Distributed Nash Equilibrium Seeking for Games in Systems with
Bounded Control Inputs

arXiv:1901.09333v2 [math.OC] 27 Sep 2020

Maojiao Ye
Abstract— Noticing that actuator limitations are ubiquitous
in practical engineering systems, this paper considers Nash
equilibrium seeking for games in systems where the control
inputs are bounded. More specifically, first-order integratortype systems with bounded control inputs are firstly considered
and two saturated control strategies are designed to seek the
Nash equilibrium of the game. Then, second-order integratortype systems are further considered. In this case, a centralized
seeking strategy is firstly proposed without considering the
boundedness of the control inputs, followed by a distributed
counterpart. By further adapting a saturation function into the
distributed Nash equilibrium seeking strategy, the boundedness
of the control input is addressed. In the proposed distributed
strategies, consensus protocols are included for information
sharing and the saturation functions are utilized to construct
bounded control inputs. The convergence results are analytically
studied by Lyapunov stability analysis. Lastly, by considering
the connectivity control of mobile sensor networks, the proposed
methods are numerically verified.
Index Terms— Nash equilibrium seeking; bounded control
inputs; distributed networks; games.

I. I NTRODUCTION
Games are attracting growing interests from researchers
in the multi-agent communities for the analysis of multiagent systems in recent years [1]. For example, consensus
was accomplished by utilizing cooperative game theory in
[2]. Differential games were applied to solve distributed
optimal tracking control of multi-agent systems with external
disturbance in [3]. The works in [4]-[6] linked games to
cooperative control and optimization of multi-agent systems,
respectively. In [7], the consensus analysis for a class of
hybrid multi-agent systems was conducted based on a noncooperative game. These works motivate us to take the physical
constraints of multi-agent systems into consideration for
Nash equilibrium seeking problems. The concerned constraints include but are not limited to communication issues,
input saturation, system dynamics and action constraints.
Recent years witnessed the trials made by researchers to
accommodate system dynamics (see, e.g., [8][9][40][41]), the
communication issues for games in distributed networks (see,
e.g., [10][11]) and action constraints (see, e.g., [11][27]). For
example, following the ideas presented in [23]-[26] to establish distributed Nash equilibrium seeking strategies by utilizing consensus algorithms and the gradient search, communication constraints were accommodated in [10]. Moreover,
M. Ye is with the School of Automation, Nanjing University of Science
and Technology, Nanjing 210094, P.R. China (Email: mjye@njust.edu.cn).
This work is supported by the National Natural Science Foundation of
China (NSFC), No. 61803202, the Natural Science Foundation of Jiangsu
Province, No. BK20180455 and the Fundamental Research Funds for the
Central Universities, No. 30920032203.

weight-balanced digraphs were considered in [11]. Games in
linear systems and Euler-Lagrange systems were considered
in [8] and [40], respectively. Un-modeled dynamics and
disturbances were addressed in [37]. In addition, [9] and [41]
focused on second-order dynamics. High-order games were
considered in [42][43], where internal-model-based seeking
strategies were proposed to achieve distributed Nash equilibrium seeking. Generalized Nash equilibrium seeking, which
concerns with action constraints among the players, was
handled in [30][32][33][38]. Besides, the extremum seeking
based perspectives in [22], the gossip algorithms in [28][29],
the passivity perspectives in [36] and the integral dynamics
in [38][39] also provided insightful ideas to achieve Nash
equilibrium seeking. However, actuation limitations are not
considered in these works.
As many engineering systems are subject to actuator
limitations (e.g., robotic manipulators [12], spacecraft [13],
hard disk drive servo systems [14], just to name a few), the
boundedness of control inputs appears to be a problem that
is both practically and theoretically concerned. The study
for systems with bounded control inputs has a rich history.
For example, input-saturated linear systems were considered in [15] based on an anti-windup design. Backstepping
approaches were employed for developing robust adaptive
control strategies to accommodate uncertain nonlinear systems subject to input saturation [16]. Two-player zero-sum
games with non-quadratic payoffs were employed to solve
the H∞ control of systems with bounded control inputs
in [17]. Moreover, with the development of multi-agent
systems, consensus problems in input-saturated multi-agent
systems have attracted a lot of attention. The authors in [18]
dealt with leader-following consensus of linear multi-agent
systems with input saturation. Global consensus of saturated
discrete-time systems was addressed in [19]. Optimal consensus for multi-agent systems with bounded control inputs was
investigated in [20]-[21]. However, Nash equilibrium seeking
for games in systems with bounded controls has not been
addressed yet, though it is a problem of great interest.
Inspired by the above observations, we intend to design
Nash equilibrium seeking strategies for games in both firstorder and second-order integrator-type systems in which
the controls are bounded. The considered problem is challenging as the saturation function would introduce high
nonlinearity into the closed-loop system. Moreover, the
nonlinearity would result in difficulties on the design of
the Nash equilibrium seeking algorithms, the establishment
of the Lyapunov functions and the corresponding stability
analyses. In summary, with part of the manuscript presented

