Adv Syst Sci Appl 2024; 02:40–53
Published online at https://ijassa.ipu.ru.

Stability Analysis of Switched Positive Persidskii Systems
with Distributed and Unbounded Delays

Alexander Aleksandrov*, Natalya Andriyanova

Saint Petersburg State University, Saint Petersburg, Russia

Abstract: Switched positive Persidskii systems with distributed and unbounded delays are
studied. Right-hand sides of these systems are linear combinations of nonlinearities of a sector
type. Special constructions of diagonal Lyapunov–Krasovskii functionals are proposed and
conditions are derived under which the absolute stability of the considered systems can be proved
with the aid of such functionals. The developed approaches are applied to the stability analysis
of a mechanical system with switched nonlinear positional forces and to a problem of mobile
agent deployment. Results of numerical simulations are presented confirming the theoretical
conclusions.

Keywords: switched positive system, sector nonlinearity, Lyapunov–Krasovskii functional,
distributed and unbounded delays, mechanical system, formation control

1. INTRODUCTION

Studying the stability of nonlinear time-delay systems is an actual problem of the
contemporary control theory due to wide applications of these systems (see, e.g., [1, 2] and
the bibliography therein). Moreover, it is worth noticing that, in numerous models of practical
systems, state variables are restricted to be nonnegative [3, 4]. Therefore, an important class
of time-delay systems is that of positive systems. Methods for stability analysis of positive
systems with delay are well developed for linear systems [3,4]. In the nonlinear case, the basic
approaches are the comparison method and the Lyapunov direct method [5–8]. However,
under the constructing comparison systems, usually it is required that nonlinearities satisfy
linear estimates (see [9]). As regards to the Lyapunov direct method, its application to
time-delay systems is based on the using Lyapunov–Razumikhin functions or Lyapunov–
Krasovskii functionals [1, 5]. At the same time, a general constructive technique to finding
such functions and functionals is still lacking. This problem becomes even more difficult if the
model under consideration needs to take into account uncertainties and switching of operation
modes. As a result, stability conditions are well investigated only for special classes of
nonlinear time-delay systems, for instance, for homogeneous systems, Lurie control systems,
etc. [5, 10–13].

One of the interesting and important such classes is that of Persidskii systems [3]. Right-
hand sides of these systems are represented as linear combinations of separable nonlinearities
satisfying sector conditions. Persidskii systems are widely used for modeling automatic
control systems, neural networks, opinion dynamics, digital filters, etc. [3, 14].

First, stability conditions for delay-free Persidskii systems were obtained by Barbashin
via constructing a Lyapunov function in the form of linear combination of integrals from
nonlinearities [15]. In [16], linear Lyapunov functions were proposed for the stability analysis

∗Corresponding author: a.u.aleksandrov@spbu.ru



STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 41

of positive delay-free Persidskii systems. In a number of subsequent works, various canonical
forms of Lyapunov functions were used to derive stability conditions for both switched
and nonswitched Persidskii systems (see [3, 17, 18] and the references therein). Stability of
switched positive Persidskii systems with constant delay was studied in [19] with the aid
of a special construction of Lyapunov–Krasovskii functionals. In [8, 9], positive Persidskii
systems with distributed delay were investigated. In recent paper [14], generalized Persidskii
systems with constant delays were considered and conditions of input-to-state stability
formulated in terms of LMIs were derived.

In [20], an approach to constructing Lyapunov–Krasovskii functionals for switched
positive Persidskii systems with constant delay was developed. The proposed functionals
depend on a positive tuning parameter. It was shown that, via an appropriate parameter choice,
less conservative stability conditions can be obtained compared to known ones. The objective
of the present paper is an extension of the above approach to systems with distributed
and unbounded delays. In addition, we will apply our results to the stability analysis of a
mechanical system with nonlinear positional forces and to a problem of formation control.

Through the paper we will use the following notation:
• R is the field of real numbers, Rn and Rn×n are the n-dimensional Euclidean space and

the vector space of n× n matrices, respectively.
• Let ∥ · ∥ be the Euclidean norm of a vector.
• A matrix C ∈ Rn×n is called nonnegative if all its entries are nonnegative.
• A matrix C ∈ Rn×n is called Metzler if all its off-diagonal entries are nonnegative.
• The identity matrix is denoted by I .
• Inequalities for vectors are understood componentwise.

