8432

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS, VOL. 55, NO. 11, NOVEMBER 2025

Finite-Time Distributed Aggregative Optimal
Consensus of Multivehicle Systems With
Multiple Time-Varying Constraints
Wenbo Zhu

and Qingling Wang , Senior Member, IEEE

Abstract— In this article, the finite-time distributed aggregative
optimal consensus (DAOC) problems for multivehicle systems
(MVSs) with multiple time-varying constraints are investigated.
First, we formulate a new distributed optimization model, called
finite-time DAOC (FT-DAOC), where each cost function contains
an extra aggregative variable. Then, a class of new finite-time
distributed algorithms is designed for MVSs with time-varying
cost functions under time-varying digraphs. Moreover, as vehicles
may work in settings with time-varying unknown control gains
and unknown disturbances, we extend the newly presented
distributed algorithms to solve finite-time aggregative optimal
consensus issues for MVSs with multiple time-varying constraints. Finally, the validity of the newly illustrated algorithms is
analyzed theoretically, and two simulation examples are provided.
Index Terms— Aggregative optimal consensus, distributed optimization, finite-time convergence, multivehicle systems (MVSs),
time-varying constraints.

I. I NTRODUCTION

I

N THE past decades, the distributed optimization issues [1]
have received enormous attention due to their broad applications (e.g., autonomous aerial vehicles [2], smart grids [3],
resource allocation [4], etc.). The goal of distributed optimization is to establish a distributed control framework by utilizing
local information such that global optimization is realized.
As is well known, a typical application of multivehicle systems
(MVSs) is to accomplish cooperative tasks [5], [6], [7]. For an
MVS, where each vehicle has a local cost function, if vehicles
can cooperatively accomplish control tasks in a distributed
manner while minimizing total cost, we say that the distributed
optimal consensus of MVSs is realized. Furthermore, it is
worth noting that the MVSs are modeled as multiagent systems
(MASs) in some works [8].
A vast number of distributed algorithms [9], [10], [11],
[12], [13], [14], [15] were developed to address distributed
Received 28 December 2024; revised 6 June 2025; accepted 27 August
2025. Date of publication 5 September 2025; date of current version
17 October 2025. This work was supported in part by the National Natural
Science Foundation of China under Grant 62373102 and in part by the Natural
Science Foundation of Jiangsu Province under Grant BK20221455. This
article was recommended by Associate Editor H. Su. (Corresponding author:
Qingling Wang.)
The authors are with the School of Automation, Southeast University,
Nanjing 210096, China, and also with the Key Laboratory of Measurement
and Control of Complex Systems of Engineering, Ministry of Education,
Nanjing 210096, China (e-mail: wb_zhu@seu.edu.cn; csuwql@gmail.com).
Color versions of one or more figures in this letter are available at
https://doi.org/10.1109/TSMC.2025.3604807.
Digital Object Identifier 10.1109/TSMC.2025.3604807

optimal consensus problems. The existing distributed optimal
consensus algorithms can be loosely divided into two categories: discrete-time optimal consensus algorithms [9], [10]
and continuous-time optimal consensus algorithms [11], [12],
[13], [14], [15]. As for discrete-time algorithms, Shi et al.
[9] first formulated a new distributed optimization framework called distributed parametric consensus optimization
problems (DPCOPs) and then designed the distributed optimal consensus algorithms by using model predictive control
techniques. For reducing computational complexity, a class
of distributed optimal consensus algorithms with finite steps
was proposed in [10]. With regard to the distributed optimal consensus of continuous-time systems, work [11] first
designed a weight-balancing strategy and then illustrated the
distributed optimal consensus methods under fixed digraphs.
In the settings of vehicles with high-order dynamics, the
distributed optimal consensus was realized by using embedded
techniques in literature [12]. Moreover, the distributed optimal consensus issues for MVSs with uncertainties, such as
unknown control gains, nonlinearity, and so on, were investigated in surveys [13], [14], [15]. Note that the distributed
algorithms [11], [12], [13], [14], [15] are with asymptotic
convergence, which means that the global optimal state is
reached as time goes to infinity. Nevertheless, as practical
tasks are generally expected to be accomplished within a finite
time, the distributed algorithms with finite-time convergence
are more desirable.
To achieve distributed optimal consensus in finite time,
technicians have developed various control strategies [16],
[17], [18], [19], [20], [21], [22], [23], [24], [25]. Feng et al.
[16] and Zhu et al. [17] designed finite-time distributed optimal
consensus algorithms for MVSs under undirected graphs and
directed graphs, respectively, where the graphs and cost functions are both time-invariant. However, the communication
topologies may suffer network attacks [18], communication
disturbance [19], and so on, resulting in time-varying graphs.
For adapting to such a situation, the finite-time distributed
optimal consensus issues for time-varying undirected graphs
were investigated in [18]. As for time-varying digraphs, the
embedded framework-based finite-time distributed optimal
consensus algorithms were presented in [19]. It is worth noting
that the cost functions of vehicles are probably time-varying
as well due to the existence of dynamic factors [20], [21]
(e.g., dynamic external signals, the variation of mechanical

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ZHU AND WANG: FINITE-TIME DISTRIBUTED AGGREGATIVE OPTIMAL CONSENSUS OF MVSs

characteristics, etc.). For eliminating the dynamic effect caused
by time-varying cost functions, Shi et al. [20] proposed the
distributed optimal consensus algorithms by utilizing Hessian
information and binary design. In literature [21], a twolayer hierarchical control framework is first constructed and
then the primary–secondary judgment technologies are conducted based on finite-time distributed time-varying optimal
consensus algorithms. As a further exploration, work [22]
considered MVSs with two time-varying constraints, that is,
the undirected graphs and cost functions of MVSs are both
time-varying, and the optimal consensus was realized in finite
time. In addition, the finite-time distributed optimal consensus methods designed for MVSs with time-varying complex
dynamics were illustrated in [23], [24], and [25].
Above-mentioned results [9], [10], [11], [12], [13], [14],
[15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]
were conducted for distributed optimal consensus, where local
cost function of each vehicle only related to its own state.
Inspired by aggregative games, Li et al. [26] first constructed
a novel distributed aggregative optimization (DAO) framework, where each cost function is determined by the states
of multiple vehicles. In the DAO framework, the gradient
observer-based asymptotic distributed control algorithms [27]
were proposed. Moreover, the finite-time DAO (FT-DAO) was
investigated in [28]. However, to our knowledge, the consensus
is not taken into account in the existing DAO results, and the
considered MVSs are generally with time-invariant constraints
or a few time-varying constraints.
Motivated by aforementioned observations, in this study,
we construct a new distributed aggregative optimal consensus
(DAOC) framework (see Section II-C), and the finite-time
DAOC (FT-DAOC) is realized under multiple time-varying
constraints. To be specific, by adopting finite-time stability
theory, a time-varying gradient observer is first designed.
Then, the gradient estimation-based FT-DAOC algorithms are
proposed, where the digraphs and aggregative cost functions
can be both time-varying. In addition, the new proposed
algorithms are extended to realize FT-DAOC of MVSs with
multiple time-varying constraints. The main contributions of
this article are summarized as follows.
1) A new distributed optimization framework called DAOC
is first constructed in this article, that is, each local cost
function contains an extra aggregative variable, and vehicles reach consensus at the global optimal state. Then,
we investigate FT-DAOC of MVSs under dynamic scenarios characterized by multiple time-varying constraints,
for example, time-varying aggregative cost functions, timevarying digraphs, time-varying unknown control gains, and
time-varying unknown disturbances.
2) The newly illustrated FT-DAOC methods do not utilize
the information related to time, for example, the derivatives of
weights of graphs, the partial derivatives of the gradient with
respect to time, the derivatives of control gains, and so on.
Thus, the newly presented distributed algorithms can reduce
the computational consumption of MVSs.
3) In contrast to the distributed optimal consensus methods [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19],
[20], [21], [22], [23], [24], [25] which require cost functions to

