Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method

Accepted Manuscript

Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method Ying Yang, Yong He, Yong Wang, Min Wu PII: DOI: Reference:

S1007-5704(16)30318-5 10.1016/j.cnsns.2016.09.009 CNSNS 3983

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Communications in Nonlinear Science and Numerical Simulation

Received date: Revised date: Accepted date:

1 April 2016 13 September 2016 15 September 2016

Please cite this article as: Ying Yang, Yong He, Yong Wang, Min Wu, Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method, Communications in Nonlinear Science and Numerical Simulation (2016), doi: 10.1016/j.cnsns.2016.09.009

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Highlights • Variational Lyapunov method is employed to handle the hybrid term in the system. • A fractional variational comparison principle is established. • Stability and instability criteria in terms of two measures are obtained.

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• Results generalize corresponding theory of impulsive fractional differential systems.

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Stability analysis for impulsive fractional hybrid

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systems via variational Lyapunov method Ying Yanga , Yong Heb,∗, Yong Wanga , Min Wub a

School of Information Science and Engineering, Central South University, Changsha 410083, China

School of Automation, China University of Geosciences, Wuhan 430074, China

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b

Abstract: This paper investigates the stability properties for a class of impulsive Caputo fractional-order hybrid systems with impulse effects at fixed moments. By utilizing the variational Lyapunov method, a fractional variational comparison principle is established. Some stability and instability criteria in terms of two measures are obtained. These results generalize the known ones, extending the corresponding thetheir effectiveness.

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ory of impulsive fractional differential systems. An example is given to demonstrate

Keywords: Variational Lyapunov method; Impulsive fractional hybrid systems; Sta-

Introduction

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bility; Two measures.

In recent years, fractional calculus has played a vital role in different areas such as physics, control theory, electrical circuits, and fractal media [1–3]. Fractional-order models have been

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found to be an excellent tool for describing complex real-world systems with hereditary and memory properties. Consequently, the study of fractional-order systems has attracted consid-

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erable attention, and significant progress has been made [4–7], particularly in stability theory [8–12].

Numerous real models or modern complex engineering systems usually influence each other

mutually and physically. By operating a number of constraints with information and communication networks, they become highly interconnected and interdependent. Normally, we need to use a sequence of abstract decision-making units to decide operation mode and to specify sub controllers to be activated, which will make controlled dynamical systems more complex and to be hierarchical. These multi-echelon systems are classified as hybrid systems [13]. Sometimes, ∗

Corresponding author. Tel.: +86-27-87175080. Email address: [email protected] (Yong He).

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the systems need to be switched to a new set of differential equations with momentary perturbations in form of impulses. Such systems are defined as impulsive hybrid systems (IHSs). IHSs have a wide range of applications in practical situations, such as computer science, mathematical programming, modeling and simulation [14, 15]. For more details about such systems, one can be referred to the references [16–19]. Stability analysis is one of the most essential and fundamental issues for control systems

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including IHSs. It must be noticed that much attention has been mainly focused on IHSs of integer order. However, few theoretical results for stability analysis of impulsive fractional hybrid systems (IFHSs) are reported in the literature. In [20], the authors presented some existence results of a two-point boundary value problem for nonlinear impulsive fractional hybrid differential equations. In [21], fixed point theorems were used to investigate the problem of the existence of solutions for IHSs involving Caputo-Hadamard type fractional multi-orders with

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nonlinear integral boundary conditions. Despite the great potential applications, the stability theory of IFHSs has not yet been fully developed, which inspires us to conduct the current work. As a commonly-used approach, the Lyapunov direct method provide a convenient way to obtain sufficient conditions for the stability of a system without explicitly requiring the behaviors of solutions. However, if we treat the hybrid term as a perturbation of IFHSs and use the

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Lyapunov direct method to judge the stability, dynamic characteristics of the perturbation often lead to a more complex structure of derivative of the Lyapunov function and complicated calculation. Even in some cases, it will fail due to an indefinite derivative of the Lyapunov

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function. Moreover, boundedness or precise measurement of perturbation is required in certain circumstances.