2. STATEMENT OF THE PROBLEM

Consider the following switched Persidskii system with distributed delay:

ẋ(t) = PσΦ(x(t)) +Qσ

∫ t

t−τ

f(ξ − t)Φ(x(ξ))dξ. (1)

Here x(t) ∈ Rn, Φ(x) is a separable vector function, i.e., Φ(x) = (φ1(x1), . . . , φn(xn))
⊤,

where the functions φi(xi), which are said to be admissible nonlinearities, are continuous and
locally Lipschitz for |xi| < ∆ (0 < ∆ ≤ +∞) and satisfy the sector constraints xiφi(xi) > 0
for xi ̸= 0, i = 1, . . . , n, τ = const > 0, f(ζ) is nonnegative and continuous for ζ ∈ [−τ, 0]
scalar kernel, σ = σ(t) is a piecewise constant function defining the switching law, σ(t) :
[0,+∞) 7→ {1, . . . , N}, Ps, Qs are constant matrices, s = 1, . . . , N .

According to the standard assumption [21], we will consider the case where the function
σ(t) admits only finitely many discontinuities on every bounded interval. Such switching
laws will be called admissible.

Initial functions for solutions of (1) are chosen from the space C([−τ, 0], Rn) of
continuous functions θ(ξ) : [−τ, 0] 7→ Rn with the uniform norm

∥θ∥τ = max
ξ∈[−τ,0]

∥θ(ξ)∥.

For a solution x(t), xt denotes the restriction of this solution to the interval [t− τ, t], i.e.,
xt : ξ 7→ x(t+ ξ) for ξ ∈ [−τ, 0].
Assumption 2.1:
The matrices P1, . . . , PN are Metzler and the matrices Q1, . . . , QN are nonnegative.

Remark 2.1:
Under Assumption 2.1, the system (1) is positive (see [6]).

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42 A. ALEKSANDROV, N. ANDRIYANOVA

From the continuity of Φ(x) and the sector restrictions it follows that Φ(0) = 0. Hence,
there exists the zero solution of (1).

Definition 2.1:
The system (1) is called absolutely stable if its zero solution is asymptotically stable for any
admissible switching signal and any admissible nonlinearities.

To derive the absolute stability conditions, we will use a special construction of diagonal
Lyapunov–Krasovskii functional. This functional is a counterpart of that proposed in [20] for
switched positive Persidskii systems with constant delays.

Furthermore, we will study the problem of absolute stability for a switched positive
Persidskii system with unbounded delay.

Finally, we will provide applications of the developed approaches to the stability analysis
of a mechanical system with switched nonlinear positional forces and to a problem of mobile
agent deployment on a line segment.

3. ABSOLUTE STABILITY CONDITIONS FOR THE SYSTEM
WITH DISTRIBUTED DELAY

Construct a Lyapunov–Krasovskii functional candidate for (1) as follows:

V1(xt) =
n∑

i=1

αi

∫ xi(t)

0

φν
i (u)du+

n∑
i=1

βi

∫ t

t−τ

∫ ξ−t

−τ

f(ζ)dζφν+1
i (xi(ξ))dξ, (2)

where αi, βi are positive coefficients, ν is a positive rational number with odd numerator
and denominator. We will look for conditions under which the absolute stability of (1) can be
proven with the aid of such a functional.

Theorem 3.1:
Let Assumption 2.1 be fulfilled. If there exist a positive rational number ν with odd numerator
and denominator, positive vectors λ, η and numbers ω1, ω2 such that

νω1 + ω2 < 0, (3)

(
Ps +

∫ 0

−τ

f(ζ)dζQs

)
λ ≤ ω1λ, s = 1, . . . , N,

(
Ps +

∫ 0

−τ

f(ζ)dζQr

)⊤

η ≤ ω2η, s, r = 1, . . . , N,

then one can choose coefficients αi, βi for which the functional (2) guarantees the absolute
stability of the system (1).

