A Continuous-Time Distributed Optimization Algorithm for Multi-Agent Systems with Parametric Uncertainties over Unbalanced Digraphs

Abstract

This paper investigates distributed optimization problems for multi-agent systems with parametric uncertainties over unbalanced directed communication networks. To settle this class of optimization problems, a continuous-time algorithm is proposed by integrating adaptive control techniques with an output feedback tracking protocol. By systematically employing Lyapunov stability theory, perturbed system analysis, and input-to-state stability theory, we rigorously establish the asymptotic convergence property of the proposed algorithm. A numerical simulation further demonstrates the effectiveness of the algorithm in computing the global optimal solution.

1. Introduction

Distributed optimization in multi-agent systems has attracted substantial interest in recent years due to its broad applicability in diverse domains such as smart grids [1], distributed machine learning [2,3], resource allocation [4,5], and cooperative robotics [6]. Specifically, distributed optimization can be utilized to achieve state-of-charge balancing control at the battery-pack level in grid-connected battery energy storage systems, thereby enhancing overall system efficiency and effectively preventing potential damage caused by overcharging or discharging during operation [7]. Compared to centralized approaches, distributed optimization offers notable advantages in terms of preserving data privacy, computational scalability, and robustness against node or link failures [8,9].

Research on distributed optimization problems (DOPs) can generally be categorized into discrete-time and continuous-time frameworks. Following the decentralized optimization paradigm introduced in [10], a wide range of discrete-time algorithms have been developed [11,12,13]. In parallel, continuous-time distributed algorithms have also gained considerable attention. For scenarios where the communication network is undirected [14,15,16,17], several methods have been proposed. To address complex setups that involve both consensus constraints and nonsmooth cost functions, the authors of [14] proposed an adaptive method based on a distance penalty function, while the authors of [15] developed a subgradient algorithm exhibiting exponential convergence. Furthermore, zero-gradient-sum algorithms that are initialization-free and based on sliding mode control were proposed in [16,17] to address DOPs with consensus constraints.

Nevertheless, algorithms designed under the assumption of undirected graphs often encounter difficulties in accommodating more general network topologies. To overcome these limitations, recent research has explored distributed methods applicable to weight-balanced digraphs [18,19,20,21]. The authors of [18] proposed a primal–dual consensus algorithm that removes the dependency on initial states. For solving systems of linear equations in a distributed setting, a predefined-time consensus protocol was introduced in [19]. Moreover, to accommodate parametric uncertainties in multi-agent systems, an adaptive control strategy was developed in [20]. In the presence of external disturbances, the authors of [21] introduced a distributed algorithm that ensures finite-time state convergence for all agents.

Despite these advances, the above-mentioned algorithms are not directly applicable under unbalanced digraphs, which frequently arise in practice. To address this gap, several continuous-time algorithms have been proposed specifically for such networks [22,23,24]. For example, the authors of [22] introduced an auxiliary variable to counteract the imbalance effects inherent in directed topologies. In addressing resource allocation problems with nondifferentiable cost functions, the authors of [23] proposed an adaptive protocol that estimates a positive right eigenvector of the out-Laplacian matrix to facilitate convergence to the optimal solution. Furthermore, a gradient tracking method specifically designed for consensus-constrained problems was proposed in [24]; however, its applicability is restricted to scenarios where the Laplacian matrix is asymmetric, rendering it unsuitable for undirected graphs.

Leveraging insights from existing methods, this paper investigates the design of distributed consensus algorithms for multi-agent systems with parametric uncertainties. In practical implementations, such systems are likely to exhibit unbalanced directed networks due to inherent limitations in communication links or nodes. To this end, we develop a continuous-time algorithm capable to work over unbalanced digraphs, with particular emphasis on handling consensus constraints and parametric uncertainties. The primary contributions of this study are summarized as follows:

• A novel distributed continuous-time optimization algorithm is developed for multi-agent systems over unbalanced digraphs. Unlike existing methods such as [17,19,20], which are restricted to undirected or weight-balanced graphs, the proposed adaptive algorithm is applicable to more general directed graphs.
• This protocol employs a virtual vector to construct a reference signal, enabling the vector to approach the optimal solution of the considered problem. Moreover, the proposed approach can effectively handle parametric uncertainties, which are not considered in some related works such as [24,25,26].
• The asymptotic convergence of the proposed algorithm is rigorously established through the integration of Lyapunov stability theory and input-to-state stability (ISS) analysis. Additionally, the method improves agent-level privacy by eliminating the need to access the cost function (sub)gradients of neighboring agents, in contrast to existing approaches such as [24,27].

To facilitate understanding, the structure of this paper is organized as follows. The second section introduces mathematical preliminaries, graph theory fundamentals, and problem formulation. The third section presents the main results, including algorithm design and convergence analysis. The fourth section validates the efficacy of the algorithm through a numerical simulation, and the last section concludes this work.

2. Preliminaries and Problem Formulation

2.1. Notations

Let $\mathbb{R}$ and ${\mathbb{R}}_{+}$ denote the sets of real numbers and positive real numbers, respectively. The symbols ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ represent the spaces of n-dimensional real vectors and $n\times m$ real matrices, respectively. The identity matrix of size n is denoted by $I_n$. The vectors $1_n$ and $0_n$ denote n-dimensional column vectors whose entries are all ones and all zeros, respectively. The Kronecker product is represented by ⊗. $\mathrm{diag}(x_1, x_2, \dots , x_n)$ denotes a diagonal matrix with scalar diagonal entries $x_1, x_2, \dots , x_n$, while $\mathrm{blkdiag}(A_1, A_2, \dots , A_n)$ denotes a block diagonal matrix composed of matrices $A_1, A_2, \dots , A_n$ on its diagonal and zero matrices elsewhere. For any vector $y\in {\mathbb{R}}^n$, the Euclidean norm is denoted by $\parallel y\parallel$.

Consider a differentiable function $f:{\mathbb{R}}^n\to \mathbb{R}$. Its gradient is denoted by $\nabla f$. The function f is said to be strongly convex on a convex set $\mathsf{\Omega }\subseteq {\mathbb{R}}^n$ if there exists a constant $C\in {\mathbb{R}}_{+}$ such that ${(x-z)}^T\left(\nabla f\left(x\right)-\nabla f\left(z\right)\right)>C{\parallel x-z\parallel }^2$ for all $x,z\in \mathsf{\Omega }$ with $x\ne z$. In addition, if the gradient $\nabla g\left(x\right)$ of a function $g:{\mathbb{R}}^n\to \mathbb{R}$ is globally l-Lipschitz-continuous on ${\mathbb{R}}^n$, then there exists a constant $l\in {\mathbb{R}}_{+}$ such that $\parallel \nabla g\left(x\right)-\nabla g\left(z\right)\parallel \le l\parallel x-z\parallel$ for all $x,z\in {\mathbb{R}}^n$.

2.2. Graph Theory

The interaction topology of a multi-agent system is modeled by a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, where $\mathcal{V}=\{v_1, v_2, \dots , v_N\}$ denotes the set of vertices, $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of directed edges, and $\mathcal{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{N\times N}$ represents the adjacency matrix corresponding to the communication topology. A directed edge $(v_i,v_j)\in \mathcal{E}$ indicates that agent $v_i$ can access the state information of agent $v_j$. A directed path from vertex $v_i$ to vertex $v_j$ is a sequence of directed edges of the form $(v_i,v_{i_1}),(v_{i_1}, v_{i_2}), \dots , (v_{i_k},v_j)$. The digraph $\mathcal{G}$ is said to be strongly connected if there exists a directed path between every pair of distinct nodes. The index set of all agents is denoted by $S=\{1, 2, \dots , N\}$. Furthermore, all entries of $\mathcal{A}$ are non-negative, and $a_{ij}>0$ if and only if $(v_i,v_j)\in \mathcal{E}$. The graph Laplacian matrix $L=\left[l_{ij}\right]\in {\mathbb{R}}^{N\times N}$ is defined by $l_{ii}={\sum }_{j=1}^N{a}_{ij}$ and $l_{ij}=-a_{ij}$ for $i\ne j$. The graph $\mathcal{G}$ is said to be weight-balanced if $1_N^{T}L=0_N^T$; otherwise, it is unbalanced.

Lemma 1 

([28]). Let $L\in {\mathbb{R}}^{N\times N}$ be the Laplacian matrix of a strongly connected digraph $\mathcal{G}$. Then, the following properties hold:

1. 

L has a simple zero eigenvalue with the associated right eigenvector $1_N$, and all other eigenvalues have positive real parts.

