Automatica 177 (2025) 112313

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Automatica
journal homepage: www.elsevier.com/locate/automatica

Fully distributed resource allocation over unbalanced digraphs in
prescribed time: A relaxed time-base generator approachI
Meng Luan a,c , Guanghui Wen b , Xiaohua Ge c , Qing-Long Han c ,

∗

a

Department of Systems Science, School of Mathematics, Southeast University, Nanjing 211189, PR China
School of Automation, Southeast University, Nanjing 210096, PR China
c
School of Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia
b

article

info

Article history:
Received 12 August 2024
Received in revised form 13 December 2024
Accepted 23 February 2025
Available online 10 April 2025
Keywords:
Fully distributed optimization
Resource allocation
Prescribed-time convergence
Time-base generator

a b s t r a c t
This paper focuses on three critical aspects of designing distributed optimization algorithms in realworld scenarios: feasibility, convergence time, and applicability to unbalanced networks. A specific
class of resource allocation problems (RAPs) are addressed with these challenges in mind. These
RAPs occur over unbalanced digraphs, have strict time constraints, and require continuous resourcedemand balance. To tackle these challenges, a relaxed framework of time-base generators (TBGs)
is presented. Sufficient conditions are then derived to guarantee that TBGs achieve prescribed-time
convergence for distributed optimization. Secondly, leveraging the designed TBGs, a new prescribedtime distributed continuous-time optimization algorithm specifically suited for RAPs over unbalanced
digraphs is firstly developed. This algorithm excels at solving these problems within user-defined and
arbitrary timeframes. Thirdly, a fully distributed design of the above prescribed-time optimization
algorithm is put forward to eliminate the stringent global topology knowledge. It is formally proved
that both algorithms achieve the optimal resource allocation within prescribed-time convergence.
Furthermore, both algorithms ensure continuous feasibility by maintaining resource-demand balance
throughout operation and offer a significant advantage in simplicity compared to existing primal–dual
methods and their variants. The proposed algorithms exclude the need for Lagrangian multipliers in
the design process, leading to a more straightforward and convenient approach to handling the coupled
resource-demand constraint. Finally, the effectiveness of the proposed algorithms is substantiated
through numerical simulations.
© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).

1. Introduction
Distributed optimization has become a major research topic
due to its wide-ranging applicability in areas such as load sharing
and economic dispatch (Yang et al., 2019). A critical focus in this
field is the optimal resource allocation of large-scale industrial
systems, as discussed in Doostmohammadian et al. (2025) and
Zhou, Wen, Lv, Yang, and Chen (2024). The fundamental goal of
resource allocation problems (RAPs) is to minimize the total cost
while satisfying resource-demand constraints. To address this,
several distributed optimization algorithms have been proposed
I This work was supported in part by the National Key Research and
Development Program of China under Grant No. 2022YFA1004702, National
Natural Science Foundation of China through Grant Nos. 62325304, U22B2046,
62088101, and in part by the Jiangsu Provincial Scientific Research Center of
Applied Mathematics, PR China under Grant No. BK20233002. The material in
this paper was not presented at any conference. This paper was recommended
for publication in revised form by Associate Editor Yongcan Cao under the
direction of Editor Christos G. Cassandras.
∗ Corresponding author.
E-mail addresses: 228433lm@seu.edu.cn (M. Luan), ghwen@seu.edu.cn
(G. Wen), xge@swin.edu.au (X. Ge), qhan@swin.edu.au (Q.-L. Han).

in the literature, which can be generally categorized into discretetime methods (Yuan, Wang, Proutiere, & Shi, 2024; Zhou, Lv, Wen,
& Wen, 2022) and continuous-time methods (Chen & Li, 2018;
Zhu, Ren, Yu, & Wen, 2021).
Distributed continuous-time optimization algorithms are increasingly preferred due to their adept integration of control
techniques into optimization decisions, and the emergence of
powerful digital signal processors has further enabled their implementation, driving significant progress in the field. Early
examples employed Laplacian-gradient dynamics (Cherukuri &
Cortés, 2015) and primal–dual methods (Guo, Shi, Cao, & Wang,
2024). However, these approaches often achieve only asymptotic
or exponential convergence in infinite time, unsuitable for realworld industrial systems demanding rapid responses and finitetime decision making. In response, finite-time and fixed-time
distributed optimization algorithms gained traction. These algorithms address optimization problems with various constraints
(Chen, Yang, Song, & Lewis, 2022; Shi, Xu, Yang, Lin, & Wang,
2022; Wang, Fei, & Wu, 2020). However, finite-time algorithms
exhibit convergence times influenced by network connectivity
and initial values. Fixed-time algorithms, while offering predictable or deterministic convergence times, tend to provide

https://doi.org/10.1016/j.automatica.2025.112313
0005-1098/© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313

overly conservative estimates of this duration. Additionally, the
implicit relationship between algorithm parameters and convergence time in fixed-time approaches makes presetting a specific
time difficult. Consequently, these existing algorithms struggle
with real-world RAPs with strict time constraints where achieving optimal allocation within a precise, predefined timeframe is
crucial.
To address these limitations, prescribed-time distributed optimization algorithms have emerged. Notably, piecewise algorithms with prescribed-time convergence were developed in Gong,
Cui, Shen, Xiong, and Huang (2021) and Ma, Hu, Yu, Wang,
and Jiang (2023) for unconstrained problems over undirected
graphs and balanced digraphs, respectively. Advances in timebase generator (TBG)-based techniques (Ning, Han, Zuo, Ding, Lu
et al., 2023) have further promoted the development of this field.
For example, a TBG-based approach was put forward in Liu, Xia,
and Gui (2023) to tackle multi-objective optimization problems
on undirected graphs. Additionally, predefined-time optimization
algorithms for undirected graphs addressing RAPs with resourcedemand constraint were proposed in Zhang, Xu et al. (2023).
Furthermore, in Guo and Chen (2022a), prescribed-time algorithms for handling RAPs with equality and inequality constraints
were presented. However, most of the existing prescribed-time
algorithms achieve resource-demand satisfaction only at the settling time and are limited to undirected graphs or balanced
digraphs.
Maintaining resource-demand balance at any time is critical
for real-world applications. For example, in economic dispatch
problems, if the algorithm is halted before convergence and the
allocated resources fail to meet the desired demand, it may
lead to power delivery issues, including service interruptions.
Conversely, over-supply may cause substantial voltage and frequency fluctuations, compromising system stability (Doostmohammadian et al., 2025). Additionally, as discussed in Rivera and
Jacobsen (2014), real-time electric vehicle charging control is a
time-critical application that demands rapid responses within
milliseconds to prevent grid overload and avoid the activation
of protection devices. Given this urgency, it may be necessary
to terminate the solution process prematurely prior to reaching
optimality, making it essential that the decision variables remain feasible at any given time. Moreover, real-time satisfaction
of resource-demand balance makes online implementation possible. Unfortunately, developing distributed optimization algorithms for RAPs with hard resource-demand and time constraints
over unbalanced digraphs faces significant challenges. Specifically, most of the current distributed optimization algorithms,
including Laplacian-gradient, finite-time, fixed-time, and aforementioned prescribed-time approaches, rely heavily on
symmetric and balanced network topologies to maintain resourcedemand balance (Chen & Li, 2018; Cherukuri & Cortés, 2015;
Guo & Chen, 2022a; Liu et al., 2023). As a result, they are only
applicable to undirected graphs or balanced digraphs. Additionally, while techniques exist for eliminating topology imbalance
through eigenvalue estimation, the additional exponential error
term introduced in this process hinders the incorporation of
TBGs (Zhu et al., 2021). Besides, the existing literature tends to
adopt relatively fixed and rigid designs for TBGs guaranteeing
algorithm convergence within a predefined time, underscoring
the need for developing more diverse and flexible TBG formulations. To tackle these challenges, it is desirable to design
prescribed-time distributed optimization algorithms applicable to
unbalanced digraphs for solving RAPs with hard resource-demand
constraints.
Despite the progress made, parameter design in distributed
optimization usually relies on global information such as Laplacian matrix eigenvalues, posing challenges for scalable applications in practice. Therefore, significant research focuses on