Proof
Differentiating V1(xt) along the solutions of (1), we obtain

V̇1 =
n∑

i,j=1

αip
(σ)
ij φν

i (xi(t))φj(xj(t)) +
n∑

i,j=1

αiq
(σ)
ij φν

i (xi(t))

∫ t

t−τ

f(ξ − t)φj(xj(ξ))dξ

−
n∑

i=1

βi

∫ t

t−τ

f(ξ − t)φν+1
i (xi(ξ))dξ + F

n∑
i=1

βiφ
ν+1
i (xi(t)),

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 43

where F =
∫ 0

−τ
f(ζ)dζ , p(σ)ij and q

(σ)
ij are entries of the matrices Pσ and Qσ, respectively.

Let
αi = ηi/λ

ν
i , i = 1, . . . , n, (4)

yi(t) = φi(xi(t))/λi, i = 1, . . . , n. (5)

Here λi and ηi are components of the vectors λ and η, respectively. Then the derivative of
the functional can be rewritten in the form

V̇1 =
n∑

i,j=1

ηiλjp
(σ)
ij yνi (t)yj(t) +

n∑
i,j=1

ηiλjq
(σ)
ij yνi (t)

∫ t

t−τ

f(ξ − t)yj(ξ)dξ

−
n∑

i=1

βiλ
ν+1
i

∫ t

t−τ

f(ξ − t)yν+1
i (ξ)dξ + F

n∑
i=1

βiλ
ν+1
i yν+1

i (t).

Using the Young inequality [1], we arrive at the estimate

V̇1 ≤
ν

ν + 1

n∑
i=1

ηiy
ν+1
i (t)

n∑
j=1

(
p
(σ)
ij + Fq

(σ)
ij

)
λj

+
n∑

i=1

λiy
ν+1
i (t)

(
1

ν + 1

n∑
j=1

ηjp
(σ)
ji + Fβiλ

ν
i

)

+
1

ν + 1

n∑
i=1

λi

∫ t

t−τ

f(ξ − t)yν+1
i (ξ)dξ

n∑
j=1

ηjq
(σ)
ji

−
n∑

i=1

βiλ
ν+1
i

∫ t

t−τ

f(ξ − t)yν+1
i (ξ)dξ.

Choose the coefficients βi as follows:

βi =
1

(ν + 1)λν
i

(
max

r=1,...,N

n∑
j=1

ηjq
(r)
ji + δ

)
, (6)

i = 1, . . . , n,

where δ is a positive parameter. If δF < −(νω1 + ω2)/2, then

V̇1 ≤
νω1 + ω2

2(ν + 1)

n∑
i=1

ηiλiy
ν+1
i (t)− δ

ν + 1

n∑
i=1

λi

∫ t

t−τ

f(ξ − t)yν+1
i (ξ)dξ.

Hence (see [1]), the zero solution of the system (1) is asymptotically stable. This completes
the proof.

Remark 3.1:
It is worth mentioning that, under Assumption 2.1, the conditions of Theorem 3.1 are fulfilled
iff one of the following conditions is satisfied:

(a) there exists a vector λ > 0 such that

(Ps + FQs)λ < 0, s = 1, . . . , N, (7)

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44 A. ALEKSANDROV, N. ANDRIYANOVA

(b) there exists a vector η > 0 such that

(Ps + FQr)
⊤η < 0, s, r = 1, . . . , N.

Really, in the case (a) the conditions of Theorem 3.1 hold for sufficiently large values of
ν, whereas in the case (b) they hold for sufficiently small values of ν. However, in some
problems, e.g., estimation of attraction domain or the convergence rate of solutions, analysis
of the impact of external perturbations, etc., to derive less conservative results, it is useful to
have opportunity for an appropriate choice of the parameter ν (see, for instance, [20]). The
presented statement of Theorem 3.1 permits us to obtain the domain of admissible values for
this parameter.

4. ABSOLUTE STABILITY CONDITIONS FOR THE SYSTEM
WITH UNBOUNDED DELAY

Next, consider the system

ẋ(t) = PσΦ(x(t)) +Qσ

∫ t

0

g(t− ξ)Φ(x(ξ))dξ, (8)

where g(ζ) is nonnegative and continuous for ζ ≥ 0 scalar kernel and the remaining notation
is the same as for (1). Thus, we will study switched Persidskii system with unbounded delay.