2. 

Let $q={[q_1, \dots , q_N]}^T$ with $q_i\in {\mathbb{R}}_{+}$ for $i\in S$ denote the left eigenvector of L corresponding to the zero eigenvalue. Define $Q=\mathrm{diag}(q_1, \dots , q_N)$. Then the matrix $\widehat{L}=QL+L^{T}Q$ satisfies

$$\underset{{\vartheta }^{T}x=0,x\ne 0}{min}{\displaystyle \frac{x^T\widehat{L}x}{x^{T}x}}>{\displaystyle \frac{{\lambda }_2\left(\widehat{L}\right)}{N}},$$

where ϑ is any vector with positive entries and ${\lambda }_2\left(\widehat{L}\right)$ represents the second smallest eigenvalue of $\widehat{L}$. Moreover, $q=1_N$ if and only if $\mathcal{G}$ is weight-balanced.

2.3. Problem Formulation

Consider a multi-agent system comprising N agents. The continuous-time dynamics can be described by

$$\begin{array}{cc}\hfill {\dot{x}}_i\left(t\right)& =u_i\left(t\right)+{\psi }_i^T\left(x_i\left(t\right)\right){\overline{\omega }}_i,\hfill \\ \hfill y_i\left(t\right)& =x_i\left(t\right),\forall i\in S,\hfill \end{array}$$

(1)

where $x_i\left(t\right)\in {\mathbb{R}}^n$ is the state associated with agent i, $u_i\left(t\right)\in {\mathbb{R}}^n$ is the control input, and $y_i\left(t\right)\in {\mathbb{R}}^n$ is the control output. The vector ${\overline{\omega }}_i\in {\mathbb{R}}^{m_i}$ contains unknown system parameters, and ${\psi }_i\left(x_i\right):{\mathbb{R}}^n\to {\mathbb{R}}^{m_i\times n}$ denotes a matrix-valued regressor, which can be constructed using the output $y_i$ and the known system dynamics.

The objective of this study is to formulate a distributed continuous-time optimization algorithm such that each agent, operating under an unbalanced directed communication topology, can drive $y_i\left(t\right)$ to track the solution that optimally solves the following problem:

$$\underset{x\in {\mathbb{R}}^n}{min}f\left(x\right)=\sum _{i=1}^N{f}_i\left(x\right),$$

(2)

where $f_i:{\mathbb{R}}^n\to \mathbb{R}$ is a private objective function known only to the agent i, and $f\left(x\right)$ represents the global objective function. The optimization problem above can be equivalently reformulated as

$$\underset{x_i\in {\mathbb{R}}^n}{min}\sum _{i=1}^N{f}_i\left(x_i\right),\mathrm{subject}\mathrm{to}{x}_i=x_j,\forall i,j\in S.$$

(3)

Assumption 1. 

Each individual cost function $f_i\left(x_i\right)$ is differentiable and strongly convex, with its gradient $\nabla f_i\left(x_i\right)$ being ${\ell }_i$-Lipschitz.

Assumption 2. 

The unbalanced directed communication graph of agents is strongly connected.

Remark 1. 

The distributed problem formulated in this paper is of practical significance and has been previously investigated in works such as [16,24,29]. Moreover, Assumptions 1 and 2 are commonly adopted in distributed optimization [16,17,30]. Under Assumption 1, the Problem (3) is guaranteed to admit a unique globally optimal solution, with both existence and uniqueness being mathematically provable.

Define $y^{*}$ as the optimal output of the dynamical system (1). Then the objective of this paper is to develop a control protocol that ensures each agent’s control output $y_i\left(t\right)$ asymptotically approaches $y^{*}$; in other words,

$$\underset{t\to \infty }{lim}\parallel y_i\left(t\right)-y^{*}\parallel =0,\forall i\in S.$$

(4)

3. Main Results

3.1. Algorithm Design

In this section, we propose a distributed continuous-time optimization algorithm with adaptive coupling gains for unbalanced directed communication networks. The algorithm is designed as follows:

$$\begin{array}{}\mathrm{(5a)}& \hfill {\dot{y}}_i& =-c_i(y_i-r_i)+{\psi }_i^T\left(x_i\right)({\overline{\omega }}_i-{\tilde{\omega }}_i),\hfill \mathrm{(5b)}& \hfill {\dot{r}}_i& =-{\theta }_1{\displaystyle \frac{\nabla f_i\left(y_i\right)}{z_i^i}}-{\theta }_2({\alpha }_i+{\beta }_i)\sum _{j\in S}{a}_{ij}(r_i-r_j)-\sum _{j\in S}{a}_{ij}({\upsilon }_i-{\upsilon }_j),\hfill \mathrm{(5c)}& \hfill {\dot{\upsilon }}_i& ={\theta }_2({\alpha }_i+{\beta }_i)\sum _{j\in S}{a}_{ij}(r_i-r_j),\hfill \mathrm{(5d)}& \hfill {\dot{z}}_i& =-\sum _{j\in S}{a}_{ij}(z_i-z_j),{\dot{\tilde{\omega }}}_i={\mathsf{\Lambda }}_i{\psi }_i\left(x_i\right)(y_i-r_i),{\dot{\alpha }}_i=e_i^T{e}_i,\hfill \end{array}$$

where ${\beta }_i=e_i^T{e}_i$ and $e_i={\sum }_{j\in S}{a}_{ij}(r_i-r_j)$. Here, $r_i\in {\mathbb{R}}^n$, ${\upsilon }_i\in {\mathbb{R}}^n$, ${\tilde{\omega }}_i\in {\mathbb{R}}^{m_i}$, and $z_i\in {\mathbb{R}}^N$ are auxiliary variables, with $z_i^k$ representing the k-th entry of $z_i$. The initial condition of $z_i$ satisfies $z_i^i\left(0\right)=1$ and $z_i^k\left(0\right)=0$ for $k\ne i$. The parameters $c_i$, ${\theta }_1$, and ${\theta }_2$ are positive constants and ${\mathsf{\Lambda }}_i\in {\mathbb{R}}^{m_i\times m_i}$ is positive definite. The adaptive coupling gain ${\alpha }_i$ is initialized with a positive value, i.e., ${\alpha }_i\left(0\right)\in {\mathbb{R}}_{+}$. In Equation (5), each agent i transmits only its local variables $r_i$, ${\upsilon }_i$, and $z_i$ to neighbors, without sharing (sub)gradient information, as conducted in [24,27], thereby enhancing agent-level privacy.

Remark 2. 

Equation (5) introduces the virtual vector $r_i$ to generate a reference signal that converges to the minimizer of Problem (3). The variable ${\tilde{\omega }}_i$ is introduced to estimate the unknown parameter vector, with the estimation process regulated by the matrix ${\mathsf{\Lambda }}_i$, while the adaptive coupling gains ${\alpha }_i$ and ${\beta }_i$ are designed to drive the agents toward the minimizer of Problem (3). The auxiliary variable ${\upsilon }_i$ is introduced to ensure that $r_i$ eventually reaches consensus, while the parameters ${\theta }_1$, ${\theta }_2$, and $c_i$ are incorporated to facilitate the subsequent rigorous convergence analysis. Additionally, the variable $z_i$ is introduced to facilitate the estimation of the left eigenvector of L associated with the zero eigenvalue, without requiring prior knowledge of this information, as is needed in [31].

Remark 3. 

Rather than being predetermined, the coupling gains ${\alpha }_i$ and ${\beta }_i$ are adaptively tuned during the algorithm’s execution. Their updates depend exclusively on the local error $e_i$ arising from the discrepancy between virtual vectors of neighboring agents. In particular, ${\alpha }_i$ asymptotically converges to a finite constant as the system outputs achieve consensus, which will be established in the subsequent convergence analysis.