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be determined by local state, the new proposed algorithms can
realize FT-DAOC of MVSs, that is, each cost function can be
determined by the states of multiple vehicles. Compared with
the DAO framework-based techniques [26], [27], [28], in this
study, vehicles can achieve consensus while realizing globally
aggregative optimization. Furthermore, the considered MVSs
of this article are with multiple time-varying constraints,
which are typically distinct from the finite-time distributed
optimization results designed for MVSs with time-invariant
constraints [16], [17] and few time-varying constraints [18],
[19], [20], [21], [22], [23], [24], [25], [28].
The rest of this article is organized as follows. Some
basic preliminaries and problem formulation are presented in
Section II. In Section III, we investigate the FT-DAOC of
MVSs under dynamic scenarios, and a class of FT-DAOC
algorithms is presented. In Section IV, the simulation examples are provided to validate the newly proposed distributed
algorithms, and the conclusions are given in Section V.
II. P RELIMINARIES AND P ROBLEM F ORMULATION
Notations: Let notations R, R+ , and R++ be the set of
real numbers, nonnegative real numbers, and positive real
numbers, respectively. The column vector is represented by
vec(y1 , y2 , . . . , y N ) ∈ R N , and the row vector is denoted by
vecT (y1 , y2 , . . . , y N ), where yi ∈ R. The matrix is denoted
by Z = [z i j ] ∈ R N ×N with z i j ∈ R.
A. Graph Theory
The communication topologies of MVSs are modeled as
graphs G = (V, E, A), where notations V, E, and A denote
the vehicles set, communication channels set, and adjacency
matrix, respectively. The ith row and jth column element of
A is denoted by ai j . The element ai j > 0 if there exists
a directed channel from vehicle j to vehicle i. If there is
no directed channel from vehicle j to vehicle i, then ai j =
0. Digraphs G are called strongly connected if there exists
a directed path between any two vehicles. Moreover, if the
communication channels set E(t) and the weights ai j (t) are
both time-varying, then G(t) = (V, E(t), A(t)) is said to be
time-varying digraphs.
B. Preliminaries
Definition 1 [29]: If a function 2(·) has following
properties:

Z θ



2 (τ ) dτ/θ = −∞
 lim inf
θ →∞
Z0 θ

(1)


 lim sup
2 (τ ) dτ/θ = +∞
θ →∞

0

function 2(·) is called Nussbaum-type function.
Remark 1: Suppose that α(·) is a bounded function, where
the upper boundary is c ∈ R, the lower boundary is c ∈ R,
and α(·) ̸ = 0. The function 2(θ ) = α(·)2(θ) + β(·) is a
Nussbaum-type function if 2(θ ) is a Nussbaum-type function
and continuous function β(·) ∈ L∞ . The usual Nussbaum-type
2
functions contain θ 2 sin(θ ), eθ cos(θ ), and so on.

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Lemma 1 [30]: Let 2(·) be a Nussbaum-type function.
If inequality
V̇ (t) ≤ 2 (θ (t)) θ̇ (t) + υ (t)
(2)
Rt
holds for t ∈ [0, ∞), then V (t), θ(t), and 0 2(θ(τ ))θ̇ (τ )dτ
are all bounded for t ∈ [0, ∞), where V (t) and θ (t) are
smooth functions on interval [0, ∞), V (t) ≥ 0, and
Rboth
t
υ(τ
)dτ < ∞ for t ∈ [0, ∞).
0
Rt
Lemma 2 [31]: If 0 η2 (τ )dτ and η̇(t) are both bounded
for t ∈ [0, ∞), then limt→∞ η(t) = 0.
Lemma 3 [32]: For the system ẋ(t) = h(x(t)) with
h(x(t)) : R → R, where 0 is the equilibrium point of
system. If inequality V̇ (x(t)) ≤ −κ1 V κ2 (x(t)) + κ3 holds for
x(t) ̸ = 0, where κ1 > 0, κ2 ∈ (0, 1), and κ3 > 0, then the
system ẋ(t) = h(x(t)) can be semi-globally practically finitetime stable, which is, ∥x(t)∥ ≤ (2κ3 /κ1 (1 − γ ))1/(2κ2 ) with
γ ∈ (0, 1). In addition, if inequality V̇ (x(t)) ≤ −κ1 V κ2 (x(t))
holds for x(t) ̸ = 0 with κ1 > 0 and κ2 ∈ [0, 1), then
the system ẋ(t) = h(x(t)) is globally finite-time stable.
In particular, the settling time T is not larger than constant
T , where T = (V 1−κ2 (0)/(1 − κ2 )κ1 ).
C. Problem Formulation
In this article, we first construct a new distributed optimization model for MVSs, called DAOC. The formulation of
DAOC issues is
min f (x (t) , t) =

x∈RN

s.t.