It should be noted that models with unbounded delays are widely used in various
applications such as PID controller design, population dynamics, networked control, social
science, etc. (see [1, 5, 22, 23]).

Every solution x(t) of (8) is defined by an initial time instant t0 ≥ 0 and an initial
function θ(ξ) ∈ C([0, t0], R

n), where C([0, t0], R
n) is the space of continuous functions

θ(ξ) : [0, t0] 7→ Rn with the uniform norm

∥θ∥[0,t0] = max
ξ∈[0,t0]

∥θ(ξ)∥.

Let xt be the restriction of a solution x(t) of (8) to the interval [0, t], i.e., xt : ξ 7→ x(t+ ξ)
for ξ ∈ [−t, 0].

Remark 4.1:
It is known [7] that, under Assumption 2.1, the system (8) is positive.

Definition 4.1:
The system (8) is called absolutely stable if its zero solution is asymptotically stable for any
admissible switching signal and any admissible nonlinearities.

To obtain absolute stability conditions for (8) we impose additional constraints on the
kernel g(ζ).

Assumption 4.1:
Let
∫ +∞
0

g(ζ)dζ < +∞.

Assumption 4.2:
Let
∫ +∞
0

G(ζ)dζ < +∞, where G(ζ) =
∫ +∞
ζ

g(ξ)dξ.

Remark 4.2:
In particular, Assumptions 4.1 and 4.2 are satisfied for the function g(ζ) = exp(−cζ), where
c = const > 0. It is worth mentioning that PID-controllers with exponential kernels are
widely used in the modern control theory (see [1]).

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 45

We will choose a Lyapunov–Krasovskii functional candidate for (8) in the form

V2(xt) =
n∑

i=1

αi

∫ xi(t)

0

φν
i (u)du+

n∑
i=1

βi

∫ t

0

G(t− ξ)φν+1
i (xi(ξ))dξ, (9)

where αi, βi are positive coefficients, ν is a positive rational number with odd numerator and
denominator.

Theorem 4.1:
Let Assumptions 2.1, 4.1, 4.2 be fulfilled. If there exist a positive rational number ν with odd
numerator and denominator, positive vectors λ, η and numbers ω1, ω2 such that

(Ps +G(0)Qs)λ ≤ ω1λ, s = 1, . . . , N, (10)

(Ps +G(0)Qr)
⊤ η ≤ ω2η, s, r = 1, . . . , N, (11)

and the inequality (3) hold, then one can choose coefficients αi, βi for which the functional
(9) guarantees the absolute stability of the system (8).

Proof
Consider the derivative of the functional (9) along the solutions of (8). We obtain

V̇2 =
n∑

i,j=1

αip
(σ)
ij φν

i (xi(t))φj(xj(t))

+
n∑

i,j=1

αiq
(σ)
ij φν

i (xi(t))

∫ t

0

g(t− ξ)φj(xj(ξ))dξ

−
n∑

i=1

βi

∫ t

0

g(t− ξ)φν+1
i (xi(ξ))dξ +G(0)

n∑
i=1

βiφ
ν+1
i (xi(t)).

Define the coefficients αi and βi by the formulae (4) and (6), where λi and ηi are
components of the positive vectors λ and η satisfying the inequalities (10) and (11),
respectively.

Similarly to the proof of Theorem 3.1, using the substitution (5) and the Young inequality,
we arrive at the estimate

V̇2 ≤ −c1

n∑
i=1

yν+1
i (t)− c2

n∑
i=1

∫ t

0

g(t− ξ)yν+1
i (ξ)dξ,

where c1 > 0, c2 > 0. Hence (see [5]), the zero solution of the system (8) is asymptotically
stable. This completes the proof.

It is worth noticing that, regarding Theorem 4.1, a remark similar to Remark 3.1 can be
formulated.

5. APPLICATIONS

Consider two applications of the proposed approaches.