The control Equation (5) admits a compact representation as

$$\begin{array}{}\mathrm{(6a)}& \hfill \dot{y}& =-(c\otimes I_n)(y-r)+{\psi }^T\left(x\right)(\overline{\omega }-\tilde{\omega }),\hfill \mathrm{(6b)}& \hfill \dot{r}& =-{\theta }_1({\mathsf{\Phi }}_3^{-1}\otimes I_n)\nabla f\left(y\right)-{\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)r-(L\otimes I_n)\upsilon ,\hfill \mathrm{(6c)}& \hfill \dot{\upsilon }& ={\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)r,\hfill \mathrm{(6d)}& \hfill \dot{z}& =-(L\otimes I_N)z,\dot{\tilde{\omega }}=\mathsf{\Lambda }\psi \left(x\right)(y-r),{\dot{\alpha }}_i=e_i^T{e}_i,\hfill \end{array}$$

where $y={[y_1^T, y_2^T, \dots , y_N^T]}^T\in {\mathbb{R}}^{Nn}$, $r={[r_1^T, r_2^T, \dots , r_N^T]}^T\in {\mathbb{R}}^{Nn}$, $\upsilon ={[{\upsilon }_1^T, {\upsilon }_2^T, \dots , {\upsilon }_N^T]}^T\in {\mathbb{R}}^{Nn}$, $\nabla f\left(y\right)={[\nabla f_1{\left(y_1\right)}^T,\nabla f_2{\left(y_2\right)}^T,\dots ,\nabla f_N{\left(y_N\right)}^T]}^T\in {\mathbb{R}}^{Nn}$, $c=\mathrm{diag}(c_1, c_2, \dots , c_N)\in {\mathbb{R}}^{N\times N}$, $\tilde{\omega }={[{\tilde{\omega }}_1^T, {\tilde{\omega }}_2^T, \dots , {\tilde{\omega }}_N^T]}^T\in {\mathbb{R}}^{{\sum }_{i=1}^N{m}_i}$, $\mathsf{\Lambda }=\mathrm{blkdiag}({\mathsf{\Lambda }}_1, {\mathsf{\Lambda }}_2, \dots , {\mathsf{\Lambda }}_N)\in {\mathbb{R}}^{{\sum }_{i=1}^N{m}_i\times {\sum }_{i=1}^N{m}_i}$, $\psi \left(x\right)=$ $\mathrm{blkdiag}({\psi }_1\left(x_1\right),\dots ,{\psi }_N\left(x_N\right))\in {\mathbb{R}}^{{\sum }_{i=1}^N{m}_i\times Nn}$, ${\mathsf{\Phi }}_1=\mathrm{diag}({\alpha }_1, {\alpha }_2, \dots , {\alpha }_N)\in {\mathbb{R}}^{N\times N}$, ${\mathsf{\Phi }}_2=$$\mathrm{diag}({\beta }_1, {\beta }_2, \dots , {\beta }_N)\in {\mathbb{R}}^{N\times N}$, and ${\mathsf{\Phi }}_3^{-1}=\mathrm{diag}\left({\displaystyle \frac{1}{z_1^1}},{\displaystyle \frac{1}{z_2^2}},\dots ,{\displaystyle \frac{1}{z_N^N}}\right)\in {\mathbb{R}}^{N\times N}$.

Lemma 2. 

Under Assumptions 1 and 2, define variables $\delta =y-r$ and $\widehat{\omega }=\overline{\omega }-\tilde{\omega }$. If $\delta =\widehat{\omega }\equiv$ 0, and $(\tilde{y},\tilde{\upsilon })$ satisfy

$$\begin{array}{}\mathrm{(7a)}& \hfill \mathbf{0}& =-{\theta }_1({\mathsf{\Xi }}^{-1}\otimes I_n)\nabla f\left(\tilde{y}\right)-{\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\tilde{y}-(L\otimes I_n)\tilde{\upsilon },\hfill \mathrm{(7b)}& \hfill \mathbf{0}& ={\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\tilde{y},\hfill \end{array}$$

where $\tilde{y}={[{\tilde{y}}_1^T,{\tilde{y}}_2^T,\dots ,{\tilde{y}}_N^T]}^T$, ${\mathsf{\Xi }}^{-1}=\mathrm{diag}\left({\displaystyle \frac{1}{{\xi }_1}}, {\displaystyle \frac{1}{{\xi }_2}}, \dots , {\displaystyle \frac{1}{{\xi }_N}}\right)$, with $\xi ={[{\xi }_1, {\xi }_2, \dots , {\xi }_N]}^T$ being the left eigenvector of L corresponding to its zero eigenvalue, then $\tilde{y}=1_N\otimes y^{*}$, where $y^{*}$ is the optimal solution to Problem (3).

Proof of Lemma 2. 

Since $\delta =\widehat{\omega }\equiv \mathbf{0}$, it follows from (6) that the point $(\tilde{y},\tilde{\upsilon })$ satisfying (7) constitutes an equilibrium of the dynamical system described by Equation (6). From (7b), we obtain $\tilde{y}=1_N\otimes d$, where $d\in {\mathbb{R}}^n$. Left-multiplying both sides of (7a) by ${\xi }^T\otimes I_n$ gives ${\sum }_{i=1}^N\nabla f_i\left({\tilde{y}}_i\right)=\mathbf{0}$. Under Assumption 1, the optimality condition ${\sum }_{i=1}^N\nabla f_i\left(y^{*}\right)=\mathbf{0}$ holds. Therefore, it follows that $\tilde{y}=1_N\otimes y^{*}$. □

Remark 4. 

Lemma 2 establishes the optimality condition for Equation (5), deriving the necessary equations for the optimal solution $y^{*}$, which forms the basis for the subsequent convergence analysis.

3.2. Convergence Analysis

In this section, we prove the convergence property of Equation (5) by employing Lemma 2 in conjunction with global uniform asymptotic stability and ISS theory.

Based on Lemma 2, to prove ${lim}_{t\to \infty }{y}_i\left(t\right)=y^{*}$, we must show the convergence of $(y,\upsilon )$ to $(\tilde{y},\tilde{\upsilon })$. Through coordinate transformations, we define new variables $\delta =y-r$, $\widehat{\omega }=\overline{\omega }-\tilde{\omega }$, $\eta =r-\tilde{y}$, and $\mu =\upsilon -\tilde{\upsilon }$. Taking the difference of (6) and (7), the transformed dynamics of $(\delta ,\eta ,\mu ,\widehat{\omega },z,{\alpha }_i)$ derived from (6) are given by the following differential equations:

$$\begin{array}{}\mathrm{(8a)}& \hfill \dot{\delta }& =-(c\otimes I_n)\delta +{\psi }^T\left(x\right)\widehat{\omega }-\dot{r},\hfill & \hfill \dot{\eta }& =-{\theta }_1({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi +{\theta }_1(({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1})\otimes I_n)\nabla f\left(\tilde{y}\right)\hfill \mathrm{(8b)}& \hfill & -{\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\eta -(L\otimes I_n)\mu ,\hfill \mathrm{(8c)}& \hfill \dot{\mu }& ={\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\eta ,\hfill \mathrm{(8d)}& \hfill \dot{z}& =-(L\otimes I_N)z,\dot{\widehat{\omega }}=-\mathsf{\Lambda }\psi \left(x\right)\delta ,\dot{{\alpha }_i}=e_i^T{e}_i,\hfill \end{array}$$

where $\phi =\nabla f\left(y\right)-\nabla f\left(\tilde{y}\right)$ and $e_i={\sum }_{j\in S}{a}_{ij}({\eta }_i-{\eta }_j)$.

Based on Lemmas 2 and (8), we can conclude that to prove the convergence of variable $y_i$ in (5) to the optimal point $y^{*}$ of Problem (3), we need to show that $\delta$, $\widehat{\omega }$, and $\eta$ converge to $\mathbf{0}$ as time tends to infinity. Since ${\mathsf{\Xi }}^{-1}\ne {\mathsf{\Phi }}_3^{-1}$, the origin is not an equilibrium point of (8). For subsequent convergence analysis, we introduce additional coordinate transformations for $\eta$ and $\mu$, defining new variables $\widehat{\eta }=(L\otimes I_n)\eta$ and $\widehat{\mu }=(L\otimes I_n)\mu$. Therefore, we first need to prove ${lim}_{t\to \infty }\delta \left(t\right)=\mathbf{0}$, ${lim}_{t\to \infty }\widehat{\eta }\left(t\right)=\mathbf{0}$, ${lim}_{t\to \infty }\widehat{\mu }\left(t\right)=\mathbf{0}$, and ${lim}_{t\to \infty }\widehat{\omega }\left(t\right)=\mathbf{0}$ and then prove ${lim}_{t\to \infty }\eta \left(t\right)=1_N\otimes {\sigma }_1$ and ${lim}_{t\to \infty }\mu \left(t\right)=1_N\otimes {\sigma }_2$, where ${\sigma }_1,{\sigma }_2\in {\mathbb{R}}^n$ are two constant vectors. Finally, by proving ${\sigma }_1=\mathbf{0}$ and ${\sigma }_2<\infty$, we ultimately prove ${lim}_{t\to \infty }{y}_i\left(t\right)=y^{*}$.