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An alternative method to study the stability property of IFHSs is the variational Lyapunov method. In perturbation theory, the variational Lyapunov method [22–24] is proved to be an effective instrument in the investigation of perturbed systems. This technique combines

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the method of variation of parameters and the Lyapunov second method to provide a flexible mechanism for studying the effect of perturbations on differential systems. The advantage is that it is not necessary for the perturbations to be measured by means of a norm. Additionally,

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Lyapunov-like functions and differential inequalities are employed to connect the solutions of the systems with and without perturbation in terms of the maximal solution of a comparison problem. This method has been promoted from integer-order systems to fractional-order systems [25]. Due to the flexibility and strength of the variational Lyapunov method, this paper aims to extend the promoted variational Lyapunov method to the area of IFHSs and to give stability conditions of the systems. Most previous studies about stability analysis for fractional systems are concerned with stability or asymptotical stability in the sense of Lyapunov [11, 12], MittagLeffler stability [8–10], and practical stability [26, 27]. However, few results on stability analysis 3

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in terms of two measures [28] are available. In [29], practical stability in terms of two measures with initial time difference for nonlinear fractional equations was studied. The concepts of stability in terms of two measures can unify many known stability notions such as Lyapunov stability, eventual stability and partial stability in a single setup, and offer a general framework for investigation of stability theory in the qualitative analysis [28]. Motivated by the aforementioned discussions, we investigate the stability properties in terms

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of two measures for a class of IHSs involving Caputo’s fractional order and impulse effects at fixed moments via the promoted variational Lyapunov method. The main contributions of the paper can be summarized as follows. Comparing with the conventional way, we treat the hybrid term of IFHSs as a perturbation to avoid considering the specific characteristics of the hybrid term directly. Then, a fractional variational comparison principle is established by utilizing differential inequalities and the Lyapunov-like function in Caputo’s sense, which is an extension

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of the commonly-used one for impulsive fractional systems and provides a link between the perturbed system and the unperturbed one. Furthermore, based on the comparison principle, some stability properties in terms of two different measures are discussed for the system under consideration. Finally, an example is given to illustrate the effectiveness of the results. The rest of paper is organized as follows. In Section 2, some preliminaries are introduced,

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and the research problem is formulated. The fractional variational comparison principle is established and corresponding results are shown in Section 3. Section 4 discusses some stability and instability properties in terms of two measures for the systems. In Section 5, an example is

Systems description and preliminaries

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presented to illustrate the results. Finally, conclusions are made in Section 6.

In this section, some useful notations, definitions and lemmas are presented.

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Definition 2.1. [4] The fractional integral of order α for a function x(t) is defined as Z t 1 −α (t − τ )α−1 x(τ )dτ, t0 Dt x(t) = Γ(α) t0

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where t > t0 and α > 0, Γ(·) is the Gamma function.

Definition 2.2. [4] The Riemann-Liouville derivative of fractional order α of function x(t) is given as

RL α t0 Dt x(t)

dn = n dt





−(n−α) x(t) t 0 Dt

dn 1 = Γ(n − α) dtn

Z

t

t0

n−α−1

(t − τ )

 x(τ )dτ ,

in which n − 1 < α < n ∈ Z+ . Definition 2.3. [4] The Caputo derivative of fractional order α of function x(t) is defined as   n Z t 1 −(n−α) d C α x(t) = (t − τ )n−α−1 x(n) (τ )dτ, t0 Dt x(t) = t0 Dt dtn Γ(n − α) t0 4

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in which n − 1 < α < n ∈ Z+ . Lemma 2.4. [30] If x(t) ∈ C m [t0 , ∞) and m − 1 < α < m, m ∈ Z+ , m = 1, then C α t0 Dt x(t)

α = RL t0 Dt x(t) −

m−1 X tk k=0

k!


x(k) (0) .

Definition 2.5. [25] m ∈ Cp ([t0 , T ], R) means that m ∈ C([t0 , T ], R) and (t − t0 )p m(t) ∈

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C([t0 , T ], R) with p + q = 1.

Definition 2.6. [25] For m ∈ Cp ([t0 , T ], R), the Riemann-Liouville derivative of m(t) is defined

as

RL q t0 Dt m(t)

=

Z

1 d Γ(p) dt

t

t0

(t − τ )p−1 m(τ )dτ,

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with p + q = 1.

Lemma 2.7. [25] Let m ∈ Cp ([t0 , T ], R). Suppose that for any t1 ∈ [t0 , T ], if m(t1 ) = 0 and

m(t) < 0 for t0 6 t < t1 , then it follows that

RL q t0 Dt m(t1 )

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with p + q = 1.

> 0,

Based on the lemma above, a similar result can be acquired as follows, which will be used

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in the next section.

