key: cord-0947917-461eq1j6
authors: Benfatah, Youssef; El Bhih, Amine; Rachik, Mostafa; Tridane, Abdessamad
title: On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems
date: 2021-09-02
journal: Int J Control Autom Syst
DOI: 10.1007/s12555-020-0486-6
sha: af403026008da84e0790c6b4c8ca7e9be1fe0535
doc_id: 947917
cord_uid: 461eq1j6

Given a discrete-time controlled bilinear systems with initial state x(0) and output function y(i), we investigate the maximal output set Θ(Ω) = {x(0) ∈ ℝ(n), y(i) ∈ Ω, ∀ i ≥ 0} where Ω is a given constraint set and is a subset of ℝ(p). Using some stability hypothesis, we show that Θ(Ω) can be determined via a finite number of inequations. Also, we give an algorithmic process to generate the set Θ(Ω). To illustrate our theoretical approach, we present some examples and numerical simulations. Moreover, to demonstrate the effectiveness of our approach in real-life problems, we provide an application to the SI epidemic model and the SIR model.

Output admissible sets arise in many essential practical applications such as stability analysis and control of constrained systems (see [1] [2] [3] [4] [5] [6] ). This concept has been considerably investigated (see [7, 8] and the references therein), but not in the case of the bilinear systems. Consequently, its applicability is limited.

Bilinear systems were introduced into control theory in the 1960s. They were a distinct type of nonlinear systems, in which nonlinear terms are developed by way of multiplication of control vector and state vector. These type of systems has attracted a large community of researchers in almost half a century (see [9] ). Such systems' significance lies in the fact that many necessary engineering processes [10] can be modeled through bilinear systems [11] .

As we know, a nonlinear discrete-time finitedimensional system can be represented by the following dynamic equation:

where x i ∈ R n and u i ∈ R q , for i ≥ 0, are respectively the state and the control.

If we assume that f is linear with respect to the state or the control, we obtain, respectively.

and

we obtain an another form of bilinear systems given as follows:

where for each i ≥ 0, A is a n × n matrix, B is a n × q matrix.Bx i u i is a bilinear form in the x i , u i variables that can be expressed as

where u i = {u j i } q j=1 and B j is a n × n matrix. The use of bilinear systems has two significant advantages: First, they provide better modeling of a nonlinear phenomenon than a linear system [12] ). Second, many real-life problems can be model using bilinear systems.

The maximal output admissible (MOA) sets are an important concept for analyzing controlled systems with constraints. The MOA set has been well studied particularly for linear systems with state and control constraints [13, 14] . This concept provides an understanding of the analysis of constrained control systems. Besides, it is widely used in control system design methods [15] .

The MOA set for a class of nonlinear systems has recently been studied by Rachik et al. in [16] . Later on, Hirata and Ohta [17] considered a special class of nonlinear systems, the so-called polynomial systems, and discussed the finite determinability of the MOA set.

Some other resources that provide information about the study of the concept MOA sets are given in the references [18] [19] [20] [21] [22] [23] [24] . In [18] , Ossareh considered a Lyapunovstable periodic system and gave the development of the MOA set theory. He also investigated some of its geometric and algebraic properties. In [19] , Yamamoto presented MOA's computational procedure for a trajectory tracking control of biped robots.

Various algorithms have been added in the literature for determining the maximal state constraint sets [13, 25] . In [25] the authors considered autonomous linear discretetime systems subject to linear constraints, and proposed an effective procedure to determine the maximal set of admissible initial states. Other models from population dynamics and optimal control problems can be found in [26, 27] .

This work aims to present a new approach to studying the MOA set for a class of bilinear discrete-time systems. To the best of our knowledge, the MOA sets for this type of system have not been investigated before. More specifically, we characterize the initial states of a controlled bilinear system whose resulting trajectory verify a pointwise constraint. Moreover, our approach is applicable to epidemiological, sociological, and physiological models.

