# Stability of Fuzzy Dynamical Systems via Lyapunov Functions

Authors: Abdelati El Allaoui; Said Melliani; Lalla Saadia Chadli
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

The purpose of this paper is to introduce the concept of fuzzy Lyapunov functions to study the notion of stability of equilibrium points for fuzzy dynamical systems associated with fuzzy initial value problems, through the principle of Zadeh. Our contribution consists in a qualitative characterization of stability by a study of the trajectories of fuzzy dynamical systems, using auxiliary functions, and they will be called fuzzy Lyapunov functions. And, among the main results that have been proven is that the existence of fuzzy Lyapunov functions is a necessary and sufficient condition for stability. Some examples are given to illustrate the obtained results.


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## Body

## 1. Introduction

The topics of fuzzy dynamical systems have been rapidly growing in recent years, and the first characterization of this concept is presented in [1]. Fuzzy dynamical systems have been dealt with different approaches. Some authors use the extension principle in order to extend deterministic systems of differential equations to the fuzzy case [2–7]. Others construct the fuzzy dynamical systems by using a family of differential inclusions [8–10].The notion of stability for this type of dynamical systems has been studied by many researchers [3, 7, 8, 10–15]. In [12], the authors introduced the concept of fuzzy equilibrium point stability of fuzzy initial value problems defined ℱℝn, where ℱℝn is the fuzzy set space on ℝn, by using equilibrium points. The authors in [10] have studied the stability of invariant sets for dynamical systems. According to them, equilibrium points have been considered as a special case of fuzzy invariant sets. These approaches have some shortcomings because they require knowledge of the explicit form of the solution of the fuzzy differential equation, which is not always possible to find.The aim of this paper is to present an alternative approach to these methods to prove the stability of an equilibrium point by introducing fuzzy Lyapunov functions, which are defined onℱℝn and obtained by the Zadeh’s extension of a Lyapunov function on ℝn. Moreover, without having the explicit solution of the fuzzy problem. This is an important point because the fuzzy space ℱℝn is bigger than the space ℝn. Thus, the case of Lyapunov functions on ℝn will be particular cases of fuzzy Lyapunov functions because ℝn is a classic subset of ℱℝn.


