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.
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 , where is the fuzzy set space on , 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 on and obtained by the Zadeh’s extension of a Lyapunov function on . Moreover, without having the explicit solution of the fuzzy problem. This is an important point because the fuzzy space is bigger than the space . Thus, the case of Lyapunov functions on will be particular cases of fuzzy Lyapunov functions because is a classic subset of .
2. Preliminaries
In this section, we recall some basic tools of fuzzy set theory.
Let denote the family of all nonempty compact convex subsets of .
The distance between two nonempty bounded subsets and of is defined by the Hausdorff metric:where and denotes the usual Euclidean norm in .
is a complete and separable metric space (see [16]).
Remember that a fuzzy subset of a classical set is characterized by a mapping called the membership function of , and means the degree of membership of in .
In the following, to simplify, we denote by the membership function .
The -cuts of a fuzzy set are defined byand the support of is defined by
Denote by the set of fuzzy subsets of with nonempty and compact cuts. We are only interested here in , 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 be a function, and let be a fuzzy subset of . Zadeh’s extension of is the function which applied to gives us the fuzzy subset of with the membershipwhere .
The following result is very useful for fuzzy differential equations (see [17–21]).
Theorem 1 (see [5, 6]). Let be a continuous function, and let be a fuzzy subset of . Then, for all ,
The idea of the Zadeh’s extension approach is as follows.
We consider the following fuzzy initial value problem:where is the Zadeh’s extension of a continuous function .
A solution of (7) is defined as Zadeh’s extension of the deterministic solution of the initial value problem associated:
The fuzzy solution of equation (7) is denoted by .
The family satisfies the properties of a flow, and the result is given in the following theorem.
Theorem 2 (see [11]). The fuzzy solution verifies the properties:(1)(2), for all and
So, the family defines a flow , which associates each with a point . The phase space of is the metric space is continuous with respect to the initial condition, so is also continuous
Thus, the family is a dynamical system in , for that it is called a fuzzy dynamical system.
Example 1. Consider the following nonlinear differential equation:where .
This system determines the flow given byWe consider the fuzzy initial value problem:where is the Zadeh’s extension of defined byfor and .
For , and . The fuzzy solution of problem (11) is the family given byAccording to Theorem 2, 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 is a fuzzy equilibrium point for when , for every .
Definition 3. Let be an equilibrium point of :(1) is said to be Lyapunov stable, if and only if for every , there exists such that, if , then for every , we have (2) is said to be asymptotically stable if it is Lyapunov stable and there exists such that, if , then (3) is said to be exponentially stable if it is asymptotically stable and there exist , such that, if , then For more details on stability, we refer to [11, 12, 21–25].
Stability of equilibrium points for in is characterized by the following result.
Theorem 3 (see [10]). Let be an equilibrium point of . So, the following statements are satisfied: is stable for if and only if is stable for is asymptotically stable for if and only if is asymptotically stable for
Remark 1. (see [10]). Let be a neighborhood of , then is a neighborhood of .
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.
Let be a function, and let be the Zadeh’s extension of . We denote by (with 0 being the null element of which is also an element of with the membership function if ).
Definition 4. A function is a fuzzy Lyapunov-candidate function, if is Zadeh’s extension of a deterministic function : such that:(1)(2) for a neighborhood of the origin is called the Lyapunov-candidate function associated with .
Remark 2. Let be a fuzzy Lyapunov-candidate function and be the Lyapunov-candidate function associated. Then,Indeed,From Remark 1, Remark 2, and the previous definition, a fuzzy Lyapunov-candidate function satisfies the following properties:(1)(2) for a neighborhood U of the origin
Definition 5. If a fuzzy Lyapunov-candidate function satisfiesfor a neighborhood of the origin, where is the Lyapunov-candidate function associated with .
We say that is a fuzzy Lyapunov function, and in this case, is called the Lyapunov function associated.
