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

http://dx.doi.org/10.1007/s12555-020-0486-6 http://www.springer.com/12555

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

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

Abstract:Given a discrete-time controlled bilinear systems with initial statex0and output functionyi, we investi- gate the maximal output setΘ(Ω) ={x₀∈Rⁿ,y_i∈Ω,∀i≥0}whereΩis a given constraint set and is a subset ofR^p. Using some stability hypothesis, we show thatΘ(Ω)can be determined via a finite number of inequations.

Also, we give an algorithmic process to generate the setΘ(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.

Keywords:Asymptotic stability, bilinear systems, constraint set, discrete-time systems, output admissible set.

1. INTRODUCTION

Output admissible sets arise in many essential practi- cal applications such as stability analysis and control of constrained systems (see [1–6]). This concept has been considerably investigated (see [7,8] and the references therein), but not in the case of the bilinear systems. Con- sequently, 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 multi- plication 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 finite- dimensional system can be represented by the following dynamic equation:

xi+1= f(xi,ui), (1)

wherexi∈Rⁿandui∈R^q, fori≥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.

xi+1=h1(u_i)x_i+f1(u_i), (2)

and

xi+1=h2(x_i)u_i+f2(x_i), (3) we obtain an another form of bilinear systems given as follows:

xi+1=Axi+Bx˜ iui+Bui, (4) where for each i≥0, A is a n×n matrix,B is a n×q matrix. ˜Bxiuiis a bilinear form in thexi,uivariables that can be expressed as

Bx˜ iui=

q

∑

j=1

u_i^jBjxi, (5)

whereui={u_i^j}^q_j=1andBjis an×nmatrix.

The use of bilinear systems has two significant advan- tages: 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.

1.1. Related work

The maximal output admissible (MOA) sets are an important concept for analyzing controlled systems with constraints. The MOA set has been well studied particu- larly 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].

Manuscript received July 13, 2020; revised November 27, 2020 and January 6, 2021; accepted January, 31, 2021. Recommended by Associate Editor Ohmin Kwon under the direction of Editor Jessie (Ju H.) Park. The authors would like to thank the reviewer for his time to help improve this paper. Research reported in this paper was supported by the Moroccan Systems Theory Network.

Youssef Benfatah, Amine El Bhih, and Mostafa Rachik are with the Laboratory of Analysis Modeling and Simulation,Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University, Casablanca, Sidi Othman, BP 7955 (e-mails:

{youssef.benfatah, elbhihamine}@gmail.com, m-rachik@yahoo.fr). Abdessamad Tridane is with the Department of Mathematical Sci- ences,United Arab Emirates University, Al Ain P.O. Box 15551, UAE (e-mail: a-tridane@uaeu.ac.ae).

* Corresponding author.

ICROS, KIEE and Springer 2021c

The MOA set for a class of nonlinear systems has re- cently been studied by Rachiket al. in [16]. Later on, Hi- rata 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 ref- erences [18–24]. In [18], Ossareh considered a Lyapunov- stable periodic system and gave the development of the MOA set theory. He also investigated some of its geomet- ric 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 discrete- time 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].

1.2. Problem statement

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 specif- ically, we characterize the initial states of a controlled bi- linear system whose resulting trajectory verify a point- wise constraint. Moreover, our approach is applicable to epidemiological, sociological, and physiological models.

In these examples, the MOA set represents the set of ini- tial 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:







xi+1=Axi+

q

∑

j=1

u_i^jBjxi+Bvi, i≥0, x₀∈Rⁿ,

(6)

the corresponding output is

yi=Cxi, i∈N, (7)

whereA∈ L(Rⁿ,Rⁿ)is the matrix of the state,R,is the set of real numbers,L(Rⁿ,Rⁿ),is the set of real matrices of ordern×n,B∈ L(R^m,Rⁿ)is the matrix of the input, C∈ L(Rⁿ,R^p)is the matrix of the output,xi∈Rⁿ is the state variable,Bj∈ L(Rⁿ,Rⁿ)for all j∈ {1, 2, ...,q},N, is the set of nonnegative integers,n,m, p andq are the nonnegative integer andu_i^j,vi are the feedback controls defined by

u_i^j=K^jyi and vi=Lyi (8)

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

K^j=h

k₁^j k₂^j k₃^j . . . k_p−1^j kp^j

i

. (9)

Our purpose right here is to determine all vectors x0

(the initial states) such that the corresponding signal out- put(y_i)_iverify the conditiony_i∈Ω,∀i∈N, whereΩ⊂R^p is a given constraint set. The set of all such vectors is the maximal output admissible setΘ(Ω).

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

Θ^λ(Ω) =Θ(Ω)∩B(0,λ), (10) whereB(0,λ)is the ball of center 0 and radiusλ>0, this means we study the set of all vectors x0∈Rⁿ such that kx0k ≤λ andy_i∈Ω,∀ i∈N, whereλ 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 outputλ-admissible set. More- over, in Section 4 we give some sufficient conditions to ensure the finite determination of the maximal outputλ- admissible set. In the following section, we propose an algorithm for determining the output admissibility index and, consequently the set Θ^λ(Ω). Various numerical ex- amples 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 ap- proach in Section 7. The last section includes conclusion.