N
X

f i (xi (t) , σ (x (t)) , t)

i=1

x (t) = x ∗ (t) ⊗ 1 N

(3)

with
σ (x(t)) =

N
X

φi (xi (t))

(4)

i=1

where x(t) = vec(x1 (t), x2 (t), . . . , x N (t)) ∈ R N , xi (t) ∈ R is
the state of vehicle i(i = 1, 2, . . . , N ), x ∗ (t) ∈ R is the global
optimal state, f i (xi (t), σ (x(t)), t) : R × R × R+ → R is the
local cost function, and σ (x(t)) : R N → R is the aggregative
variable. Moreover, φi (xi (t)) : R → R is a function which can
be accessed by vehicle i only. The objective of this article is
to design control laws to realize DAOC (3) within finite-time
t ∗ ∈ R+ , that is,
lim x (t) = x ∗ (t) ⊗ 1 N

t→t ∗

(5)

called FT-DAOC, where the dynamics of each vehicle i(i =
1, 2, . . . , N ) is
ẋi (t) = u i (t)

(6)

and u i (t) is the controller input.
Remark 2: Note that the FT-DAOC (5) is first proposed
here, and the inspirations of FT-DAOC originate from the
following aspects.
1) The cooperative task is a typical application of MVSs.
For reducing energy consumption, minimizing the total
cost of cooperative tasks for MVSs has become an
important issue.

2) The cost of each vehicle may be influenced by other
vehicles. For example, the physical sensing capabilities of a vehicle decrease as the number of vehicles
increases [26]. Thus, the DAOC of MVSs is worth
discussing.
3) The time-varying factors [20], [21] (e.g., temperature,
humidity, etc.) in real scenarios may affect the cost of
a vehicle by influencing its mechanical characteristics,
which results in time-varying cost functions. Moreover, the variation of energy prices can also cause
time-varying cost functions.
4) Notice that asymptotic stability generally means the
control objective can be realized as time goes to infinity.
By contrast, the finite-time convergence indicates the
setting time T is not larger than some constant (see
Lemma 3). Thus, the finite-time algorithms have faster
convergence speed, which is more desirable in practice.
For brevity, let f i and φi represent f i (xi (t), σ (x(t)), t) and
φi (xi (t)), respectively. To each local cost function f i which
removes the terms containing σ (t), notation ∇xs f i denotes its
gradient with respect to xs (t). Moreover, the gradient of f i
with respect to σ (t) and the gradient of φi with respect to
xs (t) are denoted by ∇σ f i and ∇xs φi , respectively. Based on
the above definitions, the partial derivative of the cost function
f i with respect to xs (t) can be written as
∂ fi
= ∇xs f i + ∇σ f i × ∇xs φs
∂ xs

(7)

where ∇xs f i = 0 for s ̸ = i and i, s = 1, 2, . . . , N . Take
f 1 = t × x12 (t) + σ 2 (t) = t × x12 (t) + (x1 (t) + x2 (t))2 and
f 2 = x22 (t) + t × σ 2 (t) = x22 (t) + t × (x1 (t) + x2 (t))2 as an
example to illustrate above definitions. It is clear that
∂ f1
= ∇x1 f 1 + ∇σ f 1 × ∇x1 φ1 = 2t x1 (t) + 2σ (t) × 1
∂ x1
∂ f1
= ∇x2 f 1 + ∇σ f 1 × ∇x2 φ2 = 0 + 2σ (t) × 1
∂ x2
∂ f2
= ∇x1 f 2 + ∇σ f 2 × ∇x1 φ1 = 0 + 2tσ (t) × 1
∂ x1
∂ f2
= ∇x2 f 2 + ∇σ f 2 × ∇x2 φ2 = 2x2 (t) + 2tσ (t) × 1.
∂ x2
Move on, we denote the gradient of ∇xs f i , ∇σ f i , and ∇xs φs
with respect to xk (t) by ∇xs xk f i , ∇σ xk f i , and ∇xs xk φs , respectively. Obviously
∂ fi
= ∇xs xk f i + ∇σ xk f i × ∇xs φs
∂ xs xk
+ ∇ σ f i × ∇ x s x k φs

(8)

where ∇xs xk f i = 0 for s ̸ = i or k ̸ = i, ∇σ xk f i = 0 for
k ̸ = i, and ∇xs xk φs = 0 for s ̸ = k and i, s, k = 1, 2, . . . , N .
In addition, the partial derivative of ∇xs f i and ∇σ f i with
respect to time are denoted by ∇xs t f i and ∇σ t f i , respectively.
In the rest of this article, we investigate FT-DAOC of MVSs
under the following assumptions.
Assumption 1: Digraphs G(t) are strongly connected at any
given time, where the edges and weights of G(t) can be both
time-varying.

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ZHU AND WANG: FINITE-TIME DISTRIBUTED AGGREGATIVE OPTIMAL CONSENSUS OF MVSs

Remark 3: Due to the potential existence of network
attacks [18], communication disturbance [19], and so on,
in realistic deployments, the communication channels may
be arbitrarily connected and disconnected, which leads to a
time-varying edge set E(t). Moreover, the weight ai j (t) is
also probably time-varying since the priority of a neighbor’s
state may change over time. Therefore, the edges and weights
of G(t) are assumed to be both time-varying in this article.
As a note, Assumption 1 is weaker than the assumptions that
only assume the edges or weights are time-varying in the
literature [18], [19], [33], [34], and so on.
Assumption 2: The assumptions of time-varying cost functions are given as follows.
PN
1) The global cost function i=1
f i is ω1 -strongly convex,
where ω1 is some positive constant.
2) ∇xi xi f i , ∇σ σ f i , and ∇xi xi φi are all bounded, that is,
|∇xi xi f i | ≤ ω2 , |∇σ σ f i | ≤ ω3 , and |∇xi xi φi | ≤ ω4 , where
ω2 , ω3 , and ω4 are all positive constants.
3) The partial derivative of ∇xi f i and ∇σ f i with respect to
time are both bounded, that is, |∇xi t f i | ≤ ω5 and |∇σ t f i | ≤ ω6 ,
where ω5 and ω6 are both nonnegative constants.
4) ∇xi φi is bounded, that is, |∇xi φi | ≤ ω7 , where ω7 is
some positive constant.
Remark 4: According to Assumptions 1 and 2, it is known
that the digraphs and cost functions of considered MVSs can
be both time-varying, which are typically different from the
time-varying optimal consensus methods [20], [21], [35], [36],
[37] for fixed P
graphs. In particular, we only assume the global
N
cost function i=1
f i is ω1 -strongly convex, that is, local cost
function f i can be nonconvex, which relaxes the assumptions
on convexity in works [34], [35], [36], [37], [38].
III. M AIN R ESULTS
This part studies the FT-DAOC of MVSs with multiple timevarying constraints. First, a class of FT-DAOC algorithms (see
Section III-A) is proposed, where the digraphs and aggregative
cost functions can be both time-varying. As a further exploration, we consider the settings with time-varying unknown
control gains and time-varying unknown disturbances, then
the FT-DAOC issues of MVSs with multiple time-varying
constraints are solved (see Section III-B).
A. FT-DAOC of MVSs With Time-Varying Aggregative Cost
Functions and Time-Varying Digraphs
In this section, we first establish a finite-time distributed
observer to estimate the sum of gradients, then a class of
FT-DAOC algorithms is proposed.
Theorem 1: For the MVSs with dynamics (6) under
Assumptions 1 and 2. The FT-DAOC (5) can be reached with
distributed algorithms as follows:


