4}.

Note that

\|A_{i}\|^{2}\leq 6\|w_{\gamma}\|^{2}\|S_{\gamma}^{-1}(t)p_{i}\|^{2}+3\|\dot{w}_{\gamma}\|^{2}.

Since $p_{i}(t)=W_{\gamma}^{t}S_{\gamma}^{t}x_{i}(t)+S_{\gamma}^{t}\dot{x}_{i}(t)$ ,

\|S_{\gamma}^{-1}(t)q_{i}\|^{2}\leq 2\|w_{\gamma}\|^{2}+2\|v_{i}\|^{2}

and

\|A_{i}\|^{2}\leq 12(C^{w}_{\gamma})^{2}\|v_{i}\|^{2}+12(C^{w}_{\gamma})^{4}+3(C^{w}_{\gamma})^{2}.

Therefore, we have

\displaystyle\left|\langle x_{i}-x_{\gamma},A_{i}\rangle\right|\leq\|x_{i}-x_{\gamma}\|^{2}+3(C^{w}_{\gamma})^{2}\|v_{i}\|^{2}+3(C^{w}_{\gamma})^{4}+\frac{3(C^{w}_{\gamma})^{2}}{4}.

(5.4)

Similarly, we have

\displaystyle\left|\langle v_{i},A_{i}\rangle\right|\leq\|v_{i}\|^{2}+3(C^{w}_{\gamma})^{2}\|v_{i}\|^{2}+3(C^{w}_{\gamma})^{4}+\frac{3(C^{w}_{\gamma})^{2}}{4}.

(5.5)

By (5.4)-(5.5) and the above argument, if $\displaystyle\max_{1\leq i\leq N}\|x_{i}-x_{\gamma}\|<\frac{2\sqrt{C_{1}-1}}{\sqrt{C_{1}}}<2$ , then

	$\displaystyle\|F_{i}^{1}\|$	$\displaystyle=0,$
	$\displaystyle\|F_{i}^{2}\|$	$\displaystyle\leq(2+2\sigma C_{1}+5C_{1}+3(C^{w}_{\gamma})^{2})X_{\infty}+3(C^{w}_{\gamma})^{4}C_{1}^{2}+\frac{3(C^{w}_{\gamma})^{2}C_{1}^{2}}{4},$
	$\displaystyle\|F_{i}^{3}\|$	$\displaystyle\leq\left(2+2\sigma+2\sigma C_{1}+6(C^{w}_{\gamma})^{2}\right)X_{\infty}+6(C^{w}_{\gamma})^{4}C_{1}^{2}+\frac{6(C^{w}_{\gamma})^{2}C_{1}^{2}}{4}+4X_{\infty}^{3/2}.$

We conclude that

\|F_{i}\|\leq\left(4+2\sigma+4\sigma C_{1}+5C_{1}+9(C^{w}_{\gamma})^{2}\right)X_{\infty}+9(C^{w}_{\gamma})^{4}C_{1}^{2}+\frac{9(C^{w}_{\gamma})^{2}C_{1}^{2}}{4}+4X_{\infty}^{3/2}.

Therefore, we obtain that

\displaystyle\dot{X}_{\infty}\leq-\mu_{\infty}X_{\infty}+\left(4+2\sigma+4\sigma C_{1}+5C_{1}+9(C^{w}_{\gamma})^{2}\right)X_{\infty}+9(C^{w}_{\gamma})^{4}C_{1}^{2}+\frac{9(C^{w}_{\gamma})^{2}C_{1}^{2}}{4}+4X_{\infty}^{3/2}.

(5.6)

We choose $c_{q}$ and $c_{p}$ sufficiently large and take

X_{\infty}(0)=\frac{\sqrt{C_{1}-1}}{\sqrt{C_{1}}}.

Let $T>0$ be a maximal number such that on $t\in[0,T)$ ,

\displaystyle\max_{1\leq i\leq N}\|x_{i}(t)-x_{\gamma}(t)\|<2X_{\infty}(0),\quad\mbox{$t\in[0,T)$.}

(5.7)

By the initial condition and the continuity of the solution, there is a positive number $T>0$ satisfying (5.7). We claim that if $c_{q}$ and $c_{p}$ are sufficiently large, then $T=\infty$ . We note that for a given initial data, $\sigma$ , $C_{1}$ , $C_{\gamma}^{w}$ are fixed constants. Therefore, on $t\in[0,T)$ ,

\displaystyle\dot{X}_{\infty}\leq-\mu_{\infty}X_{\infty}+CX_{\infty}+C.