Let ${\widehat{\eta }}_i\in {\mathbb{R}}^n$ represent the column vector consisting of elements from the $\left(\right(i-1)\times n+1)$-th to the $(i\times n)$-th of the vector $\widehat{\eta }$. Employing $e_i={\sum }_{j\in S}{a}_{ij}({\eta }_i-{\eta }_j)$, we obtain ${\widehat{\eta }}_i=e_i$, and the dynamics of $(\delta ,\widehat{\eta },\widehat{\mu },\widehat{\omega },z,{\alpha }_i)$ can thus be rewritten as

$$\begin{array}{}\mathrm{(9a)}& \hfill \dot{\delta }& =-(c\otimes I_n)\delta +{\psi }^T\left(x\right)\widehat{\omega }-\dot{r},\hfill & \hfill \dot{\widehat{\eta }}& =-{\theta }_1({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi +{\theta }_1(({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1})\otimes I_n)\nabla f\left(\tilde{y}\right)\hfill \mathrm{(9b)}& \hfill & -{\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\widehat{\eta }-(L\otimes I_n)\widehat{\mu },\hfill \mathrm{(9c)}& \hfill \dot{\widehat{\mu }}& ={\theta }_2(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n)\widehat{\eta },\hfill \mathrm{(9d)}& \hfill \dot{z}& =-(L\otimes I_N)z,\dot{\widehat{\omega }}=-\mathsf{\Lambda }\psi \left(x\right)\delta ,\dot{{\alpha }_i}={\widehat{\eta }}_i^T{\widehat{\eta }}_i.\hfill \end{array}$$

For the subsequent convergence analysis, we define $w=col(\widehat{\eta },\widehat{\mu },\delta )$. Then the dynamics of w can be written as

$$\begin{array}{cc}\hfill \underset{\dot{w}}{\underbrace{\left(\begin{array}{c}\dot{\widehat{\eta }}\\ \dot{\widehat{\mu }}\\ \dot{\delta }\end{array}\right)}}=& \underset{h_1\left(w\right)}{\underbrace{\left(\begin{array}{c}-{\theta }_1(L{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\theta }_2(L({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n)\widehat{\eta }-(L\otimes I_n)\widehat{\mu }\\ {\theta }_2(L({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n)\widehat{\eta }\\ -(c\otimes I_n)\delta +{\psi }^T\left(x\right)\widehat{\omega }-\dot{r}\end{array}\right)}}\hfill \\ \hfill & +\underset{h_2\left(t\right)}{\underbrace{\left(\begin{array}{c}{\theta }_1(L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1})\otimes I_n)\nabla f\left(\tilde{y}\right)\\ \mathbf{0}\\ \mathbf{0}\end{array}\right)}},\hfill \end{array}$$

(10)

where $\dot{w}=h_1\left(w\right)$ represents the unperturbed system.

Remark 5. 

Under Assumption 1, it can be verified that $h_1\left(w\right)$ is locally Lipschitz-continuous for $w\in \mathsf{\Omega }$, where $\mathsf{\Omega }\subset {\mathbb{R}}^{3Nn}$ is any compact set. Furthermore, $h_2\left(t\right)$ maintains boundedness for all $t\ge 0$. According to Theorem 4.19 in [32], to verify the asymptotic convergence of system (10) to the origin when $h_2\left(t\right)\ne$  0, we must first demonstrate the global uniform asymptotic stability of the unperturbed system and the ISS of system (10).

Theorem 1. 

Suppose Assumptions 1 and 2 hold, and the initial adaptive coupling gains satisfy ${\alpha }_i\left(0\right)>0$ for all $i=1,2,\dots ,N$. Then the proposed initialization-free continuous-time optimization algorithm described by Equation (5) can ensure that the control output $y_i\left(t\right)$ tracks the optimal trajectory $y^{*}$ through the appropriate selection of parameters $c_i$, ${\theta }_1$, and ${\theta }_2$.

Proof of Theorem 1. 

The proof consists of three main steps:

• Prove the global uniform asymptotic stability of the unperturbed system $\dot{w}=h_1\left(w\right)$;
• Establish the ISS property of the system (10);
• Demonstrate the convergence of variables $\eta$ and $\mu$ in (8) to $\mathbf{0}$ and $1_N\otimes {\sigma }_2$, respectively, as $t\to \infty$.

The detailed proof proceeds as follows.

Step 1. Consider $w=$ 0 as the equilibrium point of the unperturbed system $\dot{w}=h_1\left(w\right)$. Accordingly, the following Lyapunov function candidate is considered:

$$V=V_1+s_1{V}_2+V_3+s_2{V}_4,$$

(11)

where $s_1={\displaystyle \frac{33N{\overline{\lambda }}_N\left(L^{T}L\right)}{{\theta }_2{\lambda }_2^2\left(\overline{L}\right)}}$, with ${\overline{\lambda }}_N\left(L^{T}L\right)$ denoting the maximum eigenvalue of $L^{T}L$ and ${\lambda }_2\left(\overline{L}\right)$ representing the second smallest eigenvalue of the matrix $\overline{L}=\mathsf{\Xi }\mathrm{L}+{\mathrm{L}}^T\mathsf{\Xi }$ (abbreviated as ${\overline{\lambda }}_N$ and ${\lambda }_2$ for simplicity). Moreover, the components of the Lyapunov function are defined as

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill V_1& ={\displaystyle \frac{1}{2}}\sum _{i=1}^N{({\alpha }_i-\tilde{\alpha })}^2+{\displaystyle \frac{1}{2}}\sum _{i=1}^N{\xi }_i(2{\alpha }_i+{\beta }_i){\widehat{\eta }}_i^T{\widehat{\eta }}_i,\hfill \\ \hfill V_2& ={\displaystyle \frac{1}{2}}{(\widehat{\eta }+\widehat{\mu })}^T(\mathsf{\Xi }\otimes I_n)(\widehat{\eta }+\widehat{\mu }),\hfill \\ \hfill V_3& ={\displaystyle \frac{1}{2}}{\delta }^T\delta +{\displaystyle \frac{1}{2}}{\widehat{\omega }}^T{\mathsf{\Lambda }}^{-1}\widehat{\omega },\hfill \\ \hfill V_4& ={\displaystyle \frac{1}{2}}{X}^T{P}_{1}X,\hfill \end{array}\right.\end{array}$$

where $\tilde{\alpha }$ and $s_2$ are positive constants, $X={[{\eta }^T,{\mu }^T]}^T$, $P_1=\left[\begin{array}{cc}{\tau }_{11}& {\tau }_{12}\\ {\tau }_{12}& {\tau }_{22}\end{array}\right]\otimes I_{Nn}$, and ${\tau }_{11}$, ${\tau }_{12}$, and ${\tau }_{22}$ are positive constants satisfying the positive definiteness condition ${\tau }_{11}{\tau }_{22}>{\tau }_{12}^2$.

The derivative of $V_1$ along the trajectories of the unperturbed system $\dot{w}=h_1\left(w\right)$ is given by

$$\begin{array}{cc}\hfill {\dot{V}}_1& ={\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1-\tilde{\alpha }{I}_N)\otimes I_n\right)\widehat{\eta }+\sum _{i=1}^N{\xi }_i(2{\alpha }_i+2{\beta }_i){\widehat{\eta }}_i^T{\widehat{\eta }}_i+\sum _{i=1}^N{\xi }_i{\beta }_i{\dot{\alpha }}_i\hfill \\ \hfill & \le -{\theta }_2{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)(\mathsf{\Xi }L+L^T\mathsf{\Xi })({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n\right)\widehat{\eta }+{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+\mathsf{\Xi }{\mathsf{\Phi }}_2-\tilde{\alpha }{I}_N)\otimes I_n\right)\widehat{\eta }\hfill \\ \hfill & -{\theta }_1{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)(\mathsf{\Xi }L+L^T\mathsf{\Xi }){\mathsf{\Phi }}_3^{-1}\otimes I_n\right)\phi -{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)(\mathsf{\Xi }L+L^T\mathsf{\Xi })\otimes I_n\right)\widehat{\mu }.\hfill \end{array}$$

(12)