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

2. PRELIMINARY RESULTS

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

We replaceu_i^jandviby its values in system (6) we ob- tain







xi+1= (A+BLC)xi+

q

∑

j=1

K^jCxiBjxi, i≥0, x0∈Rⁿ.

(11) Notations: In what follows we will denote the matrix A+BLCby ˜A,K^jCby ˜K^j and{i,i+1, ...,k},i≤k, the finite subset ofNbyσ_i^k.

Remark 1: In caseq=1 the system (11) becomes (xi+1=Ax˜ i+δxiB1xi, i≥0

x0∈Rⁿ

(12)

whereδ=K¹C∈ L(Rⁿ,R).

Hence and by taking into account the previous nota- tions, we consider in this paper the class of discrete sys- tems defined by







xi+1=Ax˜ _i+

q

∑

j=1

K˜^jx_iB_jx_i, i≥0, x0∈Rⁿ,

(13)

increased by the output

y_i=Cx_i, ∀i≥0, (14)

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

Θ^λ(Ω) ={x₀∈B(0,λ)∩Rⁿ|yi∈Ω, ∀i∈N}. (15) Indeed, letψ be the function defined as follows:

ψ: Rⁿ→Rⁿ x 7→ψ(x) =

q

∑

j=1

K˜^jxBjx. (16) LetKbe the matrix defined by

K=











k¹₁ k¹₂ · · · k¹_p k²₁ k²₂ · · · k²_p ... ... . .. ... k^q₁ k^q₂ · · · k^qp











. (17)

Definition 1: LetΩ⊂R^pandx0∈Rⁿ. The initial state x0is said to beΩ−output admissible if

yi∈Ω, ∀i∈N.

The set of all such initial states is defined by Θ(Ω) ={x₀∈Rⁿ|yi∈Ω, ∀i∈N}. The general solution of system (6) is given by

xi=Φⁱ(x₀), ∀i∈N, (18) where the functionΦis defined by

Φ: Rⁿ→Rⁿ

x 7→Φ(x) =Ax˜ +ψ(x), and

Φⁱ=

i−times

z }| { Φ◦Φ◦...◦Φ.

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

Θ(Ω) =

x0∈Rⁿ|CΦⁱ(x₀)∈Ω, ∀i∈N .

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

Θ^λ(Ω) =Θ(Ω)∩B(0,λ)

={x0∈B(0,λ)∩Rⁿ|CΦⁱ(x0)∈Ω,∀i∈N}.

(19) The restriction to the setΘ^λ(Ω)will not reduce the value of the work and this for the following reason. Having an initial state x0∈Rⁿ, we would like to know if x0 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)whereλ is a real that verifies kx0k ≤λ and we check ifx0∈Θ^λ(Ω)or not.

Remark 2: 1) The general solution of system (6) is given by

x_i=

i−1

∏

k=0

A˜+

q

∑

j=1

u_k^jB_j

!

x0,∀i≥1. (20) 2) ∀x,y∈Rⁿ,

kψ(x)−ψ(y)k

≤ ( _q

∑

j=1

k(K˜^j)^>kkB_jk(kxk+kyk) )

× kx−yk, (21) where>denotes the transpose.

3) Forx0∈B(0,λ), we have

kx_ik ≤Fⁱ(λ), ∀i∈N (22) whereFis defined by

F(t) =αt+C_Kt², ∀t∈R, (23) with

α=kAk˜ and C_K=

q

∑

j=1

k(K˜^j)^>kkB_jk.

Now, we give some conditions which are sufficient to ensure, for convenient initial statesx0,the asymptotic sta- bility 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 kAk˜ ^k ≤ τ θ^k,∀k∈N( the matrixLcan be chosen such that the hypothesis (1) holds ),

2) θ+C_Kλ τ²<1,where CK=

q

∑

j=1

k(K˜^j)^>kkBjk. (24)

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

i→∞limkx_ik=lim

i→∞kΦⁱ(x0)k=0,∀x0∈B(0,λ). (25) Proof: Letx0∈B(0,λ). Then the general solution of system (6) can be written as

xi=A˜ⁱx0+

i−1

∑

k=0

A˜^i−k−1ψ(xk), ∀i≥1 (26)

=⇒ kx_ik ≤ kA˜ⁱkkx0k+

i−1 k=0

∑

kA˜^i−k−1kkψ(x_k)k

=⇒ kx_ik ≤τ θⁱkx₀k+

i−1

∑

k=0

τ θ^i−k−1kψ(x_k)k.

On the other hand, we have kψ(xk)k=k

q

∑

j=1

K˜^jxkBjxkk (27)

≤

q

∑

j=1

k(K˜^j)^>kkxkkkBjkkxkk (28)

=

q

∑

j=1

k(K˜^j)^>kkB_jk

!

kx_kk² (29)

=C_Kkx_kk², (30) we have used Cauchy-Schwarz inequality. This leads to

kx_ik ≤τ θⁱkx₀k +C_Kτ θⁱ⁻¹

i−1 k=0

∑

θ^−kkx_kk², ∀i≥1.

