Synchronization in a multilevel network using the Hamilton-Jacobi-Bellman HJB technique Thierry Njougouo1 2 3Victor Camargo4 5Patrick Louodop1 3

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Synchronization in a multilevel network using the Hamilton-Jacobi-Bellman (HJB)

technique

Thierry Njougouo,1, 2, 3 Victor Camargo,4, 5 Patrick Louodop,1, 3

Fernando Fagundes Ferreira,4, 5 Pierre K. Talla,6and Hilda A. Cerdeira7

1Research Unit Condensed Matter, Electronics and Signal Processing,

University of Dschang, P.O. Box 67 Dschang, Cameroon.

2Faculty of Computer Science, University of Namur, Rue Gandgagnage 21, 5000 Namur, Belgium

3MoCLiS Research Group, Dschang, Cameroon

4Center for Interdisciplinary Research on Complex Systems,

University of Sao Paulo, Av. Arlindo Bettio 1000, 03828-000 S˜ao Paulo, Brazil.

5Department of Physics-FFCLRP, University of S˜ao Paulo, Ribeirao Preto-SP, 14040-901, Brasil.

6L2MSP, University of Dschang, P.O. Box 67 Dschang, Cameroon.

7S˜ao Paulo State University (UNESP), Instituto de F´ısica Te´orica,

Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 01140-070 S˜ao Paulo, Brazil.

Author to whom correspondence should be addressed: thierrynjougouo@ymail.com

(Dated: October 18, 2022)

This paper presents the optimal control and synchronization problem of a multilevel network

of R¨ossler chaotic oscillators. Using the Hamilton-Jacobi-Bellman (HJB) technique, the optimal

control law with three-state variables feedback is designed such that the trajectories of all the

R¨ossler oscillators in the network are optimally synchronized in each level. Furthermore, we provide

numerical simulations to demonstrate the eﬀectiveness of the proposed approach for the cases of

one and three networks. A perfect correlation between the MATLAB and the PSPICE results

was obtained, thus allowing the experimental validation of our designed controller and shows the

eﬀectiveness of the theoretical results.

I. INTRODUCTION

The history of the synchronization of dynamical sys-

tems goes back to Christiaan Huygens in 1665 [26] and,

in the past three decades, it has become a subject of

intensive research due to their various domains of appli-

cations in ﬁelds like mathematics, physics, biology, eco-

nomics, technology, engineering [1, 4, 5, 13, 19, 26]. This

phenomenon exists in the case of two coupled systems as

well as a network [10, 17, 26]. In recent decades, several

works based on the study of synchronization in complex

networks have focused on the problem of orienting the

network towards a collective state shared by all the units,

but for the most part considering the coupling coeﬃcient

as the control parameter used to achieve this dynamic

[10, 14, 25, 35].

After an initial period of characterization of the com-

plex networks in terms of local and global statistical prop-

erties, attention was turned to the dynamics of their in-

teracting units. A widely studied example of such behav-

ior is synchronization of coupled oscillators arranged into

complex networks [10]. Synchronization can found appli-

cations in communication systems, system’s security and

secrecy or cryptography[3, 11].

The investigations on the behavior of the network co-

operative systems (or multi-agent systems) has received

extensive attention, mainly due to its widespread appli-

cations such as mobile robots, spacecraft, networked au-

tonomous team, sensor networks, etc. [24, 29, 38]. In all

these applications, whatever the ﬁeld, the main idea is

control. Based on the literature of the control, a wide

variety of approaches have been developed to control the

behaviour of the systems in a network. Several meth-

ods have been proposed to achieve chaos synchronization

such as impulsive control, adaptive control, time-delay

feedback approach, active control, sliding mode, pinning

control, compound synchronization, nonlinear control,

[2, 7, 20, 21, 30–32, 36] etc. Most of the above methods

were used to synchronize two identical chaotic systems

using adaptive methods.

To control a system is to be able to perform the appropri-

ate modiﬁcation on its inputs in order to place the out-

puts in a desired state. Most studies conducted in com-

plex systems and particularly in the control of network

dynamics use linear (usually diﬀusive) coupling models

to study network dynamics [14, 17, 22, 25, 26, 35]. This

method is limited because it takes too long to achieve

synchronization thus rendering simulations practically

useless. To solve this problem, we propose to build an

optimal controller in the case of a network of chaotic os-

cillators that will not only reduce the transient phase to

achieve the desired behaviour but it also reduces con-

siderably the simulation time. It is important to men-

tion that this work completes the work of Raﬁkov and

Balthazar[28] who initially presented the synchronization

of two R¨ossler chaotic systems based on the HJB tech-

niques.