In these examples, the MOA set represents the set of initial data that guarantee for these systems do not violate certain constraints. This paper considers discrete-time controlled bilinear systems. More precisely, the system has the following form:

the corresponding output is

where A ∈ L(R n , R n ) is the matrix of the state, R, is the set of real numbers, L(R n , R n ), is the set of real matrices of order n × n, B ∈ L(R m , R n ) is the matrix of the input, C ∈ L(R n , R p ) is the matrix of the output, x i ∈ R n is the state variable, B j ∈ L(R n , R n ) for all j ∈ {1, 2, ..., q}, N, is the set of nonnegative integers, n, m, p and q are the nonnegative integer and u j i , v i are the feedback controls defined by

with L ∈ L(R p , R m ) and K j ∈ L(R p , R), j ∈ {1, ..., q} are matrices with single row, and are given by

Our purpose right here is to determine all vectors x 0 (the initial states) such that the corresponding signal output (y i ) i verify the condition y i ∈ Ω, ∀ i ∈ N, where Ω ⊂ R p is a given constraint set. The set of all such vectors is the maximal output admissible set Θ(Ω).

For technical consideration we restrict our own selves in the current work to the study of Θ λ (Ω), the maximal output λ -admissible set, which defined by

where B(0, λ ) is the ball of center 0 and radius λ > 0, this means we study the set of all vectors x 0 ∈ R n such that x 0 ≤ λ and y i ∈ Ω, ∀ i ∈ N, where λ is any positive number.

Moreover, we give some useful properties of Θ λ (Ω), and we investigate its determination by way of a finite number of functional inequalities.

The rest of this paper is structured as follows: In Section 2, we present some preliminary results. In Section 3, we characterize the maximal output λ -admissible set. Moreover, in Section 4 we give some sufficient conditions to ensure the finite determination of the maximal output λadmissible set. In the following section, we propose an algorithm for determining the output admissibility index and, consequently the set Θ λ (Ω). Various numerical examples to illustrate our results are given in Section 6. An application to a controlled SI epidemic model and SIR model is given, to demonstrate the effectiveness of our approach in Section 7. The last section includes conclusion.

In the next section, we will give some necessary mathematical preliminaries that will be used in this paper.

In this section we consider the discrete-time controlled bilinear systems described by (6)- (7) .

We replace u j i and v i by its values in system (6) we obtain

Notations: In what follows we will denote the matrix A + BLC byÃ, K j C byK j and {i, i + 1, ..., k}, i ≤ k, the finite subset of N by σ k i . Remark 1: In case q = 1 the system (11) becomes

On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems 3 where δ = K 1 C ∈ L(R n , R).

Hence and by taking into account the previous notations, we consider in this paper the class of discrete systems defined by

increased by the output

and we are interested in the theoretical and numerical characterization of the set

Indeed, let ψ be the function defined as follows:

Let K be the matrix defined by

Definition 1: Let Ω ⊂ R p and x 0 ∈ R n . The initial state x 0 is said to be Ω−output admissible if

The set of all such initial states is defined by

The general solution of system (6) is given by

where the function Φ is defined by

and

Using (18) the set Θ(Ω) could be rewritten as follows:

In this paper we restrict our-selves to the study of the set Θ λ (Ω) defined by

The restriction to the set Θ λ (Ω) will not reduce the value of the work and this for the following reason. Having an initial state x 0 ∈ R n , we would like to know if x 0 is Ω−output admissible or not. In order to give an answer to such a question, we first identify the set Θ λ (Ω) = Θ(Ω) ∩ B(0, λ ) where λ is a real that verifies x 0 ≤ λ and we check if x 0 ∈ Θ λ (Ω) or not. Remark 2: 1) The general solution of system (6) is given by

2) ∀x, y ∈ R n ,

where denotes the transpose.

3) For x 0 ∈ B(0, λ ), we have

where F is defined by

with

Now, we give some conditions which are sufficient to ensure, for convenient initial states x 0 , the asymptotic stability of system (6) . Our main result in this direction is the following.

Proposition 1: Suppose the following hypothesis to hold.

1) There exists τ ≥ 1 and θ ∈ ]0, 1[ such that à k ≤ τθ k , ∀k ∈ N ( the matrix L can be chosen such that the hypothesis (1) holds ), 2) θ +C K λ τ 2 < 1, where

Then the system (6) is asymptoticaly stable in the ball of center 0 and radius λ , i.e.,

Proof: Let x 0 ∈ B(0, λ ). Then the general solution of system (6) can be written as

On the other hand, we have

we have used Cauchy-Schwarz inequality. This leads to

Then

Since z 0 ≤ τ, we prove that

Indeed, if we assume that

then using (31) it follows

Take

where α = τ z 0 and β = λC K τ 2 θ −1 .