developing fully distributed optimization algorithms based on
local adaptation ideas. The node-based and edge-based adaptive schemes include examples applied to optimization problems in multi-agent systems over undirected graphs (Zhao, Liu,
Wen, & Chen, 2017). A distributed adaptive algorithm for solving
optimization problems with local constraints over undirected
graphs was proposed in Zhou, Zeng, and Hong (2019). More
recently, a fully distributed approach leveraging the primal–
dual idea and state feedback was introduced in Deng and Liu
(2024) for nonsmooth RAPs over undirected graphs in high-order
multi-agent systems. Moreover, a class of uncertain saddle-point
dynamics, suitable for weight-balanced digraphs and achieving
global asymptotic convergence, was presented in Yue, Baldi, Cao,
Li, and De Schutter (2023). Furthermore, distributed adaptive
optimization algorithms applicable to unbalanced digraphs, employing eigenvalue estimation-based methods, adaptive control
and topology balancing techniques, were developed in Li, Ding,
Sun, and Li (2017) and Zhang, Liu et al. (2023), respectively. These
algorithms broaden our understanding of fully distributed approaches for RAPs. However, their limitation to achieving global
asymptotic convergence prevents them from satisfying the strict
time constraints of distributed RAPs. In Jia, Ye, and Ding (2022),
some exponential terms were employed to achieve fully distributed Nash Equilibrium seeking within a prescribed time for
unconstrained networked games. Nevertheless, the algorithm
struggles with guaranteeing the boundedness of adaptive parameters. How to devise fully distributed optimization algorithms for
RAPs with hard time and resource-demand constraints remains a
significant challenge.
Motivated by the observations above, in this paper, we aim to
address a class of distributed RAPs over unbalanced digraphs with
hard resource-demand and time constraints. Specifically, this requires the resource-demand constraint involved in the RAPs to
be satisfied at any time, and the optimal resource allocation to be
achieved within arbitrarily explicit and easily preset times. To the
best of the authors’ knowledge, this paper is among the first few
attempts to address distributed RAPs over unbalanced digraphs
within an arbitrarily preset time, particularly in a fully distributed
manner. The novelty of this paper lies in two novel prescribedtime distributed optimization algorithms that are featured with
a relaxed form of TBGs. The main technical contributions are
threefold.
(i) Relaxed TBGs: We present a novel approach to construct
TBGs, and derive sufficient conditions to ensure their
prescribed-time convergence for distributed optimization
algorithms. Compared to existing results (Guo & Chen, 2022a,
2022b; Liu et al., 2023; Liu & Yi, 2023; Ning, Han, & Zuo,
2019; Wang, Song, Hill, & Krstic, 2019), the proposed TBGs
can be viewed as a relaxed version, where the gain designs are no longer restricted to fixed formats, providing an
arbitrarily close solution within user-defined convergence
times. Furthermore, compared with Gong et al. (2021), Jia
et al. (2022), Ma et al. (2023), Wang et al. (2019) and
Zhou et al. (2022), the designed TBGs are easier to design and implement in a distributed manner, as they are
well-defined over the entire time domain without requiring
global information, while also preventing singularities and
Zeno behavior associated with discrete sampling. Thus, this
design is particularly beneficial for real-world, large-scale
systems with strict time constraints.
(ii) A novel prescribed-time distributed optimization algorithm for
RAPs: Building upon the proposed TBGs, we devise a novel
prescribed-time distributed optimization algorithm for RAPs
with hard time and resource-demand constraints over unbalanced digraphs. Users can flexibly preset the settling
2

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Automatica 177 (2025) 112313

∑n

with dout
=
i
j=1 aji being the diagonal elements. It is evident
that LTO 1 = 0 holds. A directed path in a digraph is a sequence
of directed and no repeated edges connecting distinct agents. G
is strongly connected if there exists a directed path between any
two distinct agents.

time, independent of algorithm parameters and initial values. Furthermore, the variable substitution based on the
out-Laplacian matrix ensures real-time satisfaction of the
resource-demand constraint, enhancing both feasibility and
online implementation. Notably, unlike existing Laplaciangradient and primal–dual based methods (Chen, Jiang et al.,
2024; Cherukuri & Cortés, 2015; Guo & Chen, 2022a, 2022b;
Ji, Yu, Zhang, Guo, & Li, 2023; Liu et al., 2023; Zhang, Xu
et al., 2023), our algorithm tackles unbalanced digraphs and
consistently maintains resource-demand balance. Additionally, it simplifies the design process by eliminating the need
for Lagrangian multipliers.
(iii) A novel prescribed-time fully distributed optimization algorithm for RAPs: To further eliminate the algorithm’s reliance
on global information (topology-related matrix parameters),
we further develop a novel fully distributed optimization
algorithm with the prescribed-time convergence using an
adaptive approach. Similar to the algorithm in (ii), this fully
distributed version guarantees satisfaction of hard resourcedemand and time constraints. This represents the first study
of prescribed-time and fully distributed optimization over
unbalanced digraphs for resource-constrained RAPs. The
prescribed-time convergence is rigorously analyzed using
the time transformation method and a novel Lyapunov
function construction.

Assumption 1. G is directed and strongly connected.
2.2. Problem formulation
Consider a multi-agent system consisting of n autonomous
agents for collective decision making. Each agent i ∈ V sets its
local decision variable xi (t) ∈ R at time t to find an optimal
solution to the following RAP:
min f (x(t)) =

n
∑

fi (xi (t)),

(1a)

i=1

s. t.

n
∑

xi (t) =

i=1

n
∑

xi (0) = D, ∀t ∈ [0, tf ),

(1b)

i=1

where f : Rn →R and fi : R→R are global and local cost functions, respectively, x = [x1 , . . . , xn ]∈Rn , constant D∈R is the
total demand, and tf is an arbitrary user-determined time. Our
aim is to address problem (1) with hard resource-demand and
time constraints over unbalanced digraphs, which requires the
resource-demand constraint (1b) to be satisfied at any time and
the optimal resource allocation (1a) to be achieved within the
preset time tf .
Let x∗ = [x∗1 , . . . , x∗n ]∈Rn denote the optimal solution to the
problem (1). The following mild assumption is then made, which
is widely utilized in the existing works to guarantee the existence
and uniqueness of the optimal solution x∗ ; see, e.g Guo and Chen
(2022a, 2022b) and Zhou et al. (2022).

The organization of the remaining article is as follows. Section 2 introduces the optimization problem. Section 3 gives a
relaxed TBG design and presents two prescribed-time distributed
optimization algorithms as well as their convergence analysis.
Section 4 conducts simulation validations. Section 5 summarizes
the entire paper.
Notations: R, R+ , Rn and Rm×n represent the sets of real
numbers, positive real numbers, n-dimensional vectors and m×n
matrices. Given a vector x∈Rn , xi , xT , and ∥x∥ represent its ith
entry, transpose, and Euclidean norm, respectively. For a matrix
A∈Rn×n , Aij and ∥A∥ denote its ijth entry and Frobenius norm.
Let λmax (A), λmin (A), and λ2 (A) be its maximum, minimum, and
second smallest eigenvalue, respectively. Symbol diag{ai } ∈ Rn×n
denotes a diagonal matrix with ai as its diagonal elements for i ∈
{1, . . . , n}. Symbols 1 and 0 denote the vector with each element
being 1 and 0, respectively. In represents an identity matrix of
n × n dimensions. For a differentiable function f , ∇ f and ∇ 2 f
denote its gradient and Hessian matrix, respectively. Symbol ⊗
represents the Kronecker product. For a piecewise differentiable
function r(t , τ ) with t ≥ 0 and a constant τ , defined as r(t , τ ) =
r1 (t , τ ) for 0 ≤ t < tf and r(t , τ ) = r2 (t , τ ) for t ≥ tf , where
functions r1 (t , τ ) and r2 (t , τ ) are differentiable with respect to
dr (t ,τ )
t, its derivative is given by: r ′ (t , τ ) = 1dt
for 0 ≤ t < tf ,
′
r (t , τ ) = (r2 (t , τ ) − r1 (tf − , τ ))δ (t − tf − ) for t = tf , where δ (t − tf − )
dr (t ,τ )
is the Dirac delta function, and r ′ (t , τ ) = 2dt for t > tf .

Assumption 2. The global cost function f (·) is twice continuously differentiable. Moreover, it is also l1 -strongly convex and
l2 -smooth, i.e., l1 I ≤ ∇ 2 f (·) ≤ l2 I.
The following lemmas play an important role in the subsequent convergence analysis.
Lemma 1. For problem (1) under Assumptions 1 and 2, x(t) =
x∗ holds if and only if the Karush-Kuhn–Tucker (KKT) optimality
conditions are satisfied, namely, LTO ∇ f (x∗ ) = 0 and 1T x∗ = D.
Lemma 2 (Zhou et al., 2022). Let A ∈ Rn×n be a symmetric matrix
with eigenvalues λ1 ≤ · · · ≤ λn and corresponding eigenvectors
u1 , . . . , un . For any vector x orthogonal to uj for j = 1, · · ·, i − 1, the
inequality xT Ax ≥ λi xT x holds for i = 1, · · ·, n.
Before ending this section, the definition regarding prescribedtime optimization is recalled.

2. Problem statement

Definition 1 (Guo & Chen, 2022b). The problem (1) is characterized as prescribed-time optimization at time tf if the following conditions are satisfied: (i) limt →tf ∥x(t) − x∗ ∥ ≤ ε ; (ii)
∥x(t) − x∗ ∥ ≤ ε , for ∀t > tf ; (iii) limt →∞ ∥x(t) − x∗ ∥ = 0, where
tf can be arbitrary preset, and error ε>0 can be tuned to the
desired value.