The structure of the article is as follows: In Sect.II,

the control of the dynamics of one network (sometimes

called patch) of 50 R¨ossler chaotic oscillators based on

the formulation of the problem is introduced a theorem

illustrating how to design the controllers is proven also in

this section. In Sect. III, the HJB technique presented

in Sect.II is extended to three networks of 50 R¨ossler

arXiv:2210.09200v1 [math.OC] 14 Oct 2022

chaotic oscillators. Then, in Sec. IV we illustrate the

implementation of the technique using electronic circuits

for a small number of oscillators.

II. SYNCHRONIZATION OF A NETWORK OF

R¨

OSSLER CHAOTIC OSCILLATORS

The purpose of this section is to introduce a devel-

opment optimal control law to resolve for the optimal

synchronization of R¨ossler chaotic oscillators. The opti-

mal control law is obtained using the Hamilton-Jacobi-

Bellman (HJB) technique[15, 28].

A. Problem Formulation

First we present the model of a single network. Fig.1

shows the topology of connections between the nodes of

the network.

FIG. 1: Representation of the model of a single network.

Let us consider the well-known R¨ossler system [22, 34]

as the node dynamics with the following mathematical

description Eq.1.







˙x1

i=−x2

i−x3

˙x2

i=x1

i+ax2

i, i = 1,2, ..., N

˙x3

i=bx1

i+x3

i(x1

i−c).

(1)

where a= 0.36, b= 0.4 and c= 4.5.

The system has a zero bounded volume, globally at-

tracting set [8, 18]. Hence, for all time t > 0, the

state trajectories Xi(t)=(x1

i(t), x2

i(t), x3

i(t)) are globally

bounded and continuously diﬀerentiable with respect to

time t. Thereby, Npositive constants Lifor all the N

nodes of the network exist such that:

||Xi|| ≤ Li≤Lmax, i = 1,2, ..., N. (2)

where ||Xi|| is the norm of the system identiﬁed by the

index i,Liis maximum constant for the node iand

Lmax <∞is the maximum constant for all nodes in

the network.

Our goal is to develop an optimal control ui(t) to guar-

antee the complete synchronization of all systems in the

network. We assume that the controlled model is deﬁned

by Eq.3.

Xi=f(Xi) + Bui.(3)

where f(Xi): <n→ <nrepresents the self-dynamics of

node i(see Eq.1) of the network and B∈ <n×m. Taking

into account the controller ui∈ <mthe dynamics of the

network becomes:







˙x1

i=−x2

i−x3

i+u1

˙x2

i=x1

i+ax2

i+u2

i, i = 1,2, ..., N

˙x3

i=bx1

i+x3

i(x1

i−c) + u3

(4)

As mentioned previously, the goal is to design an ap-

propriate optimal controller uk

i(k= 1,2,3 and i=

1,2, ..., N) such that for any initial condition, we have:

lim

t→∞ keij k= lim

t→∞ kXi(t)−Xj(t)k= 0.(5)

where k.krepresents the Euclidean norm and eij the error

between system iand system jdeﬁned by eij (t) = Xi(t)−

Xj(t). Therefore, the dynamical system error between

node iand node jis calculated as follows:







˙e1

ij =−e2

ij −e3

ij +u1

ij ,

˙e2

ij =e1

ij +ae2

ij +u2

ij ,

˙e3

ij =be1

ij +x1

je3

ij +x3

je1

ij +e1

ij e3

ij −ce3

ij +u3

ij .

(6)

Clearly, the optimal synchronization problem is now

replaced by the equivalent problem of optimally stabiliz-

ing the error system Eq.6 using a suitable choice of the

controllers u1

ij , u2

ij and u3

ij . In order to generalize Raﬁkov

and Balthazar’s work[28] to apply for a network we prove

that:

Theorem II.1 The controlled R¨ossler chaotic oscil-

lators presented by Eq.4 will asymptotically synchronize

provided the optimal controller u∗found minimizes the

performance functional deﬁned by Eq.7.

∞

Ω(ek

ij , U)dt

∞

k=1

i,j=1,i6=jαk

ij ek

ij 2+ηk

ij uk

ij 2dt.

(7)

Let uk

ij =−λk

ηk

ij be the feedback controllers that

minimize the above integral measure with λk

ij ,ηk

ij and

αk

ij being the weight of the links which satisfy the

relationship αk

ij =−λk

ηk

The dynamical system error Eq.6 converge to equilibrium

ij = 0 (k= 1,2,3and i, j = 1,2, ..., N.).

Proof: Let us assume that the mini-

mum of Eq.7 is obtained with U=U∗=

{u1∗

1, u2∗

1, u3∗

1;u1∗

2, u2∗

2, u3∗

2;...;u1∗

N, u2∗

N, u3∗

N}. So, we

have:

V(ek

ij , U∗, t) = minU

∞

Ω(ek

ij , U∗, t)dt. (8)

The function Vmay be treated as the Lyapunov function

candidate.