N
X

u i (t) = −c1 sign 
ai j (t) xi (t) − x j (t) 
j=1

−c2 sign

N
X
s=1

!
µis (t)

(9)

8435

µis (t) = ϑis (t) + ςis (t) ×

N
X

ξik (t)

(10)

k=1



N


X
ai j (t) ϑis (t) − ϑ sj (t) 
ϑ̇is (t) = −c3 sign 
−c4 bis sign

j=1

ϑis (t) − ∇xs f s

(11)



N


X
ς̇is (t) = −c5 sign 
ai j (t) ςis (t) − ς sj (t) 
j=1


− c6 bis sign ςis (t) − ∇σ f s


N


X
ξ̇is (t) = −c7 sign 
ai j (t) ξis (t) − ξ sj (t) 
−c8 bis sign
(
bis =

j=1

ξis (t) − ∇xs φs

1, if i = s
0, if i ̸ = s

(12)

(13)
(14)

where xi (t) is the state, µis (t) is the optimization variable, and
i, s = 1, 2, . . . , N . Moreover, variables ϑis (t), ςis (t), and ξis (t)
are the estimation of ∇xs f s , ∇σ f s , and ∇xs φs , respectively.
The design of controller parameters should satisfy c1 > c2 >
(ω5 + ω6 ω7 )/ω1 , c3 > c4 > ω2 (c1 + c2 ) + ω5 , c5 > c6 >
N ω3 ω7 (c1 + c2 ) + ω6 , and c7 > c8 > ω4 (c1 + c2 ).
Proof: The whole proof consists of finite-time gradient
estimation (see Step 1), finite-time consensus (see Step 2),
and finite-time aggregative optimization (5) (see Step 3).
Step 1: In this step, we prove that each vehicle i can estimate
the gradient ∇xs f s , ∇σ f s , and ∇xs φs within finite time. The
proof of limt→tϑ ϑ s (t) = ∇xs f s ⊗ 1 N (s = 1, 2, . . . , N ) is first
given, where tϑ is the finite setting time. By defining auxiliary
vectors

s
ϑ (t) = max ϑ1s (t) , ϑ2s (t) , . . . , ϑ Ns (t)

ϑ s (t) = min ϑ1s (t) , ϑ2s (t) , . . . , ϑ Ns (t)
and Lyapunov function
s

Vϑ (t) = ϑ (t) − ϑ s (t)

(15)

we can deduce that
V̇ϑ (t) = ϑ̇ (t) − ϑ̇ (t) ≤ −2c3 + 2c4

(16)

for Vϑ (t) ̸ = 0. Therefore, ϑ(t) = ϑ(t) can be achieved within
finite time tϑ1 . When t > tϑ1 , it is clear that

(17)
ϑ̇ss (t) = −c4 sign ϑss (t) − ∇xs f s .
Choose a Lyapunov function
2
1 s
ϑs (t) − ∇xs f s
(18)
2
the time derivative of (18) is



V̇1 (t) = ϑss (t) − ∇xs f s −c4 sign ϑss (t) − ∇xs f s

−∇xs xs f s × ẋs (t) − ∇xs t f s
≤ − [c4 − (c1 + c2 ) ω2 − ω5 ] |ϑss (t) − ∇xs f s |
√ 1
= − [c4 − (c1 + c2 ) ω2 − ω5 ] 2V 2 (t) .
(19)
V1 (t) =

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Thus, the finite-time convergence of ϑss (t) − ∇xs f s can be
realized within tϑ2 . Base on the aforementioned analysis, it is
derived that
ϑ s (t) = ∇xs f s ⊗ 1 N

(20)

holds for t ≥ tϑ = tϑ1 + tϑ2 , where s = 1, 2, . . . , N . With
similar analysis of above proof process, it is obtained that
limt→tς ς s (t) = ∇σ f s ⊗ 1 N , and limt→tξ ξ s (t) = ∇xs φs ⊗ 1 N ,
where tς and tξ are both positive constants.
Step 2: In this step, the finite-time consensus is proved, that
is, xi (t) → x j (t) for i, j = 1, 2, . . . , N . Define Lyapunov
function as
Vx (t) = x (t) − x (t)

(21)

where x(t) and x(t) represent the maximal state and minimal
state at time t, respectively. Then, we have from (21) that
V̇x (t) = ẋ (t) − ẋ (t) ≤ −2 (c1 − c2 )

Based on the above analysis, it is known that the proposed
distributed algorithms (9)–(14) can realize FT-DAOC. Actually, the DAO can also be realized within a finite time by
extending algorithms (9)–(14), and the objective of FT-DAO
is represented as

lim∗ x (t) = vec x1∗ (t) , x2∗ (t) , . . . , x N∗ (t)
t→t

s.t. min f (x, t) =
x∈RN

u i (t) = −c1 sign (µi (t))

holds when Vx (t) ̸ = 0. Therefore, x(t) = x(t) can be reached
within finite-time tx , which means, limt→tx xi (t) = x j (t) holds
for i, j = 1, 2, . . . , N , and the finite-time consensus is proved.
Step 3: In this step, the finite-time aggregative optimization
is proved. When t ≥ t1 = max{tϑ , tς , tξ , tx }, the dynamics of
vehicle i can be rewritten as
" N N
#
XX

ẋi (t) = −c2 sign
∇xs f i + ∇σ f i ×∇xs φs
(23)

N X
N
X


∇xs f i + ∇σ f i ×∇xs φs .

(25)

i=1 s=1

i=1 s=1 k=1
N X
N
X

#


∇xs t f i + ∇σ t f i × ∇xs φs

+

i=1 s=1

≤ − (c2 ω1 − ω5 − ω6 ω7 ) N 1 (t) sign (1 (t))
√
1
≤ − (c2 ω1 − ω5 − ω6 ω7 ) 2N V 2 (t) .
(26)
It is obvious from (26) and Lemma 3 that the aggregative
optimization is reached within finite time t2 . By now, we can
say that the FT-DAOC can be realized, where the finite time
is t = t1 + t2 . The whole proof is completed.
Remark 5: The finite-time distributed gradient observer
(11) is constructed for each vehicle i under time-varying
digraphs as assumed in Assumption 1. Actually, the new
proposed gradient observer can also be regarded as a bounded
dynamic signals tracker, that is, the bounded dynamic signals
can be tracked by using finite-time distributed algorithms (11).