(5.8)

(5.8) implies

\displaystyle\dot{X}_{\infty}\leq-\frac{\mu}{2}X_{\infty}+C,

(5.9)

if $c_{q}$ and $c_{p}$ are sufficiently large. Therefore, by the Gronwall inequality and (5.9),

X_{\infty}(t)\leq e^{-\frac{\mu}{2}t}X_{\infty}(0)+e^{-\frac{\mu}{2}t}\frac{2Ce^{\frac{\mu}{2}t}-2C}{\mu}=e^{-\frac{\mu}{2}t}\left(X_{\infty}(0)-\frac{2C}{\mu}\right)+\frac{2C}{\mu}.

If $c_{q}$ and $c_{p}$ are sufficiently large, then $\mu$ is sufficiently large and $X_{\infty}\leq X_{\infty}(0)$ . These imply that on $t\in[0,T)$ ,

\max_{1\leq i\leq N}\|x_{i}(t)-x_{\gamma}(t)\|\leq X_{\infty}\leq X_{\infty}(0)<2X_{\infty}(0).

By the continuity of the solution, we obtain that

T=\infty.

Finally, by the above, we obtain the following practical rendezvous estimate.

X_{\infty}(t)\leq e^{-\frac{\mu}{2}t}\left(X_{\infty}(0)-\frac{2C}{\mu}\right)+\frac{2C}{\mu}.

Thus, we complete the proof of Theorem 3.

6. Simulation results

In this section, we present several numerical simulations for the target tracking problem on the unit sphere and the flat space to verify the asymptotic complete rendezvous and practical rendezvous. We use the fourth-order Runge-Kutta method. We consider six $\alpha$ -agents $\{(q_{i},p_{i})\}_{i=1}^{6}$ chasing one target $(q_{\gamma},p_{\gamma})$ . We assume that the control law for the target $(q_{\gamma},p_{\gamma})$ is given by

u_{\gamma}(t)=a(\cos t,\sin t,1),

where $a$ is a constant. Throughout this section, we assume that the inter-particle bonding force parameter is given by

\sigma=1.

With the extra control law for agents

U_{i}=2\langle w_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{w}_{\gamma}(t)\times q_{i},

the initial positions and velocities for the agents are randomly chosen in

(q_{i}(0),p_{i}(0))\in T\mathbb{S}^{2}\cap[-1,1]^{3}\times[-1,1]^{3}

as follows: -

	$\displaystyle q_{1}(0)=(\phantom{-}0.8132,\phantom{-}0.4989,-0.2993),$	$\displaystyle q_{2}(0)=(\phantom{-}0.7198,\phantom{-}0.4908,\phantom{-}0.4908),$
	$\displaystyle q_{3}(0)=(-0.6758,-0.6991,\phantom{-}0.2330),$	$\displaystyle q_{4}(0)=(-0.7878,\phantom{-}0.5627,-0.2501),$
	$\displaystyle q_{5}(0)=(-0.5440,-0.7504,\phantom{-}0.3752),$	$\displaystyle q_{6}(0)=(-0.8599,-0.3608,\phantom{-}0.3608),$

and

	$\displaystyle p_{1}(0)=(\phantom{-}0.1028,-0.1884,-0.0347),$	$\displaystyle p_{2}(0)=(-0.1168,\phantom{-}0.5118,-0.3405),$
	$\displaystyle p_{3}(0)=(-0.0821,\phantom{-}0.0857,\phantom{-}0.0191),$	$\displaystyle p_{4}(0)=(-0.1454,-0.1506,\phantom{-}0.1189),$
	$\displaystyle p_{5}(0)=(\phantom{-}0.2220,-0.1040,\phantom{-}0.1137),$	$\displaystyle p_{6}(0)=(-0.0003,\phantom{-}0.3768,\phantom{-}0.3759).$

The initial data for the target is

q_{\gamma}(0)=(-0.6451,0.6605,-0.3840)\quad\text{and}\quad p_{\gamma}(0)=(0.1761,0.3646,0.3311).