in [35], this paper contributes in the following aspects: 1).
Distributed Nash equilibrium seeking for games in systems
with bounded control inputs is considered in this paper. Firstorder integrator-type systems are firstly considered, in which
both the saturated gradient play and a distributed strategy
are investigated. Then, second-order integrator-type systems
are explored. A centralized algorithm is firstly proposed
without considering the boundedness of the control inputs,
followed by two distributed seeking schemes. 2). The convergence results of the proposed Nash equilibrium seeking
strategies are analytically investigated. It is proven that the
proposed seeking strategies would enable the players’ actions
to asymptotically converge to the Nash equilibrium under the
given conditions.
Notations: In the remainder, we use R to denote the
set of real numbers. The notation [hi ]vec is defined as
[hi ]vec = [h1 , h2 , · · · , hN ]T and diag{hij }(diag{hi }) for
i, j ∈ {1, 2, · · · , N } denotes a diagonal matrix whose
diagonal elements are h11 , h12 , · · · , h1N , h21 , · · · , hN N ,
(h1 , h2 , · · · , hN ), successively. For a symmetric matrix
Q ∈ RN ×N , λmin (Q) denotes the minimum eigenvalue
of Q. Moreover, ⊗ is the Kronecker product. The notation
min{a, b} = a if a ≤ b and min{a, b} = b if a > b. In
addition, H̃ = [hij ] defines a matrix whose (i, j)th entry is
hij .
II. P ROBLEM F ORMULATION
Consider a game with N players whose dynamics are
governed by
xni = ui ,
(1)
where xi ∈ R is the action of player i and ui ∈ R
is the control input that satisfies |ui | ≤ Ū . Moreover,
xni denotes the nth-order time derivative of xi and in the
subsequent section, n = 1 and n = 2 will be investigated
successively. Let fi (x), where x = [x1 , x2 , · · · , xN ]T , be
the cost function of player i and {1, 2, · · · , N } denotes the
set of N players. This paper aims to design the bounded
controls to seek the Nash equilibrium x∗ = (x∗i , x∗−i ) on
which
fi (x∗i , x∗−i ) ≤ fi (xi , x∗−i ),
(2)
for xi ∈ R, i ∈ {1, 2, · · · , N } and x−i =
[x1 , x2 , · · · , xi−1 , xi+1 , · · · , xN ]T .
The following conditions will be utilized to establish the
convergence results.
Assumption 1: The players’ cost functions are C 2 functions.
Assumption 2: The players are equipped with a communication graph G, which is undirected and connected.
Assumption 3: [24][25] There exists a positive constant m
such that
(x − z)T (P̄(x) − P̄(z)) ≥ m||x − z||2 ,

(3)

for all x, z ∈ RN . Note that in (3), P̄(x) = [∇i fi (x)]vec
i (x)
.
and ∇i fi (x) = ∂f∂x
i

4: The elements in H(x), defined as H(x) =
i
h Assumption
∂ 2 fi (x)
,
are
bounded for x ∈ RN .
∂xi ∂xj
Remark 1: From Assumption 3, it can be obtained that
for each fixed x−i , fi (xi , x−i ) is strongly convex in xi
and H T (x) + H(x) ≥ 2mI for x ∈ RN by Proposition
2.3.2 in [34]. Moreover, under Assumption 3, the game
admits a unique Nash equilibrium by Theorem 2.3.3 in
[34] and the players’ actions are at the Nash equilibrium
if and only if P̄(x) = 0N [24]. Assumption 4 indicates that
for each i ∈ {1, 2, · · · , N }, ∇i fi (x) is globally Lipschitz.
Moreover, it’s worth noting that Assumption 4 is utilized for
the development of global convergence results, and without
this condition, weaker convergence results can be obtained.
Remark 2: Note that compared with our previous works in
[10][23]-[26], it is required that |ui | ≤ Ū in this paper. Due
to the high nonlinearity introduced by the boundedness of
controls, the establishments of the seeking strategies and the
associated Lyapunov stability analysis would be challenging.
Moreover, the graph related definitions utilized in the paper
follow those in [24] and are omitted directly in this paper
due to space limitation.
III. M AIN R ESULTS
In this section, Nash equilibrium seeking for games in
which the players are of first-order integrator-type dynamics
and second-order integrator-type dynamics will be successively investigated.
A. First-order integrator-type systems
In this section, we consider games in which the players’
actions are governed by
ẋi = ui , i ∈ {1, 2, · · · , N }.

(4)

In the following, saturated gradient play will be firstly
considered, followed by a distributed seeking strategy.
1) Saturated gradient play: To seek the Nash equilibrium
of the game, we suppose that the players update their actions
according to
ẋi = −ρŪ (∇i fi (x)) ,
(5)
where i
∈
{1, 2, · · · , N }, and ρŪ (η i )
=
sgn(η i ) min{|η i |, Ū }.
Theorem 1: The Nash equilibrium of the game is globally
asymptotically stable under (5) given that Assumptions 1 and
3 are satisfied.

PN R ∇i fi (x)
Proof: Let V P̄(x) =
ρŪ (t)dt be the
i=1 0
Lyapunov candidate function. Then, if 0 ≤ ∇i fi (x) <
R ∇ f (x)
Ū , 0 i i ρŪ (t)dt = 21 (∇i fi (x))2 , and if ∇i fi (x) ≥

R ∇ f (x)
Ū , 0 i i ρŪ (t)dt = 12 Ū 2 + ∇i fi (x) − Ū Ū . ThereR ∇i fi (x)
fore,
ρŪ (t)dt > 0 for ∇i fi (x) > 0 and
R ∇i fi (x)0
ρŪ (t)dt → +∞ as ∇i fi (x) → +∞. In addition,
0
R ∇ f (x)
R0
if ∇i fi (x) < 0, 0 i i ρŪ (t)dt = − ∇i fi (x) ρŪ (t)dt =
R |∇i fi (x)|
R ∇ f (x)
ρŪ (t)dt. Hence, 0 i i ρŪ (t)dt > 0 as well
0
R ∇ f (x)
for ∇i fi (x) < 0, and 0 i i ρŪ (t)dt → +∞, as
∇ f (x) → −∞. Moreover, if ∇i fi (x) = 0, it is clear that
R ∇i i fi i (x)
ρŪ (t)dt = 0. Recalling that P̄(x) = [∇i fi (x)]vec ,
0