Letz_i=^xi

λ. Then kzik ≤τ θⁱkz0k

+λCKτ θⁱ⁻¹

i−1

∑

k=0

θ^−kkzkkkzkk, ∀i≥1. (31) Sincekz0k ≤τ, we prove that

kz_ik ≤τ, ∀i∈N. (32) Indeed, if we assume that

kz_ik ≤τ, ∀i∈σ₀^N, (33) then using (31) it follows

kz_ik ≤τ θⁱkz₀k +λCKτ²θⁱ⁻¹

i−1

∑

k=0

θ^−kkz_kk, ∀i∈σ₁^N+1. (34) Take

yi=θ⁻ⁱkzik and Γi=α+β

i−1

∑

k=0

yk, (35)

where

α=τkz0k and β=λCKτ²θ⁻¹. (36) From (34) we conclude

yi≤Γi, ∀i∈σ₁^N+1. (37) We haveyi=^Γi+1−Γi

β . Then Γi+1−Γi

β ≤Γi, ∀i∈σ₁^N+1, which implies

yi≤Γi≤(1+β)ⁱ⁻¹Γ1, ∀i∈σ₁^N+1. (38) If one replacesyiandΓ1by their values, we deduce from (38) that∀i∈σ₁^N+1,

kzik ≤ θ+C_Kλ τ²i

τ+C_Kλ τ²θ⁻¹

1+CKλ τ²θ⁻¹ kz0k. (39) Because,θ+CKλ τ²<1 andkz0k ≤1 it follows

kzik ≤τ, ∀i∈σ₁^N+1, and

kzik ≤τ, ∀i∈N. (40) Using (39) for alli≥0, we deduce that

i→∞limkz_ik=0 sinceθ+C_Kλ τ²≤1.

Therefore

i→∞limkΦⁱ(x₀)k=0. (41)

This completes the proof.

The following result assumes that 0∈Ω^◦ (

◦

Ωdenoted the interieur ofΩ), this assumption is satisfied in any reason- able application and has nice consequences (see [13] ). In practice, the setΩhas the form

Ω={y∈R^p| f1(y)≤0, f2(y)≤0, . . . , fs(y)≤0}, (42) wheres∈Nand fi,i∈σ₁^sare continuous functions such that fi(0)≤0,∀i∈σ₁^s, such a sets have many importance in a practical view.

Remark 3: The condition taken above fi(0)≤0,∀i∈ σ₁^simplies 0∈Θ^λ(Ω)and then the setΘ^λ(Ω)nonempty.

Indeed, the setΘ^λ(Ω)is given by Θ^λ(Ω)

={x0∈B(0,λ)∩Rⁿ|CΦⁱ(x0)∈ Ω, ∀i∈N}

={x0∈B(0,λ)|fj(CΦⁱ(x0))≤0, ∀j∈σ₁^s,∀i∈N}, and we have

fj(CΦⁱ(0)) =fj(0)≤0, ∀j∈σ₁^s,∀i∈N, (43) sinceΦ(0) =0.

Hence 0∈Θ^λ(Ω)because 0∈B(0,λ).

Imposing special conditions on ˜A, ˜K^j,Bj, j∈ {1, ..., q} and Ω which imposes corresponding conditions on Θ^λ(Ω).

Proposition 2: 1) IfΩis closed, then the setΘ^λ(Ω)is also closed.

2) If we assume the following assumptions to hold:

(a) there existsτ≥1 andθ∈]0,1[such thatkAk˜ ^k≤ τ θ^k,∀k∈N,

(b) θ+C_Kλ τ²≤1,C_K=

q

∑

j=1

k(K˜^j)^>kkB_jk, (c) 0_R^p∈Ω,^◦

then 0_Rⁿ∈

◦

Θ\^λ(Ω).

Proof: 1) Leti∈N. Then we define the functions Ψi:B(0,λ)∩Rⁿ→R^p, x07→CΦⁱ(x₀). (44) Using these functions, the setΘ^λ(Ω)can be written as

Θ^λ(Ω) =^\

i∈N

Ψ⁻¹_i (Ω). (45) SinceΩis closed andΨi,i≥0 are continuous func- tions (sinceψ(x) =

q

∑

j=1

K^jxB_jxis continuous),Ψ⁻¹_i (Ω) is closed and thenΘ^λ(Ω)is closed.

2) By hypothesis (a), (b), and (c), and from Proposition 1 we have

∀z₀∈B(0,λ), lim

i→∞kΦⁱ(z₀)k=0. (46)

This leads to

∀z0∈B(0,λ), ∀ε>0, ∃i0∈N: kΦⁱ(z₀)k ≤ε, ∀i≥i0. i.e.,

∀z0∈B(0,λ), ∀ε>0, ∃i0∈N:

Φⁱ(z0)∈B(0,ε), ∀i≥i0. On the other hand

0∈Ω^◦ =⇒ ∃η>0 : B(0,η)⊂Ω. (47) Forε=_kCk^η , we have

∀z0∈B(0,λ), ∃i0∈N

such that

CΦⁱ(z₀)∈B(0,η)⊂Ω,∀i≥i₀. (48) It remains to show that

CΦⁱ(z0)∈Ω, ∀i∈σ₀ⁱ⁰⁻¹. (49) Φⁱ⁰⁻¹continuous in 0 implies

∃ηi₀−1>0, ∀z0∈B(0,ηi₀−1),

kΦⁱ⁰⁻¹(z₀)−Φⁱ⁰⁻¹(0)k ≤ η kCk

=⇒ ∃ηi₀−1>0, ∀z0∈B(0,ηi₀−1), CΦⁱ⁰⁻¹(z₀)∈Ω.

