Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

: In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Lefﬂer stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given. supervision, G.F.-A., O.M.-F. J.F.G.-A.; G.F.-A.; funding acquisition, G.F.-A. authors


Introduction
Fractional calculus generalizes the well-known classical calculus to operators with non-integer orders. In this sense, this theory has had as a main purpose to extend classical mathematical results and develop a new formulation of calculus; this has been done along with the development of integer calculus. However, fractional calculus did not begin to have practical applications until the 1970s. The first theoretical and applied results in fractional calculus involve the Riemann-Liouville integral and the Riemann-Liouville and Caputo derivatives, which are still widely studied and used [1][2][3][4][5][6][7][8][9][10]; nevertheless, other definitions of non-integer operators have been developed. Some recent classifications of families of operators appear in [11][12][13][14][15].
Moreover, some fractional operators have been proposed, which seek to generalize several other existing operators. Some of these generalizations have been given in [16][17][18][19][20][21]. In this sense, Fernandez et al. introduced a family of operators that generalizes a wide variety of fractional operators [22], such as the classical Riemann-Liouville and Caputo operators, and the operators with a non-singular kernel; among these, the Caputo-Fabrizio [23] • I denotes fractional integrals. The left superscript of I denotes the type of integral, while the right superscript represents the order, and the right subscript indicates the lower limit of the integral: -RL I α a+ is the Riemann-Liouville fractional integral of order α. -AB I α a+ is the Atangana-Baleanu fractional integral of Riemann-Liouville type of order α.
a+ is the fractional-order integral operator with general analytic kernel with orders α and β.
• D denotes fractional derivatives. The left superscript of D denotes the type of derivative, while the right superscript represents the order, and the right subscript indicates the lower limit of the set where the operator is being applied: -RL D α a+ is the Riemann-Liouville fractional differential operator of order α. -C D α a+ is the Caputo differential operator of order α. -CF D α a+ is the Caputo-Fabrizio differential operator of order α. -ABC D α a+ is the Atangana-Baleanu differential operator of Caputo type of order α. -ABR D α a+ is the Atangana-Baleanu differential operator of Riemann-Liouville type of order α.

•
In the case of the differential operators with general analytic kernel, both left superand subscripts are considered: a+ is the Riemann-Liouville fractional differential operator with general analytic kernel with orders α and β. -A C D α,β a+ is the Caputo fractional differential operator with general analytic kernel with orders α and β.

Preliminaries
As we mentioned, in this work, we propose new tools for the stability analysis of a class of fractional-order systems, where the main characteristic of those systems is the general analytic kernel in their derivative. This fractional derivative covers a wide variety of fractional operators, and, as particular cases, we find the classical Caputo, Riemann-Liouville, and Atangana-Baleanu derivatives. Therefore, we start this section with some basic definitions from the theory of fractional calculus and, subsequently, their generalization by using the general analytic kernel operators. Definition 1. [41]. Let α ∈ R + and Γ(α) = ∞ 0 t α−1 e −t dt be the Gamma function. The Riemann-Liouville fractional integral of order α is given by RL where a ≤ t ≤ b and RL I α Definition 2. [41]. Let α ∈ R + and consider n = min{k ∈ N | k > α}. Based on the Riemann-Liouville integral, the Riemann-Liouville fractional derivative of order α is defined by where D n is the usual derivative of order n ∈ Z + . Definition 3. [41]. Let α ∈ R + and consider n = min{k ∈ N | k > α}. Whenever D n ∈ L 1 [a, b], the Caputo derivative of fractional order α is given by Definition 4. [23]. Let 0 < α < 1 and consider a function f ∈ H 1 (a, b) with b > a. The operator described by represents a fractional derivative with a nonsingular kernel called the Caputo-Fabrizio fractional derivative, where the normalization function M(α) satisfies M(0) = M(1) = 1.