it can be derived that the Lyapunov candidate function is positive definite with respect to P̄(x). Moreover, if ||P̄(x)|| →
+∞, there exists at least one player j whose
gradient value
R ∇ f (x)
satisfies |∇j fj (x)| → +∞, and hence, 0 j j ρŪ (t)dt →
R ∇ f (x)
+∞. As for each i ∈ {1, 2, · · · , N }, 0 i i ρŪ (t)dt ≥ 0,
we can conclude that V (P̄(x)) → +∞ as ||P̄(x)|| → +∞,
i.e., the Lyapunov candidate function is radially unbounded
with respect to P̄(x).
Taking
the
time
derivative
of
V
gives
T
PN
∂
(∇
f
(x))
ẋ
=
V̇
=
ρ
(∇
f
(x))
i i
i i
i=1 Ū
∂x
T
− [ρŪ (∇i fi (x))]vec H(x) [ρŪ (∇i fi (x))]vec .
By
Assumption 3, H T (x) + H(x) ≥ 2mI. Therefore,
V̇ ≤ −m ||[ρŪ (∇i fi (x))]vec ||2 . Hence, ||ρŪ (∇i fi (x))|| →
0 for all i ∈ {1, 2, · · · , N } as t → +∞. Noticing that by
Assumption 3, P̄(x) = 0N if and only if x = x∗ , we can
conclude that ||x − x∗ || → 0 as t → +∞.

In Theorem 1, the convergence property of the saturated
gradient play in (5) is investigated. However, the saturated
gradient play is not suitable for distributed games as all the
players’ actions are contained in the gradient information.
Therefore, we further investigate the Nash equilibrium seeking problem under distributed networks in the subsequent
section.
2) Consensus-based distributed Nash equilibrium seeking:
To achieve Nash equilibrium seeking in distributed networks,
we suppose that the players can communicate with each other
via communication graph G. Then, the Nash equilibrium
seeking strategy can be designed as
ẋi = − ρŪ (∇i fi (yi )) ,
ẏij = − θij

N
X

k=1

!

aik (yij − ykj ) + aij (yij − xj ) ,

(6)

for i, j ∈ {1, 2, · · · , N } and θij = θ θ̄ij , where θ is a
positive parameter to be determined and θ̄ ij is a fixed
positive constant for each i, j ∈ {1, 2, · · · , N }. Moreover,
yi = [yi1 , yi2 , · · · , yiN ]T stands for player i’s local estimate
i (x)
|x=yi .
on x and ∇i fi (yi ) is defined as ∇i fi (yi ) = ∂f∂x
i
Furthermore, aij is the element on the ith row and jth
column of the adjacency matrix of G.
Then, the concatenated-vector form of (6) is
ẋ = − [ρŪ (∇i fi (yi ))]vec

ẏ = − θΘ̄(L ⊗ IN ×N + A)(y − 1N ⊗ x),

(7)

where y = [yij ]vec , Θ̄ = diag{θ̄ij }, L is the Laplacian
matrix of G, A = diag{aij } and IN ×N is an N × N
dimensional identity matrix.
The following theorem establishes the stability result for
the seeking strategy in (6).
Theorem 2: Suppose that Assumptions 1-4 are satisfied,
and the players update their actions according to (6). Then,
there exists a θ∗ such that for each θ ∈ (θ ∗ , ∞), the Nash
equilibrium is globally asymptotically stable.

Proof: Define the Lyapunov candidate function as
V =

N Z ∇i fi (x)
X
i=1

0

ρŪ (t)dt + (y − 1N ⊗ x)T P(y − 1N ⊗ x)

(8)
where P is a symmetric positive definite matrix that satisfies
P Θ̄(L ⊗ IN ×N + A) + (L ⊗ IN ×N + A)Θ̄P = Q where
Q is a symmetric positive definite matrix by Assumption 2
[10]. Then,
T

V̇ = − [ρŪ (∇i fi (x))]vec H(x) [ρŪ (∇i fi (yi ))]vec
+ (ẏ − 1N ⊗ ẋ)T P(y − 1N ⊗ x)
+ (y − 1N ⊗ x)T P(ẏ − 1N ⊗ ẋ)
T

≤ − [ρŪ (∇i fi (x))]vec H(x) [ρŪ (∇i fi (yi ))]vec

(9)

+ 2(y − 1N ⊗ x)T P (1N ⊗ [ρŪ (∇i fi (yi ))]vec )

− λmin (Q)θ||y − 1N ⊗ x||2 ,
T

in which − [ρŪ (∇i fi (x))]vec H(x) [ρŪ (∇i fi (yi ))]vec =
− [ρŪ (∇i fi (x))]Tvec H(x) [ρŪ (∇i fi (x))]vec
+
T
[ρŪ (∇i fi (x))]vec H(x) [ρŪ (∇i fi (x)) − ρŪ (∇i fi (yi ))]vec .
Furthermore,
by
Assumption
4
|ρŪ (∇i fi (x)) − ρŪ (∇i fi (yi ))| ≤ |∇i fi (x) − ∇i fi (yi )| ≤
l̄i ||x − yi ||, where l̄i is the Lipschitz constant of ∇i fi (x).
Noticing that the elements in H(x) are bounded according
to Assumption
max{l̄i },
√ 4, let l1 = supx∈RN ||H(x)||
√
l2 = 2||P|| N max{l̄i }, and l3 = 2||P|| N , in which
max{l̄i } is the maximum value of l̄i for i ∈ {1, 2, · · · , N }.
2
Then, V̇ ≤ −m ||[ρŪ (∇i fi (x))]vec || − λmin (Q)θ||y −
1N ⊗ x||2 + l1 ||[ρŪ (∇i fi (x))]vec || ||y − 1N ⊗ x|| + l2 ||y −
1N ⊗ x||2 + l3 ||y − 1N ⊗ x|| ||[ρŪ (∇i fi (x))]vec || .
Moreover, as l1 ||[ρŪ (∇i fi (x))]vec || ||y − 1N ⊗ x|| ≤
2
l1
l1 ǫ1
2
2ǫ1 ||[ρŪ (∇i fi (x))]vec || + 2 ||y − 1N ⊗ x|| and l3 ||y −
2
l3
1N ⊗ x|| ||[ρŪ (∇i fi (x))]vec || ≤ 2ǫ2 ||[ρŪ (∇i fi (x))]vec || +
l3 ǫ 2
2
2 ||y − 1N ⊗ x|| , in which ǫ1 , ǫ2 are positive constants
that can be arbitrarily chosen,