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46 A. ALEKSANDROV, N. ANDRIYANOVA

5.1. Stability Analysis of a Mechanical System
Let motions of a mechanical system be defined by the equations

q̈(t) + hAq̇(t) +BσΦ(q(t)) + Cσ

∫ t

0

g(t− ξ)Φ(q(ξ))dξ = 0. (12)

Here q(t), q̇(t) ∈ Rn are vectors of generalized coordinates and generalized velocities,
respectively, Φ(q) = (φ1(q1), . . . , φn(qn))

⊤, where the functions φi(qi) are continuous for
|qi| < ∆ (0 < ∆ ≤ +∞) and satisfy the sector constraints qiφi(qi) > 0 for qi ̸= 0, i =
1, . . . , n, g(ζ) is nonnegative and continuous for ζ ≥ 0 scalar kernel, σ = σ(t) is an
admissible switching law, σ(t) : [0,+∞) 7→ {1, . . . , N}, A,Bs, Cs are constant matrices,
s = 1, . . . , N , h is a positive parameter. It is worth noting that the term Cσ

∫ t

0
g(t−

ξ)Φ(q(ξ))dξ can be treated as integral part of a PID-controller [5, 22].
Every solution q(t) of (12) is defined by an initial time instant t0 ≥ 0 and an

initial function θ(ξ) ∈ C1([0, t0], R
n), where C1([0, t0], R

n) is the space of continuously
differentiable functions θ(ξ) : [0, t0] 7→ Rn with the uniform norm

∥θ∥[0,t0] = max
ξ∈[0,t0]

(
∥θ(ξ)∥+ ∥θ̇(ξ)∥

)
.

The system (12) admits the equilibrium position

q = q̇ = 0. (13)

To derive stability conditions for (13), we will use a special approach based on the
decomposition method. This approach was first proposed in [24] for stability analysis of
linear delay-free mechanical systems and it was subsequently extended to some classes of
linear and nonlinear mechanical systems with constant delay [12, 25].

Consider the following auxiliary subsystems:

ẋ(t) = −A−1

(
BσΦ(x(t)) + Cσ

∫ t

0

g(t− ξ)Φ(x(ξ))dξ

)
, (14)

ẏ(t) = −Ay(t). (15)

Assumption 5.1:
The matrices

Ps = −A−1Bs, s = 1, . . . , N, (16)

are Metzler and the matrices

Qs = −A−1Cs, s = 1, . . . , N, (17)

are nonnegative.
Assumption 5.2:
The system (15) is asymptotically stable.
Assumption 5.3:
Vector function Φ(q) satisfies the Lipschitz condition for ∥q∥ < ∆ with a constant L > 0.
Theorem 5.1:
Let the matrices Ps, Qs be defined by the formulae (16), (17), respectively, and Assumptions
4.1, 4.2, 5.1–5.3 be fulfilled. If one of the following conditions is satisfied:

(a) there exists a vector λ > 0 such that

(Ps +G(0)Qs)λ < 0, s = 1, . . . , N,

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STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 47

(b) there exists a vector η > 0 such that

(Ps +G(0)Qr)
⊤η < 0, s, r = 1, . . . , N,

then one can find a number h̄ > 0 such that the equilibrium position (13) is asymptotically
stable for h ≥ h̄, for any admissible switching law and any admissible nonlinearity Φ(q).

Proof
Define new variables as follows:

x(t) = q(t) +
1

h
A−1q̇(t), y(t) = q̇(t). (18)

Then

hẋ(t) = PσΦ(x(t)) +Qσ

∫ t

0

g(t− ξ)Φ(x(ξ))dξ

+Pσ

(
Φ

(
x(t)− 1

h
A−1y(t)

)
− Φ(x(t))

)
+Qσ

∫ t

0

g(t− ξ)

(
Φ

(
x(ξ)− 1

h
A−1y(ξ)

)
− Φ(x(ξ))

)
dξ,

ẏ(t) = −hAy(t) + PσΦ

(
x(t)− 1

h
A−1y(t)

)
+Qσ

∫ t

0

g(t− ξ)Φ

(
x(ξ)− 1

h
A−1y(ξ)

)
dξ. (19)

It is worth mentioning that (19) can be considered as a complex system describing the
interaction of subsystems (14) and (15).

Under the conditions of Theorem 5.1, there exist a positive rational number ν with odd
numerator and denominator and positive numbers αi, βi, i = 1, . . . , n, such that

W (t, xt) ≤ −c1∥Φ(x(t))∥ν+1 − c2

∫ t

0

g(t− ξ)∥Φ(x(ξ))∥ν+1dξ,

where W (t, xt) is the derivative of the functional (9) along the solutions of the subsystem
(14) and c1, c2 are positive coefficients.