From (34) we conclude

We have

which implies

If one replaces y i and Γ 1 by their values, we deduce from

Because, θ +C K λ τ 2 < 1 and z 0 ≤ 1 it follows

Using (39) for all i ≥ 0, we deduce that

This completes the proof.

The following result assumes that 0 ∈

• Ω (

• Ω denoted the interieur of Ω), this assumption is satisfied in any reasonable application and has nice consequences (see [13] ). In practice, the set Ω has the form

where s ∈ N and f i , i ∈ σ s 1 are continuous functions such that f i (0) ≤ 0, ∀i ∈ σ s 1 , such a sets have many importance in a practical view.

Remark 3: The condition taken above f i (0) ≤ 0, ∀i ∈ σ s 1 implies 0 ∈ Θ λ (Ω) and then the set Θ λ (Ω) nonempty. Indeed, the set Θ λ (Ω) is given by

On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems

and we have

Imposing special conditions onÃ,K j , B j , j ∈ {1, ..., q} and Ω which imposes corresponding conditions on Θ λ (Ω).

Proposition 2: 1) If Ω is closed, then the set Θ λ (Ω) is also closed. 2) If we assume the following assumptions to hold:

Then we define the functions

Using these functions, the set Θ λ (Ω) can be written as

Since Ω is closed and Ψ i , i ≥ 0 are continuous func-

is closed and then Θ λ (Ω) is closed. 2) By hypothesis (a), (b), and (c), and from Proposition 1 we have

This leads to

i.e.,

On the other hand

For ε = η C , we have

It remains to show that

We have also

If we choose this time ζ = inf(λ , ς ), it follows

Thus B(0, ζ ) ⊂ Θ λ (Ω), and consequently 0 ∈ • Θ λ (Ω). This completes the proof.

It is more difficult to characterize the set Θ λ (Ω) defined in (19) , for that reason we define for each k ∈ N the set

and we introduce the following definition. Definition 2: The set Θ λ (Ω) is said to be finitely determined if there exists an integer k 0 such that Θ λ (Ω) = Θ λ k 0 (Ω). Remark 4: 1) For r, s ∈ N such that r ≤ s, we have

2) If Θ λ (Ω) is finitely determined, and k 0 the smallest k such that Θ λ k (Ω) = Θ λ k+1 (Ω), then we have

Conditions that demonstrate finite determinability, are examined in the following proposition.

Proof: 1) Assume that the set Θ λ (Ω) is finitely determined. Then by Definition 2 it follows

We deduce from (57) and (58) that

Therefore

This leads

it follows

By iteration

Using Remark 4, we conclude

which completes the proof.

The following two theorems are the main results that give simple conditions to guarantee the finite determination of the set Θ λ (Ω).

Theorem 1: Assume the following hypothesis to hold

Then, Θ λ (Ω) is finitely determined.

Theorem 2: If we assume that

Then, Θ λ (Ω) is finitely determined.

In this section we note k * 0 the smallest integer such that

we call k * 0 the admissibility index. We shall provide a useful algorithm (see [13] ) for identifying this admissibility index k * 0 and therefore the characterization of the set Θ λ (Ω) .

Remark 5: 1) Algorithm 1 produce k * 0 and Θ λ (Ω) if and only if Θ λ (Ω) is finitely determined. 2) If Algorithm 1 does not converge then the set Θ λ (Ω)

is not finitely determined.

Algorithm 1 is not practical due to the fact it does not describe how the test Θ λ k (Ω) = Θ λ k+1 (Ω) is implemented. To deal with this difficulty we will consider the following approach.

Let Ω be defined as

(67)

On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems

Thus, for all k ∈ N, the set Θ λ k (Ω) can be written as

On the other hand,

Therefore, the test Θ λ k (Ω) = Θ λ k+1 (Ω) leads to a set of mathematical programming problems. We will suggest another version of Algorithm 1, this new algorithm is given by Algorithm 2.

Remark 6: If the matrixà is Lyapunov stable then the supremum in Algorithm 2 is defined for all x ∈ R n . Indeed, using the notation defined in Proposition 1 it follows

On the other hand, the assumption of Lyapunov stability of A implies CÃx ≤ γ x for some positive γ. This leads

Let k ∈ N and define the function F by

Then

and

using the compactness of B and the continuity of the functions f i , the sequence f i CΦ k+1 (x) is bounded from above. Remark 7: 1) This algorithm can not be effectively used if there is not a feasible way to find global optima.