Theorem 4. (1)There exists a fuzzy Lyapunov function for the fuzzy dynamical system associated with the problem (7), if and only if is Lyapunov stable.(2) is asymptotically stable if and only if there is a fuzzy Lyapunov function verifyingfor a neighborhood of the origin, where is the Lyapunov function associated with .
Proof. (1)It is known that there exists a fuzzy Lyapunov function if and only if there is a Lyapunov function associated. In the fact is a Lyapunov function associated with system (8). Then, 0 is stable for , and according to Theorem 3, is stable for .(2)According to Theorem 3, is asymptotically stable for if and only if 0 is asymptotically stable for . And, this last point is equivalent to say that there is a Lyapunov function that verifiesfor a neighborhood of the origin.
So, let be the function constructed by Zadeh’s extension applied to , the desired fuzzy Lyapunov function.
Still using the notion of fuzzy Lyapunov functions, we have the following result concerning the exponential stability.
Theorem 5. Let and such that . Then, 0 is exponentially stable for if and only if is exponentially stable for .
Proof. It should be noted that the condition implies that for all .
Suppose that 0 is exponentially stable for , then 0 is asymptotically stable for and there exist such that, if , thenIf we havewhich means thatSo, we getwhich implies thatwhich leads us to obtainTherefore,So, we haveThus, we obtainTherefore, for all ,In the same way, for , we haveThen,From (28) and (30), we can conclude thatand thereforewhich shows that is exponentially stable for .
If is exponentially stable, then is asymptotically stable and there exist , such that, if , then, for all ,If we have , which means thatSo, by using (33) and the fact that for all , we getTherefore,Consequently,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 for and with , and is the real part of the eigenvalue associated with 0:(1)If , then there exists a fuzzy Lyapunov function verifying for a neighborhood of the origin, where is the Lyapunov function associated with . In other words, is asymptotically stable for (7).(2)If , for some , then there is no fuzzy Lyapunov function. In other words, 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):The deterministic flow is given byNote that 0 is an equilibrium point for , which is exponentially stable indeed.
Let on .
is a Lyapunov function, and we have , for .
So, 0 is asymptotically stable for . Moreover, we havewhich shows that 0 is exponentially stable for .
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:The fuzzy flow associated with problem (42) is given bywhere is a fuzzy dynamical system.
Let be the fuzzy Lyapunov function associated with problem (42) given byfor , whereNote that is an equilibrium point for .
By using Theorem 4, we conclude that is asymptotically stable for , and we haveThat is to say, is exponentially stable for .
Note that we can deduce directly from Theorem 5 the last point.
Figure 1 represents the dynamic of around , where we considered in problem (42) the following parameters: is “around 35,” which can be modeled by a triangular fuzzy number , whose cuts are given byAnd, is the symmetric triangular fuzzy number defined by , whose cuts are given by
Example 3. Consider the following system:This system can be written aswhere andWe consider the fuzzy initial value problem:where is Zadeh’s extension applied to .
We study the stability of the equilibrium point for system (50).
We define on the following function:It is easy to verify that is a Lyapunov function, and we haveAnd, is therefore asymptotically stable.
Now, we want to study the stability of system (51).
We have , then will be an equilibrium point of the fuzzy dynamical system associated with (51).
Let , for .
is the fuzzy Lyapunov function associated with (51), whose cuts areFrom Theorem 4, we deduce that is asymptotically stable for problem (52).
We can also show the asymptotic stability of using Corollary 1 indeed.
It is easy to check that and are the eigenvalues associated with system (50).
We have , and therefore is asymptotically stable for the fuzzy initial value problem (52).
Now, we show that is exponentially stable for system (51) indeed.
We haveBy using Grönwall’s inequality, for , we can obtainwhich leads us to haveThen, is exponentially stable for (50).
Using Theorem 5, we can conclude that is exponentially stable for (51).
Let , whose membership function is defined as, for ,It is easy to verify that, for all ,Figure 2 shows the stability of for problem (52).
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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.