1 Introduction

Distributed coordination for multi-agent systems involves many aspects such as consensus and flocking. Consensus is a basic problem, in which the agent uses the distributed rule to make the system reach a common state. Gao et al.^[1] studied the consensus with leader-following and without leader, respectively. Li et al.^[2] demonstrated that all agents can achieve a consensus, even if the system have arbitrarily bounded communication delays.

In recent years, there are increasing studies on optimization problems. For instance, San et al.^[3] presented a multi-objective optimization method to design the shape of membrane structures. Bi et al.^[4] proposed a constrained optimization algorithm based on double populations to improve the distribution and convergence of the constrained optimization algorithms. However, previous researches focused on the distributed optimization problems and most of them are related to discrete-time algorithms^[5-6]. Tsitsiklis et al.^[7] originally presented a model for asynchronous distributed computation to achieve optimization. Li et al.^[8] extended the optimization rule to the multi-agent systems, and directed topology played an important role in their research. Nedic et al.^[9] formulated a projected rule for every agent in different convex sets to achieve optimization with a topology. At present, the distributed optimization problem has aroused interests of many scholars. By introducing a dynamic integrator, Wang et al.^[10] presented a novel distributed rule for convex optimization. Based on the work in Ref. [10], other researchers^[11-12] studied the optimization problem by strengthening conditions. Zhao et al.^[13] formulated distribution optimization rule using the edge and adaptive design method on the basis of nodes. Studies in Refs. [14-15] achieved the finite-time consensus theoretically using the presented continuous forms of distributed non-smooth controllers. George and Subramanian^[16]extended adaptive control with multiple models to the systems where the unknown parameters vary rapidly and frequently.

It is noteworthy that all the above results are related to the homogeneous multi-agents problems. The dynamics of the agents are always different due to the constraints in practical engineering applications, whereas the heterogeneous optimization problems have aroused little attention. Thus, given the actual situation, the study of distributed optimization with flocking of different dynamics systems is necessary both theoretically and practically. Zheng and Wang^[17] created a sufficient condition of consensus for continuous-time heterogeneous systems without the measurements of velocity.

The innovations of this study are as follows. First, previous works primarily studied the consensus, while the distributed optimization problems of the heterogeneous systems have been rarely discussed. Accordingly, a distributed optimization algorithm was developed for heterogeneous system with flocking behavior to study how a group of agents can achieve optimization cooperatively. Second, this study proved the consensus and minimized the sum of the local cost functions for the heterogeneous systems.

Here is the structure of the paper. In Section 2, notations, concepts, and the preliminaries are presented, and a distributed control rule is proposed for heterogeneous multi-agent system with flocking behavior. The stability and the optimization are achieved in Section 3. The results are proved by a numerical case in Section 4, and the conclusions are drawn in Section 5.

2 Notations and Preliminaries 2.1 Algebraic Graph Theory

The notations that will be used in this paper are presented in this part. A^T denotes the transpose matrix of A, x^T represents the transpose vector of x, and I_n ∈ R^n×n represents the unit matrix. $\mathbb N$={1, 2, …, N}. A ⊗ B is denoted as the Kronecker product of matrixes A and B. Let ‖x‖_p denote the p-norm of x∈ Rⁿ. The gradient of function g is ▽g, and the Hessian is H.

In general, an undirected graph G=(V, ε) is composed of a series of nodes V={1, 2, …, N} and a set of links ε. If i and j can be joined through a link (i, j), then (i, j)∈ε. N_i={j∈V:(j, i)∈ε} denotes the neighbor set of node i. If each pair of nodes in G has a link connected to each other, then the graph G is connected. A =[a_ij]∈ R^n×n is a weighted adjacent matrix of the graph G, which meets the conditions: 1)a_ii=0; 2)a_ij=a_ji>0, if (i, j)∈ε; and 3) a_ij=0, if (i, j)≠ε.

Let Λ =diag{d₁, d₂, …, d_n}∈ R^n×n denote the degree matrix of the graph G, and $ d_{i}=\sum\limits^ N_{ j=1, j≠i }a_{ij}$ for i∈$\mathbb N$. D denotes the correlation matrix related to the graph G. Let L = Λ － A be a symmetric Laplacian matrix and L=DD^T. The eigenvalues of L are defined as λ₁(L), λ₂(L), …, λ_N(L), and then the well-known result is given for 0=λ₁(L) ≤λ₂(L) ≤…≤ λ_N(L) if the graph G is connected^[18].