2.1. Network topology
The network topology among n agents is depicted as a directed
graph (digraph) G ≜ (V , E ) with an agent set V = {1, . . . , n}, an
edge set E ⊆ V × V , and an adjacency matrix A = [aij ] ∈ Rn×n .
Let (j, i) ∈ E denote that agent i can receive information from
agent j. Let aij = 1 if (j, i) ∈ E and aij = 0 otherwise. Assume
aii = 0 for i ∈ V . The Laplacian matrix L associated with graph G
is defined as L = D − A, where the∑
in-degree matrix D ∈ Rn×n
n
in
is a diagonal matrix with di =
j=1 aij being the diagonal
elements, ∀i ∈ V . Clearly, L1 = 0 holds. Additionally, define
LO = DO − A, where DO ∈ Rn×n is a diagonal out-degree matrix

3. Main results
This section begins with a relaxed TBG design and subsequently develops two prescribed-time distributed optimization
algorithms to address the problem (1).
3

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Automatica 177 (2025) 112313

3.1. A relaxed time-base generator design

• Relaxed design conditions: The developed TBG simplifies condition requirements compared to existing ones while ensuring the prescribed-time convergence of algorithms. To
demonstrate this, an example for TBG (3) is given as

Consider the following system described by

̇ = −δ0 (T (t , tf )+1)S(t), S(0) > 0,
S(t)

(2)

{

where S : R ∪ {0} → R, T (t , tf ) : R ∪ {0} × R → (−1, +∞)
is a time-based function, tf ∈ R+ is an arbitrary predefined time,
and constant δ0 > 0.
+

+

TBG-3: r(t , τ )=

+

It is found that the design condition ‘‘limt →+∞ [γ (t , σ ) −
γ (0, σ )] = +∞’’ in Liu et al. (2023) is no longer required
under TBG-3, where γ (t , σ ) corresponds to r(t , τ ) as referred to throughout this paper. Essentially, the formulation
in (2) integrates the TBG gain and constant terms into the
system gain (Guo & Chen, 2022a; Liu & Yi, 2023; Zhang,
Xu et al., 2023), ensuring exponential convergence of the
system to the origin after the prescribed time tf .
• Enhanced implementation: To demonstrate this, another two
examples of the proposed TBG (3) satisfying (4) are provided
below:

Definition 2. System (2) achieves prescribed-time convergence
at arbitrary time tf , if the following conditions are satisfied: (i)
limt →tf |S(t)| ≤ ε , (ii) |S(t)| ≤ ε , for ∀t > tf , (iii) limt →∞ |S(t)| =
0, where error ε > 0 can be tuned to the preferred value.
Motivated by Liu et al. (2023), our first main result is presented in the following theorem.
Theorem 1. System (2) achieves the prescribed-time convergence
at time tf with gain:
T (t , tf ) = (r(t , τ ))′ ,

{

− ln(2tf − 2t + τ ), 0 ≤ t < tf ,
− ln τ , t ≥ tf ,

{

− ln(3tf2 − 3t 2 + τ ), 0 ≤ t < tf ,
− ln τ , t ≥ tf .

(3)

TBG-4: r(t , τ )=

where a time-dependent function r(t , τ ) with constant τ called a
relaxed TBG is a continuous and piecewise differentiable function,
satisfying the conditions:

TBG-5: r(t , τ )=

lim

τ →0+

[

r(tf , τ )−r(0, τ ) = +∞, 0 < τ ≪ 1,

]

(4a)

r(t , τ )−r(tf , τ ) ≥ 0, ∀t > tf .

(4b)

Compared to the distributed preassigned-time algorithms
with piecewise power-law in Ma et al. (2023) and the
cascading fractional-step stages in Gong et al. (2021), the
proposed TBG (e.g, TBG-3, -4, -5) is easier to design and
implement in a distributed manner, as the aforementioned
approaches require global information for certain parameters. Additionally, unlike the TBG in Jia et al. (2022), which
is undefined beyond tf , the proposed TBG is well-defined
for the whole domain t ∈ (0, +∞), thereby facilitating
the continuous operation of systems. In contrast to the existing time-varying-gain-based prescribed-time algorithms
in Wang et al. (2019) and Zhou et al. (2022), our TBG avoids
the occurrence of singularity and Zeno behavior associated
with discrete sampling.

̇ = −δ0 ((r(t , τ ))′ +
Proof. According to (2) and (3), one derives S(t)
1)S(t). By integrating both sides of the above equality, one obtains
S(t) = S(0)e−δ0 t e−δ0 (r(t ,τ )−r(0,τ )) .

(5)

Then, as t → tf , (i) in Definition 2 is verified from (4a) with
ε = S(0)e−δ0 tf e−δ0 (r(tf ,τ )−r(0,τ )) . It can be observed from (4a) that ε
can be arbitrarily small by adjusting τ .
When t > tf , it can be deduced from (5) that S(t) =
ε e−δ0 (t −tf ) e−δ0 (r(t ,τ )−r(tf ,τ )) . It implies that (ii) and (iii) in Definition
2 hold under (4b). Thus, the proof is completed. ■

3.2. Prescribed-time distributed resource allocation

The proposed TBG (3) in Theorem 1 offers several advantages
over existing TBG designs, particularly in the following three
aspects.

We next present a novel TBG-based prescribed-time optimization algorithm for distributed RAP (1):

• Generality: The TBG r(t , τ ) in (3) can be considered a general

xi (t) = xi (0) − dout
i ξi (t) +

version compared to the existing results, such as Guo and
Chen (2022a, 2022b), Liu and Yi (2023), and Ning et al.
(2019). Specifically, in these existing works, a TBG is defined
as a real-valued and continuously differentiable function
θ (t) if it satisfies the following conditions: i) θ (t) = 0 if
t = 0 and θ (t) = 1 if t ≥ tf ; ii) θ̇ (t) > 0 if t ∈ (0, tf )
and θ̇ (t) = 0 otherwise. Accordingly, to achieve prescribedtime convergence for algorithms, the gain is designed as
θ̇ (t)
T (t , tf ) = 1−θ (t)+τ , with the following TBGs being explored.
(1) TBG-1 (Guo & Chen, 2022b; Ning et al., 2019):

{
θ (t) =

10 6
t
tf6

1, t ≥ tf ,

θ (t)=

aij ξj (t),

ξ̇i (t) = (T (t , tf )+1)(dout
i ψii (t)−

(6a)

n
∑

aji ψij (t)),

(6b)

j=1
n
∑
ψ̇ ij (t) = −k(T (t , tf ) + 1)(
aim (ψij (t) − ψmj (t))
m=1

+ aij (ψij (t) − ∇ fj (xj (t)))), j ∈ V ,

(6c)

where ξi (t) is the auxiliary variable of agent i, ψij (t) represents
the estimation of agent i for ∇ fj (xj (t)), T (t , tf ) is defined in (3)
and (4) with any given time tf , k is the algorithm parameter to
be designed, ∇ fi (xi (t)) denotes the
of fi at xi (t), and xi (0)
∑gradient
n
is the initial condition satisfying
x
(0)
= D.
i
i=1
Define ξ = [ξ1 , . . . , ξn ]T ∈ Rn , ψ i = [ψi1 , . . . , ψin ]T ∈

f

2) TBG-2 (Liu & Yi, 2023):

{

n
∑
j=1

− 24
t 5 + 154 t 4 , 0 ≤ t < tf ,
t
t5
f

0.15(20t 2 − (t + 1) ln τ ), 0 ≤ t < tf ,
0.15(20tf2 − (tf + 1) ln τ ), t ≥ tf .

0.5 sin( tπ t − π2 −2kπ )+0.5, 0≤t <tf , k∈Z ,
f

1, t ≥ tf .

Rn , ψ = ψ T1 , . . . , ψ Tn

]T

2

∈ Rn , ∇ f (x) = [∇ f1 (x1 ), . . ., ∇ fn (xn )]T ,
Ad = diag aij ∈ R
, L̂0 = diag{(LTO )i } with (LTO )i being the
T
ith row of LO , and L̃ = L ⊗ In + Ad . Clearly, −L̃ is Hurwitz under

[
{ }

It can be seen that by setting r(t , τ ) = − ln(1 − θ (t) + τ ),
TBG-1 and TBG-2 can be derived as special cases of TBG (3).
The proposed TBG (3), which satisfies (4), thus provides a
foundation for developing more practical and flexible TBG
designs.

n2 ×n2

Assumption 1. Write (6) into the compact form:
x(t)=x(0) − LO ξ (t),
4

(7a)

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313

ξ̇ (t)=(T (t , tf ) + 1)L̂0 ψ(t),
̇ (t)= − k(T (t , tf ) + 1)L̃(ψ(t) − 1n ⊗∇f (x(t))).
ψ

along (6), one has V̇ 2 (t) = ė T (t)W e(t) + eT (t)W ė (t), which leads
to

(7b)
(7c)

V̇ 2 (t)

Some salient features of the proposed TBG-based prescribedtime optimization algorithm (6) are highlighted as follows. First,
leveraging the out-Laplacian matrix LO , problem (1) involving
the decision variable xi (t) is transformed into a problem involving
the auxiliary variable ξi (t) in (6a). Such a variable transformation
technique (Zhou et al., 2022) empowers us to skillfully incorporate the proposed TBG into the algorithm design. Second, (6b)
employs the gradient descent method to minimize the global
cost in terms of ξi (t), with gain T (t , tf ) ensuring prescribed-time
convergence. Third, agent i cannot obtain the gradient information ∇ fj (xj (t)) from its out-neighbor agent j in the digraph G ,
and thus additional estimation terms are incorporated into (6c)
by adopting a leader-following consensus scheme (Ren, Wang, &
Duan, 2022). Notably, from (6a), it can be observed that the hard
resource-demand constraint 1T x(t)=D is always satisfied at any
time, as 1T LO =0T . This characteristic indicates that the allocated
resources adequately meet the desired demand, thus providing
better quality of service and enhancing the algorithm’s suitability
for online scenarios.
We are now in a position to present our second main result regarding the design of the algorithm parameter k in the estimation
updating law (6c).