When Ω is a polyhedron (i.e., the functions f j are affine for all j), the difficulty becomes less. 2) Assumptions of our two previous results (Theorem 1 and 2 of Section 4) are sufficient but not necessary. If these assumptions are not established, then the convergence of Algorithm 2 is not guaranteed.

To illustrate our results, we should give several examples, which will be given in the upcoming section.

In case k = 0 we have

If max f i (CΦ(x)) ≤ 0, ∀i ∈ {1, . . . , s} then we stop and k * 0 = 0 else we continue.

In case k = 1 we have

Youssef Benfatah, Amine El Bhih, Mostafa Rachik, and Abdessamad Tridane

If max f i (CΦ 2 (x)) ≤ 0, ∀i ∈ {1, . . ., s} then we stop and k * 0 = 1 else we continue. For example in step k = k * we have for i ∈ {1, . . ., s}

If max f i (CΦ k * +1 (x)) ≤ 0, ∀i ∈ {1, . . . , s} then we stop and k * 0 = k * else we continue. We can use a predefined output function to be called at each iteration.

Remark 8: Let us suppose that the cost of function of maximization is Cost max , the cost of repeat loop is Cost Repeat and the cost associated to the computation of CΦ i , i ∈ {0, 1, . . ., Cost Repeat } is Cost f ct . Then the computational complexity of Algorithm 2 is given by

To illustrate our results, we present some examples in two and three dimensional cases (n = 2 and n = 3). For this reason, we construct the matrices K j , j ∈ σ q 1 (in the case when p = 1) by the following way

where the choice of the number Nb > 1 depend on the value of θ which satisfies the first hypothesis of Proposition 1. This choice of K j guarantee the verification of the second hypothesis of Proposition 1. The matrixà is chosen such that à < 1. Even if A > 1, this choice is possible by selecting some matrix L sinceà = A + BLC.

Using these matrices, we guarantee the asymptotic stability of the considered systems. Assumption Φ(B(0, λ )) ⊂ B(0, λ ) is also taken into consideration. In all examples the dotted region will denote the set Θ λ (Ω). For Example 4, we have k * 0 = ∞, which means that Algorithm 2 does not converge. Various selections of matrices determining the systems are considered.

In Examples 1-7, we construct Ω as follows:

clearly 0 ∈

• Ω, f i , i ∈ σ s 1 are continuous functions such that f i (0) ≤ 0, ∀i ∈ σ s 1 . Example 1: Let λ , q and n defined by

Let the matrixÃ, B 1 , C andK 1 defined bỹ A = x + 1 6 y,

Using Algorithm 2 with ε = 0.2 we get k * 0 = 0

and using the same algorithm with ε = 0.1 we get k * 0 = 1

On the Maximal Output Admissible Set for a Class of Bilinear Discrete-time Systems Using Algorithm 2 we obtain k * 0 = 1, and then the set Θ λ (Ω)

Example 3: Let λ , q, n and ε be defined as λ = 2, q = 2, n = 2 and ε = 0.1.

Let the matrixÃ, C ,K 1 ,K 2 , B 1 and B 2 be defined as

y.

Using Algorithm 2 we obtain k * 0 = 2, and then the set Θ λ (Ω) Then Φ x y = Using Algorithm 2 we get k * 0 = ∞. Using Algorithm 2 we get k * 0 = 1

Example 6: LetÃ, B 1 ,K 1 , C, q, ε and λ defined as Using Algorithm 2 we obtain k * 0 = 2, and then 

Example 7: To show the efficiency of the stability we take the following example.

Let λ , q, n, ε and τ defined as Let B 1 ,Ã,K 1 , C, C K and θ defined by 1.2 ∞.

In Examples 1-7, we have determined the index of admissibility k * 0 using the predefined function fmincon in Matlab [28] , which allow us to solve the maximization problems in Algorithm 2. In Figs. 1-5, we have plotted all the constraints forming the sets Θ λ (Ω), in order to visualize such sets of Examples 1, 2, 3, and 5. Example 8: In this example we take Ω in the form

It is easy to see that 0 ∈

• Ω, f is continuous function such that f (0) ≤ 0 and that Ω is closed since Ω = f −1 (] − ∞, 0]). Now, we use the results of Example 3 and Algorithm 2 it follows

the solution of this problem gives 1.7500.