Φⁱ⁰⁻²continuous in 0 implies

∃ηi0−2>0, ∀z0∈B(0,ηi0−2), CΦⁱ⁰⁻²(z₀)∈Ω by a similar way, using the continuity ofΦ^j, j∈σ₁ⁱ⁰⁻³ in 0, we obtain

∀j∈σ₁ⁱ⁰⁻³,

∃ηj>0,∀z0∈B(0,ηj),CΦ^j(z0)∈Ω. (50) We have also

∀z0∈B

0, η kCk

,Cz0∈Ω. (51)

If we takeς=inf(η_i0−1, ...,η1,_kCk^η ), we get

∀z0∈B(0,ς),CΦⁱ(z0)∈Ω, ∀i∈σ₀ⁱ⁰⁻¹. (52) If we choose this timeζ =inf(λ,ς), it follows

∀z0∈B(0,ζ),CΦⁱ(z0)∈Ω, ∀i∈N. (53) ThusB(0,ζ)⊂Θ^λ(Ω), and consequently 0∈

◦

Θ\^λ(Ω).

This completes the proof.

3. CHARACTERIZATION OF THE MAXIMAL OUTPUTλ-ADMISSIBLE SET

It is more difficult to characterize the setΘ^λ(Ω)defined in (19), for that reason we define for eachk∈Nthe set

Θ^λ_k(Ω) ={x0∈B(0,λ)∩Rⁿ:CΦⁱ(x0)∈Ω,∀i∈σ₀^k}, (54) and we introduce the following definition.

Definition 2: The setΘ^λ(Ω)is said to be finitely de- termined if there exists an integerk0such thatΘ^λ(Ω) = Θ^λ_k

0(Ω).

Remark 4: 1) Forr,s∈Nsuch thatr≤s, we have Θ^λ(Ω)⊂Θ^λ_s(Ω)⊂Θ^λ_r(Ω). (55)

2) IfΘ^λ(Ω)is finitely determined, andk0the smallestk such thatΘ^λ_k(Ω) =Θ^λ_k+1(Ω), then we have

Θ^λ(Ω) =Θ^λ_k0(Ω) =Θ^λ_k(Ω),∀k≥k0. (56) Conditions that demonstrate finite determinability, are examined in the following proposition.

Proposition 3: 1) IfΘ^λ(Ω)is finitely determined then there existsk₀∈Nsuch thatΘ^λ_k0(Ω) =Θ^λ_k0+1(Ω).

2) IfΦ(B(0,λ))⊂B(0,λ)and Θ^λ_k0(Ω) =Θ^λ_k0+1(Ω)for somek0∈NthenΘ^λ(Ω)is finitely determined.

Proof: 1) Assume that the setΘ^λ(Ω)is finitely deter- mined. Then by Definition 2 it follows

∃k0∈Nsuch thatΘ^λ(Ω) =Θ^λ_k0(Ω). (57) Using Remark 4 we have Θ^λ_k0+1(Ω)⊂Θ^λ_k0(Ω)since k0+1≥k0. On the other hand,

Θ^λ(Ω)⊂Θ^λ_k0+1(Ω). (58) We deduce from (57) and (58) that

Θ^λ_k0(Ω)⊂Θ^λ_k0+1(Ω). (59) Therefore

Θ^λ_k0(Ω) =Θ^λ_k0+1(Ω) for some k0∈N. (60) 2) Letx0∈Θ^λ_k

0(Ω), thenx0∈Θ^λ_k

0+1(Ω).

This leads

x0∈B(0,λ), CΦⁱ(x0)∈Ω, ∀i∈σ₀^k0+1. (61) it follows

CΦⁱ(Φ(x₀))∈Ω, ∀i∈σ₀^k0. (62) ThusΦ(x₀)∈Θ^λ_k0(Ω)since

Φ(x₀)∈Φ(B(0,λ))⊂B(0,λ). (63) By iteration

x0∈Θ^λ_k0(Ω) =⇒Φ^j(x0)∈Θ^λ_k0(Ω),∀j∈N

=⇒CΦⁱ Φ^j(x₀)

∈Ω, ∀i∈σ₀^k0

=⇒CΦⁱ(x₀)∈Ω, ∀i∈N

=⇒x0∈Θ^λ(Ω) sincex0∈B(0,λ). Hence

Θ^λ_k0(Ω)⊂Θ^λ(Ω). (64) Using Remark 4, we conclude

Θ^λ_k0(Ω) =Θ^λ(Ω)for somek₀∈N, (65)

which completes the proof.

4. SUFFICIENT CONDITIONS FOR FINITE DETERMINATION OFΘ^λ(Ω)

The following two theorems are the main results that give simple conditions to guarantee the finite determina- tion of the setΘ^λ(Ω).

Theorem 1: Assume the following hypothesis to hold (i) There existsτ≥1 andθ∈]0,1[such thatkAk˜ ^k≤

τ θ^k,∀k∈N,

(ii) θ+C_Kλ τ²≤1,CK=

q

∑

j=1

k(K˜^j)^>kkB_jk, (iii) 0∈Ω,^◦

(iv) Φ(B(0,λ))⊂B(0,λ).

Then,Θ^λ(Ω)is finitely determined.

Theorem 2: If we assume that

(i) kΦ(z)k ≤νkzk,∀z∈Rⁿandν∈]0,1[. (ii) 0∈Ω.^◦

Then,Θ^λ(Ω)is finitely determined.