Refer to caption — Figure 1. The time evolution of (1.2) with extra control law (6.1)

Note that all the initial positions and velocities satisfy the admissible conditions in (1.3). Since $\omega_{\gamma}=q_{\gamma}\times p_{\gamma}$ , we can check that

\displaystyle\begin{aligned} U_{i}&=2\langle\omega_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\dot{\omega}_{\gamma}(t)\times q_{i}\\ &=2\langle\omega_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+(\dot{q}_{\gamma}\times p_{\gamma}+q_{\gamma}\times\dot{p}_{\gamma})\times q_{i}\\ &=2\langle\omega_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+\left(q_{\gamma}\times\left[-\frac{\|p_{\gamma}\|^{2}}{\|q_{\gamma}\|^{2}}q_{\gamma}+\|q_{\gamma}\|^{2}u_{\gamma}-\langle u_{\gamma},q_{\gamma}\rangle q_{\gamma}\right]\right)\times q_{i}\\ &=2\langle\omega_{\gamma},q_{i}\rangle(q_{i}\times p_{i})+(q_{\gamma}\times\|q_{\gamma}\|^{2}u_{\gamma})\times q_{i}.\end{aligned}

(6.1)

We fix

\sigma=1,~{}c_{q}=5,~{}c_{p}=0.1\quad\mbox{ and}\quad a=0.5.

For this case, the time evolution of (1.2) is given in Figure 1. The red points and blue lines stand for the position $q_{i}(t)$ at $t=t_{0}$ and trajectories for the time interval $[t_{0}-3,t_{0}]$ , respectively. The yellow one is for the target agent $q_{\gamma}(t)$ . In addition, we can check that the asymptotic complete rendezvous occurs as we proved in Theorem 2. See Figure 2. Here, the exponential function is $2e^{(-c_{p}+0.05)(t-40)}$ .

For the zero extra control law, i.e. $U_{i}=0$ , we fix the parameters such that

\sigma=1,~{}c_{q}=4,~{}c_{p}=4,~{}a=0.5.

The initial data of agents are randomly chosen but near the target as follows:

	$\displaystyle q_{1}(0)=(-0.8147,-0.5366,\phantom{-}0.2193),$	$\displaystyle q_{2}(0)=(-0.4575,-0.8843,\phantom{-}0.0922),$
	$\displaystyle q_{3}(0)=(-0.4335,-0.8173,\phantom{-}0.3794),$	$\displaystyle q_{4}(0)=(-0.8645,-0.2373,\phantom{-}0.4429),$
	$\displaystyle q_{5}(0)=(-0.4420,-0.7998,\phantom{-}0.4060),$	$\displaystyle q_{6}(0)=(-0.4312,-0.6004,\phantom{-}0.6734),$

and

	$\displaystyle p_{1}(0)=(0.0228,-0.0750,-0.0987),$	$\displaystyle p_{2}(0)=(0.2519,-0.1263,\phantom{-}0.0383),$
	$\displaystyle p_{3}(0)=(0.0200,\phantom{-}0.0169,\phantom{-}0.0594),$	$\displaystyle p_{4}(0)=(0.0388,-0.1447,-0.0017),$
	$\displaystyle p_{5}(0)=(0.0365,\phantom{-}0.1109,\phantom{-}0.2583),$	$\displaystyle p_{6}(0)=(0.0081,\phantom{-}0.0050,\phantom{-}0.0097).$

The initial data for the target is given by

q_{\gamma}(0)=(-0.6324,-0.6324,0.4472)\quad\text{and}\quad p_{\gamma}(0)=(0.4712,-0.1742,0.4199).

Figure 3 shows the time evolution of (1.2) without extra control law.

We can see that the maximum distance

\max\limits_{1\leq i\leq 6}\|q_{i}(t)-q_{\gamma}(t)\|

between agents and the target is bounded by $2/\sqrt{c_{p}}$ . See Figure 4(A). Let

d(t)=\max\limits_{1\leq i\leq 6}\|q_{i}(t)-q_{\gamma}(t)\|.

Figure 4(B) displays $d(t)$ at $t=100$ with respect to $c_{p}$ . As $c_{p}$ increases, the maximum distance between agents and target decreases. Therefore, we observe that the asymptotic practical rendezvous occurs.