Maximize

the solution of this problem gives: −1.2000e −06 . Therefore, the value of index of admissiblity is k * 0 = 1. Hence

Example 9: In this example the set Ω is represented by

It is easy to see that f i (0) ≤ 0, ∀i ∈ {1, 2} and the functions f i , i = 1, 2 are continuous.

Using results of Example 5 and Algorithm 2 we obtain k * 0 = 0, and then the maximal output λ -admissible set Θ λ (Ω) is given by

In Examples 8 and 9, we have modified the set Ω which affects the mathematical programming problems that arise in Algorithm 2, we have plotted again the sets Θ λ (Ω) in Figs. 6 and 7.

In order to obtain the numerical computational (elapsed time in seconds (s)) of Examples 1, 2, 3, 5, and 6 using Algorithm 2 the predefined output function fmincon (see [28] ) is applied with different values for the coefficient λ , ε, k * 0 , n to solve the maximization problems which arise in such algorithm. The obtained results are presented in Table 1 .

In case p ≥ 2 we have to reconstruct the matrices K j such that the condition θ + λC K τ 2 < 1 of Proposition 1 is verified. 

This section aims to apply the approach developed in this paper to epidemiological and sociological models that includes control terms.

We consider a deterministic model of susceptible-infected (SI) , where S i represents the number of individuals susceptible to infection of the disease but not yet infected at the time i. I i denotes the number of individuals who have been infected with the disease, and they are infectious. These types of models are used to model many infectious diseases such as HIV. The model is based on the assumption that disease transmission happens as a result of an "effective" contact between a person who is susceptible and an infectious person. Because of the nonlinear contact dynamics within the population, the αSI incidence function is used to reflect successful disease transmission, where α is the rate of effective contact with infected individuals in compartment I.

The treatment and awareness campaigns are well known as efficient strategies to control and prevent the spread of such infectious diseases. Consequently, we introduce in the following system with two controls u i and v i . u i : refers to the effort to protect individuals who are susceptible to becoming infected. This effort to provide awareness.

v i : refers to the effort to provide treatment.

Therefore, we use the following model:

The corresponding output is

( y i can be the cost of treatment). Take x i = S i I i and u i , v i feedback controls defined by

We assume that our control depends on the level of endemicity in the population (given the observation times k andk) and the available resource to control the spread of the disease (m andm, respectively). In fact, the public health authorities are all-time facing limited resources to contain the infection. Therefore, our control is a function of the observation y i , which gives the decision-maker an optimal way to treat and vaccinate the population. Then, the system (77) can be rewritten as follows:

or equivalently

This model can be written as

wherẽ

Now we selectK 3 andK 4 as follows:

and the parameter α is taken as

The second hypothesis of Proposition 1 is verified. Indeed, θ +C K λ τ 2 = 0.3 + 19 12 λ τ 2 = 0.99271 < 1, with θ = Ã = 0.3, λ = 42 96 , τ = 1 (m =m = 0.7). Therefore, Cost max is the number of operations (×and+) applying the predefined output function fmincon from the Matlab Optimization Toolboxe [28] to solve the maximization problems which arise in each iteration. Thus, k * 0 = 1 and the set Θ λ (Ω) is given by 9 10 x + 1 2 y ≤ 0.03, 0.27x + 0.15y − 11 30 xy − 9 40 x 2 ≤ 0.03

The plot of the set Θ λ (Ω) is given in Fig. 8 .

Our next application is to consider a SIR type model of rumor [29] , where the population is divided into three categories: ignorants (people who never heard the rumor), spreaders (people who have heard the rumor and transmitted it to others), and stiflers (having heard the rumor but have not passed it to others). The variables S i , I i , and R i to denote the population densities of ignorant users, spreading users, and stifler users, respectively. This model's dynamic is based on the following assumption: Rumor transmission happens as a result of adequate contact between a person who is ignorant and a person who is a spreader. The ignorant person accepts the rumor and becoming a spreading at a rate β . We assume that the propagation of a constant rumor in a population with a constant emigration rate µ.