T (t , tf )+1

=(−kL̃e(t)+1n ⊗∇ 2 f (x(t))LO L̂0 ψ(t))T W e(t)
+eT (t)W (−kL̃e(t)+1n ⊗∇ 2 f (x(t))LO L̂0 ψ(t))
√
≤ −keT (t)Q e(t)+c2 nl2 ∥W ∥ ∥e(t)∥2
1√
2
+
nl2 ∥W ∥ ∥LO L̂0 ψ (t)∥
c2
√
≤ −(kλmin (Q )−c2 nl2 ∥W ∥)∥e(t)∥2
√
nl2 ∥W ∥
+
∥LO L̂0 ψ(t)∥2 ,
c2

where the constant c2 >0 and the Young’s inequality is utilized to
get the first ‘‘≤’’.
Moreover, according to the fact that −∥a + b∥2 ≤ ∥a∥2 −
∥b∥2 /2, ∀a, b ∈ Rn , one obtains

−∥LO L̂0 ψ(t)∥2 =−∥LO L̂0 e(t)+LO LTO ∇ f (x(t))∥2
λ2 (LTO LO ) T
≤∥LO L̂0 ∥2 ∥e(t)∥2 −
∥LO ∇ f (x(t))∥2 ,
2

2

V2 (t)

λmax (W )

≤ ∥e(t)∥2 ≤

theorem. It can be yielded from (8), (11), (12), (13) and (14) that
V̇ (t) ≤ −δ1 (T (t , tf ) + 1)V (t),

Proof. Recalling (7a) and 1T LO = 0T , it is clear that the hard
resource-demand constraint 1T x(t) = D is maintained at any
time. Then, according to Lemma 1, the remain proof will be
divided into two steps.
Step 1: Define an estimation error as e(t) = ψ (t) − 1n ⊗
∇ f (x(t)). Construct a Lyapunov function as follows:

(15)

with positive constant

(2
)
⎧
2 ⎫
2nl2 ∥W ∥2
l2
l1 
 ⎪
⎪
kλmin (Q ) − 2 l
−
+
L
L̂
O
0
⎪
⎪
l1
4
⎪
⎪
1
⎨
,⎬
λmax (W )
δ1 = min
.
( T
)
⎪
⎪
⎪
⎪
l
λ
L
L
⎪
⎪
1
2
O
O
⎩
⎭

(8)

4
Combining Theorem 1, it is evident that system (15) can achieve

where
V1 (t) =

V2 (t)

.
(14)
λmin (W )
√
Select c1 = 2l2 /l1 , c2 = 2l2 n ∥W ∥ /l1 , and k as shown in the

2

2
2
k> 2
, where W = diag{wi } ⊗ In ∈ Rn ×n
λmin (Q )
is a real positive-definite diagonal matrix with 0 < wi < 1, Q is a
symmetric positive-definite matrix satisfying L̃T W + W L̃ = Q , and
l1 , l2 are defined in Assumption 2.

V (t) ≜ V1 (t) + V2 (t),

(13)

where λ2 (LTO LO ) denotes the second smallest eigenvalue of LTO LO ,
and Lemma 2 is utilized.
Additionally, from the definition of V2 , one attains

Theorem 2.
Suppose that Assumptions 1 and 2 hold. Problem
(1) achieves the prescribed-time optimization under algorithm (6) if
2nl2 ∥W ∥2 /l1 +(l1 /4+l2 /l1 )∥LO L̂0 ∥2

(12)

∆

1



LTO

2
∇ f (x(t)) , V2 (t) = eT (t)W e(t).

2
Taking the time derivative of V1 (t), one obtains

)T

V̇ 1 (t) = LTO ∇ f (x(t)) LTO ∇ 2f (x)ẋ (t).

(

the prescribed-time convergence at the time tf with ε = ε0 =
(eT (0)W e(0) + ∥LTO ∇ f (x(0))∥2 /2)e−δ1 tf e−δ1 (r(tf ,τ )−r(0,τ )) in Definition 2.
Step 2: Define ξ ∗ ∈ Rn satisfying x∗ = x(0) − LO ξ ∗ . It is yielded
from the convexity of f (x(t)) and (7a) that

(9)

(10)

f (x(t)) ≤ f (x∗ ) − ∇ f (x(t))T LO (ξ (t) − ξ ∗ ).

Noting the relation L̂0 ψ (t)−LTO ∇ f (x(t))=L̂0 e(t), it follows (6) and
Assumption 2 that

Denote the unit orthogonal basis vectors u1 , . . . , un ∈ Rn with
u1 satisfying uT1 u1 = 1 and LO u1 = 0. Then a collection of real
numbers
c1 (t), · · ·, cn (t) at time t are available to meet ξ (t) −ξ ∗ =
∑n
c
(t)u
i , which indicates that
i=1 i

V̇ 1 (t)
T (t , tf )+1

= −(L̂0 ψ(t)−L̂0 e(t))T LTO ∇ 2f (x(t))LO L̂0 ψ(t)

∗

= −(LO L̂0 ψ(t)−LO L̂0 e(t)) ∇ f (x(t))LO L̂0 ψ(t)
2
l2 
≤ −(l1 −
) LO L̂0 ψ (t)
T

(16)

2

f (x(t)) − f (x ) ≤

√

∑

 n


2V (t)
ci (t)ui 
.

(17)

i=2

Meanwhile, noting LTO ∇ f (x∗ ) = 0, it follows the strongly convexity of function f in Assumption 2 that

2c1


l2 c1 
LO L̂0 2 ∥e(t)∥2 ,
(11)
2
where the constant c1 > 0.
Since −L̃ is Hurwitz under Assumption 1, there exist a real
2
2
positive-definite diagonal matrix W = diag{wi } ⊗ In ∈ Rn ×n
with 0 < wi < 1 and a symmetric positive-definite matrix Q
such that L̃T W + W L̃ = Q . Then, taking the time derivative of V2
+

f (x(t)) − f (x∗ )

2
l1 
≥ −∇ f (x∗ )TLO (ξ (t) − ξ ∗ ) + x(t) − x∗ 
2
∑
2
 n

l1
 .
≥ λ2 (LTO LO )
c
(t)u
i
i


2

5

i=2

(18)

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313

where pij (t) is the adaptive parameter of agent i with pij (0) >
0, θij (t) is the weight parameter, ηij (t) is an auxiliary variable
of agent i, and the other variables are defined as the same in
algorithm (6).
{ }
2
2
2
Define η = [η11 , . . . , ηnn ] ∈ Rn , Θ = diag θij ∈ Rn ×n ,

Combining (17) and (18) yields

∑
2
 n

(
)
∗ 2

2V (t)
ci (t)ui 
 ≥ f (x(t))−f (x )
i=2

≥

l21
4

λ

T
2 (L O L O )

∑
2
 n

∥x(t) − x ∥ 
ci (t)ui 
 ,
∗ 2

{ }

P = diag pij

i=2

{ }
2
2
2
2
∈ Rn ×n , R = diag rij ∈ Rn ×n with rij = ηij2 .

Write (19) into the compact form:

8
V (t). It can be easily
l1 λ2 (LTO LO )

which implies that ∥x(t) − x∗ ∥2 ≤ 2

deduced that problem (1) under algorithm (6) can achieve the
8ε0
prescribed-time optimization at time tf with ε = 2
as
T
l1 λ2 (LO LO )

defined in Definition 1. Thus, the proof is completed.

■

Remark 1. Unlike some existing finite-time distributed optimization algorithms for RAPs, e.g Firouzbahrami and Nobakhti
(2022) and Wang et al. (2020), which have convergence times
that depend on initial conditions, and some existing fixed-time
distributed optimization algorithms for RAPs, e.g Chen and Li
(2018), Chen, Yang et al. (2024a) and Shi et al. (2022), which
have predictable and fixed convergence times independent of
initial conditions, the proposed algorithm (6) can ensure userdefined convergence times that are also independent of initial
conditions, thereby offering the most flexibility for the user to
preset the convergence time. On the other hand, compared with
the existing prescribed-time distributed optimization algorithms
for RAPs in Chen, Jiang et al. (2024), Guo and Chen (2022a,
2022b), Ji et al. (2023) and Zhang, Xu et al. (2023), which are
applicable to undirected graphs or balanced digraphs, our algorithm is suitable for RAP applications over generic unbalanced
digraphs, which, to the best of the authors’ knowledge, represents a significant challenge in the existing prescribed-time
distributed optimization studies over unbalanced digraphs. More
concretely, the convergence analysis of the existing prescribedtime optimization algorithms aforementioned relies heavily on
constructing a scheme like (15), which is difficult to achieve
under unbalanced digraphs. Even though the technique of using
left eigenvalue estimation has been commonly employed to deal
with the topology unbalancing issue, it unavoidably introduces
additional exponential error terms like e−β t with a constant β >
0, as presented in Zhu et al. (2021). This potentially precludes the
derivation of (15) and restricts the direct incorporation of TBGs.
To tackle this, the out-Laplacian matrix is integrated into (6a) for
unbalanced communication topologies.

x(t)=x(0) − LO ξ (t),

(20a)

ξ̇ (t)=(T (t , tf ) + 1)L̂0 ψ(t),
̇ (t)= − (T (t , tf ) + 1)Θ (t)η(t),
ψ

(20b)

Θ (t) = P(t) + R(t),

(20d)

η(t)=L̃(ψ(t) − 1n ⊗∇f (x(t))).