N
X

ςis (t)

(29)

s=1



N


X
ς̇is (t) = −c2 sign 
ai j (t) ςis (t) − ς sj (t) 
− c3 bis sign
(
bis =

j=1

ςis (t) − ∇σ f s

(30)

1, if i = s
0, if i ̸ = s

(31)

where xi (t) is the state, µis (t) is the optimization variable, and
i, s = 1, 2, . . . , N . Moreover, variables ςis (t) is the estimation
of ∇σ f s . The design of controller parameters should satisfy
c1 > (ω5 + ω6 ω7 )/ω1 and c2 > c3 > N ω3 ω7 c1 + ω6 .
Proof: The whole proof consists of finite-time gradient
estimation (see Step 1) and FT-DAO (see Step 2).
Step 1: With a similar analysis of Step 1 in the proof of
Theorem 1, it is clear that

Then, we have from (24) and Assumption 2 that
V̇2 (t) = 1 (t) 1̇ (t)
" N N N
X X X ∂ fi
= 1 (t)
× ẋk
∂ xs xk

(27)

(28)

µi (t) = ∇xi f i + ∇xi φi ×

i=1 s=1

1 (t) =

f i (xi , σ (x) , t).

i=1

Typically different from DAOC (3), the consensus is not
considered in DAO (27). The extended distributed optimization
algorithms are given in the following corollary.
Corollary 1: Consider MVSs with dynamics (6) under
Assumptions 1 and 2. The FT-DAO (27) can be reached by
using following distributed control algorithms:

(22)

with the fact that ∇xs f i = 0 for s ̸ = i. Define Lyapunov
function as
1
(24)
V2 (t) = 12 (t)
2
where

N
X

lim ς s (t) = ∇σ f s ⊗ 1 N

(32)

t→tς

where tς is some positive constant, that is, the gradient
estimation is realized within a finite time tς .
Step 2: The FT-DAO (27) is proved in this step. When t ≥
tς , the dynamics of each vehicle i can be rewritten as
"
#
N
X
ẋi (t) = −c1 sign ∇xi f i + ∇xi φi ×
∇σ f s .
(33)
s=1

Define the Lyapunov function as
V3 (t) =

1 2
0 (t)
2

(34)

where
0 (t) = ∇xi f i + ∇xi φi ×

N
X
s=1

∇σ f s =

N
X
∂ fs
s=1

∂ xi

(35)

with the fact that ∇xi f s = 0 for s ̸ = i. Then, we have
from (34) and Assumption 2 that
V̇3 (t) = 0 T (t) 0̇ (t)

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ZHU AND WANG: FINITE-TIME DISTRIBUTED AGGREGATIVE OPTIMAL CONSENSUS OF MVSs

= 0 T (t)

" N
X ∂ fs
s=1

∂ xi xi

× ẋi (t)

#
N
X


+
∇xi t f s + ∇xi φi × ∇σ t f s
s=1

≤ − (c1 ω1 − ω5 − ω6 ω7 ) N 0 T (t) sign (0 (t))
√
1
(36)
≤ − (c1 ω1 − ω5 − ω6 ω7 ) 2N V 2 (t)
PN
which means that ∇xi f i + ∇xi φi × s=1 ∇σ f s can converge to zero within finite time t5 , and the FT-DAO (27) is
realized.
Remark 6: Note that the proposed distributed algorithms (9)–(14) and (28)–(31) can realize FT-DAOC (5) and
FT-DAO (27), respectively, where aggregative cost functions
and digraphs are both time-varying. In particular, the newly
presented distributed algorithms do not require the Hessian
information and the partial derivatives of the gradient with
respect to time. Therefore, we can say that the computational
complexity of algorithms (9)–(14) and (28)–(31) are lower
than the time-varying optimization methods proposed in [20],
[21], [39], [40], and [41].
B. FT-DAOC Algorithms for MVSs With Multiple
Time-Varying Constraints
In this section, we investigate the FT-DAOC of MVSs under
dynamic scenarios characterized by multiple time-varying constraints, where time-varying constraints contain time-varying
aggregative cost functions, time-varying digraphs, timevarying unknown control gains, and time-varying unknown
disturbances. The dynamics of vehicle i is
ẋi (t) = bi (t) u i (t) + di (t)

(37)

where xi (t), bi (t), u i (t), and di (t) are state, time-varying
unknown control gain, controller input, and time-varying
unknown disturbance, respectively. The necessary assumptions
of unknown control gains and disturbances are given as
follows.
Assumption 3: The time-varying unknown control gains
satisfy |bi (t)| ≤ b ∈ R++ and |bi (t)| ̸= 0.
Remark 7: As a note, unknown control direction [42] generally refers that the sign of the controller coefficient is
unknown. For example, ẋi (t) = bi u i (t) is a classical first-order
dynamics with unknown control direction, where bi is the
unknown constant. By contrast, the considered unknown control gains bi (t) in this article are time-varying, where bi (t) and
ḃi (t) are both unknown. Therefore, in this sense, unknown
control direction bi can be regarded as a special case of
unknown control gain bi (t).
Assumption 4: The time-varying unknown disturbances are
bounded, that is, |di (t)| ≤ d ∈ R+ .
Remark 8: In fact, apart from the time-varying
digraphs/cost functions as introduced in Remarks 2 and 3,
the time-varying unknown control gains and disturbances
widely existed in the practical settings as well. To be specific,
considering a case where that vehicle may accelerate or
decelerate at any time, where the acceleration/deceleration
cannot be real-timely sensed, it is clear that the above

8437

behavior can be formulated as the controller with time-varying
unknown control gains. In addition, as the controller of a
vehicle generally suffers uncertainties [43] caused by some
factors (e.g., actuator error, inaccurate model, unstable voltage,
etc.), there probably exists a deviation between the real control
input and the ideal control input, which can be described by
time-varying unknown disturbances. With the aforementioned
statements, it is seen that the FT-DAOC issues for MVSs
with multiple time-varying constraints are significant.
Remark 9: As shown in Assumptions 1–4, it is known that
the considered MVSs have multiple time-varying constraints.
Compared with the finite-time distributed optimal consensus
results for MVSs with time-invariant constraints [16], [17]
or few time-varying constraints [18], [19], [20], [21], [22],
[23], [24], [25], the design of finite-time distributed algorithms
become more challenging as the number of time-varying
constraints increases.
In the rest of this part, we design a class of distributed
algorithms for MVSs (37) to realize semi-globally practically
FT-DAOC, that is,
lim xi (t) − x ∗ (t) ≤ ϵ