Lemma 2.8. Let m ∈ C 1 ([t0 , T ], R). Suppose that for any t1 ∈ [t0 , T ], if m(t1 ) = 0 and

C q t0 Dt m(t)

=

1 Γ(p)

Rt

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where

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m(t) < 0 for t0 6 t < t1 , then it follows that

t0 (t

C q t0 Dt m(t1 )

> 0,

− τ )p−1 x0 (τ )dτ and p + q = 1.

Proof. Since m ∈ C 1 ([t0 , T ], R), then m(t) is continuous on [t0 , T ] and (t − t0 )p m(t) is also

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continuous on [t0 , T ]. Thus, we have m ∈ Cp ([t0 , T ], R) based on the Definition 2.5. By Definition 2.2, Lemma 2.4 and Lemma 2.7, for 0 < q < 1, we have C q t0 Dt m(t)

m(t0 ) (t − t0 )p−1 , Γ(p)

q = RL t0 Dt m(t) −

and RL q t0 Dt m(t1 )

> 0.

Replace t by t1 in (2.1), then C q t0 Dt m(t1 )

q = RL t0 Dt m(t1 ) −

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m(t0 ) (t1 − t0 )p−1 . Γ(p)

(2.1)

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For any t0 6 t < t1 , since m(t0 ) < 0 and

RL q t0 Dt m(t1 )

C q t0 Dt m(t1 )

> 0, (t1 − t0 )p−1 > 0, we obtain

> 0. 

The proof is completed.

k

k

0

0

and the fractional-order differential system   C Dα y(t) = F (t, y), t0 t y(t+ 0 ) = x0 ,

(2.3)

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C α t0 Dt m n R , R ), F

where

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Consider the impulsive fractional hybrid system with impulse effects at fixed moments,   Dα x(t) = f (t, x, λk (xk )), t ∈ (tk , tk+1 ], C   t0 t + (2.2) x(t+ x+ k = 1, 2, · · · , k ) = xk , k = xk + Ik (xk ),     x = x(t ), x(t+ ) = x ,

denotes Caputo fractional derivative of order α, 0 < α < 1, f ∈ C(R+ × Rn ×

∈ C(R+ × Rn , Rn ), f (t, x, λk (xk )) = F (t, x(t)) + R(t, x, λk (xk )), R(t, x, λk (xk )) ∈

n n C(R+ × Rn × Rm , Rn ), λk ∈ C(Rn , Rm ), 4x(tk ) = x(t+ k ) − x(tk ), Ik ∈ C(R , R ), k = 1, 2, · · · ,

t0 < t1 < t2 < · · · < tk < tk+1 < · · · are impulsive moments, and tk → ∞, k → ∞.

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Let x0 ∈ Rn . Denote by x(t) = x(t; t0 , x0 ) the solution to system (2.2) and y(t) = y(t; t0 , x0 )

the solution to system (2.3) respectively, satisfying the initial conditions y(t+ 0 ; t0 , x0 ) = x0 .

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x(t+ 0 ; t0 , x0 ) = x0 ,

In general, the solutions x(t) = x(t; t0 , x0 ) are piecewise continuous functions with points of

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discontinuity of first type at which they are left continuous, that is, at the moments tk , k =

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1, 2, · · · , the following relations are satisfied [31]: + x(t− k ) = x(tk ) and x(tk ) = x(tk ) + Ik (x(tk )),

+ where x(t− k ) = lim x(tk − ε) and x(tk ) = lim x(tk + ε). ε→0−

ε→0+

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Without loss of generality, assume that the functions f, F, Ik , k = 1, 2, · · · are smooth enough

to guarantee the existence, uniqueness and continuability of the solutions of systems (2.2) and (2.3).

Moreover, the following assumption [25] is needed relative to system (2.3).

Assumption (H) The solutions y(t) = y(t; t0 , x0 ) of system (2.3) that exist for all t > t0 , are unique and depend continuously on the initial data, and ||y(t; t0 , x0 )|| is locally Lipschitzian in x0 .

Let Gk = (tk−1 , tk ) × Rn , k = 1, 2, · · · and G =

S∞

k=1 Gk .