Remark 1.
The operators (5)-(7) use a real-valued normalization function B(α) that satisfies B(α) > 0 and B(0) = B(1) = 1. In addition, E α (·) represents the Mittag-Leffler function with one parameter α > 0 defined by the convergent series [42]: A generalization of function (8) is the Mittag-Leffler function with two parameters α > 0, β > 0 defined by the convergent series [42]: Now that the previous fractional derivatives and integrals have been defined, we may ask ourselves if there is a way to describe a family of derivatives to which operators (2) to (7) belong. In this sense, Fernández et al. proposed a simple integral model based on an analytical kernel and the Riemann-Liouville integral (1) that generalizes the already known operators [22]. Some useful definitions and results concerning these new operators are presented below. Definition 8. [22]. Let [a, b] ⊂ R, α, β ∈ C with Re(α) > 0, Re(β) > 0, and R ∈ R + such that R > (b − a) Re(β) . A general analytic kernel is a complex function analytic on the disc D(0, R) defined by the following locally uniformly convergent power series where the coefficients a k = a k (α, β) may have dependence on α and β. Definition 9. [22]. Let A be a general analytic kernel satisfying Definition 8 and f ∈ L 1 [a, b]. The fractional-order integral operator with general analytic kernel (GAK) is given by: Definition 10. [22]. Let A be a general analytic kernel satisfying Definition 8. The transformed function A Γ is defined as follows: Theorem 1. [22]. Let A be a general analytic kernel satisfying Definition 8. Then, for any function f ∈ L 1 [a, b], the integral (11) is equivalent to the locally uniformly convergent series on [a, b]: where RL I α+kβ a+ is the Riemann-Liouville integral.
Definition 11. [22]. Let a, b, α, β, A satisfying Definition 8, and let f be a function f ∈ L 1 [a, b] with sufficient differentiability properties. The differential operators of Riemann-Liouville and Caputo type with general analytic kernel (GAKRL and GAKC, respectively) are given by respectively, where m ∈ Z + , and the orders α and β depend on α and β.
On the other hand, by direct calculation employing Equation (14): Remark 2. Some classical fractional operators may be obtained using different values of α and β in definitions (11), (14), (15), as follows: , then represents the Riemann-Liouville integral operator of order α.
represent the Riemann-Liouville fractional derivative and the Caputo fractional derivative, respectively.
is the Caputo-Fabrizio derivative.
represent the ABR and ABC fractional derivatives, respectively.

Some Results with the General Analytic Kernel Operators
In this section, we relate the general analytic kernel differential operators (14) and (15) to each other, in a similar manner to the relationship between the classical Riemann-Liouville and Caputo derivatives. Moreover, the main result of the section consists of a new formula to calculate the fractional-order GAK derivative of the product of two functions.
Proof. From Definition (14) and the representation (13) for the fractional integral operator, we have: The series for A is assumed to be locally uniformly convergent, so the order of summation can be swapped, to get: i.e., Moreover, considering that the classical Riemann-Liouville and Caputo operators are related to each other [3], then Proof. From the Theorem 2, we have This concludes the proof.
with sufficient differentiability properties, and the following relationship holds:

Proof.
By hypothesis, the series is locally uniformly convergent, hence the order of integration and the summation can be changed. Considering the properties between the Riemann-Liouville fractional derivative and integral [1], one has that .
It is well known that the fractional derivative of a product of functions obtained with the classical fractional operators is difficult to calculate. Therefore, we present a result that helps to solve this problem by using the GAKRL derivative. According to the integer-order definition, the first order derivative of the function f (t) is given by Similarly, Iterating this process n−times, we get On the other hand, observe that the product of two functions can be expressed as follows: Based on the previous comments, we prove the following result.
Proof. Combining Definition (14) and expression (28), the derivative of the product between u(t) and v(t) can be expressed as follows: where By definition (14), it follows that and the integral (30) can be expressed by using the form (27), i.e., By a simple calculation in the integral I 1 , observe that lim To solve the problem with the indeterminate form, by using the L'Hôpital's rule: where we have used the fact that Finally, the integral (33) is solved by direct calculation: Therefore, Lemma 3. If the assumptions in Lemma 2 are satisfied, but replacing A RL D α,β Proof. The result can be directly obtained from Theorem 2 and Lemma 2.