With the extra control law, we observed the asymptotic complete rendezvous in Figure 1 and Figure 2. However, if we choose the parameter $c_{p}$ as zero, then the agents are not able to track the target. See Figure 5. Here, other parameters and initial data are the same as the case in Figure 1. In the absence of the velocity alignment term, the agents easily escape the sphere due to the accumulation of errors. To overcome this, as in [8], we add the following feedback term $f_{i}^{0}$ on the second equation of (1.2).

f_{i}^{0}=-k_{0}\Big{(}q_{i}-\frac{q_{i}}{\|q_{i}\|}\Big{)},

where $k_{0}=10^{4}$ . From this, we conclude that the velocity alignment operator is crucial in this target tracking algorithm.

As we mentioned in Subsection 2.2, the flocking term is negligible for the target tracking problem (1.2). With the same parameters of Figure 1 and Figure 3, the numerical results of (1.2) including the rotational flocking term

\sum_{j=1}^{N}\frac{\psi_{ij}}{N}(R_{q_{j}\to q_{i}}(p_{j})-p_{i}),

where $\psi_{ij}=1$ is given in Figure 6. It is confirmed that the flocking term does not affect the results. See also Figure 7.

Finally, we compare the target tracking problems on a sphere and flat space numerically. To compare the two cases, we impose the periodic boundary for the flat space and fix parameters such as $\sigma=1$ , $c_{q}=5$ , and $c_{p}=0.1$ . Let

u_{\gamma}=(a\cos t,a\sin t,a),

where $a=0.5$ and $u_{i}=u_{\gamma}$ . Then we can observe that the complete rendezvous occurs. See Figure 8. If $u_{i}=0$ , then we observe the practical rendezvous. See Figure 9.

7. Conclusion

In this paper, we proposed a novel model for target tracking on spherical geometry. With the target’s position, velocity, and acceleration, if the initial energy of agents is small or the bonding force between the target and each agent is larger than the one between agents, the complete rendezvous occurs. When only the information of position and velocity is known and the target’s angular velocity and its time derivative are bounded, the practical rendezvous is obtained for relatively large intra-bonding forces. The target tracking problems on $\mathbb{S}^{2}$ with time delay, white noises from the observation, and measurement are also interesting topics. These issues will be discussed in our future researches.

Appendix A Properties of the admissible rotation operator

In this part, we consider admissible rotation operators on a sphere and their properties. The rotation operator appears naturally for defining the flocking on a sphere [6]. Let $R_{\cdot\rightarrow\cdot}$ be Rodrigues’ rotation operator given by

R_{x_{k}\rightarrow x_{i}}(v_{k})=R(x_{k},x_{i})\cdot v_{k}

and for $x_{k}\neq x_{i}$ ,

\displaystyle\begin{aligned} R(x_{k},x_{i}):=\langle x_{k},x_{i}\rangle I+x_{i}x_{k}^{T}-x_{k}x_{i}^{T}+(1-\langle x_{k},x_{i}\rangle)\left(\frac{x_{k}\times x_{i}}{|x_{k}\times x_{i}|}\right)\left(\frac{x_{k}\times x_{i}}{|x_{k}\times x_{i}|}\right)^{T}.\end{aligned}

Here, $x_{k}$ , $x_{i}$ and $v_{j}$ are three dimensional column vectors. The rotation operator ${R_{\cdot\rightarrow\cdot}}$ has many good properties we desired or needed to be physically established and we can construct a flocking model by replacing the velocity difference term $v_{i}-v_{j}$ in the flat space to ${R_{x_{j}\rightarrow x_{i}}}v_{j}(t)-v_{i}(t)$ . See [6] for the details. However, there are some inconvenient points due to the presence of singularity on ${R_{\cdot\rightarrow\cdot}}$ . Therefore, we can naturally ask whether such alternatives can be found.

The idea to find the alternative is as follows. First, classify the properties that the rotation operators must satisfy, and find all the operators that satisfy the properties. Next, we will choose one of those operators that meets our needs. Our option will be the simplest of the possible operators. This form has various advantages. It is convenient to calculate, and it shares most of the good properties of the rotation operator ${R_{\cdot\rightarrow\cdot}}$ previously defined. By removing the singularity, we easily show the global-in-time existence and uniqueness of the new model in (1.2). See [6] for the existence and uniqueness of the model with ${R_{\cdot\rightarrow\cdot}}$ .