In the further considerations,

the piecewise continuous auxiliary functions [31] are used, which are similar to the classical Lyapunov functions. 6

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Definition 2.9. [32] A function V : R+ × Rn → R+ belongs to the class ν0 , if: (1) V (t, x) is continuous in G and locally Lipschitz continuous with respect to x on each of the sets Gk , k = 1, 2, · · · . (2) For each k = 1, 2, · · · and x ∈ Rn , there exist the finite limits V (t− k , x) = lim V (t, x),

V (t+ k , x) = lim V (t, x)

and the following equality is valid V (t− k , x) = V (tk , x). 0

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t→tk t>tk

t→tk t
For a function V ∈ ν0 , the Dini-like fractional order derivative in Caputo’s sense is defined

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as follows. The specific formulation can be seen in [25]. 0

Definition 2.10. [25] Given a function V ∈ ν0 . For any fixed t > t0 , any arbitrary point s ∈

(t0 , t], and x ∈ Rn , the Caputo fractional Dini derivative of the Lyapunov function V (s, y(t; s, x)) is given by

α D+ V (s, y(t; s, x)) 1 = lim sup α [V (s, y(t; s, x)) − V (s − h, y(t; s − h, x − hα F (s, x)))], h→0+ h P where V (s−h, y(t; s−h, x−hα F (s, x))) = nr=1 (−1)r+1 αCr V (s−rh, y(t; s−rh, x−hα F (s, x))),

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C

and αCr represents the binomial coefficients in the definition of the Gr¨ unwald-Letnikov fractional

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derivative.

For convenience, the following classes of functions are introduced.

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K = {u ∈ C(R+ , R+ ), u(0) = 0, u is strictly increasing},  P C = u : R+ → R+ , continuous on (tk , tk+1 ] and lim u(t) = u(t+ k ) exists }, t→t+ k

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P CK = {u : R+ × R+ → R+ , ∀s ∈ R+ , u(·, s) ∈ P C, ∀t ∈ R+ , u(t, ·) ∈ K},  Γ = h : R+ × Rn → R+ , ∀x ∈ Rn , h(·, x) ∈ P C, ∀t ∈ R+ , h(t, ·) ∈ C(Rn , R+ ), and inf h(t, x) = x 0 ,  Γ0 = h ∈ Γ : sup h(t, x) exists for x ∈ Rn . R+

Here, we shall state the relationships between the two measures h0 (t, x) and h(t, x) (or h0

and h for short), some concepts of stability in terms of two measures of system (2.2), and some features of the function V (t, x) in terms of the two measures h0 and h. Definition 2.11. [23] Let h0 , h ∈ Γ. h0 is finer than h, that is, if there exists a δ > 0 and a

function φ ∈ P CK such that h(t, x) 6 φ(t, h0 (t, x)) whenever h0 (t, x) < δ. If φ ∈ K, then we say that h0 is uniformly finer than h.

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Definition 2.12. [23] Let V ∈ ν0 , h0 , h ∈ Γ. V (t, x) is said to be (1) h-positive definite, if there exists a function b ∈ K and a constant ρ > 0 such that h(t, x) < ρ implies b(h(t, x)) 6 V (t, x);

(2) weakly h0 -decrescent, if there exists a δ > 0 and a function a ∈ P CK such that h0 (t, x) < δ

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implies V (t, x) 6 a(h0 (t, x));

(3) h0 -decrescent if a ∈ K in (2).

Definition 2.13. [22] Let h0 , h ∈ Γ. The system is said to be (h0 , h)-stable, if for any given

ε > 0 and t0 ∈ R+ , there exists a δ = δ(t0 , ε) > 0 such that h0 (t0 , x0 ) < δ implies that h(t, x(t; t0 , x0 )) < ε, t > t0 .

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Definition 2.14. [28] Let h0 , h ∈ Γ. The system is said to be (h0 , h)-attractive, if for each ε > 0 and t0 ∈ R+ , there exist positive constants δ0 = δ0 (t0 ) and T = T (t0 , ε) such that h0 (t0 , x0 ) < δ0 implies h(t, x(t)) < ε, t > t0 + T .

˜ 0 (x) = sup h0 (t0 , x), h ˜ 1 (x) = sup h1 (t0 , x). Definition 2.15. [22] Let h0 , h ∈ Γ. We define h R+ R+ ˜ ˜ Then we say that the system (2.2) is (h0 , h1 )-strictly stable if given ε1 > 0 and t0 ∈ R+ , there

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˜ 0 (x0 ) 6 δ1 implies h ˜ 0 (y(t; t0 , x0 )) < ε1 , t > t0 and for every exists a δ1 = δ1 (ε1 ) such that h ˜ 1 (x0 ) implies ε2 < h ˜ 1 (t, y(t; t0 , x0 )), t > t0 . δ2 6 δ1 , there exists an ε2 6 δ2 such that δ2 6 h

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Based on the definitions mentioned above, it is easy to formulate other kinds of (h0 , h)stability for system (2.2). For more discussions of the concepts of two measures, one can be

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referred to the references [28, 33].