Laplace Transform and Generalized Lyapunov Direct Method
As with their integer-order counterpart, Laplace transform has been one of the most useful tools for the solution of fractional differential equations. Thus, in this section, we establish some results related to the Laplace transform of the differential operators of Riemann-Liouville and Caputo type with general analytic kernels, the relationship between them, and their application to the analysis of existence and uniqueness of certain types of differential equations. In addition, some results of a generalized Lyapunov direct method for these operators are presented.
where the function A Γ satisfies Definition 10.
The Laplace transforms of (14) and (15) are determined as follows.
Proof. From Definition (14) and its representation (13), The series is locally uniformly convergent, due to 0 ≤ |(t − τ) β | ≤ b Re(β ) < R, and A is locally uniformly convergent by hypothesis on D(0, R). This allows for interchanging the order between the summation and the integration. Therefore, by using the Laplace transform for the classical Riemann-Liouville integral, we have that Finally, considering the representation given in Definition (10), the Laplace transform for the GAKRL derivative is reduced to Proof. The operator (15) can be rewritten in its integral form as follows: where * is the convolution operator. Applying the Laplace transform to the above equation, one has that The series for A is assumed to be locally uniformly convergent, which allows for interchanging the order of the integration and the summation, i.e., Finally, by Definition 10 and rewriting the summation in square brackets, the proof is completed.
where ABC D α 0+ f (t) is the Atangana-Baleanu derivative [24] in the Caputo sense, with On the other hand, from expression (37), with the parameters established previously, Now, by using a k as in the Equation (40) and the geometric series, one has that This result coincides with the Laplace transform for the ABC derivative. In a similar manner, we can obtain the Laplace transform pairs for other fractional operators.
In this paper, we consider fractional-order nonlinear systems with a general analytic kernel of the form where the initial conditions have the form The set of initial conditions is necessary to specify the unique solution to the system (43); moreover, f : [t 0 , ∞) × Ω → R n is piecewise continuous in t and locally Lipschitz in x on [t 0 , ∞) × Ω, where Ω ⊂ R n is a domain that contains the equilibrium point x = 0. The equilibrium point of (43) is defined as follows.

Definition 12.
An equilibrium point of the fractional-order system (43) is a constant x 0 that satisfies f (t, x 0 ) = 0.

Remark 3.
Without loss of generality, let the equilibrium point be x = 0. If the equilibrium point of system (43) where g(t, 0) = 0. A direct implication of this analysis is that the system with the variable y(t) satisfies Definition 12 so that its equilibrium point lies at the origin. Now, consider the following assumptions, and define a function that will be useful for the analysis of the solution x(t).

Theorem 7.
If the conditions stated in Assumption 1 hold, then the solution of system (43) can be rewritten as Proof. Under Assumption 1, we apply the Laplace transform to (43): where F(s) = L { f (t, x(t))}. Rearranging and taking inverse Laplace transforms, it follows that where * denotes the standard convolution operation. Finally, by using the FOB-function and rearranging the summation, the proof is completed.
Then, by Theorem 7, the solution of a fractional-order system C D α t 0 + x(t) = f (t, x(t)) can be determined by solving the following equivalent problem: According to the convolution operation and by the Riemann-Liouville integral (1), we obtain the following Volterra integral equation: On the other hand, for the Atangana-Baleanu-Caputo derivative, we set m = 1, α = 1, β = α; then, . (40), we have

By Definition 13 and Equation
Applying the inverse Laplace transform to the previous expression, we have that where δ(t) is the Dirac delta function. Finally, considering the convolution operator and its properties, we obtain the following Volterra integral equation: that is, and rearranging terms, where AB I α 0+ f (t, x(t)) represents the Atangana-Baleanu integral [24]. A similar analysis for different values of parameters α , β and m can be done to obtain the solutions of fractional differential equations that involve different fractional operators.
If we use the GAKRL fractional derivative instead of the GAKC operator, the solution of takes the form (46), where all the initial conditions x (j) (0), j = 0, 1, . . . , m − 1 are set to zero. An equilibrium point of a system (49) can be defined.

Definition 14.
An equilibrium point of the fractional-order system (49) is a constant x 0 that satisfies Remark 4. Similarly to Remark 3, suppose the equilibrium point for (49) isx = 0 and consider the same change of variable y(t) = x(t) −x. Then, where g(t, 0) = 0. In terms of the new variable, the system has an equilibrium at the origin.

Proof. From inequality
, it follows that there exists a nonnegative function M(t) that satisfies the following equation: Applying the Laplace transform (37) to Equation (51), we have that By hypothesis, x(0) = y(0), . . . , x (m−1) (0) = y (m−1) (0), then equality (52) is reduced to: Dividing by A Γ s −β s −α +m produces Taking the inverse Laplace transform of (54) and using the Convolution Theorem, one has that The second term of the right-hand side of (55) is non-negative because M(t) is non-negative. Based on this reasoning, the proof is completed and x(t) ≥ y(t).