5. ALGORITHMIC DETERMINATION

In this section we notek₀^∗the smallest integer such that Θ^λ(Ω) =Θ^λ_k^∗

0(Ω), (66)

we callk₀^∗the admissibility index. We shall provide a use- ful algorithm (see [13]) for identifying this admissibil- ity index k₀^∗ and therefore the characterization of the set Θ^λ(Ω).

Remark 5: 1) Algorithm 1 producek^∗₀ andΘ^λ(Ω)if and only ifΘ^λ(Ω)is finitely determined.

2) If Algorithm 1 does not converge then the setΘ^λ(Ω) is not finitely determined.

Algorithm 1 is not practical due to the fact it does not describe how the testΘ^λ_k(Ω) =Θ^λ_k+1(Ω)is implemented.

To deal with this difficulty we will consider the following approach.

LetΩbe defined as

Ω={y∈R^p/f1(y)≤0,f2(y)≤0, . . . ,fs(y)≤0}. (67)

Algorithm 1:Determination ofk^∗₀.

Require: λ>0,n,p,q∈N^∗, ˜A,C,ψ,Ω⊂R^p k←0

if Θ^λ_k(Ω) =Θ^λ_k+1(Ω) then letk^∗₀←k and stop else

letk←k+1 and continue (return to previous test ) end if

Algorithm 2Determination ofk^∗₀

Require: λ >0,n,p,q,s∈N^∗,A,C,˜ ψ,fi,i=1, ...,s k←0

Repeat fori=1,...,sdo

MaximizeJi(x) =fi(CΦ^k+1(x)) Subjet to the constraints (fj CΦ^l(x)

≤0, x∈B(0,λ),

∀j∈ {1, . . . ,s}, ∀l∈ {0, . . . ,k}.

end for

J_i^∗←max{Ji(x)}

if J_i^∗≤0,∀i=1,2, . . . ,s then k^∗₀←k

break else

k←k+1 end if

Thus, for allk∈N, the setΘ^λ_k(Ω)can be written as Θ^λ_k(Ω) =

x∈B(0,λ)|f_j(CΦⁱ(x))≤0, j=1, . . . ,s, i=0, . . . ,k . (68) On the other hand,

Θ^λ_k+1(Ω) =

x∈Θ^λ_k(Ω,K)|fj CΦ^k+1(x)

≤0, j=1, . . . ,s . (69) Now, sinceΘ^λ_k+1(Ω)⊂Θ^λ_k(Ω)for allk∈N, we deduce

Θ^λ_k+1(Ω) =Θ^λ_k(Ω,K)

⇐⇒Θ^λ_k(Ω)⊂Θ^λ_k+1(Ω)

⇐⇒ ∀x∈Θ^λ_k(Ω),fj CΦ^k+1(x)

≤0,∀j∈ {1, . . . ,s}

⇐⇒ sup

x∈B(0,λ),fi(CΦ^l(x))≤0

∀i∈{1,...,s},∀l∈{0,...,k}

fj CΦ^k+1(x)

≤0,∀j∈ {1, . . . ,s}.

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 matrix ˜Ais Lyapunov stable then the supremum in Algorithm 2 is defined for allx∈Rⁿ. Indeed, using the notation defined in Proposition 1 it follows

kCΦ(x)k=

CAx˜ +C

q

∑

j=1

K˜^jxBx

(70)

≤ CAx˜

+kCkCKkxk². (71) On the other hand, the assumption of Lyapunov stability of A˜implies

CAx˜

≤γkxkfor some positiveγ. This leads kCΦ(x)k ≤(γ+kCkC_Kkxk)kxk. (72)

Letk∈Nand define the functionFby

F(α) = (γ+kCkCKα)α, ∀α∈R. (73) Then

CΦ^k+1(x)

≤F^k+1(kxk), (74) and

CΦ^k+1(x)∈B(0,F^k+1(kxk)), (75) using the compactness ofBand the continuity of the func- tions fi, the sequence fi 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 op- tima. WhenΩ is a polyhedron (i.e., the functions fj

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 conver- gence of Algorithm 2 is not guaranteed.

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

5.1. Algorithm 2 steps In casek=0 we have fori=1

maxf₁(CΦ(x)) s.c

(f1(Cx)≤0, . . . , fs(Cx)≤0,

|x1| ≤λ, . . . , |x_n| ≤λ, fori=2

maxf2(CΦ(x)) s.c

(f1(Cx)≤0, . . . , fs(Cx)≤0,

|x₁| ≤λ, . . . , |x_n| ≤λ, we continue like this untili=s

maxfs(CΦ(x)) s.c

(f1(Cx)≤0, . . . , fs(Cx)≤0,

|x1| ≤λ, . . . , |xn| ≤λ.

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

In casek=1 we have fori=1

maxf₁(CΦ²(x)) s.c







f1(Cx)≤0, . . . , fs(Cx)≤0, f₁(CΦ(x))≤0, . . . , f_s(CΦ(x))≤0,

|x1| ≤λ, . . . , |xn| ≤λ,

fori=2

maxf2(CΦ²(x)) s.c







f₁(Cx)≤0, . . . , f_s(Cx)≤0, f1(CΦ(x))≤0, . . . , fs(CΦ(x))≤0,

|x1| ≤λ, . . . , |xn| ≤λ, we continue untili=s

maxfs(CΦ²(x)) s.c







f₁(Cx)≤0, . . . , f_s(Cx)≤0, f1(CΦ(x))≤0, . . . , fs(CΦ(x))≤0,

|x1| ≤λ, . . . , |xn| ≤λ.