To construct a unit sphere model with the Newtonian equation, we need a modification of $v_{j}-v_{i}$ terms, which is the first motivation of the operators ${R_{x_{j}\rightarrow x_{i}}}$ in [6]. As we compute the velocity difference between $v_{i}$ and $v_{j}$ at the point $x_{i}$ , we should transform $v_{j}$ into a tangential vector of the sphere at $x_{i}$ . We note that the typical ansatz for the flocking motion on a sphere is circle motions. In order to include circle motions along one great circle, the operator should coincide with a rotation operator in two dimensions, a $(x_{i},x_{j})$ -plane. In other words, an admissible rotation operator $M$ from $z_{1}$ to $z_{2}$ can be a $3\times 3$ matrix such that


		$\displaystyle Mz_{1}=z_{2},\qquad Mz_{2}=2\langle z_{1},z_{2}\rangle z_{2}-z_{1},$		(A.1a)
		$\displaystyle\langle Mv,z_{2}\rangle=0\hbox{ for any }z_{1},z_{2}\in\mathcal{D}\hbox{ and }v\in T_{z_{1}}\mathcal{D}.$		(A.1b)

In the next proposition, we can prove that the admissible choices in (A.1) for the rotation operator are equivalent to the following set.

\displaystyle{\mathcal{A}}_{z_{1}\rightarrow z_{2}}:=\left\{{P}_{z_{1}\rightarrow z_{2}}+a(z_{1}\times z_{2})(z_{1}\times z_{2})^{T}+b(z_{1}-\langle z_{1},z_{2}\rangle z_{2})(z_{1}\times z_{2})^{T}:a,b\in\mathbb{R}\right\},

(A.2)

where ${P}_{z_{1}\rightarrow z_{2}}$ is the operator defined in (1.4).

Proposition A.1.

Suppose that unit vectors $z_{1}$ and $z_{2}$ are linearly independent. Then, a $3\times 3$ matrix $M$ satisfies (A.1) if and only if $M\in{\mathcal{A}}_{z_{1}\rightarrow z_{2}}$ .

Proof.

As two vectors $z_{1}$ and $z_{2}$ are perpendicular to ${z_{1}\times z_{2}}$ , operator ${P}_{z_{1}\rightarrow z_{2}}$ satisfies (A.1) from the direct computation. Note that $\langle z_{1}\times z_{2},z_{i}\rangle=0$ for $i=1,2$ . From this motivation, we naturally define

\displaystyle M:={P}_{z_{1}\rightarrow z_{2}}+a(z_{1}\times z_{2})(z_{1}\times z_{2})^{T}+b(z_{1}-\langle z_{1},z_{2}\rangle z_{2})(z_{1}\times z_{2})^{T}

(A.3)

for any $a,b\in\mathbb{R}$ . Then $M$ satisfies (A.1a). Also, as $z_{2}$ is perpendicular to both $z_{1}\times z_{2}$ and $(z_{1}-\langle z_{1},z_{2}\rangle z_{2})$ , we conclude (A.1b).

Conversely, choose any $3\times 3$ matrix $M^{\prime}$ satisfying (A.1). As $z_{1}$ and $z_{2}$ are linearly independent, $\{z_{2},~{}z_{1}-\langle z_{1},z_{2}\rangle z_{2},~{}{z_{1}\times z_{2}}\}$ are a basis of $\mathbb{R}^{3}$ . Therefore, there are $a,b,c\in\mathbb{R}$ such that

\displaystyle M^{\prime}\frac{{z_{1}\times z_{2}}}{\|{z_{1}\times z_{2}}\|^{2}}=a(z_{1}\times z_{2})+b(z_{1}-\langle z_{1},z_{2}\rangle z_{2})+cz_{2}.

(A.4)

From (A.1b) and ${z_{1}\times z_{2}}\in T_{z_{1}}\mathcal{D}$ , it follows that $c=0$ . Therefore, we conclude that

\displaystyle M{z_{1}\times z_{2}}=M^{\prime}{z_{1}\times z_{2}}

for $M$ given in (A.3). On the other hand, (A.1a) show that

\displaystyle M(z_{2})=M^{\prime}(z_{2})\quad\hbox{ and }\quad M(z_{1}-\langle z_{1},z_{2}\rangle z_{2})=M^{\prime}(z_{1}-\langle z_{1},z_{2}\rangle z_{2}).