Remark 2.16. If h0 (t, x) = h(t, x) = ||x||, then the Definition 2.13 can be reduced to the well

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known concepts of stability of system (2.2) in the Lyapunov sense.

Remark 2.17. By choosing suitable formulation of the two measures (h0 , h), the concepts in terms of two measures (h0 , h) [28] enable us to unify a variety of stability notions, such as

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eventual stability, partial stability, relative stability, and conditional stability, which have been found in the literature but would otherwise be treated separately.

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Fractional variational comparison principle

In this section, a fractional variational comparison principle is established, which can connect the solutions of the perturbed system and the solutions of the unperturbed one through the maximal or minimal solution of a fractional scalar comparison system. As derivative results, some corollaries are obtained. For the sake of our later proofs, we present the following comparison lemma. 8

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Lemma 3.1. Let m ∈ C 1 ∈ ([t0 , ∞], R+ ), and C α t0 Dt m(t)

6 (>)g(t, m(t), σ(m(t0 ))),

where g ∈ C([t0 , ∞] × R+ × R, R+ ), σ ∈ C(R+ , R), and g(t, u, v) is nondecreasing in v, σ(u) is initial value problem (IVP) for the fractional equation   C Dα u(t) = g(t, u, σ(u0 )), t0 t 

u(t0 ) = u0 ,

existing on [t0 , ∞). If m(t0 ) 6 (>)u0 , then t ∈ [t0 , ∞].

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m(t) 6 (>)η(t),

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nondecreasing in u. Assume that η(t) = η(t; t0 , u0 ) is the maximal (minimal) solution of the

(3.1)

(3.2)

Proof. Case I: Suppose that all the inequalities are 6. Since u(t; t0 , u0 , ε) → η(t; t0 , u0 ) uniformly with ε → ∞, to prove (3.2), it is enough to prove that m(t) < u(t; t0 , u0 , ε),

t ∈ [t0 , ∞],

t0

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where u(t; t0 , u0 , ε) is any solution of IVP for   C Dα u(t) = g(t, u, σ(u0 )) + ε, t

(3.3)

(3.4)

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u(t0 ) = u0 + ε,

and ε being an arbitrary small number.

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If (3.3) is not true, then there would exist a t1 > t0 such that m(t1 ) = u(t1 ; t0 , u0 , ε), and for t0 6 t < t1 , m(t) < u(t; t0 , u0 , ε).

C α t0 Dt m(t1 )

α >C t0 Dt u(t1 ; t0 , u0 , ε)

(3.5)

= g(t1 , u(t1 ; t0 , u0 , ε), σ(u0 )) + ε.

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Now by Lemma 2.8 and the linearity of fractional operator [34], we have

Since g(t, u, v) is nondecreasing in v, σ(u) is nondecreasing in u, m(t1 ) = u(t1 ; t0 , u0 , ε) and

m(t0 ) 6 u(t0 ), this implies, from the preceding considerations, that C α t0 Dt m(t1 )

α >C t0 Dt u(t1 ; t0 , u0 , ε)

= g(t1 , u(t1 ; t0 , u0 , ε), σ(u0 )) + ε > g(t1 , m(t1 ), σ(m(t0 ))) + ε > g(t1 , m(t1 ), σ(m(t0 ))), which contradicts with

C α t0 Dt m(t1 )

6 g(t1 , m(t1 ), σ(m(t0 ))). Hence (3.3) is valid. 9

(3.6)

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Case II: For all the inequalities are >. Set m(t) e = −m(t). Similarly, to prove m(t) > η(t),

we need to prove that

m(t) e < −u(t; t0 , u0 , ε),

t ∈ [t0 , ∞],

where u(t; t0 , u0 , ε) is any solution of IVP for   C Dα u(t) = g(t, u, σ(u0 )) − ε, t



u(t0 ) = u0 − ε,

(3.8)

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t0

(3.7)

and ε being an arbitrary small number, since u(t; t0 , u0 , ε) → η(t; t0 , u0 ) uniformly with ε → ∞. If (3.7) does not hold, then there would exist a t1 > t0 such that m(t e 1 ) = −u(t1 ; t0 , u0 , ε),

and for t0 6 t < t1 , m(t) e < −u(t; t0 , u0 , ε). It follows from Lemma 2.8 and the linearity of fractional operator [34] that

α > −C t0 Dt u(t1 ; t0 , u0 , ε)

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C α e 1) t0 Dt m(t

= −g(t1 , u(t1 ; t0 , u0 , ε), σ(u0 )) + ε.