(20e)

(20c)

The rationale behind algorithm (19) is explained as follows.
First, utilizing the out-Laplacian matrix-based variable substitution (19a), the original problem (1) with xi (t) is converted
into one involving ξi (t). Second, the gradient descent method is
employed in (19b) to minimize the cost with respect to ξi (t).
Third, due to the unidirectional communication characteristic of
digraphs, for each agent i, the auxiliary variables ψij (t) and ηij (t)
are designed in (19c) and (19f), respectively, to estimate the gradient ∇ fj (xj (t)) of agent j. Finally, to make the algorithm operate
in a fully distributed manner, the adaptive gains θij (t) based on
local information are introduced via a sum-based scheme in (19d)
and (19e).
Before proceeding, the following time-based transformation
function and related lemmas are presented to facilitate subsequent convergence analysis.
Lemma 3 (Benner, Findeisen, Flockerzi, Reichl, Sundmacher et al.,
2014; Feng & Hu, 2020). For a dynamical system ẋ (t) = f (t , x(t))
with x(0) = x0 . Let ζ (t) represent the solution to this system and
tf > 0 is a user-defined time. There exists a time transformation
function t = λ(s), where λ(s) is a continuous differentiable and
strictly increasing function on s ∈ [0, +∞) satisfying conditions
λ(0) = 0, λ′ (0) = tf , lims→+∞ λ(s) = tf , and lims→+∞ λ′ (s) = 0.
Hence, for ϕ (s) ≜ ζ (t) one obtains

ϕ ′ (s) = λ′ (s)f (λ(s), ϕ (s)), ϕ (λ−1 (0)) = x0 ,

(21a)

lims→+∞ ϕ (s) = limt →tf ζ (t),

(21b)

3.3. Prescribed-time fully distributed resource allocation

with ϕ ′ (s) = dϕ (s)/ds and λ′ (s) = dλ(s)/ds.

From Theorem 2, it can be seen that the algorithm parameter
k relies on the topology-related matrix parameters (LO , L̂0 , L̃)
which are regarded as certain global graph information. To further eliminate this requirement, a novel fully distributed optimization algorithm with the prescribed-time convergence is
developed for problem (1) as follows:

According to Lemma 3, the algorithm (19) on time interval
t ∈ [0, tf ) can be transformed into the following form on a new
infinite-time interval s ∈ [0, +∞) with t = λ(s):
x̃(s)=x(0) − LO ξ̃ (s),
′

ξ̃ (s)=ϑ (s)L̂0 ψ̃(s),
′

n

xi (t)=xi (0)−dout
i ξi (t)+

∑

aij ξj (t),

(19a)

j=1
n

∑
ξ̇i (t)=(T (t , tf )+1)(dout
aji ψij (t)),
i ψii (t)−

(19b)
(19c)

θij (t)=pij (t)+ηij2 (t),

(19d)

ṗ ij (t)=(T (t , tf )+1)ηij2 (t),

(19e)

∑

aik (ψij (t)−ψkj (t))+aij (ψij (t)−∇ fj(xj (t))),

ψ̃ (s)=−ϑ (s)Θ̃ (s)η̃(s),

(22c)

Θ̃ (s) = P̃(s) + R̃(s),

(22d)

η̃(s)=L̃(ψ̃(s)−1n ⊗∇f (x̃(s))),

(22e)

Lemma 4. For algorithm (22) under Assumptions 1 and 2, there
holds lims→+∞ x̃(s)=x∗ . Moreover, p̃ij (s), ∀i, j∈V , converge to certain
finite constants as s→+∞.

n

ηij (t)=

(22b)

where ϑ (s) = λ′ (s)(T (λ(s), tf ) + 1) > 0, x̃(s) ≜ x(t), ξ̃ (s) ≜
ξ (t), ψ̃(s) ≜ ψ(t), η̃(s) ≜ η(t), Θ̃ (s) ≜ Θ (t), R̃(s) ≜ R(t) with
r̃ij (s) = η̃ij2 (s) being diagonal elements, P̃(s) ≜ P(t) with p̃ij (s) being
diagonal elements, and p̃′ij (s) = ϑ (s)η̃ij2 (s).

j=1

ψ̇ ij (t)=−(T (t , tf )+1)θij (t)ηij (t),

(22a)

(19f)

k=1

6

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313
n

Proof. Consider the following Lyapunov function:
V (s) ≜ V1 (s) + V2 (s) + V3 (s),
where
V1 (s) =

n

ϑ (s)

2

(r̃ij2 (s) + 2r̃ij (s)p̃ij (s)),

(25)
V3′ (s)=
(26)

ij

=ϑ (s)

i=1 j=1

l2
2c1
2

2

ẽ(s) .

−∥LO L̂0 ψ̃(s)∥2 ≤ ∥LO L̂0 ∥2 ∥ẽ(s)∥2
λ2 (LTO LO ) T
−
∥LO ∇ f (x̃(s))∥2 .
2

V3′ (s)

(28)

+ϑ (s)

2

c3

λmax (L̃T L̃)+(

κ l2 c1 +2M

.

2

O

2

O

]

)∥LO L̂0 ∥2 ∥ẽ(s)∥2

−ϑ (s)λmin (P ∗ )λmin (L̃T L̃)∥ẽ(s)∥2

2
M
−ϑ (s) λ2 (LT LO )LT ∇ f (x̃(s)) ,

(34)

with M = κ (l1 − l2 /2c1 ) − nl22 ∥L̃∥2 /c3 .
Selecting the positive constants κ , c1 , c3 , p∗ij that satisfy
λmin (Q )−2c3 >0, κ (l1 −l2 /2c1 )−nl22 ∥L̃∥2 /c3 > 0,
κ l2 c1
2
T
2
c λmax (L̃ L̃)+( 2 +M)∥LO L̂0 ∥
λmin (L̃T L̃)

p∗ij > 3

,

one gets V (s)<0 for ∥ ∇ f (x̃(s))∥ ̸ = 0 and
 ∥ẽ(s)∥ ̸= 0.
 According
to (23)–(26), it can be concluded that LTO ∇ f (x̃(s)), p̃ij (s) and
r̃ ij (s) remain bounded. As p̃ij (s) are monotonically increasing, they
converge to certain constants when s → +∞. By LaSalle’s Invariance Principle, it follows that lims→+∞ ∥LTO ∇ f (x̃(s))∥ = 0 and
lims→+∞ r̃ ij (s) = 0, indicating lims→+∞ ∥ẽ(s)∥ = 0. Additionally,
it follows from (22a) and Assumption 1 that the hard resourcedemand constraint 1T x̃(s) = D is maintained for ∀s. Thus, the
proof is completed. ■
′

(29)

where the facts r̃ij (s) = η̃ij2 (s) and (22d) are used for the second
‘‘=’’, and W is given in Theorem 2.
Based on (22a), (22b), and (22e), one gets
′

η̃′ (s) = L̃[ψ̃ (s) − (1n ⊗∇ 2 f (x̃(s)))x̃′ (s)],
which indicates

LTO

Now, we are in a position to present our third main result.

(30)

Theorem 3. Under Assumptions 1 and 2, the following results hold
for algorithm (19):

For the former term in (29), combining (30) and the statements
in Theorem 2 yields

(i) Problem (1) can achieve the prescribed-time optimization in a
fully distributed manner. Moreover, the hard resource-demand
constraint (1b) is always satisfied.
(ii) The dynamic gains pij (t) converge to certain finite constants,
for ∀i, j ∈ V .

2η̃ (s)Θ̃ (s)W η̃ (s)
′

ϑ (s)
T

≤−ẽ (s)L̃T Θ̃ (s)Q Θ̃ (s)L̃ẽ(s)
(31)

Proof. Based on (19a) and Assumption 1, the hard resourcedemand constraint 1T x(t) = D is maintained for ∀t. Then, it
follows Lemma 4 that there exists a sufficiently large S such that
limt →t − ∥x(t) − x∗ ∥ = lims→S ∥x̃(s) − x∗ ∥ = ε , where ε is an

with the constant c3 > 0.
For the latter term of (29), noting that 0 < wi < 1, it follows
from the Young’s inequality that
n
n
∑
∑

(33)
n2 ×n2

T

[

i=1 j=1

∥LO L̂0 ψ̃(s)∥2 ,

T

ẽ (s)L̃T L̃ẽ(s)

V ′ (s)≤−ϑ (s)(λmin (Q ) − 2c3 )ẽ (s)L̃T Θ̃ 2 (s)L̃ẽ(s)

i=1 j=1

c3

1
c3

It follows from (22d) that the inequality Θ̃ (s)>P̃ 2 (s) + R̃2 (s)
holds. Substituting (31) and (32) into (29), and combining with
(23), (27), (28), and (33) yields:

n
n
n
n
∑
∑
∑
∑
=
2wi θ̃ij(s)η̃ij(s)η̃ij′ (s)+
wi r̃ij (s)ϑ (s)η̃ij2(s)

nl22 ∥L̃∥2

η̃ij2 (s)−p∗ij η̃ij2 (s)),

2

i=1 j=1

+c3 ẽT (s)L̃T Θ̃ 2 (s)L̃ẽ(s)+

c3

where (22e) is utilized and P ∗ = diag{p∗ij } ∈ R

(27)

n
n
n
n
∑
∑
∑
∑
=
wi (r̃ij(s)+p̃ij(s))r̃ij′ (s)+
wi r̃ij (s)p̃′ij(s)

η̃′ (s)
=−L̃Θ̃ (s)η̃(s)+L̃(1n ⊗∇ 2 f (x̃(s)))LO L̂0 ψ̃(s).
ϑ (s)

1

− ẽT (s)L̃T P ∗ L̃ẽ(s),

V2′ (s)

n
n
∑
∑
=2η̃T(s)Θ̃ (s)W η̃′(s)+
wi r̃ij (s)ϑ (s)η̃ij2 (s),

(c3 p̃2ij (s)η̃ij2 (s)+

≤ c3 ẽT (s)L̃T P̃ 2 (s)L̃ẽ(s) +

ϑ (s)

Secondly, from (25) it is not difficult to obtain that

i=1 j=1

n
n
∑
∑

which leads to

From (13), there holds

i=1 j=1

(p̃ij (s) − p∗ij )ϑ (s)η̃ij2 (s)

i=1 j=1

)∥LO L̂0 ψ̃ (s)∥2

κ l2 c1 ∥LO L̂0 ∥2 


(32)

i=1 j=1


p̃ij (s) − p∗ 2 ,

≤ − κ (l1 −

T

n
n
∑
∑

n

1 ∑∑

+

η̃ij2 (s))

Thirdly, noting p̃′ij (s) = ϑ (s)η̃ij2 (s) and differentiating V3 (s) with
respect to s, one obtains

where p∗ij > 0 is a constant to be designed for i, j ∈ V , and
wi ∈ (0, 1) is defined in Theorem 2 for i ∈ V .
Let ẽ(s) = ψ̃ (s)−1n ⊗∇f (x̃(s)). Firstly, similar to (10) and (11),
one attains
V1′ (s)

1
c3

c3

n

∑ ∑ wi

2

ϑ (s)(c3 r̃ij2 (s)η̃ij2 (s) +

i=1 j=1

(24)

O

i=1 j=1

V3 (s) =

∑∑

[
]
1
=ϑ (s) c3 ẽT (s)L̃T R̃2 (s)L̃ẽ(s)+ ẽT (s)L̃T L̃ẽ(s) .