2.2 Non-smooth Analysis

Consider the differential equation with a discontinuous right-hand side^[19] as

$ \dot y = f(y,t) $

(1)

where f: R^m× R→R^m is Lebesgue measurable and bounded. x(·) is called a Filippov solution to Eq. (1) on [t₀, t₁], if vector function x(·) meets the conditions in Ref. [20].

Lemma 1^[21]    Assume Ω₁ is a compact set. Every Filippov solution to Eq. (1) beginning in Ω₁ is unique and keeps in Ω₁ for all t≥t₀. Let V:Ω₁→ R be regular function, which is independent of time with v≤0, ∀v∈ $\dot{\tilde V}$. Let S={x∈Ω|0∈ $\dot{\tilde V}$ }, and M⊂S be the largest invariant set, then any trajectory in Ω₁ asymptotically tends to M.

Assumption 1    The topology graph is undirected and connected.

Lemma 2^[22]   Considering a continuous differentiable convex function g(s): Rⁿ→ R is minimized if and only if ∂g(s)=0.

Lemma 3^[22]   The second smallest eigenvalue λ₂(L) of the Laplacian matrix L meets

$ {\lambda _2}(\mathit{\boldsymbol{L}}) = {\rm{mi}}{{\rm{n}}_{{x^{\rm{T}}}{1_N} = ,x \ne 0}}\frac{{{\mathit{\boldsymbol{x}}^{\rm{T}}}\mathit{\boldsymbol{Lx}}}}{{{\mathit{\boldsymbol{x}}^{\rm{T}}}\mathit{\boldsymbol{x}}}} $

Definition 1^[18]    g(x) is σ-strongly(σ>0) convex if and only if

$ (\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}})(\nabla g(\mathit{\boldsymbol{x}}) - \nabla g(\mathit{\boldsymbol{y}})) \ge \mathit{\boldsymbol{\sigma }}{\left\| {\mathit{\boldsymbol{x}} - \mathit{\boldsymbol{y}}} \right\|^2} $

for σ>0, ∀x, y ∈Rⁿ, and x≠y. If g(x) is σ-strongly convex and twice differentiable H(x) on x, then H(x) ≥σI_n.

N(N>2) agent is considered with n-dimensional. For N agents, the number of the linear agents is m, labelled from 1 to m(m < N), and the rest are nonlinear agents. The dynamics of the system are respectively designed by

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{v}}_i}(t)\\ {{\mathit{\boldsymbol{\dot v}}}_i}(t) = f({\mathit{\boldsymbol{v}}_i}(t)) + {\mathit{\boldsymbol{u}}_i}(t),i = 1,2, \cdots ,m \end{array} \right. $

(2)

$ \left\{ \begin{array}{l} {{\mathit{\boldsymbol{\dot x}}}_i}(t) = {\mathit{\boldsymbol{v}}_i}(t)\\ {{\mathit{\boldsymbol{\dot v}}}_i}(t) = {\mathit{\boldsymbol{u}}_i}(t),i = m + 1,m + 2, \cdots ,N \end{array} \right. $

(3)

where x_i(t) ∈ Rⁿ is the position of the agent i, v_i(t)∈ Rⁿ represents the velocity of the agent i, u_i(t) is the control law of the agent i, and f(v_i) is non-linear continuous function.

Assumption 2   There exists a constant 0 < ω≤1, for i=1, 2, …, m, such that ‖f(v_i)‖≤ω ‖ v_i‖.

The purpose of this paper is to devise a controller for Eqs. (2) and (3) that enables all agents to achieve optimization using local interactive information.

$ {\rm{min}}\sum\limits_{i = 1}^N {{g_i}} (\mathit{\boldsymbol{s}}),\mathit{\boldsymbol{s}} \in {{\bf{R}}^n} $

(4)

where g_i(s) :Rⁿ →R is a local cost function known only to agent i, and g_i(s) meets Assumptions 3 and 4.

Assumption 3   The local cost function g_i(s):Rⁿ →R is convex, differentiable, and meets

$ {g_i}(\mathit{\boldsymbol{s}}) = {\mathit{\boldsymbol{s}}^{\rm{T}}}{\mathit{\boldsymbol{A}}_i}\mathit{\boldsymbol{s}} $

(5)

where $\sum\limits^ N_{ i=1} \boldsymbol{A} _{i} = \boldsymbol{B} $ is positive definite and reversible matrix. Then the maximum eigenvalue of matrix B is denoted as λ^*.