(38)

t→t ∗

where x ∗ (t) is the optimal state as formulated by (3), ϵ ∈
R+ is some positive constant, and i = 1, 2, . . . , N . The new
proposed distributed optimization algorithm is given by the
following theorem.
Theorem 2: Consider MVSs with dynamics (37)
under Assumptions 1–4. The semi-globally practically
FT-DAOC (38) can be reached with following distributed
control algorithms
!
"
N
X
s
u i (t) = 2 (θi (t)) c1 sign (δi (t)) + c3 sign
µi (t)
s=1


N
X

+ c2 sign 
ai j (t) qi (t) − q j (t)  (39)
j=1

"
θ̇i (t) = δi (t) c1 sign (δi (t)) + c3 sign

N
X

!
µis (t)

s=1


N
X

ai j (t) qi (t) − q j (t)  (40)
+c2 sign 
j=1

δi (t) = xi (t) − qi (t)


N
X

ai j (t) qi (t) − q j (t) 
q̇i (t) = −c2 sign 

(41)

j=1

−c3 sign

N
X

!
µis (t)

(42)

s=1

µis (t) = ϑis (t) + ςis (t) ×

N
X

ξik (t)

(43)

k=1



N


X
ϑ̇is (t) = −c4 sign 
ai j (t) ϑis (t) − ϑ sj (t) 
j=1
s
− c5 bi sign

ϑis (t) − ∇qs f s



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(44)

8438

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

N


X
ς̇is (t) = −c6 sign 
ai j (t) ςis (t) − ς sj (t) 
j=1
s
−c7 bi sign

ςis (t) − ∇σ f s



(45)



N


X
ξ̇is (t) = −c8 sign 
ai j (t) ξis (t) − ξ sj (t) 
j=1
− c9 bis sign

(
bis =

ξis (t) − ∇qs φs



1, if i = s
0, if i ̸ = s

lim

t→∞

(46)
(47)

where xi (t) is the state, θi (t) is the auxiliary variable, δi (t) is
the error variable, qi (t) is the optimal reference variable, µis (t)
is the optimization variable, and i, s = 1, 2, . . . , N . Moreover,
variables ϑis (t), ςis (t), and ξis (t) are the estimation of ∇qs f s ,
∇σ f s , and ∇qs φs , respectively. The design of controller parameters should satisfy c1 > d, c2 > c3 > (ω5 + ω6 ω7 )/ω1 ,
c4 > c5 > ω2 (c2 + c3 ) + ω5 , c6 > c7 > N ω3 ω7 (c2 + c3 ) + ω6 ,
and c8 > c9 > ω4 (c2 + c3 ).
Proof: According to the proof of Theorem 1, it is clear
that qi (t) → x ∗ within finite time by using distributed
algorithms (42)–(47). In the following, the semi-globally practically FT-DAOC is illustrated by choosing the Lyapunov
function as
1
V4 (t) = δi2 (t) .
(48)
2
Then, we can derive from (39), (41), and (42) that

(49)

with
ζi (t) = [bi (t) 2 (θi (t))+ 1] q̇i (t)
ϱi (t) = c1 bi (t) 2 (θi (t)) sign (δi (t)) .

It
R t is obvious from Lemma 1 and (49) that V4 (t), θi (t) and
0 [bi (t)2(θi (τ )) + 1]θ̇i (τ )dτ are all bounded for t ∈ [0, ∞),
which indicates δi (t) is bounded as well. Thus, it is obtained
from (40) that θ̇i (t) bounded for t ∈ [0, ∞), and we can further
deduce that [bi (t)2(θi (t)) + 1]θ̇i (t) ≤ C, where C is some
positive constant. That is, inequality


V̇4 (t) = −δi (t) c1 sign (δi (t)) − di (t)
+ [bi (t) 2 (θi (t)) + 1] θ̇i (t)

≤ − c1 − d̄ |δi (t)| + C
√
 1
= − 2 c1 − d V 2 (t) + C

(50)

holds. According to Lemma 3 and (50), it is clear that
δi (t) is semi-globally practically finite-time stable, and the
semi-globally practically FT-DAOC is realized, that is,
lim xi (t) − x ∗ (t) ≤ ϵ = √

t→t ∗

The proof is completed.

2C
.

2 c1 − d̄ (1 − γ )

(52)

where x ∗ (t) is the optimal state as formulated by (3). In particular, (52) also indicates that distributed algorithms (39)–(47)
can realize DAOC asymptotically.
Corollary 2: Consider MVSs with dynamics (37) under
Assumptions 1–4. The DAOC (52) can be reached asymptotically by using distributed control algorithms (39)–(47),
where controller parameters should satisfy c1 > d, c2 >
c3 > (ω5 + ω6 ω7 )/ω1 , c4 > c5 > ω2 (c2 + c3 ) + ω5 ,
c6 > c7 > N ω3 ω7 (c2 + c3 ) + ω6 , and c8 > c9 > ω7 (c2 + c3 ).
Proof: Define the Lyapunov function as
1
(53)
V5 (t) = |δi (t)|3
3
Then, we have
V̇5 (t) = δi2 (t) sign (δi (t)) (ẋi (t) − q̇i (t))


= −δi2 (t) sign (δi (t)) ζi (t) − ϱi (t) − di (t)


= −δi2 (t) sign (δi (t)) c1 sign (δi (t)) − di (t)
+ |δi (t) | [bi (t) 2 (θi (t)) + 1] θ̇i (t)
≤ |δi (t) | [bi (t) 2 (θi (t)) + 1] θ̇i (t)
(

+ [bi (t) 2 (θi (t)) + 1] θ̇i (t)

(

xi (t) − x ∗ (t) = 0

(54)

with

V̇4 (t) = δi (t) (ẋi (t) − q̇i (t))


= −δi (t) ζi (t) − ϱi (t) − di (t)


= −δi (t) c1 sign (δi (t)) − di (t)
≤ [bi (t) 2 (θi (t)) + 1] θ̇i (t)

As illustrated in Theorem 2, the semi-globally practically
FT-DAOC (38) is realized by utilizing distributed algorithms (39)–(47), which means that there exist a bounded error
ϵ between state xi (t) and optimal state x ∗ (t). Actually, the
error ϵ decays to zero as time goes to infinity, that is,

(51)

ζi (t) = [bi (t) 2 (θi (t))+ 1] q̇i (t)
ϱi (t) = c1 bi (t) 2 (θi (t)) sign (δi (t)) .