2
L ∇ f (x̃(s)) ,

n

V2 (s) =

(23)

κ
 T
2

≤

n

f

arbitrarily small constant. Thus, condition (i) in Definition 1 holds.
Similarly, each pij (t), i, j ∈ V , can converge to the neighborhood
of certain finite constant as t → tf− .

ϑ (s)wi (r̃ij (s)η̃ij (s))η̃ij (s)

i=1 j=1

7

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313

Next, analogous to (23), we construct the following Lyapunov
function on time interval t ∈ [tf , +∞):
V (t) ≜ κ V1 (t) + V3 (t) + V4 (t),

(35)

with V1 (t) defined in (9) and
V3 (t) =

n
n
∑
∑
wi

2

i=1 j=1
n

V4 (t) =

(rij2 (t) + 2rij (t)pij (t)),

Fig. 1. Communication graph G in Case 1.
n

1 ∑∑

∗ 2

pij (t) − p



ij

2

(36)

.

(37)

i=1 j=1

knowledge, remains a challenge in the theoretic research of distributed optimization over digraphs. On the other hand, in some
practical applications such as cooperative search-rescue missions
or autonomous driving, agents are often equipped with suitable
ranging sensors (e.g, LiDAR, radar, or ultrasonic sensors) to detect
nearby agents within their communication or interaction range.
Each agent thus can determine which neighbors fall within their
out-degree range. Additionally, in practical multi-agent systems,
cooperation usually relies on predefined communication protocols (e.g, Bluetooth, Wi-Fi, or V2X) that inherently allow agents
to identify their out-neighbors based on range or connectivity
criteria. Therefore, the identifiable out-neighbor requirement is
often already integrated into the hardware or software design on
each agent. In other words, it is practically achievable for each
agent to identify all of its underlying out-neighbors.

Recalling the analysis to (27)–(34) in the proof of Lemma 4, one
deduces from (35)–(37) that
V̇ (t) ≤ −(λmin (Q ) − 2c3 )eT (t)L̃T Θ 2 (t)L̃e(t)

[
+

2

c3

λmax (L̃ L̃)+(
T

κ l2 c1 +2M
2

]
)∥LO L̂0 ∥ ∥e(t)∥2
2

− λmin (P )λmin (L̃ L̃)∥e(t)∥2
T

∗

−

M
2

λ2 (LTO LO )∥LTO ∇ f (x(t))∥2 .

(38)

With the same parameters selections in Lemma 4, one knows
that V̇ (t) < 0 for ∥LTO ∇ f (x(t))∥ ̸ = 0 and ∥e(t)∥ ̸ = 0 on
t ∈ [tf , +∞). Since V (t) is continuous at t = tf , it follows
that V (tf ) = limt →t − V (t). Combined with the LaSalle’s Invariance
f

Principle, it follows that limt →+∞ ∥LTO ∇ f (x(t))∥ = 0, which leads
to limt →+∞ x(t) = x∗ . Thus, conditions (ii) and (iii) in Definition
1 hold. Moreover, pij (t) are monotonically increasing from (19e)
and converge to some constants for ∀i, j ∈ V . Thus, the proof is
completed. ■

Remark 4. The proposed prescribed-time optimization algorithms (6) and (19) can be potentially extended to cope with
some local decision variable constraints. For example, if the box
constraints xi ∈ [xli , xui ] are considered in problem (1) for any
agent i ∈ V , where xli and xui represent the lower and upper
limitations of agent i’s decision variable. One may then incorporate additional penalty terms, such as smoothing exact penalty
functions (Pinar & Zenios, 1994), into the objective functions to
obtain the approximate optimal solutions.

Remark 2. Compared with fully distributed optimization algorithms with the global asymptotic convergence in Li et al. (2017),
Zhao et al. (2017) and Zhou et al. (2019), our algorithm (19) can
tackle RAPs with coupled equality constraint over unbalanced digraphs and achieve the prescribed-time performance. In Jia et al.
(2022), some exponential terms were introduced in the algorithm
to achieve fully distributed Nash Equilibrium seeking. Instead, our
algorithm (19) integrates TBGs to ensure the prescribed-time fully
distributed optimization while preserving the boundedness of the
adaptive gain parameters.

4. Simulations
In this section, three case studies are provided to validate
the efficacy of the proposed algorithms (6) and (19) under the
designed TBG-3. For consistency among the case studies, we
choose τ = 10−6 and the prescribed time tf = 3s, and set the
initial conditions as xi (0)=50 and pij (0)=0.02 for the following
simulations.
Case 1: Distributed energy management
Consider a distributed energy management system consisting
of n = 6 agents, where agents 1, 2, 3 are conventional generators
(CGs), agents 4, 5 are energy storage devices (ESs), and agent 6 is
a renewable generator (RG). The corresponding communication
digraph G satisfying Assumption 1 is depicted in Fig. 1. The aim
is to minimize costs with the resource-demand constraint:

Remark 3. From the proposed algorithms (6) and (19), it can be
seen that each agent needs to identify its out-neighbors
(i.e., nonzero aji ) for successful implementation. However, it is
noteworthy that data/information transmissions from these outneighbors to each agent i are not required. This ensures that
out-degree neighbors are identified without transmitting sensitive or operational data to them, maintaining a clear boundary
between identification and data privacy. Specifically, this stems
from the fact that ψij (t) are local auxiliary variables of agent i, designed to estimate ∇ fj (xj (t)), j ∈ V . While this requirement might
be perceived as a limitation, it is relatively modest. For example,
similar limitations exist in many widely studied out-degreebased optimization algorithms, such as the push-sum algorithms
utilizing column-stochastic matrices (Nedić, Olshevsky, & Shi,
2017) and Yu, Liu, Zheng, and Zhu (2021), and the push-pull
algorithms employing row-stochastic and column-stochastic matrices (Pu, Shi, Xu, & Nedić, 2021; Xin & Khan, 2020). Compared to
these algorithms, it should be mentioned that ours mitigate this
limitation by eliminating the need for agents to push weighted
information to their out-neighbors using weights derived from
a column-stochastic matrix. How to remove the requirement
of identifiable out-neighbors for each agent, to the best of our

min

6
∑
i=1

fi (xi ), s. t.

6
∑
i=1

xi (t) =

6
∑

xi (0) = 300,

i=1

where for CGs, the cost function fi (xi )=fico (xi )+fiem (xi ) consists
of the generation cost fico (xi ) and the environmental pollution
cost fiem (xi ), and for ESs and RG, fi (xi ) denotes their maintenance
or operating costs. The specific forms are shown as follows:
fico (xi ) = ai x2i + bi xi + ci , fiem (xi ) = b̂i xi , i=1, 2, 3, fi (xi ) =
ai x2i , i=4, 5, fi (xi ) = ai x2i + bi xi , i = 6, where coefficients are
set as a1 = 0.32, a2 = 0.24, a3 = 0.16, a4 = 0.16, a5 = 0.12,
a6 = 0.4, b1 = 0.18, b2 = 0.30, b3 = 0.40, b6 = 0.10,
b̂1 = 0.30, b̂2 = 0.26, b̂3 = 0.36, and c1 =c2 =c3 =0. It is evident
that Assumption 2 is satisfied.
8

M. Luan, G. Wen, X. Ge et al.

Automatica 177 (2025) 112313

Fig. 2. Simulation results under algorithm (6) in Case 1. (a) The local decision
∑6
variables xi (t). (b) The overall supply
i=1 xi (t). (c) The gradient estimations
ψij (t) (solid lines) and gradient values ∇ fi (xi (t)) (dashed lines). (d) The local
decisions xi (t) without TBGs for comparison.

Fig. 3. Simulation results under algorithm (19) in Case 1. (a) The local decision
variables xi (t). (b) The global cost function f . (c) The gradient estimations
ψij (t) (solid lines) and gradient values ∇ fi (xi (t)) (dashed lines). (d) The adaptive
parameters pij (t).