The above problem is transformed into the minimization problem of function as follows:

$ \mathop {{\rm{min}}}\limits_{\mathit{\boldsymbol{v}} \in {{\bf{R}}^n}} \sum\limits_{i = 1}^N {{g_i}} ({\mathit{\boldsymbol{v}}_i}){\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}{\rm{.t}}{\rm{.}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{v}}_i} = {\mathit{\boldsymbol{v}}_j} $

(6)

Assumption 4   Each function g_i(v_i) has a non-empty optimal solution set. Namely, there exists v_i^*∈ Rⁿ which satisfies that g_i(v_i^*) is minimum.

3 Main Results

A distributed rule is presented, which makes the velocities of the agents the same and minimizes the function g_i(v_i).

$ {\mathit{\boldsymbol{u}}_i} = - \sum\limits_{j = 1}^N {{a_{ij}}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - \sum\limits_{j = 1}^N {{a_{ij}}} {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + {\varphi _i} $

(7)

where

$ {\varphi _i} = - H_i^{ - 1}({\mathit{\boldsymbol{v}}_i})\left( {\nabla {g_i}({\mathit{\boldsymbol{v}}_i}) + \frac{{\partial \nabla {g_i}({\mathit{\boldsymbol{v}}_i})}}{{\partial t}}} \right) $

a_ij describes the communication weight between agents i and j, and sgn(·) represents the signum function. It is noteworthy that φ_i relies merely on the velocity of the agent i. The twice differentiable function of the function g_i(v_i) with respect to v_i is Hessian H_i(v_i). Since the control rule developed in this paper contains sign functions, the system (2)-(3) has Filippov solution^[19].

Theorem 1    For the multi-agent system (2)-(3) with the controller (7), all agents asymptotically reach a consensus if λ₂(L) ≥σ^－1λ^*+ω.

Proof   As can be seen from Eqs. (2)-(3), the closed-loop is

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot v}}}_i} = - \sum\limits_{i = 1}^N {{a_{ij}}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - \sum\limits_{i = 1}^N {{a_{ij}}} {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varphi _i} + f({\mathit{\boldsymbol{v}}_i}) \end{array} $

(8)

Clearly, the right-hand side of Eq. (8) is discontinuous. Therefore, differential inclusions and non-smooth analysis can be used to illustrate the stability of system (2)-(3)^[20] as

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot v}}}_i}{ \in ^{{\rm{a}}{\rm{.e}}{\rm{.}}}}\kappa [ - \sum\limits_{i = 1}^N {{a_{ij}}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - \sum\limits_{i = 1}^N {{a_{ij}}} {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\varphi _i} + f({\mathit{\boldsymbol{v}}_i})] \end{array} $

where a.e. means "almost everywhere". Define the following function as

$ {\mathit{\boldsymbol{V}}_1} = \frac{1}{2}\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\mathit{\boldsymbol{v}}_i} $

From the properties of κ in Ref. [21], the set-valued Lie derivative of V(t) with on time along Eq.(8) is described as

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot {\tilde V}}}}_1} \in \kappa [\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\mathit{\boldsymbol{u}}_i} + \sum\limits_{i = 1}^m {\mathit{\boldsymbol{v}}_i^{\rm{T}}} f({\mathit{\boldsymbol{v}}_i})] \in \\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa [ - \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } \mathit{\boldsymbol{\nu }}_i^{\rm{T}}({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } \mathit{\boldsymbol{v}}_i^{\rm{T}} {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + } \end{array}\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\varphi _i} + \sum\limits_{i = 1}^m {\mathit{\boldsymbol{v}}_i^{\rm{T}}} f({\mathit{\boldsymbol{v}}_i})] \in }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \kappa [ - \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } \mathit{\boldsymbol{v}}_i^{\rm{T}}({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - } \end{array}\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\varphi _i} + \sum\limits_{i = 1}^m {\mathit{\boldsymbol{v}}_i^{\rm{T}}} f({\mathit{\boldsymbol{v}}_i})]} \end{array} \end{array} $

By using ‖f(v_i)‖≤ω ‖v_i‖, one gets

$ \begin{array}{*{20}{l}} {\mathit{\boldsymbol{\dot {\tilde V}}} \le - \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } \mathit{\boldsymbol{v}}_i^{\rm{T}}({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } {{({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j})}^{\rm{T}}} {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} |\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\varphi _i}| + \sum\limits_{i = 1}^m \omega {{\left\| {{\mathit{\boldsymbol{v}}_i}} \right\|}^2}} \end{array} $