Define $\pi =\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n\right)\widehat{\eta }$. Since both ${\mathsf{\Phi }}_1$ and ${\mathsf{\Phi }}_2$ are positive diagonal matrices, there exists a positive vector $\vartheta ={({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^{-1}\xi \otimes 1_N$ such that ${\vartheta }^T\pi =({\xi }^T{({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^{-1}\otimes 1_N^T)(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n)\widehat{\eta }=({\xi }^{T}L\otimes 1_N^T)\eta =0$. Therefore, according to Lemma 1, we obtain

$$\begin{array}{cc}\hfill & -{\theta }_2{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)(\mathsf{\Xi }L+L^T\mathsf{\Xi })({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\otimes I_n\right)\widehat{\eta }\le -{\displaystyle \frac{{\theta }_2{\lambda }_2}{N}}{\widehat{\eta }}^T\left({({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2\otimes I_n\right)\widehat{\eta }.\hfill \end{array}$$

(13)

Since $z_i^i\left(t\right)>0$ for $t>0$, $i=1,2,\dots ,N$, and defining $\widehat{z}=min\left\{{inf}_{t>0}{z}_i^i\left(t\right),i\in S\right\}$, we can employ Young’s inequality to obtain

$$-2{\theta }_1{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\mathsf{\Xi }L{\mathsf{\Phi }}_3^{-1}\otimes I_n\right)\phi \le {\displaystyle \frac{{\theta }_2{\lambda }_2}{4N}}{\widehat{\eta }}^T\left({({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2\otimes I_n\right)\widehat{\eta }+{\displaystyle \frac{4{\theta }_1^{2}N{\overline{\lambda }}_N}{{\theta }_2{\lambda }_2{\widehat{z}}^2}}{\parallel \phi \parallel }^2,$$

(14)

$$-2{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\mathsf{\Xi }L\otimes I_n\right)\widehat{\mu }\le {\displaystyle \frac{{\theta }_2{\lambda }_2}{4N}}{\widehat{\eta }}^T\left({({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2\otimes I_n\right)\widehat{\eta }+{\displaystyle \frac{4N{\overline{\lambda }}_N}{{\theta }_2{\lambda }_2}}{\parallel \widehat{\mu }\parallel }^2.$$

(15)

Substituting (13), (14), and (15) into (12), we can further derive

$$\begin{array}{cc}\hfill {\dot{V}}_1& \le -{\displaystyle \frac{{\theta }_2{\lambda }_2}{2N}}{\widehat{\eta }}^T\left({({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2\otimes I_n\right)\widehat{\eta }+{\displaystyle \frac{4{\theta }_1^{2}N{\overline{\lambda }}_N}{{\theta }_2{\lambda }_2{\widehat{z}}^2}}{\parallel \phi \parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{4N{\overline{\lambda }}_N}{{\theta }_2{\lambda }_2}}{\parallel \widehat{\mu }\parallel }^2+{\widehat{\eta }}^T(({\mathsf{\Phi }}_1+\mathsf{\Xi }{\mathsf{\Phi }}_2-\widehat{\alpha }{I}_N)\otimes I_n)\widehat{\eta }.\hfill \end{array}$$

(16)

The derivative of $V_2$ along the trajectories of the unperturbed system is given by

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-{\theta }_1{\widehat{\eta }}^T(\mathsf{\Xi }L{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\widehat{\eta }}^T(\mathsf{\Xi }L\otimes I_n)\widehat{\mu }-{\theta }_1{\widehat{\mu }}^T(\mathsf{\Xi }L{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\widehat{\mu }}^T(\mathsf{\Xi }L\otimes I_n)\widehat{\mu }\hfill \\ \hfill & \le -{\displaystyle \frac{{\lambda }_2}{4}}\parallel \widehat{\mu }{\parallel }^2+{\displaystyle \frac{{\lambda }_2+2{\overline{\lambda }}_N}{{\lambda }_2}}\parallel \widehat{\eta }{\parallel }^2+{\displaystyle \frac{(8+{\lambda }_2){\theta }_1^2{\overline{\lambda }}_N}{4{\lambda }_2{\widehat{z}}^2}}{\parallel \phi \parallel }^2.\hfill \end{array}$$

(17)

To derive the above inequality, we employ Lemma 1 in conjunction with Young’s inequality; specifically,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\widehat{\mu }}^T(\mathsf{\Xi }L\otimes I_n)\widehat{\mu }\le -{\displaystyle \frac{{\lambda }_2}{2}}{\parallel \widehat{\mu }\parallel }^2,\hfill \\ -{\widehat{\eta }}^T(\mathsf{\Xi }L\otimes I_n)\widehat{\mu }\le {\displaystyle \frac{{\lambda }_2}{8}}\parallel \widehat{\mu }{\parallel }^2+{\displaystyle \frac{2{\overline{\lambda }}_N}{{\lambda }_2}}{\parallel \widehat{\eta }\parallel }^2,\hfill \\ -{\theta }_1{\widehat{\mu }}^T(\mathsf{\Xi }L{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi \le {\displaystyle \frac{{\lambda }_2}{8}}\parallel \widehat{\mu }{\parallel }^2+{\displaystyle \frac{2{\theta }_1^2{\overline{\lambda }}_N}{{\lambda }_2{\widehat{z}}^2}}{\parallel \phi \parallel }^2,\hfill \\ -{\theta }_1{\widehat{\eta }}^T(\mathsf{\Xi }L{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi \le \parallel \widehat{\eta }{\parallel }^2+{\displaystyle \frac{{\theta }_1^2{\overline{\lambda }}_N}{4{\widehat{z}}^2}}{\parallel \phi \parallel }^2.\hfill \end{array}\right.\end{array}$$

Next, let $\widehat{V}=V_1+s_1{V}_2$, and then the derivative of $\widehat{V}$ along the trajectories of the unperturbed system satisfies

$$\begin{array}{cc}\hfill \dot{\widehat{V}}& \le {\widehat{\eta }}^T\left(\left(-{\displaystyle \frac{{\theta }_2{\lambda }_2}{2N}}{({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2+({\mathsf{\Phi }}_1+\mathsf{\Xi }{\mathsf{\Phi }}_2)-\tilde{\alpha }{I}_N+{\displaystyle \frac{33N{\overline{\lambda }}_N({\lambda }_2+2{\overline{\lambda }}_N)}{{\theta }_2{\lambda }_2^3}}{I}_N\right)\otimes I_n\right)\widehat{\eta }\hfill \\ \hfill & -{\displaystyle \frac{17N{\overline{\lambda }}_N}{4{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2+\overline{k}{\parallel \phi \parallel }^2,\hfill \end{array}$$

(18)

where $\overline{k}={\displaystyle \frac{4{\theta }_1^{2}N{\overline{\lambda }}_N}{{\theta }_2{\lambda }_2{\widehat{z}}^2}}+{\displaystyle \frac{33N{\theta }_1^2{\overline{\lambda }}_N^2(8+{\lambda }_2)}{4{\theta }_2{\lambda }_2^3{\widehat{z}}^2}}$.

According to Assumption 1, we have $\phi =\nabla f\left(y\right)-\nabla f\left(\tilde{y}\right)\le \ell \parallel y-\tilde{y}\parallel \le \ell \parallel \delta +\eta \parallel$, where $\widehat{\ell }=max\{{\ell }_1, {\ell }_2, \dots , {\ell }_N\}$. Let ${\overline{\lambda }}_2\left(L^{T}L\right)$ denote the second smallest eigenvalue of $L^{T}L$, abbreviated as ${\overline{\lambda }}_2$. Then we obtain $\parallel \widehat{\eta }{\parallel }^2={\eta }^T(L^{T}L\otimes I_n)\eta \ge {\overline{\lambda }}_2{\parallel \eta \parallel }^2$. Therefore, by applying Young’s inequality, we can further derive

$$\begin{array}{cc}\hfill {\phi }^T\phi & \le {\widehat{\ell }}^2(1+{\gamma }_1){\parallel \delta \parallel }^2+{\widehat{\ell }}^2\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right){\parallel \eta \parallel }^2\le {\widehat{\ell }}^2(1+{\gamma }_1){\parallel \delta \parallel }^2+{\widehat{\ell }}^2\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right){\displaystyle \frac{\parallel \widehat{\eta }{\parallel }^2}{{\overline{\lambda }}_2}},\hfill \end{array}$$

(19)

where ${\gamma }_1>0$. Substituting (19) into (18) yields

$$\begin{array}{cc}\hfill \dot{\widehat{V}}& \le -{\widehat{\eta }}^T\left({\displaystyle \frac{{\theta }_2{\lambda }_2}{2N}}{\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)-{\displaystyle \frac{N}{{\lambda }_2}}{I}_N\right)}^2\otimes I_n\right)\widehat{\eta }-\left(\tilde{\alpha }-{ϵ}_1-{ϵ}_2-{\displaystyle \frac{{\theta }_{2}N}{2{\lambda }_2}}\right){\parallel \widehat{\eta }\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{17N{\overline{\lambda }}_N}{4{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2+{ϵ}_3{\parallel \delta \parallel }^2.\hfill \end{array}$$