If maxfi(CΦ²(x))≤0,∀i∈ {1,. . .,s}then we stop and k^∗₀=1 else we continue.

For example in stepk=k^∗we have fori∈ {1,. . .,s}

maxfi(CΦ^k∗+1(x))

s.c



















f1(Cx)≤0, . . . , f_s(Cx)≤0, f1(CΦ(x))≤0, . . . , fs(CΦ(x))≤0,

...

f1 CΦ^k∗(x)

≤0, . . . , fs CΦ^k∗(x)

≤0,

|x₁| ≤λ, . . . , |x_n| ≤λ.

If max fi(CΦ^k∗+1(x))≤0, ∀i∈ {1, . . . ,s}then we stop andk^∗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 Costmax, the cost of repeat loop is

CostRepeat and the cost associated to the computation of

CΦⁱ,i∈ {0, 1,. . .,Cost_Repeat}isCost_{f ct}. Then the compu- tational complexity of Algorithm 2 is given by

Costalgo= (s×Costmax+2s)×CostRepeat+Costf ct. (76) 6. NUMERICAL EXAMPLES

To illustrate our results, we present some examples in two and three dimensional cases (n=2 andn=3). For this reason, we construct the matricesK^j,j∈σ₁^q (in the case whenp=1) by the following way

K^j= 1

Nb× kB_jk × kCk ×q×λ×τ²,

where the choice of the numberNb>1 depend on the value ofθ which satisfies the first hypothesis of Propo- sition 1. This choice of K^j guarantee the verification of the second hypothesis of Proposition 1. The matrix ˜A is chosen such thatkAk˜ <1. Even ifkAk>1, this choice is possible by selecting some matrixLsince ˜A=A+BLC.

Using these matrices, we guarantee the asymptotic stabil- ity of the considered systems. AssumptionΦ(B(0,λ))⊂ B(0,λ)is also taken into consideration. In all examples the dotted region will denote the setΘ^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 constructΩas follows:

Ω={y∈R^p|fi(y)≤0, i=1, . . . ,2p},

where f_i:R^p−→Rare defined for allx= (x₁,. . .,x_p)∈ R^pby

(f2m−1(x) =xm−ε, m∈ {1,2, . . . ,p}, f2m(x) =−xm−ε, m∈ {1,2, . . . ,p},ε>0, clearly 0∈Ω,^◦ fi,i∈σ₁^sare continuous functions such that

fi(0)≤0,∀i∈σ₁^s.

Example 1: Letλ,qandndefined by λ =1

2, q=1, n=2.

Let the matrix ˜A,B1,Cand ˜K¹defined by A˜=

₁

6 1 1 6

8 0

, B1=

1 0 0 0

, K˜¹= ₆₄¹ 0

, C= 1 0

.

Then the functions Φ(x),CΦ(x)andCΦ²(x)are defined by

Φ x

y

= ₁

64x²+¹₆x+¹₆y

1 8x

,

CΦ x

y

= 1 64x²+1

6x+1 6y, CΦ²

x y

= 1 36x+ 1

36y+ 1 64

1 64x²+1

6x+1 6y

2

+ 1 384x².

Using Algorithm 2 withε=0.2 we getk^∗₀=0 Θ^λ(Ω) =

x y

∈R²| |x| ≤ 1 2,|y| ≤ 1

2,|x| ≤ε

,

and using the same algorithm withε=0.1 we getk^∗₀=1

Θ^λ(Ω) =









 x

y

∈R²|

|x| ≤ 1

2, |y| ≤1 2,

1 64x²+1

6x+1 6y

≤ε









 .

Fig. 1.The setΘ^λ(Ω)corresponding to Example 1 (ε= 0.2).

Fig. 2.The setΘ^λ(Ω)corresponding to Example 1 (ε= 0.1).

Example 2: Letλ,q,nandεbe defined as λ =1, q=1, n=2, ε=0.3.

Let the matrix ˜A,C, ˜K¹andB1defined as A˜=

1 6

1 1 4 2

1 8

, B1=

1 0 2 0

, K˜¹= ¹₈ 0

, C= 1 0

. Then the functionsΦ

x y

,CΦ x

y

andCΦ² x

y

are de- fined by

Φ x

y

= 1

8x²+¹₆x+¹₄y

1

4x²+¹₂x+¹₈y

,

CΦ x

y

=1 8x²+1

6x+1 4y,

Fig. 3.The setΘ^λ(Ω)corresponding to Example 2.

CΦ² x

y

= 1

512x⁴+ 1

192x³+ 1

128x²y+ 25 288x² + 1

96xy+11 72x+ 1

128y²+ 7 96y.

Using Algorithm 2 we obtain k^∗₀ =1, and then the set Θ^λ(Ω)

Θ^λ(Ω) =





 x

y

∈R²|

|x| ≤1,|y| ≤1,|x| ≤ε,

1 8x²+1

6x+1 4y

≤ε





 .

Example 3: Letλ,q,nandεbe defined as λ =2, q=2, n=2 andε=0.1.

