Stability of Fuzzy Dynamical Systems via Lyapunov Functions

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.


Introduction
e 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][3][4][5][6][7]. Others construct the fuzzy dynamical systems by using a family of differential inclusions [8][9][10]. e notion of stability for this type of dynamical systems has been studied by many researchers [3,7,8,[10][11][12][13][14][15]. In [12], the authors introduced the concept of fuzzy equilibrium point stability of fuzzy initial value problems defined F(R n ), where F(R n ) is the fuzzy set space on R n , by using equilibrium points. e 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. ese 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. e 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 on F(R n ) and obtained by the Zadeh's extension of a Lyapunov function on R n . Moreover, without having the explicit solution of the fuzzy problem. is is an important point because the fuzzy space F(R n ) is bigger than the space R n . us, the case of Lyapunov functions on R n will be particular cases of fuzzy Lyapunov functions because R n is a classic subset of F(R n ).

Preliminaries
In this section, we recall some basic tools of fuzzy set theory.
Let P K (R n ) denote the family of all nonempty compact convex subsets of R n . e distance between two nonempty bounded subsets A and B of R n is defined by the Hausdorff metric: where ρ(A, B) � sup a∈A inf b∈B ‖a − b‖ and ‖.‖ denotes the usual Euclidean norm in R n .
(P K (R n ), d) is a complete and separable metric space (see [16]).
Remember that a fuzzy subset u of a classical set X is characterized by a mapping μ u : X ⟶ [0, 1] called the membership function of u, and μ u (x) means the degree of membership of x in u.
In the following, to simplify, we denote by u the membership function μ u . e α-cuts of a fuzzy set u are defined by and the support of u is defined by Denote by F(X) the set of fuzzy subsets of X with nonempty and compact α−cuts. We are only interested here in F(R n ), so the metric is given by 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.
Theorem 1 (see [5,6]). Let f: X ⟶ Z be a continuous function, and let A be a fuzzy subset of X. en, for all α ∈ [0, 1], e idea of the Zadeh's extension approach is as follows. We consider the following fuzzy initial value problem: dx dt � F(x(t)), where F: F(R n ) ⟶ F(R n ) is the Zadeh's extension of a continuous function f: R n ⟶ R n . A solution of (7) is defined as Zadeh's extension of the deterministic solution φ t (x 0 ) of the initial value problem associated: dx dt � f(x(t)), e fuzzy solution of equation (7) is denoted by φ t (x 0 ). e family φ t satisfies the properties of a flow, and the result is given in the following theorem.
Theorem 2 (see [11]). e fuzzy solution φ t (x 0 ) verifies the properties: for all x 0 ∈ F(R n ) and t ∈ R + So, the family φ t defines a flow φ t : F(R n ) ⟶ F(R n ), which associates each x 0 with a point φ t (x 0 ). e phase space of φ t is the metric space (F(R n ), D) φ t is continuous with respect to the initial condition, so φ t is also continuous us, the family φ t : F(R n ) ⟶ F(R n ) is a dynamical system in F(R n ), for that it is called a fuzzy dynamical system. Example 1. Consider the following nonlinear differential equation: We consider the fuzzy initial value problem: where g is the Zadeh's extension of g defined by . e fuzzy solution of problem (11) is the family φ t given by According to eorem 2, 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 that (1) x is said to be Lyapunov stable, if and only if for every x is said to be asymptotically stable if it is Lyapunov stable and there exists r > 0 such that, if D(x 0 , x) < r, x is said to be exponentially stable if it is asymptotically stable and there exist β > 0, c > 0, and σ > 0 For more details on stability, we refer to [11,12,[21][22][23][24][25]. Stability of equilibrium points for φ t in F(R n ) is characterized by the following result.
Theorem 3 (see [10]). Let x ∈ R n be an equilibrium point of φ t . So, the following statements are satisfied:

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 eorem 3. Let g: R n ⟶ R be a function, and let g: Remark 2. Let V be a fuzzy Lyapunov-candidate function and V be the Lyapunov-candidate function associated. en, Indeed, From Remark 1, Remark 2, and the previous definition, a fuzzy Lyapunov-candidate function V: F(R n ) ⟶ F(R) satisfies the following properties: for a neighborhood W of the origin, where V is the Lyapunov-candidate function associated with V. We say that V is a fuzzy Lyapunov function, and in this case, V is called the Lyapunov function associated.