It is known (see [26]) that Assumption 5.2 implies the existence of a Lyapunov function
V3(y) with the following properties:

(i) V3(y) is continuously differentiable for y ∈ Rn;
(ii) V3(y) is positive definite;
(iii) V3(y) is a homogeneous of the order ν + 1 with respect to the standard dilation

function;
(iv) the derivative of V3(y) along the solutions of the subsystem (15) is negative definite.
Construct a Lyapunov–Krasovskii functional candidate for (19) in the form

Ṽ (xt, yt) = h
n∑

i=1

αi

∫ xi(t)

0

φν
i (u)du+

n∑
i=1

βi

∫ t

0

G(t− ξ)φν+1
i (xi(ξ))dξ

+V3(y(t)) +

∫ t

0

G(t− ξ)∥y(ξ)∥ν+1dξ.

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48 A. ALEKSANDROV, N. ANDRIYANOVA

Differentiating this functional along the solutions of (19) and using the Lipschitz
condition for Φ(q), we obtain

˙̃
V ≤ −c1∥Φ(x(t))∥ν+1 − c2

∫ t

0

g(t− ξ)∥Φ(x(ξ))∥ν+1dξ

−(hc3 −G(0))∥y(t)∥ν+1 −
∫ t

0

g(t− ξ)∥y(ξ)∥ν+1dξ

+
c4
h
∥Φ(x(t))∥ν

(
∥y(t)∥+

∫ t

0

g(t− ξ)∥y(ξ)∥dξ
)

+c5∥y(t)∥ν
(
∥Φ(x(t))∥+

∫ t

0

g(t− ξ)∥Φ(x(ξ))∥dξ

+
1

h
∥y(t)∥+ 1

h

∫ t

0

g(t− ξ)∥y(ξ)∥dξ
)
,

where c3, c4, c5 are positive constants.
With the aid of the Young inequality, it is easy to prove the existence of a number h̄ > 0

such that
˙̃
V ≤ −1

2

(
c1∥Φ(x(t))∥ν+1 + hc3∥y(t)∥ν+1

+

∫ t

0

g(t− ξ)
(
c2∥Φ(x(ξ))∥ν+1 + ∥y(ξ)∥ν+1

)
dξ

)
for h ≥ h̄. This implies the asymptotic stability of the zero solution of (19), see [5]. Taking
into account the properties of the transformation (18), we obtain that, for such values of h, the
equilibrium position (13) of the system (12) is asymptotically stable, as well. This completes
the proof.

5.2. A Problem of Mobile Agent Deployment
Let a group of n mobile agents on a line be given. The agents are interpreted as numbered
points with coordinates zi(t) ∈ R, i = 1, . . . , n, and the agent dynamics is modeled by the
first-order integrators

żi(t) = ui, i = 1, . . . , n,

where ui is a control input.
Assume that a segment [a, b] of the line is given. Consider the problem of synthesis of a

decentralized control protocol that provides a prescribed agent deployment on the segment.
In [27], this problem was solved for the case of the equidistant agent distribution under

the condition that each agent receives information about the distances between itself and its
nearest left and right neighbors. It should be noted that neighbors are understood in terms of
agent numbers.

In [28, 29], the result of [27] was extended to the case where each agent receives
information about the distances between itself and several its left neighbors and several its
right neighbors (not necessarily nearest neighbors). Furthermore, the effect of communication
delay and switching of network topology (replacing chosen neighbors by the other ones) was
investigated [28,29]. Linear control protocols were designed for which neither constant delay
values nor switching disturb the agent convergence to the equidistant distribution. In addition,
the problem of nonlinearly-uniform (uniform with respect to a given nonlinear function) agent
deployment was studied [29]. It is worth mentioning that such a problem is important for
coverage control of mobile sensors, where a cost function is introduced to evaluate how

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STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 49

well a given curve or domain is covered by the sensor network, and for synchronization
of processes relatively to certain functions of phase coordinates [30, 31]. In [29], nonlinear
control protocols were proposed and robustness of these protocols with respect to constant
communication delays and network topology switching was proved.