Applying algorithms (6) and (19), the simulation results are
demonstrated in Figs. 2 and 3, respectively. Specifically, the optimal resource allocation is achieved at x∗ = [30.3106, 40.2463,
59.7458, 62.1098, 82.8609, 24.7266] in Fig. 2(a) when tf = 3s.
∑6
Moreover, as observed in Fig. 2(b), the overall supply
i=1 xi (t)
always equals the total demand 300. Additionally, Fig. 2(c) shows
that the gradient estimations ψij (t) of agent i converge to the
gradient values ∇ fj (xj (t)), respectively, with all ∇ fj (xj (t)) reaching
consensus. These results confirm that the optimality conditions
are satisfied.
For a comparison purpose, we perform the proposed algorithm
(6) again but without TBGs. The simulation results are depicted
in Fig. 2(d), from which one can observe that the corresponding
algorithm without TBGs achieves convergence at approximately
t = 30s. In contrast, the proposed algorithm (6) with TBG3 achieves prescribed-time convergence within the user-defined
time tf = 3s.
As shown in Fig. 3, our algorithm (19) can also ensure both
the optimal allocation and the convergence of the total cost to
the optimal value 3024.53 within the prescribed time tf = 3s.
Furthermore, Fig. 3(d) demonstrates that the adaptive parameters
pij (t) converge to some constant values within the prescribed
time.
Case 2: Economic dispatch for IEEE 30-bus system
To further validate the convergence performance of algorithm
(6), we focus in this case on solving the economic dispatch problem for the widely-explored IEEE 30-bus system with 6 generators. For brevity, the cost function coefficients of generators
are borrowed from Liu and Yang (2021) (see Table II therein),
and the communication among generators is configured over a
randomly selected unbalanced digraph G . The simulation results
are illustrated in Fig. 4. It can be readily observed that the decision variable x(t) reaches the optimal solution [52.6041, 55.0240,
41.5696, 44.4025, 54.7969, 51.6029], and the global cost achieves
the optimal value 2108.06 within the prescribed time tf = 3s.
Case 3: Distributed automatic generation control
This case demonstrates the application of the proposed algorithm (19) in a distributed automatic generation control application scenario. While distributed economic dispatch primarily
focuses on optimizing resource allocation over long-term planning horizons, distributed automatic generation control manages

Fig. 4. Simulation results of algorithm (6) for IEEE-30 bus in Case 2. (a) The
local decision variables xi (t). (b) The global cost function f .

the dynamic regulation of power generation to mitigate shortterm fluctuations in demand and supply, ensuring continuous
stability and reliability of the power grid in real-time. Consider
the distributed automatic generation control problem as discussed in Section 6.3 of Doostmohammadian et al. (2025) with
Pmis = 250 and without inequality constraints. Implementing
algorithm (19), the simulation results are illustrated in Fig. 5.
More specifically, Figs. 5(a) and (b) demonstrate that the decision variable x(t) reaches the optimal solution [57.6811, 60.1247,
37.3246, 43.4245, 51.4450] and the adaptive parameters pij (t)
gradually increase and approach some finite values. Fig. 5(c)
verifies that the global cost function achieves the optimal value of
2788.07 at tf = 3s. Furthermore, Fig. 5(d) substantiates that the
auxiliary variables ψij (t) of agent i converge in prescribed-time to
the true gradient values ∇ fj (xj (t)). Meanwhile, the gradient values
∇ fj (xj (t)) also reach consensus for j ∈ V .
5. Conclusion
The prescribed-time distributed optimization has been achieved for RAPs with hard resource-demand and time constraints
over unbalanced digraphs. Two prescribed-time distributed optimization algorithms, one in a distributed fashion and the other
in a fully distributed fashion, have been developed to solve the
RAP through integrating relaxed TBGs. It has been shown that
both distributed optimization algorithms can achieve arbitrarily
close optimal resource allocation within any preset time, while
satisfying the resource-demand constraint in real-time. Several
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Automatica 177 (2025) 112313
Guo, Z., & Chen, G. (2022b). Distributed dynamic event-triggered and practical
predefined-time resource allocation in cyber–physical systems. Automatica,
142, Article 110390.
Guo, L., Shi, X., Cao, J., & Wang, Z. (2024). Exponential convergence of primal–
dual dynamics under general conditions and its application to distributed
optimization. IEEE Transactions on Neural Networks and Learning Systems,
35(4), 5551–5565.
Ji, L., Yu, L., Zhang, C., Guo, X., & Li, H. (2023). Initialization-free distributed
prescribed-time consensus-based algorithm for economic dispatch problem
over directed network. Neurocomputing, 533, 1–9.
Jia, G., Ye, M., & Ding, L. (2022). Exponentially convergent and prescribed-time
fully distributed Nash equilibrium seeking strategy design. In Proceedings of
the 2022 IEEE 61st conference on decision and control (CDC), Cancun, Mexico
(pp. 6401–6406).
Li, Z., Ding, Z., Sun, J., & Li, Z. (2017). Distributed adaptive convex optimization
on directed graphs via continuous-time algorithms. IEEE Transactions on
Automatic Control, 63(5), 1434–1441.
Liu, Y., Xia, Z., & Gui, W. (2023). Multiobjective distributed optimization via a
predefined-time multiagent approach. IEEE Transactions on Automatic Control,
68(11), 6998–7005.
Liu, L.-N., & Yang, G.-H. (2021). Distributed optimal economic environmental
dispatch for microgrids over time-varying directed communication graph.
IEEE Transactions on Network Science and Engineering, 8(2), 1913–1924.
Liu, J., & Yi, P. (2023). Predefined-time distributed Nash equilibrium seeking for noncooperative games with event-triggered communication. IEEE
Transactions on Circuits and Systems II: Express Briefs, 70(9), 3434–3438.
Ma, L., Hu, C., Yu, J., Wang, L., & Jiang, H. (2023). Distributed fixed/preassignedtime optimization based on piecewise power-law design. IEEE Transactions
on Cybernetics, 53(7), 4320–4333.
Nedić, A., Olshevsky, A., & Shi, W. (2017). Achieving geometric convergence
for distributed optimization over time-varying graphs. SIAM Journal on
Optimization, 27(4), 2597–2633.
Ning, B., Han, Q.-L., & Zuo, Z. (2019). Practical fixed-time consensus for
integrator-type multi-agent systems: A time base generator approach.
Automatica, 105, 406–414.
Ning, B., Han, Q.-L., Zuo, Z., Ding, L., Lu, Q., & Ge, X. (2023). Fixed-time and
prescribed-time consensus control of multiagent systems and its applications: A survey of recent trends and methodologies. IEEE Transactions on
Industrial Informatics, 19(2), 1121–1135.
Pinar, M. Ç., & Zenios, S. A. (1994). On smoothing exact penalty functions
for convex constrained optimization. SIAM Journal on Optimization, 4(3),
486–511.
Pu, S., Shi, W., Xu, J., & Nedić, A. (2021). Push-pull gradient methods for
distributed optimization in networks. IEEE Transactions on Automatic Control,
66(1), 1–16.
Ren, Y., Wang, Q., & Duan, Z. (2022). Optimal distributed leader-following
consensus of linear multi-agent systems: A dynamic average consensusbased approach. IEEE Transactions on Circuits and Systems II: Express Briefs,
69(3), 1208–1212.
Rivera, J., & Jacobsen, H.-A. (2014). A distributed anytime algorithm for network
utility maximization with application to real-time EV charging control. In
Proceedings of the 53rd IEEE conference on decision and control, Los Angeles,
CA, USA (pp. 947–952).
Shi, X., Xu, L., Yang, T., Lin, Z., & Wang, X. (2022). Distributed fixed-time
resource allocation algorithm for the general linear multi-agent systems. IEEE
Transactions on Circuits and Systems II: Express Briefs, 69(6), 2867–2871.
Wang, B., Fei, Q., & Wu, Q. (2020). Distributed time-varying resource allocation
optimization based on finite-time consensus approach. IEEE Control Systems
Letters, 5(2), 599–604.
Wang, Y., Song, Y., Hill, D. J., & Krstic, M. (2019). Prescribed-time consensus and
containment control of networked multiagent systems. IEEE Transactions on
Cybernetics, 49(4), 1138–1147.
Xin, R., & Khan, U. A. (2020). Distributed heavy-ball: A generalization and
acceleration of first-order methods with gradient tracking. IEEE Transactions
on Automatic Control, 65(6), 2627–2633.
Yang, T., Yi, X., Wu, J., Yuan, Y., Wu, D., Meng, Z., Hong, Y., Wang, H., Lin, Z., &
Johansson, K. H. (2019). A survey of distributed optimization. Annual Reviews
in Control, 47, 278–305.
Yu, W., Liu, H., Zheng, W. X., & Zhu, Y. (2021). Distributed discrete-time
convex optimization with nonidentical local constraints over time-varying
unbalanced directed graphs. Automatica, 134, Article 109899.
Yuan, D., Wang, L., Proutiere, A., & Shi, G. (2024). Distributed zeroth-order optimization: Convergence rates that match centralized counterpart. Automatica,
159, Article 111328.
Yue, D., Baldi, S., Cao, J., Li, Q., & De Schutter, B. (2023). Distributed adaptive
resource allocation: An uncertain saddle-point dynamics viewpoint. IEEE/CAA
Journal of Automatica Sinica, 10(12), 2209–2221.