Theorem 4
(1) ere exists a fuzzy Lyapunov function for the fuzzy dynamical system φ t associated with the problem (7) 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 function V if and only if there is a Lyapunov function V associated. In the fact V is a Lyapunov function associated with system (8). en, 0 is stable for φ t , and according to eorem 3, χ 0 { } is stable for φ t . (2) According to eorem 3, χ 0 { } is asymptotically stable for φ t if and only if 0 is asymptotically stable for φ t . And, this last point is equivalent to say that there is a Lyapunov function V that verifies for a neighborhood W of the origin. So, let V 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.
Proof. It should be noted that the condition (⇒) Suppose that 0 is exponentially stable for φ t , then 0 is asymptotically stable for φ t and there exist β > 0, c > 0, and σ > 0 such that, if ‖x 0 ‖ < β, then If we have which means that International Journal of Differential Equations So, we get which implies that ∀α ∈ [0, 1], which leads us to obtain ∀α ∈ [0, 1], erefore, So, we have us, we obtain erefore, for all α ∈ [0, 1], In the same way, for α ∈ [0, 1], we have en, From (28) and (30), we can conclude that and therefore which shows that χ 0 { } is exponentially stable for φ t . (⇐) If χ 0 { } is exponentially stable, then χ 0 { } is asymptotically stable and there exist β > 0, c > 0, and If we have ‖x 0 ‖ < β, which means that So, by using (33) and the fact that Consequently, which proves the second part.
□ As a consequence of the previous results, we have the following corollary. (1) If R(λ i ) < 0, ∀i, then there exists a fuzzy Lyapunov for a neighborhood W of the origin, where V is the Lyapunov function associated with V.
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): x ′ (t) � −λx(t), λ > 0, Note that 0 is an equilibrium point for φ t , which is exponentially stable indeed.
Let V(x) � x 2 on R. V is a Lyapunov function, and we have So, 0 is asymptotically stable for φ t . Moreover, we have which shows that 0 is exponentially stable for φ t . 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: e fuzzy flow associated with problem (42) is given by where (φ t ) t≥0 is a fuzzy dynamical system. Let V(x) � V(x) be the fuzzy Lyapunov function associated with problem (42) given by Note that χ 0 { } is an equilibrium point for φ t . By using eorem 4, we conclude that χ 0 { } is asymptotically stable for φ t , and we have at is to say, χ 0 { } is exponentially stable for φ t . Note that we can deduce directly from eorem 5 the last point. Figure 1 represents the dynamic of φ t around χ 0 { } , where we considered in problem (42) the following parameters: x 0 is "around 35," which can be modeled by a triangular fuzzy number x 0 � (30; 35; 40), whose α−cuts are given by And, λ is the symmetric triangular fuzzy number defined by λ � (0.4; 0.5; 0.6), whose α−cuts are given by Example 3. Consider the following system: is system can be written as X ′ (t) � G(X(t)), where X(t) � (x(t), y(t)) ∈ R 2 and G(X(t)) � (−x(t) + y(t), −x(t) − y(t)), for t ∈ R + . (51) We consider the fuzzy initial value problem: , where G is Zadeh's extension applied to G.
We study the stability of the equilibrium point (0, 0) for system (50).
We define on R 2 the following function: It is easy to verify that V is a Lyapunov function, and we have And, (0, 0) is therefore asymptotically stable. Now, we want to study the stability of system (51). We have X 0 ∈ F(R 2 ), then χ (0,0) { } will be an equilibrium point of the fuzzy dynamical system associated with (51).
It is easy to check that λ 1 � −1 + i and λ 2 � − 1 − i are the eigenvalues associated with system (50).

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.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.