(A.5)

From (A.4) and (A.5), we obtain that $M=M^{\prime}$ . ∎

The set ${\mathcal{A}}_{z_{1}\rightarrow z_{2}}$ includes the rotation operators ${R_{z_{1}\rightarrow z_{2}}}$ and ${P}_{z_{1}\rightarrow z_{2}}$ given in [6] and (1.4), respectively. Here, if we take the following values in (A.3):

a=\frac{1-\langle z_{1},z_{2}\rangle}{\|{z_{1}\times z_{2}}\|^{2}}\quad\mbox{and}\quad b=0,

then the matrix coincides with ${R_{z_{1}\rightarrow z_{2}}}$ , which preserves the modulus of each vectors. See Lemma 2.3 in [6]. Among several choices in the admissible set in (A.2), ${P}_{z_{1}\rightarrow z_{2}}$ can be regarded as the simplest choice such that $a=b=0$ in (A.2). Moreover, there is no singularity compared to the previous rotation operator ${R_{\cdot\rightarrow\cdot}}$ . In addition to this simplicity, the rotation operator ${P}_{z_{1}\rightarrow z_{2}}$ also share the following desired transport properties.

Lemma A.2.

For $z_{1},z_{2}\in\mathcal{D}$ , ${P}_{z_{1}\rightarrow z_{2}}$ given in (1.4) satisfies (A.1). Furthermore, we have

\displaystyle{P}_{z_{1}\rightarrow z_{2}}^{T}={P}_{z_{2}\rightarrow z_{1}}

(A.6)

and

\displaystyle{P}_{z_{1}\rightarrow z_{2}}^{T}{P}_{z_{1}\rightarrow z_{2}}(z_{1})=z_{1},\quad{P}_{z_{1}\rightarrow z_{2}}^{T}{P}_{z_{1}\rightarrow z_{2}}(z_{2})=z_{2}.

Proof.

As two vectors $z_{1}$ and $z_{2}$ are perpendicular to ${z_{1}\times z_{2}}$ , the properties in (A.1) follow from the direct computation. Also, since the transpose is the linear operator, we have

\displaystyle{P}_{z_{1}\rightarrow z_{2}}^{T}

\displaystyle=\langle z_{1},z_{2}\rangle I-z_{2}z_{1}^{T}+z_{1}z_{2}^{T},

and we conclude (A.6). From (A.1) and (A.6), it holds that

\displaystyle{P}_{z_{1}\rightarrow z_{2}}^{T}{P}_{z_{1}\rightarrow z_{2}}(z_{1})={P}_{z_{1}\rightarrow z_{2}}^{T}(z_{2})=z_{1}

and

\displaystyle{P}_{z_{1}\rightarrow z_{2}}^{T}{P}_{z_{1}\rightarrow z_{2}}(z_{2})={P}_{z_{1}\rightarrow z_{2}}^{T}(2\langle z_{1},z_{2}\rangle z_{2}-z_{1})=2\langle z_{1},z_{2}\rangle z_{1}-(2\langle z_{1},z_{2}\rangle z_{1}-z_{2})=z_{2}.

∎

While the two operators ${R_{z_{1}\rightarrow z_{2}}}$ and ${P}_{z_{1}\rightarrow z_{2}}$ coincide on the $(z_{1},z_{2})$ -plane from Lemma A.2, the following lemma gives us one difference between the two operators. We can show that ${{P}_{\cdot\rightarrow\cdot}}$ gives a map between two tangent spaces although the operator is not a bijection if $\langle z_{1},z_{2}\rangle=0$ .

Lemma A.3.

${P}_{z_{1}\rightarrow z_{2}}|_{T_{z_{1}}\mathcal{D}}$ is a map from $T_{z_{1}}\mathcal{D}$ to $T_{z_{2}}\mathcal{D}$ . Furthermore, if $\langle z_{1},z_{2}\rangle\neq 0$ , then ${P}_{z_{1}\rightarrow z_{2}}|_{T_{z_{1}}\mathcal{D}}$ is a bijection from $T_{z_{1}}\mathcal{D}$ to $T_{z_{2}}\mathcal{D}$ .

Proof.