l1
l3
2
V̇ ≤ − m −
||[ρŪ (∇i fi (x))]vec ||
−
2ǫ1
2ǫ2


l3 ǫ2
l1 ǫ 1
||y − 1N ⊗ x||2 .
−
− λmin (Q)θ − l2 −
2
2
(10)
Choose ǫ1 , ǫ2 such that m− 2ǫl11 − 2ǫl32 > 0. Then, for fixed
ǫ1 , ǫ2 , let
2l2 + l1 ǫ1 + l3 ǫ2
θ∗ =
,
(11)
2λmin (Q)
l1
−
and θ > θ∗ . Subsequently, let l4 = min{m − 2ǫ
1
l3 ǫ2
l1 ǫ 1
l3
2
2ǫ2 , λmin (Q)θ − l2 − 2 − 2 }, then, V̇ ≤ −l4 ||χ|| ,
iT
h
where χ = [ρŪ (∇i fi (x))]Tvec , (y − 1N ⊗ x)T . Moreover, following the proof of Theorem 1, it can be shown that
V is positive definite and radially unbounded with respect
to χ. Hence, ||χ|| → 0 as t → +∞. Recalling that by
Assumption 3, P̄(x) = 0N if and only if x = x∗ , we see
that y → 1N ⊗ x → 1N ⊗ x∗ as t → +∞. To this end, we
arrive at the conclusion.


Remark 3: The seeking strategy in (6) is adapted from the
seeking strategy in [24] in which the saturation function is
included to ensure that |ui | ≤ Ū .
B. Second-order integrator-type systems
In this section, we consider Nash equilibrium seeking
for games in second-order integrator-type systems in which
player i’s action is governed by
ẋi = ν i , ν̇ i = ui ,

(12)

for i ∈ {1, 2, · · · , N }. More specifically, in Section III-B.1,
a centralized algorithm will be proposed without considering
the boundedness of the control inputs. Moreover, the problem is reconsidered under distributed networks in Section
III-B.2. Lastly, the boundedness of the control inputs will be
addressed in Section III-B.3.
1) Centralized Nash equilibrium seeking without considering the boundedness of the control inputs: Let the Nash
equilibrium seeking strategy be
ẋ = ν , ν̇ = −α[∇i fi (x)]vec − βν − H(x)ν,

(13)

where ν = [ν i ]vec and α, β are positive control gains to be
determined. Then, the following result can be obtained.
Theorem 3: Suppose that Assumptions 1 and 3 are satisfied and the players update their actions according to (13).
Then, there exists a positive constant α∗ such that for each
α ∈ (0, α∗ ), there exists a positive constant β ∗ (α) such
that for each β ∈ (0, β ∗ ), the Nash equilibrium is globally
asymptotically stable under (13).
Proof: Define the Lyapunov candidate function as
1
V = ν T ν + [∇i fi (x)]Tvec [∇i fi (x)]vec +ν T [∇i fi (x)]vec . (14)
2

Then,

V

=

√

2
1
6 ||[∇i fi (x)]vec ||
2

1
2
4 ||ν||

+

3
√1 [∇i fi (x)]
vec + 2 ν
3

+

, and it can be easily concluded
that the Lyapunov candidate function ishpositive definite and
i

radially unbounded with respect to
Moreover, by Assumption 3,

T

ν T , [∇i fi (x)]vec

T

.

V̇ =2ν T (−βν − α [∇i fi (x)]vec − H(x)ν)

ẋi = ν i , ν̇ i = −(xi − zi ) − (ν i − żi ), żi = −K̄i ∇i fi (yi )

N
X
ẏij = −θij (
aik (yij − ykj ) + aij (yij − zj )),
k=1

(16)
where j ∈ {1, 2, · · · , N } and zi , yij are auxiliary variables.
Moreover, K̄i = θ1 Ki , θ ij = θθ1 θ̄ ij in which θ, θ1 are
positive parameters to be determined and Ki , θ̄ ij are fixed
positive constants.
The concatenated vector form of (16) is
ẋ = ν, ν̇ = −(x − z) − (ν − ż)
ż = −K̄[∇i fi (yi )]vec

(17)

ẏ = −Θ(L ⊗ IN ×N + A)(y − 1N ⊗ z),

where K̄ = diag{K̄i }, Θ = diag{θij } and z = [zi ]vec .
The following theorem establishes the stability of the
equilibrium in (17).
Theorem 4: Suppose that Assumptions 1-4 are satisfied
and the players update their actions according to (17). Then,
there exists a positive constant θ∗ such that for each θ ∈
(θ ∗ , ∞), there exists a positive constant θ ∗1 (θ) such that
for each θ1 ∈ (0, θ∗1 ), the Nash equilibrium is globally
asymptotically stable.
Proof: Consider
1
V (η) = (z − x∗ )T K −1 (z − x∗ )
2
(18)
+ (y − 1N ⊗ z)T P(y − 1N ⊗ z)
1
1
+ (x − z)T (x − z) + (ν − ż)T (ν − ż),
2
2
where P is defined in the proof of Theorem 2, η = [(z −
x∗ )T , (y−1N ⊗z)T , (x−z)T , (ν−ż)T ]T and K = diag{Ki }
as the Lyapunov candidate function. Then,
V̇ ≤ − θ1 (z − x∗ )T [∇i fi (z)]vec