However, it should be noted that, in various formation control models, instead of discrete
delays, distributed ones are used (see [1,32]). In particular, an example of such a model is the
traffic flow dynamics [32]. Therefore, in the present subsection, we will consider the problem
of nonuniform agent deployment under distributed delay in the communication channels.

Let a continuous, locally Lipschitz and strictly increasing for z ∈ (−∞,+∞) scalar
function ϱ(z) be given. Our objective is to design a decentralized control protocol providing
the convergence of agents to the positions z̄i for which the corresponding points ϱ(z̄i) are
uniformly distributed on the segment [ϱ(a), ϱ(b)]. Hence,

ϱ(z̄i) = ϱ(a) +
i

n+ 1
(ϱ(b)− ϱ(a)), i = 1, . . . , n.

Denote z̄ = (z̄1, . . . , z̄n)
⊤. In what follows, the points a and b will be interpreted as static

agents, i.e., z0(t) = a, zn+1(t) = b for t ∈ [0,+∞).
We will consider the scenario where each agent receives information from some its

neighbors, and connections between agents can be switched on and off at any time instant.
Let σ(t) : [0,+∞) 7→ {1, . . . , N} be an admissible switching law defining the operating
order of communication topologies, Ξ(σ(t))

il and Ξ
(σ(t))
ir be the sets of indices of left and right

neighbors, respectively, from which the ith agent receives information at the instant t, and
Ξ
(σ(t))
i = Ξ

(σ(t))
il ∪ Ξ

(σ(t))
ir , i = 1, . . . , n.

Assumption 5.4:
Let Ξ(s)

il ̸= ∅ and Ξ
(s)
ir ̸= ∅ for i = 1, . . . , n, s = 1, . . . , N .

Assumption 5.5:
Each ith agent at each time instant knows the values ϱ(zi(t))− c

∫ t

t−τ
f(ξ − t)ϱ(xj(ξ))dξ

for j ∈ Ξ
(σ(t))
i , where τ = const > 0, f(ζ) is a nonnegative and continuous for ζ ∈ [−τ, 0]

kernel,
∫ τ

0
f(ζ)dζ > 0, c = 1/

∫ τ

0
f(ζ)dζ .

Assumption 5.6:
Each agent at each time instant knows how many agents are located between itself and agents
from which the signals are received.

Define the coefficients a(s)ij by the formulae

a
(s)
ij =

δ
(s)
i

(i− j)M
(s)
il

for j ∈ Ξ
(s)
il ,

a
(s)
ij =

δ
(s)
i

(j − i)M
(s)
ir

for j ∈ Ξ
(s)
ir ,

where M
(s)
il and M

(s)
ir are the numbers of elements of the sets Ξ(s)

il and Ξ
(s)
ir , respectively, and

δ
(s)
i =

 1

M
(s)
il

∑
j∈Ξ(s)

il

1

i− j
+

1

M
(s)
ir

∑
j∈Ξ(s)

ir

1

j − i


−1

, i = 1, . . . , n, s = 1, . . . , N.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



50 A. ALEKSANDROV, N. ANDRIYANOVA

Construct a control protocol as follows:

ui =
∑

j∈Ξ(σ(t))
i

a
(σ(t))
ij

(
c

∫ t

t−τ

f(ξ − t)ϱ(zj(ξ))dξ − ϱ(zi(t))

)
, i = 1, . . . , n.

Then the corresponding closed-loop system takes the form

żi(t) =
∑

j∈Ξ(σ(t))
i

a
(σ(t))
ij

(
c

∫ t

t−τ

f(ξ − t)ϱ(zj(ξ))dξ − ϱ(zi(t))

)
, i = 1, . . . , n. (20)

Theorem 5.2:
Let Assumptions 5.4–5.6 be fulfilled. Then the system (20) admits the equilibrium position
z̄ that is globally asymptotically stable for any admissible function ϱ(z) and any admissible
switching law.

Proof
It is easy to verify that z̃ is an equilibrium position for (20). Hence, the system (20) can be
rewritten in the form

żi(t) = ϱ(z̄i)− ϱ(zi(t)) + c
∑

j∈Ξ(σ(t))
i , 1≤j≤n

a
(σ(t))
ij

∫ t

t−τ

f(ξ − t) (ϱ(zj(ξ))− ϱ(z̄j)) dξ, i = 1, . . . , n.