Our model is described by

where u i represent the awareness and refers to the effort to protect individuals who are ignorants against becoming spreaders. The input u i is taken as feedback of y i , i.e.,

where y i is the corresponding output, i.e.,

Using the same approach developed before it follows 

We choose λ , µ, β and k as s.c |y| ≤ 0.9, |x| ≤ 1, |y| ≤ 1, since J * 1 , J * 2 ≤ 0, we stop. The computational complexity of applying this algorithm (see Remark 8 ) is described as follows:

Cost algo = (s ×Cost max + 2s) ×Cost Repeat +Cost f ct ,

where Cost Repeat = 1 and s = 2.

Therefore k * 0 = 0 and Θ λ (Ω) is given by

The plot of the set Θ λ (Ω) is given in Fig. 9 . In order to obtain the numerical computational (elapsed time in seconds (s)) of two applications (Apps) using Algorithm 2, the predefined output function fmincon (see [28] ) is applied with different values for the coefficient λ , ε, k * 0 , n to solve the maximization problems which arise in such algorithm. The obtained results are presented in Table 2 . Remark 10: The computational complexity of Algorithm 2 depend on the following factors:

1) The maximization problems which occur in such algorithm, 2) the value of index of admissibility k * 0 , 3) the dimension n, 4) the parameters λ , ε and s.

In the following section, we will present a conclusion of our work.

This paper presented a new problem of maximal output admissible set for discrete-time bilinear controlled systems. First, we gave a new sufficient condition that assures the asymptotic stability of the system. Then, we characterized the maximal output λ -admissible set Θ λ (Ω), and we showed that, under certain specific conditions, the finite determinations of Θ λ (Ω). Moreover, we proposed an algorithmic approach to determine the admissibility index k * 0 and, consequently, the set Θ λ (Ω). Our approach was illustrated with several numerical examples with different dimensions and parameters. In these examples, we were able to determine the maximal output and to sketch it in the 2 and 3 dimensions.

To give more applications to our approach, we applied our technique to an SI epidemic model. Our goal was to find the initial data that guarantees the control of the spread of the epidemic via awareness and treatment. In addition, we also used our approach of a rumor spread model. To contain the spread of a rumor, the control term aimed to increase the awareness of the population. In the future and inspired by the work of Larrache et al. [30] , we investigate the problem of the maximal output set for discrete-time bilinear distributed systems. using (A.1) we conclude that z 0 ∈ Θ λ i 0 (Ω). Thus Θ λ i 0 −1 (Ω) = Θ λ i 0 (Ω) and Θ λ (Ω) is finitely determined (using this time hypothesis (iv) and Proposistion 3).

It is easy to verify that Φ i (z 0 ) ≤ ν i z 0 , ∀i ∈ N. From (A.5) we have CΦ i 0 (z 0 ) ∈ B(0, η) ⊂ Ω. Therefore z 0 ∈ Θ λ i 0 (Ω).

The authors declare that they have no conflicts of interest.

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Youssef Benfatah is a Ph.D. student. He received his M.S. degree in applied mathematics in 2017 from Faculty of Sciences and Techniques of Settat, Hassan I University, Morocco. Since 2019, he is a member of the Analysis, Modeling and Simulation Laboratory at Faculty of Sciences Ben M'sik, University of Hassan II. His research interests include optimal control, dynamical systems, mathematical modeling and distributed systems.Amine El Bhih is a Ph.D. student. He received his M.S. degree in applied mathematics in 2017 from Faculty of Sciences Ben Mśik, Hassan II University, Casablanca, Morocco. Since 2019, he is a member of the Analysis, Modeling and Simulation Laboratory at the same faculty. His research interests include optimal control, dynamical systems, mathematical modeling distributed systems.Mostafa Rachik is a Professor of Mathematics and Computer Science at Faculty of Sciences Ben Mśik, Casablanca (Morocco). He received his Ph.D. degree in control systems. Professor Rachik wrote many papers in the area of systems analysis and control. Now he is the Head of the research team LAMS (Analysis, Modeling and Simulation Laboratory) at the same faculty. His main research interests are dynamical systems, mathematical modelling, stochastic epidemic systems, robotics, optimization, distributed systems, optimal control, and its applications.Abdessamad Tridane is an Associate Professor in the Department of Mathematical Sciences at United Arab Emirates University, Al Ain, UAE. He received his Ph.D. degree in control systems theory. Dr. Tridane wrote many papers in the area of control systems, mathematical epidemiology and mathematical medicine. His main research interests are dynamical systems, control systems, mathematical epidemiology and medicine.