(20)

where ${ϵ}_1={\displaystyle \frac{33N{\overline{\lambda }}_N({\lambda }_2+2{\overline{\lambda }}_N)}{{\theta }_2{\lambda }_2^3}}$, ${ϵ}_2={\displaystyle \frac{\overline{k}{\widehat{\ell }}^2}{{\overline{\lambda }}_2}}\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right)$, and ${ϵ}_3=\overline{k}{\widehat{\ell }}^2(1+{\gamma }_1)$. By choosing $\tilde{\alpha }\ge 1+{ϵ}_1+{ϵ}_2+{\displaystyle \frac{{\theta }_{2}N}{2{\lambda }_2}}$, we get

$$\dot{\widehat{V}}\le -\parallel \widehat{\eta }{\parallel }^2-{\displaystyle \frac{17N{\overline{\lambda }}_N}{4{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2+{ϵ}_3{\parallel \delta \parallel }^2.$$

(21)

Subsequently, we compute the derivative of $V_3$ along the trajectories of the unperturbed system $\dot{w}=h_1\left(w\right)$ and apply Young’s inequality to obtain

$$\begin{array}{cc}\hfill {\dot{V}}_3& =-{\delta }^T(c\otimes I_n)\delta -{\delta }^T\dot{r}\le -{\delta }^T(c\otimes I_n)\delta +{\displaystyle \frac{{\gamma }_2}{2}}{\parallel \delta \parallel }^2+{\displaystyle \frac{1}{2{\gamma }_2}}{\parallel \dot{r}\parallel }^2,\hfill \end{array}$$

(22)

where ${\gamma }_2>0$. Further processing $\parallel \dot{r}{\parallel }^2$ yields

$$\begin{array}{cc}\hfill \parallel \dot{r}{\parallel }^2={\parallel \dot{\eta }\parallel }^2& =\parallel -{\theta }_1({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\theta }_2\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)L\otimes I_n\right)\eta -(L\otimes I_n){\mu \parallel }^2\hfill \\ \hfill & \le \left((1+{\gamma }_3){\theta }_1^2{\phi }^T({\mathsf{\Phi }}_3^{-1}{\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi +\left(1+{\displaystyle \frac{1}{{\gamma }_3}}\right)X^T{P}_{2}X\right)\hfill \\ \hfill & \le \left({\displaystyle \frac{(1+{\gamma }_3){\theta }_1^2}{{\widehat{z}}^2}}{\phi }^T\phi +\left(1+{\displaystyle \frac{1}{{\gamma }_3}}\right)X^T{P}_{2}X\right),\hfill \end{array}$$

(23)

where ${\gamma }_3>0$ and $P_2=\left[\begin{array}{cc}{\theta }_2^2{({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2& {\theta }_2({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\\ {\theta }_2({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)& 1\end{array}\right]L^{T}L\otimes I_n$. Substituting (19) and (23) into (22), we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_3& \le -{\epsilon \parallel \delta \parallel }^2+{\displaystyle \frac{1}{2{\gamma }_2}}\left(1+{\displaystyle \frac{1}{{\gamma }_3}}\right)X^T{P}_{2}X+{\displaystyle \frac{(1+{\gamma }_3)}{2{\gamma }_2{\widehat{z}}^2}}\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right){\theta }_1^2{\widehat{\ell }}^2{\parallel \eta \parallel }^2,\hfill \end{array}$$

(24)

where $\epsilon =c_{min}-{\displaystyle \frac{{\gamma }_2}{2}}-{\displaystyle \frac{(1+{\gamma }_3)(1+{\gamma }_1){\theta }_1^2{\widehat{\ell }}^2}{2{\gamma }_2{\widehat{z}}^2}}$, with $c_{min}={min}_{i\in S}\left\{c_i\right\}$.

The derivative of $V_4$ along the trajectories of the unperturbed system is

$$\begin{array}{cc}\hfill {\dot{V}}_4& =-X^T{P}_{3}X-{\theta }_3{\tau }_{12}{\mu }^T\mu -{\theta }_1{\tau }_{11}{\eta }^T({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\theta }_1{\tau }_{12}{\mu }^T({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi ,\hfill \end{array}$$

(25)

where $P_3=\left[\begin{array}{cc}{\theta }_2({\tau }_{11}-{\tau }_{12})({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)& {\tau }_{11}\\ {\theta }_2({\tau }_{12}-{\tau }_{22})({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)& (1-{\theta }_3){\tau }_{12}\end{array}\right]L\otimes I_n$, with $0<{\theta }_3<1$. Applying Young’s inequality and (19), we derive

$$\begin{array}{cc}\hfill & -{\theta }_1{\tau }_{11}{\eta }^T({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi -{\theta }_1{\tau }_{12}{\mu }^T({\mathsf{\Phi }}_3^{-1}\otimes I_n)\phi \hfill \\ \hfill \le & {\displaystyle \frac{{\theta }_1{\tau }_{11}{\gamma }_4}{2}}{\parallel \eta \parallel }^2+\left({\displaystyle \frac{{\theta }_1{\tau }_{11}}{2{\gamma }_4}}+{\displaystyle \frac{{\theta }_1{\tau }_{12}}{2{\gamma }_5}}\right){\displaystyle \frac{{\widehat{\ell }}^2}{{\widehat{z}}^2}}\left((1+{\gamma }_1){\parallel \delta \parallel }^2+\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right){\parallel \eta \parallel }^2\right)\hfill \\ \hfill & +{\displaystyle \frac{{\theta }_1{\tau }_{12}{\gamma }_5}{2}}{\parallel \mu \parallel }^2.\hfill \end{array}$$

(26)

Substituting (26) back into (25) yields

$$\begin{array}{cc}\hfill {\dot{V}}_4& \le -X^T{P}_{3}X+b_1{\parallel \eta \parallel }^2-b_2{\parallel \mu \parallel }^2+b_3{\parallel \delta \parallel }^2,\hfill \end{array}$$

(27)

where ${\gamma }_4>0$, ${\gamma }_5>0$, $b_1={\displaystyle \frac{{\theta }_1{\tau }_{11}{\gamma }_4}{2}}+{\displaystyle \frac{{\widehat{\ell }}^2}{{\widehat{z}}^2}}\left({\displaystyle \frac{{\theta }_1{\tau }_{11}}{2{\gamma }_4}}+{\displaystyle \frac{{\theta }_1{\tau }_{12}}{2{\gamma }_5}}\right)\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right)$, $b_2={\theta }_3{\tau }_{12}-{\displaystyle \frac{{\theta }_1{\tau }_{12}{\gamma }_5}{2}}$, $b_3={\displaystyle \frac{{\widehat{\ell }}^2}{{\widehat{z}}^2}}\left({\displaystyle \frac{{\theta }_1{\tau }_{11}}{2{\gamma }_4}}+{\displaystyle \frac{{\theta }_1{\tau }_{12}}{2{\gamma }_5}}\right)\left(1+{\gamma }_1\right)$, and ${\theta }_2({\tau }_{12}-{\tau }_{22})={\tau }_{11}$. Appropriately choosing ${\theta }_1$ and ${\theta }_3$, we can ensure $b_2>0$.

Combining (21), (24), and (27), the derivative of V along the trajectories of the unperturbed system is obtained as

$$\begin{array}{cc}\hfill \dot{V}& \le -\parallel \widehat{\eta }{\parallel }^2-{\displaystyle \frac{17N{\overline{\lambda }}_N}{4{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2-s_2{b}_2{\parallel \mu \parallel }^2+({ϵ}_3-\epsilon +s_2{b}_3){\parallel \delta \parallel }^2+{\displaystyle \frac{1}{2{\gamma }_2}}\left(1+{\displaystyle \frac{1}{{\gamma }_3}}\right)X^T{P}_{1}X\hfill \\ \hfill & -s_2{X}^T{P}_{3}X+s_3{\parallel \eta \parallel }^2,\hfill \end{array}$$

(28)

where $s_3={\displaystyle \frac{(1+{\gamma }_3)}{2{\gamma }_2{\widehat{z}}^2}}\left(1+{\displaystyle \frac{1}{{\gamma }_1}}\right){\theta }_1^2{\widehat{\ell }}^2+s_2{b}_1$. Since $P_1$ is positive definite, by appropriately selecting $s_2$ and ensuring that $c_{min}\ge {ϵ}_3+s_2{b}_3+{\displaystyle \frac{{\gamma }_2}{2}}+{\displaystyle \frac{(1+{\gamma }_3)(1+{\gamma }_1){\theta }_1^2{\widehat{\ell }}^2}{2{\gamma }_2{\widehat{z}}^2}}$, together with choosing ${\tau }_{11}-{\tau }_{12}$ sufficiently large, it is guaranteed that $\dot{V}$ is negative definite. This shows that the origin is a globally uniformly asymptotically stable equilibrium point of the unperturbed system $\dot{w}=h_1\left(w\right)$, and the adaptive control gain ${\alpha }_i$ converges to a positive constant.