As $\mathcal{D}$ is a unit sphere, $v\in T_{y}\mathcal{D}$ if and only if $\langle v,y\rangle=0$ for any $y\in\mathbb{R}^{3}$ . Thus, we have

\displaystyle\langle v,z_{1}\rangle=0\quad\hbox{ for any vector }v\in T_{z_{1}}\mathcal{D}.

(A.7)

From (A.1a) and (A.6), it holds that for any $v\in\mathbb{R}^{3}$ ,

\displaystyle({P}_{z_{1}\rightarrow z_{2}}v)\cdot z_{2}=v^{T}{P}_{z_{1}\rightarrow z_{2}}^{T}z_{2}=v^{T}{P}_{z_{1}\rightarrow z_{2}}rz_{2}=v^{T}z_{1}=\langle v,z_{1}\rangle.

(A.8)

By (A.7) and (A.8), we conclude that

\displaystyle({P}_{z_{1}\rightarrow z_{2}}v)\cdot z_{2}=0\hbox{ and thus }{P}_{z_{1}\rightarrow z_{2}}v\in T_{z_{2}}\mathcal{D}\hbox{ for any vector }v\in T_{z_{1}}\mathcal{D}.

We now assume that $\langle z_{1},z_{2}\rangle\neq 0$ and show that ${P}_{z_{1}\rightarrow z_{2}}|_{T_{z_{1}}\mathcal{D}}$ is bijective between two tangent spaces. First, if $z_{1}=z_{2}$ or $z_{1}=-z_{2}$ , we get ${P}_{z_{1}\rightarrow z_{2}}=I$ and ${P}_{z_{1}\rightarrow z_{2}}=-I$ . If not, $z_{1}$ and $z_{2}$ are linearly independent. From the assumption, ${P}_{z_{1}\rightarrow z_{2}}(z_{1}\times z_{2})=\langle z_{1},z_{2}\rangle(z_{1}\times z_{2})$ is a nonzero vector. Combining this with (A.1a), we conclude that ${P}_{z_{1}\rightarrow z_{2}}|_{T_{z_{1}}\mathcal{D}}$ is surjective in $T_{z_{2}}\mathcal{D}$ and thus the determinant of ${P}_{z_{1}\rightarrow z_{2}}$ is nonzero. As the inverse function of ${P}_{z_{1}\rightarrow z_{2}}$ exists, we conclude that this lemma holds. ∎

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	$\displaystyle\frac{d}{dt}\sum_{i=1}^{N}\|\langle p_{i},q_{i}\rangle\|^{2}$	$\displaystyle=2\sum_{i=1}^{N}(\langle\dot{p}_{i},q_{i}\rangle+\langle p_{i},\dot{q}_{i}\rangle)\langle p_{i},q_{i}\rangle$
		$\displaystyle=2\sum_{i=1}^{N}(\langle\dot{p}_{i},q_{i}\rangle+\\|p_{i}\\|^{2})~{}\langle p_{i},q_{i}\rangle$
		$\displaystyle=-2c_{p}\sum_{i=1}^{N}\|\langle p_{i},q_{i}\rangle\|^{2}.$

	$\displaystyle\\|F_{d}\\|$	$\displaystyle\leq\\|q_{d}\\|\\|u_{\gamma}\\|+2\\|p_{d}\\|\\|u_{\gamma}\\|$
		$\displaystyle\leq\frac{\epsilon\\|q_{d}\\|^{2}}{2}+\frac{1}{2\epsilon}\\|u_{\gamma}\\|^{2}+\frac{\epsilon\\|p_{d}\\|^{2}}{2}+\frac{2}{\epsilon}\\|u_{\gamma}\\|^{2}$
		$\displaystyle\leq\epsilon\\|X_{d}\\|+\frac{5}{2\epsilon}\\|u_{\gamma}\\|^{2}.$

	$\displaystyle\\|p_{i}\\|^{2}$	$\displaystyle=\\|W_{\gamma}^{t}S_{\gamma}^{t}x_{i}\\|^{2}+2\langle W_{\gamma}^{t}S_{\gamma}^{t}x_{i},S_{\gamma}^{t}\dot{x}_{i}\rangle+\\|S_{\gamma}^{t}\dot{x}_{i}\\|^{2}$
		$\displaystyle=\\|W_{\gamma}^{t}q_{i}\\|^{2}+2\langle W_{\gamma}^{t}q_{i},p_{i}-W_{\gamma}q_{i}\rangle+\\|v_{i}\\|^{2}$
		$\displaystyle=\langle q_{i},W_{\gamma}^{t}W_{\gamma}^{t}q_{i}\rangle-2\langle q_{i},W_{\gamma}^{t}p_{i}\rangle+\\|v_{i}\\|^{2}.$