T

+ [∇i fi (x)]vec H(x)ν + ν T H(x)ν

+ (−βν − α [∇i fi (x)]vec − H(x)ν)T [∇i fi (x)]vec

≤ − (2β + m)||ν||2 − α ||[∇i fi (x)]vec ||2

− λmin (Q)θθ 1 ||y − 1N ⊗ z||2

− ||ν − ż||2 + θ 1 (z − x∗ )T [∇i fi (z) − ∇i fi (yi )]vec

− 2(y − 1N ⊗ z)T P1N ⊗ ż − (ν − ż)T z̈.

+ (2α + β)||ν|| ||[∇i fi (x)]vec ||

≤ − (2β + m − (2α + β)/(2ǫ1 )) ||ν||2

difficult to be achieved. In the following, we consider Nash
equilibrium seeking in distributed networks from another
perspective.
2) Distributed Nash equilibrium seeking without considering the boundedness of control inputs: Suppose that in the
considered game, each player i, i ∈ {1, 2, · · · , N } updates
their own action according to

2

− (α − (ǫ1 (2α + β))/2) ||[∇i fi (x)]vec || ,

(15)
where ǫ1 is a positive constant that√can be arbitrarily chosen.
√
2α+β
2α
< ǫ1 < 2α+β
, 2α−2 αm < β < 2α+2 αm.
Let 2(2β+m)
Then, V̇ is negative definite. Hence, the
√ conclusion can be
drawn with α∗ = m and β ∗ = 2α + 2 αm.

The seeking strategy in (13) achieves the Nash equilibrium
seeking in a centralized fashion. However, as it is challenging
for the players to simultaneously estimate H(x) and x in a
distributed fashion, the distributed implementation of (13) is

(19)
By Assumption 3, −(z − x∗ )T [∇i fi (z)]vec
≤
−m||z − x∗ ||2 . Moreover, by Assumption 1,
¯li
there
exists
positive
constant
such
that
||∇i fi (z) − ∇i fi (yi )|| ≤ ¯li ||y−1N ⊗z||, and ||∇i fi (yi )|| =
||∇i fi (yi ) − ∇i fi (z) + ∇i fi (z) − ∇i fi (x∗ )||
≤
l̄i ||yi − z|| + l̄i ||z − x∗ ||.
In addition, z̈ = θθ21 K H̄(y)

 Θ̄(L ⊗ IN ×N + A)(y −
h̄11 h̄12 · · · h̄1N
 h̄21 h̄22 · · · h̄2N 


1N ⊗ z), where H̄(y) =  .

..

 ..
.
h̄N 1

h̄N 2

· · · h̄N N

and h̄ij ∈ Rh1×N . Moreover, h̄ij = 0TN for i i6=
2
2
2
fi
fi
fi
(yi ), ∂x∂i ∂x
(yi ), · · · , ∂x∂i ∂x
(yi ) ,
j and h̄ii = ∂x∂i ∂x
1
2
N
2

2

(x)
fi
where ∂x∂i ∂x
(yi ) = ∂∂xfii∂x
|x=yi . Noticing that H̄(y) is
j
j
bounded according to Assumption
√ 4, let l1 = max{l̄i } +
2||P||N max{Ki l̄i }, l2 = 2||P|| N max{Ki l̄i } and l3 =
||K|| supy ||H̄(y)||||Θ̄(L ⊗ IN ×N + A)||. Then,

V̇ ≤ − θ1 m||z − x∗ ||2 − λmin (Q)θθ 1 ||y − 1N ⊗ z||2
− ||ν − ż||2 + θ 1 l1 ||z − x∗ ||||y − 1N ⊗ z||

2

+ θθ21 l3 ||ν −

then, V̇ ≤ −θ1 λmin (A1 )||E1 || − ||ν − ż||
ż||||y − 1N ⊗ z||, where λmin (A1 ) > 0 and E1 =
∗ T
T T
[(z
" − x ) , (y − 1N2 ⊗ #z) ] . Moreover, define A2 =
θθ1 l3
θ1 λmin (A1 ) − 2
. Then, λmin (A2 ) > 0 given that
θθ2 l
− 21 3
1
θ1 < θ ∗1 , where
θ∗1 =



4λmin (A1 )
θ2 l32



1
3

.

(22)

V̇ ≤ −λmin (A2 )||E||2 ,

(23)

If this is the case,

where E = [(z − x∗ )T , (y − 1N ⊗ z)T , (ν − ż)T ]T . Hence,
z = x∗ , y = 1N ⊗ z, ν = ż at V̇ = 0, which indicates that
ẋ = ν, ν̇ = −(x − z), ż = 0N , ẏ = 0N 2 . Recalling that
ν = ż at V̇ = 0, we have ν = 0. Hence, ẋ = 0N and x =
C1 , z = C2 at V̇ = 0, where C1 , C2 are constant vectors.
Therefore, ν̇ = −C1 + C2 , at V̇ = 0. Recalling that ν = 0,
we can get that C1 = C2 , i.e., x = z. Hence, the conclusion
can be derived by utilizing the LaSalle’s invariance principle.