Step 2. Prove the ISS of system (10) and the convergence of $w\left(t\right)=\mathrm{col}(\widehat{\eta },\widehat{\mu },\delta )$ to zero as $t\to \infty$.

From Step 1, the derivative of V with respect to time along trajectories of system (10) satisfies

$$\begin{array}{cc}\hfill \dot{V}\le & -{\widehat{\eta }}^T\left({\displaystyle \frac{{\theta }_2{\lambda }_2}{2N}}{\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)-{\displaystyle \frac{N}{{\lambda }_2}}{I}_N\right)}^2\otimes I_n\right)\widehat{\eta }-\left(\tilde{\alpha }-{ϵ}_1-{ϵ}_2-{\displaystyle \frac{N}{2{\lambda }_2}}\right){\parallel \widehat{\eta }\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{17N{\overline{\lambda }}_N}{4{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2+{ϵ}_3{\parallel \delta \parallel }^2+{\dot{V}}_3+s_2{\dot{V}}_4\hfill \\ \hfill & +2{\theta }_1{\widehat{\eta }}^T\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\mathsf{\Xi }L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1}\left(t\right))\otimes I_n\right)\nabla f\left(\tilde{y}\right)\hfill \\ \hfill & +s_4{\theta }_1{(\widehat{\eta }+\widehat{\mu })}^T(\mathsf{\Xi }L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1}\left(t\right))\otimes I_n)\nabla f\left(\tilde{y}\right),\hfill \end{array}$$

(29)

where $s_4={\displaystyle \frac{33N{\overline{\lambda }}_N}{{\lambda }_2^2}}$. Defining the input disturbance $v\left(t\right)=(\mathsf{\Xi }L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1}\left(t\right))\otimes I_n)\nabla f\left(\tilde{y}\right)$, we can obtain

$$\begin{array}{c}2{\theta }_1{\widehat{\eta }}^T(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)\mathsf{\Xi }L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1}\left(t\right))\otimes I_n)\nabla f\left(\tilde{y}\right)\hfill \\ \le {\displaystyle \frac{{\theta }_2{\lambda }_2}{4N}}{\widehat{\eta }}^T({({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)}^2\otimes I_n)\widehat{\eta }+{\displaystyle \frac{4{\theta }_1^{2}N}{{\theta }_2{\lambda }_2}}{\parallel v\left(t\right)\parallel }^2,\hfill \end{array}$$

(30)

$$\begin{array}{c}{\theta }_1{(\widehat{\eta }+\widehat{\mu })}^T(\mathsf{\Xi }L({\mathsf{\Xi }}^{-1}-{\mathsf{\Phi }}_3^{-1}\left(t\right))\otimes I_n)\nabla f\left(\tilde{y}\right)\hfill \\ \le {\displaystyle \frac{{\lambda }_2}{8{\theta }_2}}\parallel \widehat{\mu }{\parallel }^2+\parallel \widehat{\eta }{\parallel }^2+{\displaystyle \frac{8{\theta }_2+{\lambda }_2}{4{\lambda }_2}}{\theta }_1^2{\parallel v\left(t\right)\parallel }^2.\hfill \end{array}$$

(31)

Substituting (30) and (31) into (29), and appropriately selecting $s_2$, we further get

$$\begin{array}{cc}\hfill \dot{V}\le & -{\widehat{\eta }}^T\left({\displaystyle \frac{{\theta }_2{\lambda }_2}{4N}}{\left(({\mathsf{\Phi }}_1+{\mathsf{\Phi }}_2)-{\displaystyle \frac{2N}{{\lambda }_2}}{I}_N\right)}^2\otimes I_n\right)\widehat{\eta }-\left(\tilde{\alpha }-{ϵ}_1-{ϵ}_2-s_4-{\displaystyle \frac{{\theta }_{2}N}{{\lambda }_2}}\right){\parallel \widehat{\eta }\parallel }^2\hfill \\ \hfill & -{\displaystyle \frac{N{\overline{\lambda }}_N}{8{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2+({ϵ}_3-\epsilon +s_2{b}_3){\parallel \delta \parallel }^2+\left({\displaystyle \frac{4{\theta }_1^{2}N}{{\theta }_2{\lambda }_2}}+{\displaystyle \frac{s_4{\theta }_1^2(8+{\lambda }_2)}{4{\lambda }_2}}\right){\parallel v\left(t\right)\parallel }^2.\hfill \end{array}$$

(32)

Define $\kappa =min\left\{1,{\displaystyle \frac{N{\overline{\lambda }}_N}{8{\theta }_2{\lambda }_2}},\epsilon -{ϵ}_3-s_2{b}_3\right\}$ and $\rho ={\displaystyle \frac{4{\theta }_1^{2}N}{{\theta }_2{\lambda }_2}}+{\displaystyle \frac{s_4{\theta }_1^2(8+{\lambda }_2)}{4{\lambda }_{2}N}}$. By selecting $\tilde{\alpha }\ge 1+{ϵ}_1+{ϵ}_2+s_4+{\displaystyle \frac{{\theta }_{2}N}{{\lambda }_2}}$ and ensuring $c_{min}\ge {ϵ}_3+s_2{b}_3+{\displaystyle \frac{{\gamma }_2}{2}}+{\displaystyle \frac{(1+{\gamma }_3)(1+{\gamma }_1){\theta }_1^2{\widehat{\ell }}^2}{2{\gamma }_2{\widehat{z}}^2}}$, one obtains

$$\begin{array}{cc}\hfill \dot{V}& \le -\parallel \widehat{\eta }{\parallel }^2-{\displaystyle \frac{N{\overline{\lambda }}_N}{8{\theta }_2{\lambda }_2}}\parallel \widehat{\mu }{\parallel }^2-(\epsilon -{ϵ}_3-s_2{b}_3){\parallel \delta \parallel }^2+\rho {\parallel v\left(t\right)\parallel }^2\hfill \\ \hfill & \le -{\kappa \parallel w\left(t\right)\parallel }^2+\rho {\parallel v\left(t\right)\parallel }^2.\hfill \end{array}$$

(33)

For any $0<\zeta <1$, when $\parallel w\left(t\right)\parallel \ge \sqrt{{\displaystyle \frac{\rho }{\kappa \zeta }}}\parallel v\left(t\right)\parallel$, it holds that

$$\dot{V}\le -(1-\zeta )\kappa {\parallel w\left(t\right)\parallel }^2.$$

(34)

According to Theorem 4.19 in [32], it has been established that system (10) is input-to-state stable. Furthermore, we can show that ${lim}_{t\to \infty }z\left(t\right)=1_N\otimes \xi$, which implies ${lim}_{t\to \infty }{\mathsf{\Phi }}_3^{-1}\left(t\right)={\mathsf{\Xi }}^{-1}$. Following the ISS definition, we prove the convergence of system (10) to zero as $t\to \infty$; i.e., for the perturbed system $w\left(t\right)=\mathrm{col}(\widehat{\eta },\widehat{\mu },\delta )$, we have ${lim}_{t\to \infty }\delta \left(t\right)=\mathbf{0}$, ${lim}_{t\to \infty }\widehat{\eta }\left(t\right)=\mathbf{0}$, ${lim}_{t\to \infty }\widehat{\mu }\left(t\right)=\mathbf{0}$, ${lim}_{t\to \infty }\widehat{\omega }\left(t\right)=\mathbf{0}$ and ${lim}_{t\to \infty }{\alpha }_i\left(t\right)=\tilde{\alpha }$.

Step 3. Prove ${lim}_{t\to \infty }\eta \left(t\right)=\mathbf{0}$ and ${lim}_{t\to \infty }\mu \left(t\right)=1_N\otimes {\sigma }_2$ with ${\sigma }_2<\infty$.