Let V=[ v₁^T, v₂^T, …, v_N^T]^T, we have

$ \begin{array}{l} \begin{array}{*{20}{l}} {-\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } \mathit{\boldsymbol{v}}_i^{\rm{T}}({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) - \frac{1}{2}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_{ij}}} } {{({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j})}^{\rm{T}}} \times }\\ { {\rm{sgn}} ({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) = - {\mathit{\boldsymbol{V}}^{\rm{T}}}(\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{V}} - \frac{1}{2}{{\left\| {{\mathit{\boldsymbol{D}}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}\mathit{\boldsymbol{V}}} \right\|}_1}} \end{array}\\ |\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} {\varphi _i}| = |\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} ( - H_i^{ - 1}({\mathit{\boldsymbol{v}}_i})\nabla {g_i}({\mathit{\boldsymbol{v}}_i}))| = \\ \begin{array}{*{20}{l}} {|\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} (\mathit{\boldsymbol{H}}_i^{ - 1}({\mathit{\boldsymbol{v}}_i})\nabla {g_i}({\mathit{\boldsymbol{v}}_i}))| \le }\\ {|\sum\limits_{i = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} ({\sigma ^{ - 1}}{\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{v}}_i})| \le |\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\mathit{\boldsymbol{v}}_i^{\rm{T}}} } ({\sigma ^{ - 1}}{\mathit{\boldsymbol{A}}_i}{\mathit{\boldsymbol{v}}_i})| = }\\ {{\sigma ^{ - 1}}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{BV}}}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{i = 1}^m \mathit{\boldsymbol{\omega }} {{\left\| {{\mathit{\boldsymbol{v}}_i}} \right\|}^2} \le \sum\limits_{i = 1}^N \mathit{\boldsymbol{\omega }} {{\left\| {{\mathit{\boldsymbol{v}}_i}} \right\|}^2} = \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{V}}} \end{array} \end{array} $

Then, it yields

$ \begin{array}{l} \mathit{\boldsymbol{\dot {\tilde V}}} \le - {\mathit{\boldsymbol{V}}^{\rm{T}}}(\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{V}} - \frac{1}{2}{\left\| {{\mathit{\boldsymbol{D}}^{\rm{T}}} \otimes {\mathit{\boldsymbol{I}}_n}\mathit{\boldsymbol{V}}} \right\|_1} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\sigma ^{ - 1}}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{BV}} + \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{V}} \le \\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{V}}^{\rm{T}}}(\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{V}} - \frac{1}{2}\sqrt {\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}} \left\| \mathit{\boldsymbol{V}} \right\| + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\sigma ^{ - 1}}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{BV}} + \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{V}} = } \end{array}\\ \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{V}}^{\rm{T}}}((\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}) - {\sigma ^{ - 1}}\mathit{\boldsymbol{B}} - \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{I}}_n})\mathit{\boldsymbol{V}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}\sqrt {\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}} \left\| \mathit{\boldsymbol{V}} \right\|} \end{array} \end{array} $

Since it follows Lemma 3 and the maximum eigenvalue λ^* of matrix B, then

$ \begin{array}{l} \mathit{\boldsymbol{\dot {\tilde V}}} \le {\mathit{\boldsymbol{V}}^{\rm{T}}}( - {\lambda _2}(\mathit{\boldsymbol{L}}) + {\sigma ^{ - 1}}{\lambda ^*} + \mathit{\boldsymbol{\omega }})\mathit{\boldsymbol{V}} - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{2}\sqrt {\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}} \left\| \mathit{\boldsymbol{V}} \right\| \end{array} $

Apparently, if λ₂(L) ≥σ^－1λ^*+ ω, then

$ \mathit{\boldsymbol{\dot {\tilde V}}} \le - \frac{1}{2}\sqrt {\mathit{\boldsymbol{L}} \otimes {\mathit{\boldsymbol{I}}_n}} \left\| \mathit{\boldsymbol{V}} \right\| $

From the characteristics of L we can get $ \dot{\tilde{\boldsymbol{V}}} ≤0$. In accordance with Lemma 1, every solution beginning in Ω₁ is asymptotic to the largest invariant set M. Then v₁=v₂=…=v_N, suggesting that velocities of all agents in the system (2)-(3) reach a consensus.