The strategy in (17) addressed the Nash equilibrium
seeking problem for games in second-order integrator-type
systems without considering the boundedness of the controls.
In the upcoming section, the seeking strategy in (17) will be
adapted for systems where the controls are bounded.
3) Distributed Nash equilibrium with bounded control
inputs: Let the Nash equilibrium seeking strategy be
ẋ = ν, ν̇ = −ρŪ ((x − z) + (ν − ż))

ż = −K̄[∇i fi (yi )]vec
ẏ = −Θ(L ⊗ IN ×N + A)(y − 1N ⊗ z).

1
(z − x∗ )T K −1 (z − x∗ ) + (ν − ż)T (ν − ż)
2
+ (y − 1N ⊗ z)T P(y − 1N ⊗ z)+
N Z xi −zi +vi −żi
N Z xi −zi
X
X
ρŪ (t)dt,
ρŪ (t)dt +

V (η) =

i=1

+ θ1 l2 ||y − 1N ⊗ z||2 + θθ 21 l3 ||ν − ż||||y − 1N ⊗ z||.
(20)


m
− l21
, and choose θ > θ∗
Define A1 =
− l21 λmin (Q)θ − l2
where
l12
l2
θ∗ =
+
,
(21)
4mλmin (Q) λmin (Q)
2

given that ||(ν(0) − ż(0))T , (x(0) − z(0))T , (y(0) − 1N ⊗
T
z(0))T , (z(0) − x∗ ) || ≤ ∆.
Proof: Define the Lyapunov candidate function as

(24)

Then, the following result can be derived.
Theorem 5: Suppose that Assumptions 1-3 are satisfied.
Then, for any positive constant ∆, there exists a positive
constant θ∗ such that for each θ ∈ (θ∗ , ∞), there exists a positive constant θ∗1 (∆, θ) such that for each θ 1 ∈
(0, θ∗1 ), x generated by (24) converges asymptotically to x∗

0

i=1

0

(25)
where P is defined in the proof of Theorem 1 and η = [(z −
x∗ )T , (y − 1N ⊗ z)T , (x − z)T , (ν − ż)T , (x − z + ν − ż)T ]T .
Then, it can be easily derived that the Lyapunov candidate
function is positive definite and radially unbounded. Moreover, following the analysis in the proof of Theorem 4, it
can be derived that
V̇ ≤ − θ1 m||z − x∗ ||2 − λmin (Q)θθ 1 ||y − 1N ⊗ z||2

+ θ1 l1 ||z − x∗ ||||y − 1N ⊗ z|| + θ1 l2 ||y − 1N ⊗ z||2

+ ρŪ (x − z)T (ν − ż) − 2(ν − ż)T ρŪ (x − z + ν − ż)

− 2(ν − ż)T z̈ + ρŪ (x − z + ν − ż)T (ν − ż)

− ρŪ (x − z + ν − ż)T ρŪ (x − z + ν − ż)
− ρŪ (x − z + ν − ż)T z̈,

(26)
where√l1 = max{l̄i } + 2||P||N max{Ki ¯li } and l2 =
2||P|| N max{Ki ¯li }. Since −(ν − ż)T (ρŪ (x − z + ν −
ż) − ρŪ (x − z)) ≤ 0, we have −(ν − ż)T (ρŪ (x − z + ν −
ż)−ρŪ (x−z))−ρŪ (x−z+ν − ż)T ρŪ (x−z+ν − ż) = 0 if
and only if (ν − ż)T (ρŪ (x−z+ν − ż)−ρŪ (x−z)) = 0 and
ρŪ (x−z+ν− ż) = 0. Moreover, from ρŪ (x−z+ν− ż) = 0,
we have x−z+ν − ż = 0, by which (ν − ż)T ρŪ (x−z) = 0.
Therefore, (ν − ż)T ρŪ (ν − ż) = 0, from which we can get
that ν − ż = 0 and x−z = 0. Hence, −(ν − ż)T (ρŪ (x−z+
ν − ż) − ρŪ (x − z)) − ρŪ (x − z + ν − ż)T ρŪ (x − z + ν − ż)
is negative definite with respect to [(x − z)T , (ν − ż)T , (x −
z + ν − ż)T ]T .
By further following the proof of Theorem 4, we can
conclude that by choosing θ > θ ∗ , where
θ∗ =

l12 + 4ml2
,
4mλmin (Q)

(27)

we have V̇
≤ −θ1 λmin (A1 )||E1 ||2 − W2 (E2 ) +
2
l3 θθ 21 ||ρŪ (x−z+ν − ż)||||y−1
N ⊗z||+2l3 θθ 1 ||ν −

 ż||||y−
l1
m
−2
1N ⊗ z||, where A1 =
, E1 =
− l21 λmin (Q)θ − l2
[(z − x∗ )T , (y − 1N ⊗ z)T ]T , W2 (E2 ) = (ν − ż)T (ρŪ (x −
z+ν − ż)−ρŪ (x−z))+ρŪ (x−z+ν − ż)T ρŪ (x−z+ν − ż),
E2 = [(x − z)T , (ν − ż)T , (x − z + ν − ż)T ]T , and
l3 = ||K|| supy ||H̄(y)||||Θ̄(L ⊗ IN ×N + A)||.
To facilitate the subsequent analysis, define W (η) =
λmin (A1 )||E1 ||2 + W2 (E2 ). Then, it is clear that W (η)
is positive definite and there exists a class K function γ
such that γ(||η||) ≤ W (η). Hence, if we choose θ1 ≤
1, one can obtain that −θ1 λ(A1 )||E1 ||2 − W2 (E2 ) ≤
−θ1 W (η) ≤ −θ1 γ(||η||). Similarly, if θ 1 > 1, one has
−θ1 λ(A1 )||E1 ||2 − W2 (E2 ) ≤ −γ(||η||). Therefore, V̇ ≤