It is noticed from theR proof of Theorem 2 that δi (t),
t
δi2 (t), θi (t), θ̇i (t), and 0 [bi (t)2(θi (τ )) + 1]θ̇i (τ )dτ are all
bounded for t ∈ [0, ∞) by using the Lyapunov function V4 (t) [see (48)]. Therefore, |δi (t)|[bi (t)2(θi (t)) +
1]θ̇i (t) is a Nussbaum-type function, and we can further
derive from Lemma 1 and (54) that V5 (t) and
Rt
|δ
(t)|[b
i (t)2(θi (τ
0 i
R t)) + 1]θ̇i (τ )dτ are both bounded for t ∈
[0, ∞). Obviously, 0 δi2 (τ )dτ is bounded as wellR according
t
to (54). By now, we know that δi (t), δ̇i (t), and 0 δi2 (τ )dτ
are all bounded on t ∈ [0, ∞), then limt→∞ δi (t) =
limt→∞ ∥xi (t) − x ∗ (t)∥ = 0 is obtained according to Lemma
2. The proof is completed.
Remark 10: According to Theorem 2 and Corollary 2,
we know that the proposed distributed algorithms (39)–(47)
can achieve semi-globally practically FT-DAOC (38) and
asymptotic DAOC (52), which means that ∥xi (t) − x ∗ (t)∥ ≤ ϵ
can be reached in finite time, and the error ϵ decays to zero as
time goes to infinity. Moreover, it is obtained from (51) that ϵ
can be small enough by selecting a sufficiently large parameter
c1 . The aforementioned conclusions imply that distributed
algorithms (39)–(47) are with fast convergence speed and high
control accuracy.
Remark 11: The novelty of algorithm design is illustrated
as follows.
1) To eliminate the effect caused by aggregative variable
σ (x(t)), a new finite-time gradient observer (44)–(47)

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ZHU AND WANG: FINITE-TIME DISTRIBUTED AGGREGATIVE OPTIMAL CONSENSUS OF MVSs

Fig. 1.

Time-varying topologies of MVSs.

Fig. 2. Trajectories of gradient estimation ϑis (t) (s = 1, 2) of vehicle i.
(a) Trajectories of ϑi1 (t). (b) Trajectories of ϑi2 (t).

inspired by bounded dynamic signals tracking is first
constructed, and a novel Lyapunov function defined by
the maximal error of gradient estimations is constructed
to show that the time-varying digraphs do not affect the
finite-time convergence of observer (see Step 1 of Proof
Section III-A).
2) We develop a new method that the partial derivative of
the gradient with respect to time can be regarded as
unknown disturbances, then the sign function is adopted
to reject the effect caused by time-varying aggregative
cost functions.
3) By introducing the two-layer design framework [i.e.,
optimal reference signal generator (42) and controller (39)] and Nussbaum function, the FT-DAOC
algorithms for MVSs with multiple time-varying constraints are proposed, where the fast convergence speed
and the high control accuracy are both proved theoretically.
Remark 12: Actually, due to the existence of cost functions, the finite-time consensus algorithms cannot be directly
extended to solve optimal consensus issues [44], [45]. In particular, the DAOC problems become more challenging as an
extra aggregative variable is contained in each cost function.
Moreover, since the proposed FT-DAOC algorithms (39)–(47)
are designed for MVSs with multiple time-varying constraints,
and suitable for the case that time derivatives of constraints
cannot be obtained, we can say that algorithms (39)–(47) have
great robustness to the dynamic scenarios.
IV. S IMULATION
In this simulation, two examples are illustrated to verify
the proposed distributed algorithms (9)–(14) and (39)–(47).
The considered MVS consists of six vehicles, where the
time-varying directed topologies as shown in Fig. 1, and the
switching logic is set by the following equation:


 G1 , if t % 3 ∈ [0, 0.6)
G (t) = G2 , if t % 3 ∈ [0.6, 2.1)


G3 , if t % 3 ∈ [2.1, 3) .

(55)

The cost function for each vehicle i(i = 1, 2, . . . , 6) is
given as follows:
f 1 (x1 , σ, t) = x12 + 5x1 + 2sin (t) σ
f 2 (x2 , σ, t) = 3x22 + 2cos (t) x2 + σ + 6
f 3 (x3 , σ, t) = x32 + 6sin (t) x3 + 2σ

8439

Fig. 3. Trajectories of gradient estimation ςis (t) (s = 1, 2) of vehicle i.
(a) Trajectories of ςi1 (t). (b) Trajectories of ςi2 (t).

Fig. 4. Trajectories of gradient estimation ξis (t) (s = 1, 2) of vehicle i.
(a) Trajectories of ξi1 (t). (b) Trajectories of ξi2 (t).

f 4 (x4 , σ, t) = 1.5x42 + σ + 16
f 5 (x5 , σ, t) = x52 + tanh (t + 0.1) σ
f 6 (x6 , σ, t) = x62 + 3sin (t) x6 + σ
P
where σ = 6k=1 φk , φ1 = x12 , φ2 = 0.5x22 , φ3 = x3 , φ4 =
0.05x42 + 0.1x4 , φ5 = 0.3x52 + x5 + 7, φ6 = 0.6x62 + x6 .
A. Simulation for Finite-Time Distributed
Algorithms (9)–(14) in Theorem 1
In this simulation example, with controller parameters c1 =
8, c2 = 7, c3 = c5 = c7 = 13, and c4 = c6 = c8 = 10,
the FT-DAOC algorithms (9)–(14) proposed in Theorem 1 is
validated, where the dynamics of vehicle i(i = 1, 2, . . . , 6)
is (6). The initial condition is randomly set as x(0) =
vec(−2, 0, −8, 6.6, −5, 3) and ϑ s (0) = ς s (0) = ξ s (0) =
vec(6.7, −5, −0.7, 2, −2.9, 1.5) for s = 1, 2, . . . , 6.
As illustrated in Figs. 2–4, vectors ϑ s (t), ς s (t), and
s
ξ (t)(s = 1, 2) reach consensus at ∇xs f s , ∇σ f s , and ∇xs φs ,
respectively, which means that the gradient can be well estimated by each vehicle within finite time. It is shown from
Fig. 5(a) that the state xi (t) reach consensus on a trajectory x ∗ (t). Furthermore, the sum of optimization variables

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Fig.
P N 5. s Trajectories of state xi (t) and the sum of optimization variables
µi (t) of vehicle i. (a) Trajectories of xi (t). (b) Trajectories of
Ps=1
N µs (t).
s=1 i

Fig. 6. Case 1: the trajectoriesPof optimal reference variable qi (t) and the
N µs (t) of vehicle i. (a) Trajectories of
sum of optimization variables s=1
i
PN
s
qi (t). (b) Trajectories of s=1 µi (t).