Remark 5.
The A − FOB-function represents a family of Mittag-Leffler functions and their generalizations. Note that, for α = 1 − α, β = 0, m = 1, and γ = 0 and the FOB-stability coincides with the classical Mittag-Leffler stability [43]: Based on the previously established definitions and results, we can then analyze the behavior of fractional-order systems and, in particular, prove asymptotic stability by means of an extension of the Lyapunov direct method, which is directly related to FOB-stability. Theorem 8. Let x = 0 be an equilibrium point for system (43) and D ⊂ R n be a domain containing the origin. Let V(t, x(t)) : [0, ∞) × D → R be a continuously differentiable function and locally Lipschitz with respect to x such that V (j) (0, x(0)) > 0, 0 ≤ j ≤ m − 1, and where t ≥ 0, x ∈ D, α, β ∈ C with Re(α) > 0, Re(β) > 0, α 1 , α 2 , α 3 , a and b are arbitrary positive constants. Then, x = 0 is FOB-stable and asymptotically stable.

Proof. From inequalities (60) and (61), one has that
Let M(t) be a non-negative function. Based on this function, the inequality (62) can be rewritten as with λ = α 3 α −1 2 . Applying the Laplace transform to Equation (63), we have that Rearranging this equation and solving for V(s), we have Therefore, Once the algebraic problem is solved, we apply the inverse Laplace transform of (64) to obtain Considering A λ (t; α , β , m) ≥ 0, ∀ t ≥ 0, then A λ (t; α , β , m) * M(t) ≥ 0 and thus Combining (66) with condition stated in inequality (60) yields where K j = 0 if and only if x(0) = 0. Because V(t, x(t)) is locally Lipschitz with respect to x, its derivatives are bounded and V (j) (0, x(0)) = 0 if and only if x(0) = 0, then it follows that K j is also Lipschitz with respect to x(0) and K j (0) = 0; this implies the FOB stability of system (43). Furthermore, by using the final value theorem on the right-hand side of (66) for m − α − j > 0, we get Combining inequalities (60), (68), and considering that V(t, x(t)) ≥ 0 for all t yields It follows from (69) that lim t→∞ α 1 x a ≤ 0. Finally, due to α 1 , a > 0, then lim t→∞ x(t) = 0.
This proves that the origin of system (43) is asymptotically stable.

Remark 6.
Note that FOB-stability implies asymptotic stability. , then the origin x = 0 of system (49) is asymptotically stable.
Proof. By using the inequality of Theorem 3 and V(t, x(t)) ≥ 0, we obtain Following the proof of Theorem 8, the proof is completed.

Useful Inequalities for Lyapunov Stability Analysis
In this section, some inequalities are established. These results help develop tools for the stability analysis of fractional-order nonlinear systems, employing the generalized Lyapunov direct method shown in the previous section. We start with some significant lemmas that will help us prove the main results. Lemma 5. [44]. Let u(t) be a continuous and differentiable real-valued function. Then, for any time instant t ≥ t 0 and for all 0 < α < 1: (a) If α = 1 − α, β = 0, m = 1 and A (t − τ) 0 = Following the same idea for the function x(t), we have that Thus, to justify the proposed inequality, we can rewrite inequality (85) as follows: . By hypothesis, the function V(t, x(t)) is convex; then, according to Definition 17, it follows that ϕ(τ) ≤ 0. Now, integration by parts can be applied in the second term of (89) with Considering that α + β k + 1 − m > 0, then the limit is calculated by applying the L'Hôpital's rule, i.e., Evaluating ϕ(t 0 ) and considering that V(t 0 , x(t 0 ) = 0, we have Remark 9. If we set different values of α and β , the conclusions of Theorems (14) and (15) hold for different fractional-order derivatives. For example, for α = β = 1, m = 1 and A((t − τ)) = M(α) , then, for all t ≥ t 0 , we obtain the conclusion for systems with a Caputo-Fabrizio derivative [50].