Let the matrix ˜A,C, ˜K¹, ˜K²,B1andB2be defined as A˜=

1 2 −¹₂

2 3

1 3

, B1=

1 2 0

1 2 0

, B2=

−1 0

−1 0

, K˜¹= ₃₂¹ 0

, K˜²= ₆₄¹ 0

, C= 1 0

.

Then Φ

x y

= 1

2x−¹₂y

2 3x+¹₃y

, CΦ

x y

=1 2x−1

2y, Φ²

x y

=

−₁₂¹x−₁₂⁵y

5 9x−²₉y

, CΦ²

x y

=−1 12x−5

12y, Φ³

x y

=

−²³₇₂x−₇₂⁷y

7 54x−¹⁹₅₄y

, CΦ³

x y

=−23 72x−7

72y.

Using Algorithm 2 we obtain k^∗₀ =2, and then the set Θ^λ(Ω)

Θ^λ(Ω) =













 x

y

∈R²|

|x| ≤λ,|y| ≤λ,|x| ≤ε,

1 2x−1

2y

≤ε,

−1 12x− 5

12y

≤ε













 .

Fig. 4.The setΘ^λ(Ω)corresponding to Example 3.

Example 4: Letλ,q,nandεdefined by λ =2, q=2, n=2, ε=0.3.

Let the matrix ˜A,B1,B2,C,K˜¹and ˜K²defined as A˜=

1 2

1 1 2 3

2 3

, B1=

1 2 0

1 2 0

,B2=

−1 0

−1 0

, C= 1 0

, K˜¹= ₃₂¹ 0

,K˜²= ₆₄¹ 0 .

Then Φ

x y

= ₁

2x+¹₂y

1 3x+²₃y

,CΦ

x y

=1 2x+1

2y, CΦ²

x y

= 5 12x+ 7

12y,CΦ³ x

y

=29 72x+43

72y, CΦ⁴

x y

=173 432x+259

432y, CΦ⁵

x y

=1037

2592x+1555 2592, CΦ⁶

x y

= 6221

15552x+ 9331 15552y, CΦ⁷

x y

=37325

93312x+55987 93312y.

Using Algorithm 2 we getk^∗₀=∞.

Example 5: Letλ,q,nandεdefined by λ =2, q=2, n=2, ε=0.1.

Let the matrix ˜A,B1,B2,C, ˜K¹and ˜K²defined as A˜=

₁

8 1 1 4 3

1 24

,B1=

3 0 1 0

, B2= 2 0

3 0

, C= 1 0

,K˜¹= ₆₄¹ 0

,K˜²= ₁₂₈¹ 0 .

Fig. 5.The setΘ^λ(Ω)corresponding to Example 5.

Then, the functionsΦ x

y

,CΦ x

y

andCΦ² x

y

are de- fined by

Φ x

y

= 1

16x²+¹₈x+¹₄y

5

128x²+¹₃x+₂₄¹y

,

CΦ x

y

= 1 16x²+1

8x+1 4y, CΦ²

x y

= 1

4096x⁴+ 1

1024x³+ 1

512x²y+ 19 1024x² + 1

256xy+ 19 192x+ 1

256y²+ 1 24y.

Using Algorithm 2 we getk^∗₀=1

Θ^λ(Ω) =





 x

y

∈R²|

|x| ≤2,|y| ≤2,|x| ≤ε,

1 16x²+1

8x+1 4y

≤ε





 .

Example 6: Let ˜A,B1, ˜K¹,C,q,εandλ defined as C= 1 0 0

,λ=2,q=1, ε=0.05, B1=



 1 0 4 2 3 0 1 0 1



,A˜=





1 6

1 4

1 1 5 3

1 8

1 7

0 ₁₀¹ 0



,K˜¹= ₄₀¹ 0 0 .