Using the Hamilton-Jacobi-Bellman technique, we ﬁnd

the optimal controller Usuch that the systems Eq.6 is

stabilized to equilibrium points and the integral Eq.7 is

minimum. Therefore we have:

∂V

∂e1

˙e1

ij +∂V

∂e2

˙e2

ij +∂V

∂e3

˙e3

ij +

k=1 αk

ij ek

ij 2+ηk

ij (u∗k

ij )2= 0.

(9)

Replacing Eq.6 into Eq.9 we ﬁnd:

∂V

∂e1

ij −e2

ij −e3

ij +u∗1

ij +∂V

∂e2

ij e1

ij +ae2

ij +u∗2

ij 

+∂V

∂e3

ij be1

ij +x1

je3

ij +x3

je1

ij +e1

ij e3

ij −ce3

ij +u∗3

ij 

k=1 αk

ij ek

ij 2+ηk

ij (u∗k

ij )2= 0.

(10)

The Minimization of the Eq.10 with respect to U∗gives

the following optimal controllers:

∂V

∂ek

+ 2ηk

ij u∗k

ij =0=⇒u∗k

ij =−1

2ηk

∂V

∂ek

.(11)

with k= 1,2,3 and i, j = 1,2, ..., N

Replacing Eq.11 into Eq.10 we obtain the following Equa-

tion:

∂V

∂e1

ij −e2

ij −e3

ij +∂V

∂e2

ij e1

ij +ae2

ij 

+∂V

∂e3

ij be1

ij +x1

je3

ij +x3

je1

ij +e1

ij e3

ij −ce3

ij 

k=1 αk

ij ek

ij 2−1

2ηk

∂V

∂ek

ij = 0.

(12)

Now considering

V(ek

ij ) =

k=1

λk

ij ek

ij 2.(13)

The Hamilton-Jacobi-Bellman relation described by

Eq.12 is satisﬁed. Thereby the optimal controllers can

be derived as follows:

u∗k

ij =−λk

ηk

ij , k = 1,2,3; i, j = 1,2, ..., N. (14)

where the constants λand ηare positive. Diﬀerentiating

the function in Eq.13 along the optimal trajectories we

have:

V(ek

ij ) = −2

k=1

αk

ij ek

ij 2≤0.(15)

Therefore, we can select Vas a Lyaponuv function.

According to [12, 21], this shows the solutions of the sys-

tem Eq.6 are asymptotically stable in the Lyapunov sense

via optimal control.

B. Numerical simulation of the optimal

synchronization in a single network

In order to demonstrate the eﬀectiveness and valid-

ity of the proposed results in an optimal controller in

the case of the network (patch) described in Eq.4, we

present and discuss the numerical results. We use MAT-

LAB software with fourth order Runge-Kutta integration

method for numerical resolution of the non-linear diﬀer-

ential equations.

We consider a network constituted by N= 50 R¨ossler

chaotic oscillators with the optimal controllers obtained

in Theorem 2.1. According to Ref[37] the synchroniza-

tion error of the whole network can be calculated using

the relation given by:

e(t) = 1

i,j=1 

xk

i(t)−xk

j(t)

.(16)

In Fig.2 the we present the dynamics of the systems in

the network, without control. In Fig.2(a) we can observe

the dynamics of each oscillator of the network and we

conclude that synchronization does not exist here. This

situation is conﬁrmed in Fig.2(b) by non-zero synchro-

nization error in this network. According to the litera-

ture, synchronization between chaotic oscillators is due

to the presence of the coupling or control between these

systems. Therefore, the results presented in Fig.2 are

normal because in the absence of any type of interaction

or control the existence of synchronization is a random

fact.

Now we proceed to demonstrate the eﬀectiveness of

the optimal control obtained in Theorem 2.1. In Fig.3(a)

we show the time series of synchronized elements for a

network of R¨ossler chaotic oscillators for the constant

parameters in the optimal controller: λi= 1 and ηi= 10

with i= 1,2, ..., N. This chaotic synchronization is con-

ﬁrmed by the synchronization error plotted in Fig.3(b).

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SynchronizationinamultilevelnetworkusingtheHamilton-Jacobi-Bellman(HJB)techniqueThierryNjougouo,1,2,3VictorCamargo,4,5PatrickLouodop,1,3FernandoFagundesFerreira,4,5PierreK.Talla,6andHildaA.Cerdeira71ResearchUnitCondensedMatter,ElectronicsandSignalProcessing,UniversityofDschang,P.O.Box67Dschang,Camer...

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Synchronization in a multilevel network using the Hamilton-Jacobi-Bellman HJB technique Thierry Njougouo1 2 3Victor Camargo4 5Patrick Louodop1 3

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