## 2. Preliminaries

In this section, we recall some basic tools of fuzzy set theory.LetPKℝn denote the family of all nonempty compact convex subsets of ℝn.The distance between two nonempty bounded subsetsA and B of ℝn is defined by the Hausdorff metric:(1)dA,B=maxρA,B,ρB,A,where ρA,B=supa∈Ainfb∈Ba−b and . denotes the usual Euclidean norm in ℝn.PKℝn,d is a complete and separable metric space (see [16]).Remember that a fuzzy subsetu of a classical set X is characterized by a mapping μu:X⟶0,1 called the membership function of u, and μux means the degree of membership of x in u.In the following, to simplify, we denote byu the membership function μu.Theα-cuts of a fuzzy set u are defined by(2)uα=x∈X:ux≥α,for0<α≤1,and the support of u is defined by(3)Suppu=x∈X:ux>0¯=u0.Denote byℱX the set of fuzzy subsets of X with nonempty and compact α−cuts. We are only interested here in ℱℝn, so the metric is given by(4)Du,v=sup0≤α≤1duα,vα.Now, we recall some properties of the extension principle.Definition 1.
(Zadeh’s extension principle, see [2, 5, 6]). Let f:X⟶Z be a function, and let A be a fuzzy subset of X. Zadeh’s extension of f is the function f^:ℱX⟶ℱZ which applied to A gives us the fuzzy subset f^A of Z with the membership(5)μf^Az=supx∈f−1zμAx,iff−1z≠∅,0,iff−1z=∅,where f−1z=x:fx=z.The following result is very useful for fuzzy differential equations (see [17–21]).Theorem 1 (see [5, 6]).
Letf:X⟶Z be a continuous function, and let A be a fuzzy subset of X. Then, for all α∈0,1,(6)f^Aα=fAα.The idea of the Zadeh’s extension approach is as follows.We consider the following fuzzy initial value problem:(7)dxdt=Fxt,x0=x0∈ℱRn,where F:ℱℝn⟶ℱℝn is the Zadeh’s extension of a continuous function f:ℝn⟶ℝn.A solution of (7) is defined as Zadeh’s extension of the deterministic solution φtx0 of the initial value problem associated:(8)dxdt=fxt,x0=x0∈ℝn.The fuzzy solution of equation (7) is denoted by φ^tx0.The familyφ^t satisfies the properties of a flow, and the result is given in the following theorem.Theorem 2 (see [11]).
The fuzzy solutionφ^tx0 verifies the properties:(1)
φ^0x0=x0(2)
φ^t+sx0=φ^tφ^sx0, for all x0∈ℱℝn and t∈ℝ+So, the familyφ^t defines a flow φ^t:ℱℝn⟶ℱℝn, which associates each x0 with a point φ^tx0.The phase space ofφ^t is the metric space ℱℝn,Dφt is continuous with respect to the initial condition, so φ^t is also continuousThus, the familyφ^t:ℱℝn⟶ℱℝn is a dynamical system in ℱℝn, for that it is called a fuzzy dynamical system.Example 1.
Consider the following nonlinear differential equation:(9)x′=gx,x0=x0∈ℝ+,where gx=δx1−x.
This system determines the flowφt:ℝ⟶ℝ given by(10)φtx0=x0x0−x0−1e−δt,fort≥0.
We consider the fuzzy initial value problem:(11)x′=g^x,x0=x0∈ℱℝ+,where g^ is the Zadeh’s extension of g defined by(12)g^xα=gxα=δxα−1−xα+,δxα+1−xα−,for xα=xα−,xα+ and α∈0,1.
Forx0∈ℱℝ, α∈0,1 and x0α=x0α−,x0α+. The fuzzy solution of problem (11) is the family φ^t given by(13)φ^tx0α=φtx0α=x0α−x0α−−x0α−−1e−δt,x0α+x0α+−x0α+−1e−δt.
According to Theorem2, φ^t is a fuzzy dynamical system.We will define now an equilibrium point for the fuzzy initial value problem (7) through the extended flow.Definition 2 (see [12]).
We say thatx¯∈ℱℝn is a fuzzy equilibrium point for φ^t when φ^tx¯=x¯, for every t≥0.Definition 3.
Letx¯∈ℱℝn be an equilibrium point of φ^t:(1)
x¯ is said to be Lyapunov stable, if and only if for every ε>0, there exists η>0 such that, if Dx0,x¯<η, then for every t≥0, we have Dφ^tx0,x¯<ε(2)
x¯ is said to be asymptotically stable if it is Lyapunov stable and there exists r>0 such that, if Dx0,x¯<r, then limt⟶+∞Dφ^tx0,x¯=0(3)
x¯ is said to be exponentially stable if it is asymptotically stable and there exist β>0, γ>0,andσ>0 such that, if Dx0,x¯<β, then Dφ^tx0,x¯≤γDx0,x¯e−σt,∀t≥0
For more details on stability, we refer to [11, 12, 21–25].
Stability of equilibrium points forφ^t in ℱℝn is characterized by the following result.Theorem 3 (see [10]).
Letx¯∈ℝn be an equilibrium point of φt. So, the following statements are satisfied:x¯ is stable for φt if and only if χx¯ is stable for φ^tx¯ is asymptotically stable for φt if and only if χx¯ is asymptotically stable for φ^tRemark 1.
(see [10]). Let U be a neighborhood of x¯, then U=x∈ℱℝn:x0⊆U is a neighborhood of χx¯.