From the established results ${lim}_{t\to \infty }\widehat{\eta }\left(t\right)=\mathbf{0}$ and ${lim}_{t\to \infty }\widehat{\mu }\left(t\right)=\mathbf{0}$, we obtain ${lim}_{t\to \infty }\eta \left(t\right)=1_N\otimes {\sigma }_1$ and ${lim}_{t\to \infty }\mu \left(t\right)=1_N\otimes {\sigma }_2$, where ${\sigma }_1,{\sigma }_2\in {\mathbb{R}}^n$. Taking limits on both sides of (8b) yields

$$\begin{array}{cc}\hfill \mathbf{0}& =\underset{t\to \infty }{lim}\left((-{\mathsf{\Phi }}_3^{-1}\left(t\right)\otimes I_n)(\nabla f\left(y\right)-\nabla f\left(\tilde{y}\right))\right)\hfill \\ \hfill & =({\mathsf{\Xi }}^{-1}\otimes I_n)(\nabla f\left(\tilde{y}\right)-\nabla f(\eta +\tilde{y}))\hfill \\ \hfill & =({\mathsf{\Xi }}^{-1}\otimes I_n)(\nabla f\left(\tilde{y}\right)-\nabla f(1_N\otimes {\sigma }_1+\tilde{y})),\hfill \end{array}$$

(35)

Thereby we obtain

$$({\mathsf{\Xi }}^{-1}\otimes I_n)\nabla f\left(\tilde{y}\right)=({\mathsf{\Xi }}^{-1}\otimes I_n)\nabla f(1_N\otimes {\sigma }_1+\tilde{y}).$$

(36)

Left-multiplying both sides of (36) by ${\xi }^T\otimes I_n$ gives ${\sum }_{i=1}^N\nabla f_i\left({\tilde{y}}_i\right)={\sum }_{i=1}^N\nabla f_i({\tilde{y}}_i+{\sigma }_1)$. Under Assumption 1, the optimal solution to Problem (3) is unique, which implies ${\sigma }_1=\mathbf{0}$. Meanwhile, from (8c), it is straightforward to verify that ${\sigma }_2<\infty$. Thus, we have proved ${lim}_{t\to \infty }\eta \left(t\right)=\mathbf{0}$ and ${lim}_{t\to \infty }\mu \left(t\right)=1_N\otimes {\sigma }_2$, thereby establishing the convergence of $(y,\upsilon )$ to $(\tilde{y},\tilde{\upsilon })$. In conclusion, the states $y_i$ in Equation (5) converge to the optimal solution $y^{*}$ of Problem (2). □

Remark 6. 

For (28) and (33), it suffices to select parameters $c_i$ such that $c_{min}$ is sufficiently large, ensuring that the coefficient of ${\parallel \delta \parallel }^2$ remains negative, thus guaranteeing global uniform asymptotic stability and ISS.

Remark 7. 

While the introduction of adaptive gains ${\alpha }_i$ and ${\beta }_i$ inevitably increases the computational burden to some extent, it avoids the need for computing the inverse of Hessian matrices, as required in [16,17,29], thereby maintaining a relatively low overall computational complexity. In addition, the adaptive gains allow for dynamic adjustment of the control law, which helps improve the convergence performance of the system. To further reduce communication overhead, especially in bandwidth-constrained scenarios, our future work will consider integrating event-triggered communication mechanisms into the proposed framework.

4. Numerical Simulation

This section verifies the effectiveness of Equation (5) through a numerical simulation. The simulation considers a strongly connected but unbalanced directed communication network composed of six agents, where each directed edge has a communication weight of 1. The topology is shown in Figure 1. All local objective functions $f_i\left(x\right)$ ($x\in {\mathbb{R}}^2$, $i=1, 2, \dots , 6$) are defined as follows:

Figure 1. An unbalanced digraph of six agents, identified by numbers 1 to 6.

$f_1\left(x\right)=0.4{(x_1-0.6)}^2+0.6x_1{x}_2+0.2x_2^2$,

$f_2\left(x\right)=0.6{(x_1-2.4)}^2+0.4x_1{x}_2+0.2{(x_2+0.2)}^2$,

$f_3\left(x\right)=1.2{(x_1-1.5)}^2+0.8x_2^2+ln(3+0.6x_1^2)+ln(1.5+1.2x_2^2)$,

$f_4\left(x\right)=2.5{(x_1-1.5)}^2+0.4{(x_2-1.6)}^2+ln(4+0.8x_1^2)+ln(2.5+2x_2^2)$,

$f_5\left(x\right)=2.2{(0.5x_1-2.5)}^2+{(0.5x_2-3)}^2+{\displaystyle \frac{0.8x_1}{\sqrt{5x_1^2+2.5}}}+0.4exp\left(-0.7x_2^2\right)$,

$f_6\left(x\right)=1.2{(0.6x_1-3)}^2+{(0.6x_2-5)}^2+{\displaystyle \frac{0.2x_1}{\sqrt{6x_1^2+1}}}+0.8exp\left(-0.6x_2^2\right)$.

First, the optimal control output $y^{*}={[1.916,1.603]}^T$ corresponding to the optimization problem is determined by other efficient gradient algorithms to verify the effectiveness of Equation (5). The variables $x_i\left(0\right)$, $y_i\left(0\right)$, $r_i\left(0\right)$, and ${\upsilon }_i\left(0\right)$ are randomly initialized with parameter settings ${\theta }_1=0.5$, ${\theta }_2=1$, $c_i=2$, ${\alpha }_i\left(0\right)=1$, ${\psi }_i^T\left(x_i\right)=x_i$, ${\overline{\omega }}_i=1.5$, and ${\mathsf{\Lambda }}_i=1$ and a step size of 0.001. Figure 2 and Figure 3 demonstrate that both components of the outputs of all six agents converge asymptotically to $1.916$ and $1.603$, respectively. The evolution of the global cost function $f\left(x\right)={\sum }_{i=1}^6{f}_i\left(x_i\right)$ is shown in Figure 4, confirming convergence to the optimal value $f\left(x^{*}\right)=46.04$, where $x^{*}=y^{*}$ denotes the exact optimal solution of Problem (2). Figure 5 presents the adaptive coupling gains ${\alpha }_i\left(t\right)$, which converge to positive constants. Therefore, the obtained results validate the effectiveness of the proposed algorithm in solving Problem (2) over unbalanced digraphs.

Figure 2. The trajectories of the system control outputs $y_{ij}$ over time, where $y_{ij}$ represents the j-th component of the control output $y_i$, with $i\in \{1, 2, \dots , 6\}$ and $j\in \{1,2\}$.

Figure 3. The trajectory of the error of system control outputs ${\sum }_{i=1}^6{\parallel y_i\left(t\right)-y^{*}\parallel }^2$ over time.

Figure 4. The trajectory of the global cost function $f\left(x\right)={\sum }_{i=1}^6{f}_i\left(x\right)$ over time, with $f\left(x^{*}\right)$ denoting the optimal value of this function.

Figure 5. The trajectories of the adaptive coupling gains ${\alpha }_i\left(t\right)$ over time.

Meanwhile, the trajectories of the estimated parameters ${\tilde{\omega }}_i\left(t\right)$ for each agent over time are depicted in Figure 6, illustrating that ${\tilde{\omega }}_i\left(t\right)$ eventually converges to the true unknown parameter ${\overline{\omega }}_i$ for $i=1, 2, \dots , 6$. Consequently, the proposed algorithm is capable of addressing DOPs for control systems with unknown parameters.

Figure 6. The trajectories of the unknown parameter estimation variables ${\tilde{\omega }}_i\left(t\right)$ over time.

In contrast to its guaranteed convergence to the optimal solution over weight-balanced digraphs, Algorithm (4) proposed in [20] fails to achieve convergence under the same unbalanced directed network, as evidenced by the trajectories depicted in Figure 7.

Figure 7. The trajectories of the state variables over time in Algorithm (4) of [20] under the same unbalanced directed communication network, where $y_1^{*}$ and $y_2^{*}$ denote the first and second components of $y^{*}$, respectively.

5. Conclusions

In this paper, an adaptive continuous-time algorithm has been proposed, which is able to solve the DOPs of multi-agent systems with unknown parameters over unbalanced directed communication networks. The proposed approach is fully distributed and each agent only exchanges auxiliary variable information with neighboring agents rather than gradient information of agents. By utilizing Lyapunov stability theory, the developed approach has been proved to be globally convergent. Additionally, a simulation example has been provided to demonstrate the effectiveness and superiority of our scheme. In future work, we will further study DOPs with local constrains and unknown system parameters over unbalanced digraphs.