Scalar Systems
In this first example, we study scalar systems. Consider the system given by [51]: where the Caputo derivative of general analytic kernel (15) is considered. In system (102), the parameter p > 0 is odd, a < R − and g(x) is bounded as follows: where the inequality (103) Hence, due to the bound (103), the fractional-order derivative of V(x) is bounded as follows: Near the origin, the term ax p+1 is dominant, and then A C D α,β 0+ V ≤ k|x| p+2 . This implies that the origin is Fernandez-Özarslan-Baleanu stable and therefore asymptotically stable.

Second Order Systems
There is a wide class of systems of second order. In particular, consider the following system modeled using the Caputo derivative of general analytic kernel (15): It is not difficult to show that the equilibrium point of system (104) is (0, 0). To analyze stability, consider V(x) = 1 2 x 2 1 + x 2 2 . Applying the inequality given in Theorem 11 and considering system (104), one has that To analyze the last inequality, consider the ball x 2 ≤ r 2 where |x 1 | ≤ r. If we restrict the analysis to this set, then The right-hand side of the above inequality can be rewritten by using a matrix form, so that By a simple calculation, it is not difficult to show that the associated eigenvalues of M ; then, if we choose r < 2 and applying Theorem 8, it follows that the origin is asymptotically stable.

A Spacecraft Modeled by Generalized Dynamics
A rotating rigid spacecraft is studied in [52] by using the well-known Euler equations. The differential equations can be generalized considering the Caputo (or Riemann-Liouville (14)) derivative of general analytic kernel (15). This allows for considering different kinds of analysis in engineering applications. For this case, consider the following set of differential equations: where the scalar components of the vector ω are, respectively, ω 1 , ω 2 and ω 3 . On the other hand, some torque inputs, denoted by u 1 , u 2 and u 3 , are considered and applied about the principal axes. In addition to the dynamical analysis, the components J 1 , J 2 and J 3 represent the moments of inertia. The stability analysis will be done by considering the controlled system (105). Suppose that the torque inputs apply the feedback control u i = −k i ω i , where k i > 0 and 1 ≤ i ≤ 3. Then, the close-loop system is given by Now, taking V(ω) = 1 2 J 1 ω 2 1 + J 2 ω 2 2 + J 3 ω 2 3 as a Lyapunov function candidate and using the inequality (75), it is not difficult to obtain Thus, by applying Theorem 8, the origin is Fernandez-Özarslan-Baleanu stable and therefore asymptotically stable.

Financial Analysis
Financial systems are employed to analyze different situations in society, and many points of view have been proposed to model financial scenarios [53]. Consider, for example, the following fractional-order system modeled by using the Caputo derivative with general analytic kernel (15): A C D α,β 0+ x 1 (t) = x 3 (t) + (x 2 (t) − a)x 1 (t) + u 1 (t), where x 1 (t) represents the interest rate, x 2 (t) the investment demand, and x 2 (t) the price index, and the constants a = 1 (the saving amount), b = 0.1 (the cost per investment), and c = 1 (the elasticity of demand of commercial markets). The stability analysis considers the equilibrium point of the closed-loop system considering the control laws given by On the other hand, let V(x(t)) be the candidate Lyapunov function described by V(x(t)) = x 4/3 1 (t) + x 4/3 2 (t) + x 4/3 3 (t).

Conclusions
In this paper, we studied some properties of fractional-order derivatives and integrals with general analytic kernels. Some results concerning the Laplace transform were proved and used to establish some remarks on the solutions of fractional-order differential equations that involve these fractional operators, along with a generalized comparison principle. We proposed the FOB-function and the concept of FOB stability, which generalizes Mittag-Leffler stability for a comprehensive family of systems with different fractional-order derivatives.
Moreover, one of the main results presented consists of the generalization of the Lyapunov direct method, which is directly related to the FOB-stability and the boundedness of solutions. In addition, some inequalities for quadratic forms have been proposed. These results allow using the Lyapunov direct method for the stability analysis of fractional-order systems with the operators considered.
Furthermore, since the inequalities are established for operators with general analytic kernels, we have shown that the inequalities previously presented in the literature emerge from our work as particular cases. In addition, as an extension of the stability analysis, we have treated the stability problem and its solution through convex Lyapunov functions; some theorems are obtained directly from this part. Finally, we provided some illustrative examples to demonstrate the applicability of the proposed approach.