## 3. Main Results

Before establishing the stability results via Lyapunov functions, we introduce the notion of the fuzzy Lyapunov function, inspired by the definition of Lyapunov functions in the classical case and the relation between the stability of the equilibrium points of the problem (7) and that of the problem (8) given in Theorem 3.Letg:ℝn⟶ℝ be a function, and let g^:ℱℝn⟶ℱℝ be the Zadeh’s extension of f. We denote by g^x≻0,∀x∈ℱℝn (with 0 being the null element of ℝ which is also an element of ℱℝ with the membership function χ0 if gx>0,∀x∈ℝn).Definition 4.
A functionV:ℱℝn⟶ℱℝ is a fuzzy Lyapunov-candidate function, if V is Zadeh’s extension of a deterministic function V: ℝn⟶ℝ such that:(1)
V0=0(2)
∀x∈U\0,Vx>0 for a neighborhood U of the origin
V is called the Lyapunov-candidate function associated with V.Remark 2.
LetV be a fuzzy Lyapunov-candidate function and V be the Lyapunov-candidate function associated. Then,(14)V0=0if and only ifVχ0=χ0.
Indeed,(15)Vχ0=χ0⇔Vχ0α=χ0α,∀α∈0,1,⇔Vχ0α=χ0α,∀α∈0,1,⇔V0=0.
From Remark1, Remark 2, and the previous definition, a fuzzy Lyapunov-candidate function V:ℱℝn⟶ℱℝ satisfies the following properties:(1)
Vχ0=χ0(2)
∀x∈U\χ0,Vx≻0 for a neighborhood U of the origin χ0Definition 5.
If a fuzzy Lyapunov-candidate functionV satisfies(16)V˙x≤0,∀x∈W\0,for a neighborhood W of the origin, where V is the Lyapunov-candidate function associated with V.
We say thatV is a fuzzy Lyapunov function, and in this case, V is called the Lyapunov function associated.Theorem 4.
(1)
There exists a fuzzy Lyapunov function for the fuzzy dynamical systemφ^t associated with the problem (7), if and only if χ0 is Lyapunov stable.(2)
χ0 is asymptotically stable if and only if there is a fuzzy Lyapunov function V verifying(17)V˙x<0,∀x∈W\0,for a neighborhood W of the origin, where V is the Lyapunov function associated with V.Proof.
(1)
It is known that there exists a fuzzy Lyapunov functionV if and only if there is a Lyapunov function V associated. In the fact V is a Lyapunov function associated with system (8). Then, 0 is stable for φt, and according to Theorem 3, χ0 is stable for φ^t.(2)
According to Theorem3, χ0 is asymptotically stable for φ^t if and only if 0 is asymptotically stable for φt. And, this last point is equivalent to say that there is a Lyapunov function V that verifies(18)V˙x<0,∀x∈W\0,for a neighborhood W of the origin.
So, letV be the function constructed by Zadeh’s extension applied to V, the desired fuzzy Lyapunov function.Still using the notion of fuzzy Lyapunov functions, we have the following result concerning the exponential stability.Theorem 5.
Letx0∈ℝn and x0∈ℱℝn such that x0∈x01. Then, 0 is exponentially stable for φt if and only if χ0 is exponentially stable for φ^t.Proof.
It should be noted that the conditionx0∈x01 implies that x0∈x0α for all α∈0,1.
⇒ Suppose that 0 is exponentially stable for φt, then 0 is asymptotically stable for φt and there exist β>0,γ>0,andσ>0 such that, if x0<β, then(19)φtx0<γx0e−σt,∀t≥0.
If we have(20)Dx0,χ0<β,which means that(21)maxρx0α,χ0,ρχ0,x0α<β,∀α∈0,1.