Fig. 5. Simulation results of algorithm (19) in Case 3. (a) The local decision
variables xi (t). (b) The adaptive parameters pij (t). (c) The global cost function
f . (d) The gradient estimations ψij (t) (solid lines) and gradient values ∇ fi (xi (t))
(dashed lines).

case studies have been presented to validate the proposed optimization algorithms. Compared to traditional primal–dual algorithms, the proposed algorithms stand out for their simplicity and convenience. As a potential future work, we will focus on designing prescribed-time and fully distributed optimization algorithms that are applicable to RAPs with general coupled
inequality constraints.
References
Benner, P., Findeisen, R., Flockerzi, D., Reichl, U., Sundmacher, K., & Benner, P.
(2014). Large-scale networks in engineering and life sciences. Springer.
Chen, S., Jiang, H., & Yu, Z. (2024). Distributed predefined-time optimization
algorithm: Dynamic event-triggered control. IEEE Transactions on Control of
Network Systems, 11(1), 486–497.
Chen, G., & Li, Z. (2018). A fixed-time convergent algorithm for distributed
convex optimization in multi-agent systems. Automatica, 95, 539–543.
Chen, J., Yang, Y., & Qin, S. (2024a). A distributed optimization algorithm for
fixed-time flocking of second-order multiagent systems. IEEE Transactions on
Network Science and Engineering, 11(1), 152–162.
Chen, G., Yang, Q., Song, Y., & Lewis, F. L. (2022). Fixed-time projection algorithm
for distributed constrained optimization on time-varying digraphs. IEEE
Transactions on Automatic Control, 67(1), 390–397.
Cherukuri, A., & Cortés, J. (2015). Distributed generator coordination for initialization and anytime optimization in economic dispatch. IEEE Transactions on
Control of Network Systems, 2(3), 226–237.
Deng, Z., & Liu, C. (2024). Distributed algorithm design for nonsmooth and
nonlinear resource allocation problems of autonomous high-order agents
and its application to smart grids. IEEE Transactions on Industrial Electronics,
71(12), 16473–16484.
Doostmohammadian, M., Aghasi, A., Pirani, M., Nekouei, E., Zarrabi, H., Keypour, R., Rikos, A. I., & Johansson, K. H. (2025). Survey of distributed
algorithms for resource allocation over multi-agent systems. Annual Reviews
in Control, 59, Article 100983.
Feng, Z., & Hu, G. (2020). Prescribed-time fully distributed Nash equilibrium
seeking in noncooperative games. arXiv preprint arXiv:2009.11649.
Firouzbahrami, M., & Nobakhti, A. (2022). Cooperative fixed-time/finite-time
distributed robust optimization of multi-agent systems. Automatica, 142,
Article 110358.
Gong, X., Cui, Y., Shen, J., Xiong, J., & Huang, T. (2021). Distributed optimization
in prescribed-time: Theory and experiment. IEEE Transactions on Network
Science and Engineering, 9(2), 564–576.
Guo, Z., & Chen, G. (2022a). Predefined-time distributed optimal allocation of
resources: A time-base generator scheme. IEEE Transactions on Systems, Man,
and Cybernetics: Systems, 52(1), 438–447.
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Automatica 177 (2025) 112313
Xiaohua Ge received the Ph.D. degree in Computer
Engineering from Central Queensland University, Rockhampton, QLD, Australia, in 2014.
From 2015 to 2017, he was a Research Fellow with
the Griffith School of Engineering, Griffith University,
Gold Coast, QLD, Australia. He is currently a Senior
Lecturer with the School of Engineering, Swinburne
University of Technology, Melbourne, VIC, Australia.
His research interests include networked, secure, and
intelligent control and estimation theories, and their
applications in mobile robots, electric vehicles, autonomous road and rail vehicles, connected vehicles, intelligent transportation
systems, and so on.
Dr. Ge is currently an Associate Editor of IEEE Transactions on Industrial
Informatics, IEEE Transactions on Systems, Man, and Cybernetics: Systems, IEEE
Transactions on Intelligent Vehicles, and IEEE/CAA Journal of Automatica Sinica.
He was an Associate Editor of IEEE Transactions on Circuits and Systems-II:
Express Briefs and a Guest Editor of several prestigious journals such as Vehicle
System Dynamics and Control Engineering Practice.

Zhang, J., Liu, L., Wang, X., & Ji, H. (2023). Fully distributed algorithm for resource allocation over unbalanced directed networks without global Lipschitz
condition. IEEE Transactions on Automatic Control, 68(8), 5119–5126.
Zhang, K., Xu, L., Yi, X., Ding, Z., Johansson, K. H., Chai, T., & Yang, T.
(2023). Predefined-time distributed multiobjective optimization for network
resource allocation. Science China. Information Sciences, 66(7), Article 170204.
Zhao, Y., Liu, Y., Wen, G., & Chen, G. (2017). Distributed optimization for
linear multiagent systems: Edge- and node-based adaptive designs. IEEE
Transactions on Automatic Control, 62(7), 3602–3609.
Zhou, J., Lv, Y., Wen, C., & Wen, G. (2022). Solving specified-time distributed
optimization problem via sampled-data-based algorithm. IEEE Transactions
on Network Science and Engineering, 9(4), 2747–2758.
Zhou, J., Wen, G., Lv, Y., Yang, T., & Chen, G. (2024). Distributed resource
allocation over multiple interacting coalitions: A game-theoretic approach.
IEEE Transactions on Automatic Control, 69(11), 8128–8135.
Zhou, H., Zeng, X., & Hong, Y. (2019). Adaptive exact penalty design for
constrained distributed optimization. IEEE Transactions on Automatic Control,
64(11), 4661–4667.
Zhu, Y., Ren, W., Yu, W., & Wen, G. (2021). Distributed resource allocation over
directed graphs via continuous-time algorithms. IEEE Transactions on Systems,
Man, and Cybernetics: Systems, 51(2), 1097–1106.

Qing-Long Han received the B.Sc. degree in Mathematics from Shandong Normal University, Jinan, China,
in 1983, and the M.Sc. and Ph.D. degrees in Control
Engineering from East China University of Science and
Technology, Shanghai, China, in 1992 and 1997, respectively.
Professor Han is currently Pro Vice-Chancellor
(Research Quality) and a Distinguished Professor at
Swinburne University of Technology, Melbourne, Australia. He held various academic and management
positions at Griffith University and Central Queensland
University, Australia. His research interests include networked control systems,
multi-agent systems, time-delay systems, smart grids, unmanned surface vehicles, and neural networks.
Professor Han was awarded the 2024 IEEE Dr.-Ing. Eugene Mittelmann
Achievement Award (the Highest Award in Industrial Electronics), the 2021
Norbert Wiener Award (the Highest Award in Systems Science and Engineering,
and Cybernetics) and the 2021 M. A. Sargent Medal (the Highest Award of the
Electrical College Board of Engineers Australia). He was the recipient of the IEEE
Systems, Man, and Cybernetics Society Andrew P. Sage Best Transactions Paper
Award in 2019, 2020, and 2022, respectively, the IEEE/CAA Journal of Automatica
Sinica Norbert Wiener Review Award in 2020, and the IEEE Transactions on
Industrial Informatics Outstanding Paper Award in 2020.
Professor Han is a Member of the Academia Europaea (The Academy of
Europe). He is a Fellow of the International Federation of Automatic Control
(FIFAC), a Fellow of the Institute of Electrical and Electronics Engineers (FIEEE),
an Honorary Fellow of the Institution of Engineers Australia (HonFIEAust), and
a Fellow of the Chinese Association of Automation (FCAA). He is a Highly
Cited Researcher in both Engineering and Computer Science (Clarivate). He
has served as an AdCom Member of IEEE Industrial Electronics Society (IES),
a Member of IEEE IES Fellows Committee, a Member of IEEE IES Publications
Committee, Chair of IEEE IES Technical Committee on Network-Based Control
Systems and Applications, and the Co-Editor-in-Chief of IEEE Transactions on
Industrial Informatics. He is currently the President-Elect, an Executive Board
Member, and a Steering Committee Member of Asian Control Association (ACA).
He is currently the Editor-in-Chief of IEEE/CAA Journal of Automatica Sinica and
the Co-Editor of Australian Journal of Electrical and Electronic Engineering.

Meng Luan received the B.Sc. degree in information
and computing science from Yanshan University, Qinhuangdao, China, in 2019, and the M.S. degree in
Mathematics from Southeast University, Nanjing, China,
in 2022. She is currently pursuing the Ph.D. degree
with Southeast University, Nanjing, China. Her current research interests include distributed optimization,
networked game and their applications.

Guanghui Wen received the Ph.D. degree in mechanical systems and control from Peking University,
Beijing, China, in 2012. He is currently an Endowed
Chair Professor at the School of Automation, Southeast University, Nanjing, China. His current research
interests include coordination control of autonomous
intelligent systems, analysis and synthesis of complex
networks, cyber–physical systems, resilient control, and
distributed reinforcement learning.
Prof. Wen was the recipient of the National Science Fund for Distinguished Young Scholars, Australian
Research Council Discovery Early Career Researcher Award, and Asia Pacific
Neural Network Society Young Researcher Award. He is a reviewer for American
Mathematical Review and is an active reviewer for many journals. He currently
serves as an Associate Editor of the IEEE Transactions on Control of Network
Systems, the IEEE Transactions on Industrial Informatics, the IEEE Transactions
on Neural Networks and Learning Systems, the IEEE Journal of Emerging and
Selected Topics in Industrial Electronics, the IEEE Transactions on Systems, Man
and Cybernetics: Systems, the IEEE Open Journal of the Industrial Electronics
Society, and the Asian Journal of Control. Prof. Wen has been named a Highly
Cited Researcher by Clarivate Analytics since 2018. He is an IET Fellow.

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