As v_i －v_j=0 for i, j=1, 2, …, N, it yields

$ \frac{{\rm{d}}}{{{\rm{d}}t}}{\left\| {{\mathit{\boldsymbol{x}}_i} - {\mathit{\boldsymbol{x}}_j}} \right\|^2} = 2{({\mathit{\boldsymbol{x}}_i} - {\mathit{\boldsymbol{x}}_j})^{\rm{T}}}({\mathit{\boldsymbol{v}}_i} - {\mathit{\boldsymbol{v}}_j}) = 0 $

which shows that the distance is invariant. Hence, all agents become consensus and collision is avoided.

Theorem 2    Under the Assumptions 1-4, for system (2)-(3) with the controller (7), the agents velocities are optimal and they minimize the function g_i(v_i), thus the problem (4) is solved as t→∞.

Proof    Define the following function as

$ {\mathit{\boldsymbol{V}}_2} = \frac{1}{2}{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))^{\rm{T}}}(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i})) $

Then

$ {\mathit{\boldsymbol{\dot V}}_2} = {(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))^{\rm{T}}}\sum\limits_{i = 1}^N {\left( {{\mathit{\boldsymbol{H}}_i}({\mathit{\boldsymbol{v}}_i}){{\mathit{\boldsymbol{\dot v}}}_i} + \frac{{\partial \nabla {g_i}({\mathit{\boldsymbol{v}}_i})}}{{\partial t}}} \right)} $

By summing both sides of the system (2)-(3) with control rule(7) for i=1, 2, …, N, $\sum\limits^m_{i=1} \dot{\boldsymbol{v}} _{i}= \sum\limits^m_{ i=1} (φ_{i}+f( \boldsymbol{v} _{i})) , $ then $\sum\limits^ N_{ i=m+1} \dot{\boldsymbol{v}} _{i}= \sum\limits^ N_{i=m+1} φ_{i} $ is obtained. Therefore,

$ \begin{array}{l} {{\mathit{\boldsymbol{\dot V}}}_2} = {(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})({\varphi _i} + f({\mathit{\boldsymbol{v}}_i}))) + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\sum\limits_{i = m + 1}^N {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i}){\varphi _i} + \sum\limits_{i = 1}^N {\frac{{\partial \nabla {g_i}({\mathit{\boldsymbol{v}}_i})}}{{\partial t}}} = }\\ {{{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})f({\mathit{\boldsymbol{v}}_i}) + \sum\limits_{i = 1}^N {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i}){\varphi _i} + } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {\left. {\sum\limits_{i = 1}^N {\frac{{\partial \nabla {g_i}({\mathit{\boldsymbol{v}}_i})}}{{\partial t}}} } \right) = }\\ {{{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})f({\mathit{\boldsymbol{v}}_i}) - \sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i})) = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i})) + }\\ {{{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})f({\mathit{\boldsymbol{v}}_i})) \le } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i})) + }\\ {\left\| {{{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})f({\mathit{\boldsymbol{v}}_i}))} \right\| \le } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i})) + }\\ {\left\| {{{(\sum\limits_{i = 1}^N \nabla {g_i}({\mathit{\boldsymbol{v}}_i}))}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{H}}_i}} ({\mathit{\boldsymbol{v}}_i})\mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{v}}_i})} \right\| = } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i})}^{\rm{T}}}(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i}) + }\\ {\left\| {{{(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i})}^{\rm{T}}}(\sum\limits_{i = 1}^m {{\mathit{\boldsymbol{A}}_i}} \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{v}}_i})} \right\| \le } \end{array}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i})}^{\rm{T}}}(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i}) + }\\ {\left\| {{{(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} {\mathit{\boldsymbol{v}}_i})}^{\rm{T}}}(\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{A}}_i}} \mathit{\boldsymbol{\omega }}{\mathit{\boldsymbol{v}}_i})} \right\|} \end{array} \end{array} $

By $ \sum\limits^ N_{ i=1} \boldsymbol{A} _{i}= \boldsymbol{B} $, then

$ \begin{array}{*{20}{l}} { - {{(\sum\limits_{i = 1}^N {{A_i}} {v_i})}^{\rm{T}}}(\sum\limits_{i = 1}^N {{A_i}} {v_i}) + \left\| {{{(\sum\limits_{i = 1}^N {{A_i}} {v_i})}^{\rm{T}}}(\sum\limits_{i = 1}^N {{A_i}} \omega {v_i})} \right\| = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \mathit{\boldsymbol{v}}_i^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{v}}_i} + \omega \mathit{\boldsymbol{v}}_i^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{v}}_i} = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - \mathit{\boldsymbol{v}}_i^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{v}}_i}(1 - \omega )} \end{array} $