So, we get(22)supy0∈x0αy0<β,∀α∈0,1,which implies that(23)∀α∈0,1,∀y0∈x0α,y0<β,which leads us to obtain(24)∀α∈0,1,∀y0∈x0α,φty0≤γy0e−σt.
Therefore,(25)supz0∈x0αφtz0≤γy0e−σt,∀y0∈x0α.
So, we have(26)ρφtx0α,χ0α≤γy0e−σt,∀y0∈x0α.
Thus, we obtain(27)ρφ^tx0α,χ0α≤γ‖y0‖e−σt,∀y0∈x0α≤γsupy0∈x0αy0e−σt≤γρx0α,χ0αe−σt≤γmaxρx0α,χ0α,ρχ0α,x0αe−σt.
Therefore, for allα∈0,1,(28)ρφ^tx0α,χ0α≤γDx0,χ0e−σt.
In the same way, forα∈0,1, we have(29)ρχ0α,φ^tx0α=infy0∈x0αφty0≤φtx0≤γx0e−σt≤γsupy0∈x0αy0e−σt.
Then,(30)ρχ0α,φ^tx0α≤γDx0,χ0e−σt.
From (28) and (30), we can conclude that(31)dφ^tx0α,χ0α≤γDx0,χ0e−σt,and therefore(32)Dφ^tx0,χ0≤γDx0,χ0e−σt,which shows that χ0 is exponentially stable for φ^t.
⇐ If χ0 is exponentially stable, then χ0 is asymptotically stable and there exist β>0, γ>0,andσ>0 such that, if Dx0,χ0<β, then, for all t≥0,(33)Dφ^tx0,χ0<γDx0,χ0e−σt.
If we havex0<β, which means that(34)Dχx0,χ0<β.
So, by using (33) and the fact that x0∈χx0α for all α∈0,1, we get(35)Dφ^tχx0,χ0≤γDχx0,χ0e−σt.
Therefore,(36)sup0≤α≤1dφtχx0α,χ0α≤γsup0≤α≤1dχx0α,χ0αe−σt.
Consequently,(37)φtx0<γx0e−σt,∀t≥0,which proves the second part.As a consequence of the previous results, we have the following corollary.Corollary 1.
If  0 is an equilibrium point forφt and f:ℝn⟶ℝn with f′0=0, and ℜλi is the real part of the eigenvalue λi associated with 0:(1)
Ifℜλi<0,∀i, then there exists a fuzzy Lyapunov function V verifying(38)V˙x<0,∀x∈W\0,for a neighborhoodW of the origin, where V is the Lyapunov function associated with V.In other words,χ0 is asymptotically stable for (7).(2)
Ifℜλi>0, for some i, then there is no fuzzy Lyapunov function. In other words, χ0 is unstable for (7).To illustrate the elaborate results, we take as application the real model which describes a population, and it is the Malthusian model.Example 2.
We consider the deterministic Malthusian model with a negative variation rate (population in retraction):(39)x′t=−λxt,λ>0,x0=x0∈ℝ.
The deterministic flow is given by(40)φtx0=x0e−λt.
Note that 0 is an equilibrium point forφt, which is exponentially stable indeed.
LetVx=x2 on ℝ.
V is a Lyapunov function, and we have V˙x=V′x⋅fx=−2λx2<0, for x≠0.
So, 0 is asymptotically stable forφt. Moreover, we have(41)φtx0=x0e−λt≤x0e−λt,which shows that 0 is exponentially stable for φt.
But, when we do statistics, we focus on simple and then we generalize the property studied on the entire population, so it is more realistic to consider the initial condition as a fuzzy quantity. And, in this case, the model is of the following form:(42)x′t=−λxt,λ∈ℱℝ+,x0=x0∈ℱℝ.
The fuzzy flow associated with problem (42) is given by(43)φ^tx0=x0e−λt,x0∈ℱℝ,where φ^tt≥0 is a fuzzy dynamical system.
LetVx=V^x be the fuzzy Lyapunov function associated with problem (42) given by(44)Vxα=Vxα=minPα,maxPα,for xα=x−α,x+α,α∈0,1, where(45)Pα=x−α2,x+α2,x−α.x+α.
Note thatχ0 is an equilibrium point for φ^t.
By using Theorem4, we conclude that χ0 is asymptotically stable for φ^t, and we have(46)Dφ^tx0,χ0=Dx0e−λt,χ0=supα∈0,1de−λtx0α,e−λt0≤supα∈0,1dx0α,0e−λt=Dx0,χ0e−λt.
That is to say,χ0 is exponentially stable for φ^t.
Note that we can deduce directly from Theorem5 the last point.