TABLE I
T HREE S ETTINGS OF T IME -VARYING U NKNOWN C ONTROL G AINS

Fig. 7. Case 1: the trajectories of auxiliary variable θi (t) and Nussbaum
function 2(θi (t)) of vehicle i. (a) Trajectories of θi (t). (b) Trajectories of
2(θi (t)).

TABLE II
T HREE S ETTINGS OF T IME -VARYING U NKNOWN D ISTURBANCES

Fig. 8. Case 1: the trajectories of state xi (t) and error variable δi (t) of
vehicle i. (a) Trajectories of xi (t). (b) Trajectories of δi (t).

PN

s
s=1 µi (t)

as shown in Fig. 5(b) converges to 0, which
indicates that x ∗ (t) is the optimal state trajectory, and the
FT-DAOC (5) is realized.
B. Simulation for Finite-Time Distributed
Algorithms (39)–(47) in Theorem 2
In this simulation example, we verify the proposed
FT-DAOC algorithms (39)–(47) in three different settings,
where the dynamics of vehicle i(i = 1, 2, . . . , 6) is (37), the
time-varying unknown control gains for each case are shown
in Table I, and the time-varying unknown disturbances for each
case are displayed in Table II.
Case 1: In this case, the controller parameters are designed
as c1 = 4, c2 = 5, c3 = 3.5, c4 = c6 = c8 = 11,
and c5 = c7 = c9 = 9. The initial conditions are
randomly given by x(0) = vec(5.2, −5, 2.7, 0, 3.9, −4.5),
q(0)
=
vec(3.3, −3.5, 1.8, −0.9, 6, −5.9), θ (0)
=
vec(0.6, −3, 3.3, −2.5, −2, 0), and ϑ s (0) = ς s (0) =
ξ s (0) = vec(6.7, −5, −0.7, 2, −2.9, 1.5) for s = 1, 2, . . . , 6.
As displayed in Fig. 6(a) and (b), the variables qi (t) converge to optimal trajectory x ∗ (t) in finite time. The variables
θi (t) and Nussbaum-type functions 2(θi (t)) are shown in
Fig. 7(a) and (b), respectively. Moreover, the curves of error

variables δi (t) as illustrated in Fig. 8(a) is bounded and
converge to 0, which means that the state xi (t) as shown
in Fig. 8(b) reach semi-globally practically FT-DAOC (38).
In particular, the error ∥xi (t) − x ∗ (t)∥ can decay to zero.
Case 2: In this case, we set controller parameters as
c1 = 3.8, c2 = 6, c3 = 3.6, c4 = c6 = c8 = 16,
and c5 = c7 = c9 = 12.3. The initial conditions are
arbitrarily given by x(0) = vec(5, −5.9, 0.1, 3.5, −2.2, 5.5),
q(0)
=
vec(2.8, −2, −2.9, 6, −1.3, 5), θ (0)
=
vec(0.7, 0.3, 3.2, 0, −2.6, 2), and ϑ s (0) = ς s (0) = ξ s (0) =
vec(3.7, −2, −1.7, 2.8, 0, −5) for s = 1, 2, . . . , 6. Based on
the above settings, it is seen from Fig. 9(b) that the sum of
optimization variables can converge to 0, which indicates the
states as shown in Fig. 9(a) are global optimal.
Case 3: In this case, the controller parameters are
designed as c1 = 4, c2 = 5, c3 = 3, c4 = c6 = c8 = 8,
and c5 = c7 = c9 = 7. The initial conditions are
randomly set by x(0) = vec(−7, 6, −3, 2.2, 6, −2.3),
q(0)
=
vec(−5, 3.2, −5, 4.3, 5, −1.5), θ (0)
=
vec(0.7, 3.8, 3.2, 0, −2.6, 2), and ϑ s (0) = ς s (0) = ξ s (0) =
vec(0.5, −3.8, 2.7, 2.9, 0, 6.6) for s = 1, 2, . . . , 6. With
analysis similar to Cases 1 and 2, it is obtained from

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ZHU AND WANG: FINITE-TIME DISTRIBUTED AGGREGATIVE OPTIMAL CONSENSUS OF MVSs

Fig. 9. Case
P N 2: sthe trajectories of state xi (t) and the sum of optimization
variables s=1
µi (t) of vehicle i. (a) Trajectories of xi (t). (b) Trajectories
PN
of s=1 µis (t).

Fig. 10. P
Case 3: the trajectories of state xi (t) and the sum of optimization
N µs (t) of vehicle i. (a) Trajectories of x (t). (b) Trajectories
variables s=1
i
i
PN
of s=1
µis (t).

Fig. 10 that the FT-DAOC can be realized by using
algorithms (39)–(47).
According to the simulation results of the above-mentioned
three cases, where each case has different settings (e.g., initial value, time-varying unknown control gains, time-varying
unknown disturbances, etc.), it is clear that the distributed
algorithms (39)–(47) can solve FT-DAOC issues (38), and
error ϵ can decay to 0.
V. C ONCLUSION
In this article, we have investigated the FT-DAOC
framework-based distributed optimization problems for MVSs
with multiple time-varying constraints. A finite-time observer
has been first constructed to estimate the gradients by using
the inspirations of signal tracking, then the new FT-DAOC
algorithms have been proposed, where the digraphs and
aggregative cost functions are both time-varying. As an extension with respect to the number of time-varying constraints,
the FT-DAOC issues for MVSs with time-varying unknown
control gains and time-varying unknown disturbances have
been solved as well. In the final, we have provided two
simulation examples to show the validity of the new proposed
distributed algorithms. In the future, nonlinear high-order
MVSs, time-delay, robust control, and fixed-time DAOC may
be investigated.
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Wenbo Zhu received the B.S. degree in automation
from China University of Petroleum (East China),
Qingdao, China, in 2017, and the M.S. degree
in control science and engineering from Southeast
University, Nanjing, China, in 2020, where he is
currently pursuing the Ph.D. degree.
His research interests include multivehicle systems
and distributed optimization.

Qingling Wang (Senior Member, IEEE) received
the B.S. degree in automation and the M.S. degree
in control science and engineering from Central
South University, Changsha, China, in 2007 and
2010, respectively, and the Ph.D. degree in control
science and engineering from Harbin Institute of
Technology, Harbin, China, in 2014.
He was a Visiting Student with The Australian
National University, Canberra, ACT, Australia,
from 2012 to 2014, and a Visiting Scholar at the
Technical University of Berlin, Berlin, Germany,
in 2016. He is currently a Professor with the School of Automation, Southeast
University, Nanjing, China. His current research interests include distributed
optimization, constrained control, adaptive control, and cooperative control of
multiagent systems.

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