− min{θ1 , 1}γ(||η||) + 2l3 θθ21 ||ν − ż||||y − 1N ⊗ z|| +
l3 θθ 21 ||ρŪ (x − z + ν − ż)||||y − 1N ⊗ z||.
Therefore, for η that belongs to any compact set Dω that
contains the origin, V̇ ≤ − min{θ1 , 1}γ(||η||)+θθ21 l4 , where
l4 = supη∈Dω (2l3 ||ν − ż||||y −1N ⊗z||+l3 ||ρŪ (x−z+ν −
1 ,1}
γ(||η||), ∀||η|| ≥
ż)||||y−1N ⊗z||). Hence, V̇ ≤ − min{θ
2
2
2θθ1 l4
−1
γ ( min{θ1 ,1} ).
Recalling that V is positive definite, there exist γ 1 , γ 2 ∈ K
such that γ 1 (||η||) ≤ V (η) ≤ γ 2 (||η||). Take a positive
constant r such that Br ⊂ Dω , where Br denotes an origincentered ball with radius r. Moreover,
choose θ1 to be suffi2θθ2 l
ciently small such that γ −1 ( min{θ11 4,1} ) < γ −1
2 (γ 1 (r)). Then
for any initial condition that satisfies ||η(0)|| ≤ γ −1
2 (γ 1 (r)),
(i.e., ∆ = γ −1
(γ
(r))),
there
exists
a
positive
constant
1
2
2θθ21 l4
−1
T1 such that ||η(t)|| ≤ γ −1
(γ
(γ
(
)))
for all
2
1
min{θ1 ,1}
t ≥ T1 . Choosing θ 1 to be sufficiently small such that
2θθ 2 l
γ 1−1 (γ 2 (γ −1 ( min{θ11 4,1} ))) < Ū , then, the trajectory of the
system in (24) is the same as the trajectory of the system
in (17) for t ≥ T1 (with the same initial condition at
t = T1 ). Hence, further following the result in Theorem 4,
the conclusion can be derived.

Remark 4: Theorem 5 demonstrates a semi-global convergence result. That is, for any bounded initial conditions,
the proposed method can drive the players’ actions to the
Nash equilibrium of the game by suitably tuning the control
gains (possibly depend on the initial values of the variables).
Different from local convergence results that require the
initial errors to be sufficiently small, the semi-global results
only require the initial values to be bounded and the bounds
can be arbitrarily large.
Remark 5: Theorem 2 indicates that θ should be sufficiently large to ensure the convergence of (6). The lower
bound θ∗ is qualified in (11). Similarly, Theorems 4-5 illustrate that θ should be chosen to be sufficiently large while
θ1 should be chosen to be sufficiently small to ensure the
convergence of (17) and (24). In the proof of the theorems,
the lower bound of θ and upper bound θ 1 are provided in
(21)-(22) and (27), except that θ∗1 in Theorem 5 depends
on the initial errors and is hard to be explicitly quantified
without knowing the initial errors. From the quantifications
of θ ∗ and θ∗1 , it is clear that they depend on the Lipschitz
constants and strong monotonicity constant of the pseudogradient vector, the communication topology as well as the
number of players in the game. Moreover, though θ∗1 in
Theorem 5 depends on the unknown initial errors, the result
is still meaningful as it suggests that for any bounded initial
errors, we can directly choose θ1 to be sufficiently small to
ensure the convergence of the proposed method. Interested
readers are referred to the proofs of the corresponding
theorems for more details.
Remark 6: The theoretical results presented in the paper
are established for xi ∈ R. However, they can be easily
extended to the case in which xi ∈ Rp and p ≥ 2 is a
positive integer. Moreover, we suppose that the control inputs
satisfy |ui | ≤ Ū for presentation simplicity in this paper.
However, the presented strategies can be easily adapted to

1

2

3

澻濴澼

1

3

2

澻濵澼

Fig. 1: The communication graph among the sensors.

accommodate the case in which −U i ≤ ui ≤ Ūi , where U i
and Ūi are positive constants.
Remark 7: Different from [18]-[21] that considered (optimal) consensus of multi-agent systems with bounded controls, this paper accommodates distributed Nash equilibrium
seeking problems in systems with bounded controls. Compared with [18][19], the problem is challenging as not only
consensus of the players’ estimates but also the optimization
of the players’ objective functions need to be achieved. In
addition, the considered problem is challenging compared
with [20] especially for second-order systems as it is difficult
to distributively approximate H(x) in (13). Furthermore, [21]
provided a projection operator based method to deal with
distributed optimization problems in discrete-time systems
with bounded controls, and hence, the design and analyses
therein are distinct from this paper.
IV. S IMULATION STUDIES
This section verifies the effectiveness of the proposed
seeking strategies in a mobile sensor network in which xi ∈
R2 (denoted as xi1 and xi2 , respectively). More specifically,
we consider the connectivity control for a network of 3
mobile sensors in which the sensors’ objective functions
are given by [8] fi (xi , x−i ) = xTi rii xi + xTi pi + qi +
P
2
2×2
, pi ∈ R2×1 , qi ∈
j∈Ni mij ||xi − xj || , where rii ∈ R
R, mij ∈ R are constant matrices, vectors or parameters and
Ni denotes the physical neighboring set of player i. In the
subsequent simulations, we consider Example 1 of [8] in
which i = 3, rii for i ∈ {1, 2, 3} are identity matrices,
and mij = 1 except that m13 = m31 = 0. Moreover,
p1 = [2, −2]T , p2 = [−2, −2]T , p3 = [−4, 2]T , qi = 3
for i ∈ {1, 2} and q3 = 6. Through direct calculation, it
can be easily verified that the example satisfies Assumptions
1, 3-4 and the game admits a unique Nash equilibrium at
x∗ = [−0.125, 0.75, 0.75, 0.5, 1.375, −0.25]T [8].
In the following, velocity-actuated vehicles and
acceleration-actuated
vehicles
will
be
simulated,
successively.
A. Velocity-actuated vehicles
In this section, we consider velocity-actuated vehicles,
whose dynamics can be described as ẋi = ui , where
xi = [xi1 , xi2 ]T denotes the position of sensor i, ui =
[ui1 , ui2 ]T ∈ R2 , uij for i ∈ {1, 2, 3}, j ∈ {1, 2} denotes
the control input of sensor i that satisfies |uij | ≤ Ū .
1) Saturated gradient play: In this section, we suppose
that the mobile sensors can communicate with each other
via the communication graph depicted in Fig. 1 (a). With
x(0) = [10, 0, 0, 5, 0, 0]T and Ū = 5, the trajectories of the
sensors’ positions and the control inputs generated by the