Let xi(t) = zi(t)− z̄i, i = 1, . . . , n. Then

ẋi(t) = −φi(xi(t)) + c
∑

j∈Ξ(σ(t))
i , 1≤j≤n

a
(σ(t))
ij

∫ t

t−τ

f(ξ − t)φ(xj(ξ))dξ, i = 1, . . . , n. (21)

Here φi(xi) = ϱ(xi + z̄i)− ϱ(z̄i). Thus, we arrive at the system of the form (1) where
Ps = −I and the components q

(s)
ij of the matrices Qs are defined as follows: q(s)ij = ca

(s)
ij

for j ∈ Ξ
(s)
i , and q

(s)
ij = 0 for j /∈ Ξ

(s)
i , i, j = 1, . . . , n, s = 1, . . . , N .

It is known (see the proof of Theorem 1 in [29]), that, for such matrices Ps and Qs,
the corresponding inequality system (7) admits a positive solution. According to Remark
3.1 we obtain that, if ν is sufficiently large, then there exists a functional of the form (2)
guaranteeing the asymptotic stability of the zero solution of (21). It is worth noticing that in
this case the asymptotic stability is global [5]. This implies the global asymptotic stability of
the equilibrium position z̄ of the system (20). The proof is completed.

6. NUMERICAL EXAMPLE

For simulation, consider a group consisting of ten agents. Let [a, b] = [1, 6], ϱ(z) = arctan z,
τ = 1, f(ζ) = e−ζ , σ(t) = 1 for t ∈ [10k, 10k + 5) and σ(t) = 2 for t ∈ [10k + 5, 10(k +
1)), k = 0, 1, 2, . . .. Assume that initial functions for agents are constant on the interval
[−1, 0]. Their values and sets of agent neighbors are presented in Table 6.1.

In Fig. 6.1, the dependence of agent coordinates on time is given. The simulation results
demonstrate the convergence of agents to the prescribed distribution.

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



STABILITY ANALYSIS OF SWITCHED POSITIVE PERSIDSKII SYSTEMS 51

Table 6.1. Network topologies and agents initial positions

Agent number Initial position Ξ
(1)
i Ξ

(2)
i

1 0.41 {0, 4} {0, 5}
2 -0.39 {0, 5} {0, 6}
3 0.4 {0, 6} {0, 7}
4 -0.3 {1, 7} {0, 8}
5 -0.22 {2, 8} {1, 9}
6 0.8 {3, 9} {2, 10}
7 0.3 {4, 10} {3, 11}
8 -0.5 {5, 11} {4, 11}
9 -0.6 {6, 11} {5, 11}
10 1.7 {7, 11} {6, 11}

0 100 200 300 400
 t

0

1

2

3

4

5

6

Fig. 6.1. The agent time history.

7. CONCLUSION

In the present contribution, original constructions of diagonal Lyapunov–Krasovskii
functionals for switched positive Persidskii systems with distributed and unbounded delays
are proposed. With the aid of these functionals, new absolute stability conditions for
considered systems are derived. The obtained results are used for the stability analysis
of a mechanical system with switched nonlinear positional forces and for the design of
decentralized protocols ensuring a prescribed mobile agent deployment on a line segment. It
is worth noticing that the proofs of Theorems 3.1 and 4.1 provide us constructive algorithms
for finding parameters of the functionals. An interesting direction for further research is an
application of the developed approaches in stability analysis of generalized Lotka–Volterra
models of population dynamics.

ACKNOWLEDGEMENTS

The research was supported by the grant of Russian Science Foundation, N 24-21-00091,
https://rscf.ru/en/project/24-21-00091/

Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)



52 A. ALEKSANDROV, N. ANDRIYANOVA

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Copyright © 2024 ASSA. Adv Syst Sci Appl (2024)

https://dx.doi.org/doi.org/10.25728/assa.2019.19.3.676

	Introduction
	Statement of the problem
	Absolute stability conditions for the system  with distributed delay
	Absolute stability conditions for the system  with unbounded delay
	Applications
	Stability Analysis of a Mechanical System
	A Problem of Mobile Agent Deployment

	Numerical example
	Conclusion