	$\displaystyle\frac{d}{dt}X^{2}$	$\displaystyle=X^{3}-\frac{1}{N^{2}}\sum_{i,k=1}^{N}\frac{\\|v_{i}\\|^{2}+\\|v_{k}\\|^{2}}{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-\sigma X^{1}+\frac{\sigma}{4N^{3}}\sum_{i,j,k=1}^{N}\\|x_{i}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{\sigma}{4N^{3}}\sum_{i,j,k=1}^{N}\\|x_{k}-x_{j}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-c_{q}X^{1}+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{i}\\|^{2}\\|x_{i}-x_{k}\\|^{2}+\frac{c_{q}}{4N^{2}}\sum_{i,k=1}^{N}\\|x_{\gamma}-x_{k}\\|^{2}\\|x_{i}-x_{k}\\|^{2}$
		$\displaystyle\qquad-c_{p}X^{2}+\frac{1}{N^{2}}\sum_{i,k=1}^{N}\langle A_{i}-A_{k},x_{i}-x_{k}\rangle.$

	$\displaystyle\frac{d}{dt}X_{i}^{2}$	$\displaystyle=X_{i}^{3}+\frac{16\langle x_{i}-x_{\gamma},\dot{v}_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=X_{i}^{3}+16\bigg{\langle}x_{i}-x_{\gamma},~{}-\\|v_{i}\\|^{2}x_{i}+\sum_{j=1}^{N}\frac{\sigma}{N}(\\|x_{i}\\|^{2}x_{j}-\langle x_{i},x_{j}\rangle x_{i})$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+c_{q}(\\|x_{i}\\|^{2}x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i})-c_{p}v_{i}+A_{i}\bigg{\rangle}/\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}$
		$\displaystyle\quad+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}$
		$\displaystyle=X_{i}^{3}-\frac{\\|v_{i}\\|^{2}}{2}\frac{16\\|x_{i}-x_{\gamma}\\|^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{16\sigma}{N\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\sum_{j=1}^{N}\langle x_{i}-x_{\gamma},x_{j}-\langle x_{i},x_{j}\rangle x_{i}\rangle$
		$\displaystyle\quad+\frac{16c_{q}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\langle x_{i}-x_{\gamma},x_{\gamma}-\langle x_{i},x_{\gamma}\rangle x_{i}\rangle-\frac{16c_{p}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}\langle x_{i}-x_{\gamma},v_{i}\rangle$
		$\displaystyle\quad+\frac{16\langle x_{i}-x_{\gamma},A_{i}\rangle}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{2}}+\frac{64\langle x_{i}-x_{\gamma},v_{i}\rangle^{2}}{\left(4-\\|x_{i}-x_{\gamma}\\|^{2}\right)^{3}}.$

Multi-agent system for target tracking on a sphere and its asymptotic behavior

Abstract.

1. Introduction

Definition 1.1.

Theorem 1.

Theorem 2.

Remark 1.1.

Theorem 3.

2. Preliminary: Global well-posedness and Motivations

2.1. The global existence and uniqueness

Proposition 2.1.

2.2. Target tracking problem in ℝ3\mathbb{R}^{3}

3. Generalized rotation operator on sphere and reference frame decomposition

Lemma 3.1.

Proof.

Proposition 3.2.

Proof.

4. Reduction to a linearized system with a negative definite coefficient matrix

Proposition 4.1.

Proof.

Remark 4.2.

Proposition 4.3.

5. Asymptotic analysis on the target tracking models: complete and practical rendezvouses

5.1. Complete rendezvouses

Proposition 5.1.

Proposition 5.2.

Proof.

5.2. Practical rendezvouses

6. Simulation results

7. Conclusion

Appendix A Properties of the admissible rotation operator

Proposition A.1.

Proof.

Lemma A.2.

Proof.

Lemma A.3.

Proof.

References

2.2. Target tracking problem in $\mathbb{R}^{3}$