saturated gradient play in (5) are depicted in Fig. 2. Fig. 2
(a) illustrates that the control inputs are bounded by the given
value and Fig. 2 (b) shows that the sensors’ positions would
converge to the Nash equilibrium of the game asymptotically.
Hence, by the simulation results, Theorem 1 is numerically
verified.
(a)
5

4

4

(b)

5

4

4

2

3

xi2

Cotnrol inputs

(b)

6

(a)

6

0

2
(−0.125,0.75)

−2

1

−4

0

−6

−1

(0.75,0.5)
(1.375,−0.25)
3

xi2

Control inputs

2

0

0

5
Time (s)

10

0

5

10

xi1

2

Fig. 3: (a) and (b) show the control inputs and the trajectories
of the sensors’ positions generated by the method in (6),
respectively.

(−0.125,0.75)
−2

1

−4

0

−6

−1
−2

(0.75,0.5)

(1.375,−0.25)

4
6
Time (s)

8

10

0

2

4
xi1

6

8

10

Fig. 2: (a) and (b) show the control inputs and the trajectories
of the sensors’ positions generated by the saturated gradient
play in (5), respectively.
2) Consensus-based distributed Nash equilibrium seeking:
In Section IV-A.1, the physical interactions among the sensors’ objective functions coincide with their interactions in
the communication graph. However, if this is not the case,
the saturated gradient play can not be directly utilized in the
distributed sensor networks. As an alternative, the distributed
seeking strategy given in (6) can be adopted. To illustrate
this case, in this section we suppose that the sensors can
communicate with each other via the communication graph
depicted in Fig. 1 (b), which satisfies Assumption 2 as it is
undirected and connected.
Let x(0) = [10, 0, 0, 5, 0, 0]T , Ū = 5, yij (0) = 10 and
θij = 1000. By choosing Q and Θ̄ to be identity matrices,
it can be verified that θ > θ∗ , where θ ∗ is quantified in
(11). Driven by the method in (6), the control inputs are
illustrated in Fig. 3 (a) and the trajectories of the sensors’
positions are plotted in Fig. 3 (b). The control inputs stay
within the bounded region as shown in Fig. 3 (a). Moreover,
Fig. 3 (b) demonstrates that the trajectories of the sensors’
positions would converge to the Nash equilibrium. Hence, the
effectiveness of the proposed method in (6) is numerically
verified.

variables are initialized at zero. Note that in the simulation,
K̄i = 0.1 and θ ij = 200. By choosing Q, Θ̄ to be identity
matrices and Ki = 0.1, it can be verified that θ = 200 > θ ∗ ,
where θ ∗ is defined in (27). Under the communication graph
depicted in Fig. 1 (b), the simulation results are given in
Fig. 4. As plotted in Fig. 4 (a), the control inputs are
bounded by the given value. Moreover, Fig. 4 (b) depicts
the trajectories of the sensors’ positions, which shows that
the sensors’ positions asymptotically converge to the Nash
equilibrium of the game. The simulation results show that the
proposed method in (24) is effective to achieve distributed
Nash equilibrium seeking for second-order systems with
bounded controls.
(b)

(a)
6

2

5
4

1.5

3
(−0.125,0.75)

2
1
1
xi2

2

Control inputs

0

0
(0.75,0.5)

0.5
−1
−2
0

−3

(1.375,−0.25)
−4
−5

0

5

10
Time (s)

15

20

−0.5
−1

0

1

2
x

3

4

5

i1

Fig. 4: (a) and (b) show the control inputs and the trajectories
of the sensors’ positions generated by the method in (24),
respectively.

B. Acceleration-actuated vehicles
In this section, we suppose that the agents are accelerationactuated vehicles whose dynamics can be described by ẋi =
ν i , ν̇ i = ui , where xi = [xi1 , xi2 ]T ∈ R2 is the vector
containing the positions of sensor i, ν i = [ν i1 , ν i2 ]T ∈ R2
is the vector containing the velocities of sensor i and ui =
[ui1 , ui2 ]T ∈ R2 is the vector containing the control inputs
that satisfy |uij | ≤ Ū , for all i ∈ {1, 2, 3}, j ∈ {1, 2}.
Moreover, we suppose that the sensors update their positions according to (24), in which Ū = 5, and all the

V. C ONCLUSIONS
This paper considers Nash equilibrium seeking for games
in systems where the control inputs are bounded. More
specifically, first-order integrator-type systems are first considered, followed by second-order integrator-type systems.
For both situations, we first design a centralized seeking
strategy based on the gradient play, which is further adapted

to distributed networks. Based on the Lyapunov stability
analysis, the convergence properties of the designed algorithms are analytically investigated. It is shown that the proposed seeking strategies would enable the players’ actions to
converge to the Nash equilibrium under the given conditions.
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