Figure1 represents the dynamic of φ^t around χ0, where we considered in problem (42) the following parameters: x0 is “around 35,” which can be modeled by a triangular fuzzy number x0=30;35;40, whose α−cuts are given by(47)x0α=x0α−,x0α+=30+5α,40−5α,forα∈0,1.
And,λ is the symmetric triangular fuzzy number defined by λ=0.4;0.5;0.6, whose α−cuts are given by(48)λα=0.4+0.1α,0.6−0.1α,forα∈0,1.Figure 1
Dynamic ofφ^t around χ0.Example 3.
Consider the following system:(49)x′t=−x+y,y′t=−x−y,x0=x0,y0=y0.
This system can be written as(50)X′t=GXt,X0=X0=x0,y0∈ℝ2,where Xt=xt,yt∈ℝ2 and(51)GXt=−xt+yt,−xt−yt,fort∈ℝ+.
We consider the fuzzy initial value problem:(52)X′=G^Xt,X0=X0∈ℱℝ2,where G^ is Zadeh’s extension applied to G.
We study the stability of the equilibrium point0,0 for system (50).
We define onℝ2 the following function:(53)VX=Vx,y=x2+y2.
It is easy to verify thatV is a Lyapunov function, and we have(54)V˙x,y=2x−x+y+2y−x−y<0,forx≠0,y≠0.
And,0,0 is therefore asymptotically stable.
Now, we want to study the stability of system (51).
We haveX0∈ℱℝ2, then χ0,0 will be an equilibrium point of the fuzzy dynamical system associated with (51).
LetVX=V^X, for X∈ℱℝ2.
V is the fuzzy Lyapunov function associated with (51), whose α−cuts are(55)VXα=VXα,forα∈0,1.
From Theorem4, we deduce that χ0,0 is asymptotically stable for problem (52).
We can also show the asymptotic stability ofχ0,0 using Corollary 1 indeed.
It is easy to check thatλ1=−1+i and λ2=−1−i are the eigenvalues associated with system (50).
We haveℜλ1=ℜλ2<0, and therefore χ0,0 is asymptotically stable for the fuzzy initial value problem (52).
Now, we show thatχ0,0 is exponentially stable for system (51) indeed.
We have(56)V˙X=V˙x,y=−2x2+y2≤−2VX.
By using Grönwall’s inequality, forX0=x0,y0∈ℝ2, we can obtain(57)VX≤VX0e−2t,fort≥0,which leads us to have(58)X=x,y≤x0,y0e−t=X0e−t,fort≥0.
Then,0,0 is exponentially stable for (50).
Using Theorem5, we can conclude that χ0,0 is exponentially stable for (51).
LetX0∈ℱℝ2, whose membership function μ:ℝ2⟶0,1 is defined as, for x,y∈ℝ2,(59)μx,y=0,x≤300orx≥360,y≤600ory≥700,y−60050,x≤300orx≥360,600≤y≤650,700−y50,x≤300orx≥360,650≤y≤700,x−30030,300≤x≤330,y≤600ory≥700,minx−30030,y−60050,300≤x≤330,600≤y≤650,minx−30030,700−y50,300≤x≤330,650≤y≤700,360−x30,330≤x≤360,y≤600ory≥700,min360−x30,y−60050,330≤x≤360,600≤y≤650,min360−x30,700−y50,330≤x≤360,650≤y≤700.
It is easy to verify that, for allα∈0,1,(60)X0α=300+30α,360−30α×600+50α,700−50α.
Figure2 shows the stability of χ0,0 for problem (52).Figure 2
Dynamic ofφ^t around χ0,0.


## 4. Conclusion

In this work, we studied the stability of fuzzy dynamical systems using Lyapunov functions. We began by defining the fuzzy Lyapunov function in a way analogous to that of the classical case. We achieved to show some equivalence results between stability by different types whether it is stability, asymptotic stability, or exponential stability and the existence of a fuzzy Lyapunov function. Our results will be used in further works to generalize the notion of the stability of fuzzy dynamical systems.



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