Then, we have Φ



 x y z



=





1 6

1 4

1 1 5 3

1 8

1 7

0 ₁₀¹ 0







 x y z





+ 1 40x



 1 0 4 2 3 0 1 0 1







 x y z





=





1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x²

1

3x+¹₈y+¹₇z+₄₀³xy+₂₀¹x²

1

40x²+₄₀¹zx+₁₀¹y



,

CΦ



 x y z



=1 6x+1

4y+1 5z+ 1

10xz+ 1 40x²,

Φ²



 x y z



=

































































1

9x+₂₄₀₀²²³y+₄₂₀²⁹z+

1 40

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x²2

+

1 10

1

40x²+₄₀¹zx+₁₀¹y

×

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x² +

3

160xy+₆₀₀¹³xz+₆₀₀¹³x²

7

72x+₆₇₂₀⁷⁶¹y+₈₄₀⁷¹z+

1 20

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x²2

+

3 40

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x²

×

1

3x+¹₈y+¹₇z+₄₀³xy+₂₀¹x² +

3

320xy+₈₄₀³¹xz+₃₃₆₀⁶¹ x²

1

30x+₈₀¹y+₇₀¹z+

1 40

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x²² +

1 40

1

40x²+₄₀¹zx+₁₀¹y

×

1

6x+¹₄y+¹₅z+₁₀¹xz+₄₀¹x² +

3

400xy+₂₀₀¹ x²































































 ,

CΦ²



 x y z



=1

9x+ 223 2400y+ 29

420z + 1

40 1

6x+1 4y+1

5z+ 1 10xz+ 1

40x² 2

+ 1 10

1 40x²+ 1

40zx+ 1 10y

× 1

6x+1 4y+1

5z+ 1 10xz+ 1

40x²

+ 3

160xy+ 13

600xz+ 13 600x². Using Algorithm 2 we obtaink^∗₀=2, and then

Θ^λ(Ω) =

















































 x y z



∈R³|

|x| ≤λ,|y| ≤λ,

|y| ≤λ,|x| ≤ε,

1

6x+¹₄y+¹₅z +₁₀¹xz+₄₀¹x²

≤ε,

1

9x+₂₄₀₀²²³y+₄₂₀²⁹z +₄₀¹

1

6x+¹₄y+¹₅z +₁₀¹xz+₄₀¹x²

²

+₁₀¹ 1

40x²+₄₀¹zx +₁₀¹y

× ₁

6x+¹₄y+¹₅z +₁₀¹xz+₄₀¹x²

+₁₆₀³ xy+₆₀₀¹³xz+₆₀₀¹³x²

≤ε













































 .

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

Letλ,q,n,εandτdefined as

λ =1, q=1, n=2, ε=0.1, τ=1.

Fig. 6.The setΘ^λ(Ω)corresponding to Example 8.

Fig. 7.The setΘ^λ(Ω)corresponding to Example 9.

LetB1, ˜A, ˜K¹,C,CKandθdefined by B1=

0 0 0 1

,A˜=

0.1+γ 0

0 0

,C= 1 −1

, K˜¹= ₁₀¹ 0

,CK= 1

10, θ= A˜

. Now, we use Algorithm 2 we obtain

γ θ+CKλ τ² k^∗₀

0 0.2 0

0.5 0.7 1

0.6 0.8 1

0.9 1.1 3

1 1.2 ∞.

6.1. Comment

In Examples 1-7, we have determined the index of ad- missibility 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 setsΘ^λ(Ω),in order to vi- sualize such sets of Examples 1, 2, 3, and 5.

Example 8: In this example we takeΩin the form Ω=

y∈R| f(y) =y²+y−2≤0 .

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

Maximize f(CΦ(x,y)) = (1 2x−1

2y)²+(1 2x−1

2y)−2 s.c







f C x y

!!

=x²+x−2≤0,

|x| ≤2, |y| ≤2,

the solution of this problem gives 1.7500.

Maximize f(CΦ²(x,y)) =

−1 12x− 5

12y 2

+

−1 12x−5

12y

−2

s.c



















f C x y

!!

=x²+x−2≤0,

f CΦ x y

!!

= (¹₂x−¹₂y)²+(¹₂x−¹₂y)−2≤0,

|x| ≤2, |y| ≤2,

the solution of this problem gives:−1.2000e⁻⁰⁶. Therefore, the value of index of admissiblity isk^∗₀=1.

Hence Θ^λ(Ω)

= x

y

∈R²| |x| ≤2, |y| ≤2,x²+x−2≤0,

x 2−₂^y2

+ ^x₂−₂^y

−2≤0

= x

y

∈R²| |x| ≤2, |y| ≤2, x²+x−2≤0

∩ x

y

∈R²|x 2−y

2 2

+x 2−y

2

−2≤0

= [−2,1]×[−2,2]

∩ x

y

∈R²|x−2≤y≤x+4

.

Example 9: In this example the setΩis represented by Ω=

y∈R| f₁(y) =y³+y−3≤0, f2(y) =y²+2y−4≤0

.

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

Using results of Example 5 and Algorithm 2 we ob- taink^∗₀=0, and then the maximal outputλ-admissible set Θ^λ(Ω)is given by

Θ^λ(Ω) =





 x

y

∈R²|

|x| ≤2, |y| ≤2, x³+x−3≤0, x²+2x−4≤0







= [−2,2]×[−2,2]∩h

−2,√ 5−1i

×[−2,2]

∩[−2,r]×[−2,2]

= [−2,r]×[−2,2],

withr= ³ q3

2−₁₈¹√ 3√

247+ ³ q1

18

√3√

247+³₂.

6.2. Comment

In Examples 8 and 9, we have modified the setΩwhich affects the mathematical programming problems that arise in Algorithm 2, we have plotted again the setsΘ^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 functionfmincon(see [28]) is applied with different values for the coefficientλ, ε,k^∗₀,nto solve the maximization problems which arise in such algorithm. The obtained results are presented in Table 1.

Remark 9: In case p≥2 we have to reconstruct the matrices K^j such that the condition θ+λCKτ² <1 of Proposition 1 is verified.

Table 1.Parameters values and elapsed time of Examples 1-3, 5, and 6.

Examples A˜ λ s k^∗₀ ε Time (s)

Examples 1

1 6

1 6 1

8 0

!

1

2 2 0 0.2 1.660970

Examples 1

1 6

1 6 1

8 0

!

1

2 2 1 0.1 3.1711

Examples 2

1 6

1 4 1 2

1 8

!

1 2 1 0.3 3.2184

Examples 3

1 2 −¹₂

2 3

1 3

!

2 2 2 0.1 3.0743

Examples 5

1 8

1 4 1 3

1 24

!

2 2 1 0.1 3.5962

Examples 6







1 6

1 4

1 5 1 3

1 8

1 7

0 ₁₀¹ 0





 2 2 1 0.05 8.5662