Obviously, according to Assumption 2, it is known that $ \dot{\boldsymbol{V}} _{2}≤0 $ for $\sum\limits^N_{ i=1} ▽g_{i}( \boldsymbol{v} _{i} )≠0$, which guarantees that $\sum\limits^N_{ i=1} ▽g_{i}( \boldsymbol{v} _{i} ) →0$. In this study, by using an undirected graph and the proof of consensus, there is a global public variable v^* such that $ \sum\limits^ N _{i=1} ▽g_{i}( \boldsymbol{v} ^{*})→0.$ Therefore, on the basis of Lemma 2 and when g_i(v_i) is convex, it can be obtained that $\sum\limits^ N _{i=1} g_{i}( \boldsymbol{v}) $ is minimized with v=v^* using the proposed control rule as t→∞, meaning the problem (4) is solved.

Remark   The graph connection has an important impact on the study. Without graph connection, the classic optimization method can only obtain the local optimal value v_i^*, which minimizes each cost function g_i(v_i) with v_i =v_i^*. Then the function $\sum\limits^ N _{i=1} g_{i}( \boldsymbol{v} _{i}) $ is minimized at v=v^*, i=1, 2, …, N. However, $ \sum\limits^N_{ i=1} g_{i} (\boldsymbol{v}^{*}) $ is not the global optimal solution to the problem (4). Therefore, with the information exchange of undirected graph, $\sum\limits^ N _{i=1} g_{i}( \boldsymbol{v} _{i}) $ is minimized at v_i= v_i^* using the control rule.

4 Numerical Simulations

The multi-agent network formed by 5 agents is shown in Fig. 1. The agent aims to optimize velocities while minimizing the function $\sum\limits^ N _{i=1} g_{i}( \boldsymbol{v} _{i}) , $ where v_i(t)= (v_xi(t), v_yi(t))^T denotes the coordinate of agent i in 2D plane. The adjacency matrix of the considered network graph is distributed by

$ \mathit{\boldsymbol{A}} = \left( {\begin{array}{*{20}{l}} 0&1&1&0&0\\ 1&0&1&0&0\\ 1&1&0&1&1\\ 0&0&1&0&1\\ 0&0&1&1&0 \end{array}} \right) $

For the system (2)-(3), the nonlinear function is designed by f(v_i) =ωv_isinv_i. This study randomly chose the initial velocity of each agent and marked it with dots. The initial status of the agent is presented in Fig. 2, where the full line between the two dots represents the path of the adjacent agent, while the arrow represents the motion direction of the agent. The final stable state of the agents is shown in Fig. 3. The error between the different velocity components and the optimal velocity is illustrated by Fig. 4, and the error curves of different velocity components are represented by different lines. The graph shows that the velocities reached the optimal velocity.

The multi-agent network was formed by 10 agents. The adjacency matrix of the considered network graph is distributed by

$ \mathit{\boldsymbol{A}} = \left( {\begin{array}{*{20}{l}} 0&1&1&0&0&1&1&0&1&1\\ 1&0&1&0&0&1&0&1&1&0\\ 1&1&0&1&1&0&1&1&0&1\\ 0&0&1&0&1&1&0&0&1&0\\ 0&0&1&1&0&1&0&1&0&1\\ 1&1&0&1&1&0&1&0&0&1\\ 1&0&1&0&0&1&0&1&1&0\\ 0&1&1&0&1&0&1&0&0&1\\ 1&1&0&1&0&0&1&0&0&1\\ 1&0&1&0&1&1&0&1&1&0 \end{array}} \right) $

In this part, we select 10 agents which shows that the system can still achieve a stable state and velocities of the agents can reach the optimal velocity. The initial status of the agent is presented in Fig. 5, where the full line between the two dots represents the path of the adjacent agent, while the arrow represents the motion direction of the agent. The final stable state of the agents is shown in Fig. 6. The error between the different velocity components and the optimal velocity is illustrated by Fig. 7, and the error curves of different velocity components are represented by different lines.

5 Conclusions

This study presented a distributed rule for heterogeneous systems. The optimization rule solved both consensus and optimization. First, the consensus issue was achieved for the second-order heterogeneous system, which is a more general and realistic system in accordance with Lyapunov stability theory and non-smooth analysis. Subsequently, it was shown that velocities of all agents were asymptotically optimal while minimizing the total cost function. Finally, to be recognized for the theoretical results, a simulation case was introduced. It is noteworthy that undirected topology plays an essential role.