(3.9)

According to the monotonicity of functions g and σ and −m(t e 0 ) = m(t0 ) > u(t0 ), we can

obtain that

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σ(−m(t e 0 )) 6 σ(u(t0 ))

and

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−g(t1 , u(t1 ; t0 , u0 , ε), σ(u0 )) > −g(t1 , u(t1 ; t0 , u0 , ε), σ(−m(t e 0 ))).

Since m(t e 1 ) = −u(t1 ; t0 , u0 , ε), this implies that

α > −C t0 Dt u(t1 ; t0 , u0 , ε)

= −g(t1 , u(t1 ; t0 , u0 , ε), σ(u0 )) + ε

(3.10)

6 g(t1 , −m(t e 1 ), σ(−m(t e 0 ))) − ε

(3.11)

> −g(t1 , −m(t e 1 ), σ(−m(t e 0 ))) + ε,

C α t0 Dt m(t1 )

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that is,

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C α e 1) t0 Dt m(t

which shows a contradiction with the proof is completed.

< g(t1 , −m(t e 1 ), σ(−m(t e 0 ))),

C α t0 Dt m(t1 )

> g(t1 , m(t1 ), σ(m(t0 ))). Thus, (3.7) is valid and 

Remark 3.2. The discussions of the existence of extremal solutions for initial value problems (3.4) and (3.8) can be seen in [6], [35] and [36], which present an effective and reasonable result. Also, the conclusion that u(t; t0 , u0 , ε) → η(t; t0 , u0 ) uniformly with ε → ∞ can be obtained in the references mentioned above.

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Consider the following scalar comparison hybrid system  C α    t0 Dt u(t) = g(t, s, u, σk (uk )), t0 6 t, t ∈ (tk , tk+1 ],  + u(t+ u+ k ) = uk , k = ψk (uk ),     u = u(t ), ψ (u ) = u , u(t+ ) = u , 0 0 0 0 k k 0

(3.12)

and for all k ∈ N

lim

(t,s,u)→(t,t+ k ,w)

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where g : R2+ × R+ × R → R+ is continuous on (tk , tk+1 ] in s, k = 0, 1, 2, · · · , for each t ∈ R+ g(t, s, u, σk (uk )) = g(t, t+ k , w, σk (wk )),

where wk = w(tk ), and for any (t, s, u, v), g(t, s, u, v) is nondecreasing in v, ψk (u) ∈ C(R+ , R+ ) and ψk (u) is nondecreasing in u, σk (u) ∈ C(R+ , R) and σk (u) is nondecreasing in u.

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We now acquire the fractional comparison theorem as follows.

Theorem 3.3. Assume that assumption (H) holds. Suppose that 0

(1) V : R+ × Rn → R+ , V ∈ ν0 , V (t, x) is locally Lipschitzian in x with Lipschitz constant L > 0 and for t0 6 s 6 t, x ∈ S(ρ), C

α D+ V (s, y(t; s, x)) 6 g(t, s, V (s, y(t; s, x)), σk (V (tk , y(t; tk , x(tk ))))),

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 where s 6= tk , k = 1, 2, · · · , S(ρ) = x ∈ Rn ||x|| < ρ ;

ED

(2) V (s, y(t; s, x + Ik (x(s)))) 6 ψk (V (s, y(t; s, x))), s = tk , k = 1, 2, · · · ; (3) ψk (u) ∈ C(R+ , R+ ), k = 1, 2, · · · is nondecreasing in u;

PT

(4) The function g : R2+ × R+ × R → R+ is continuous in each of the sets (tk , tk+1 ] in s, k = 0, 1, 2, · · · , and for any (t, s, u, v), g(t, s, u, v) is nondecreasing in v, σk (u) ∈ C(R+ , R)

CE

and σk (u) is nondecreasing in u;

(5) The maximal solution r(t; s, t0 , u0 ) with initial value (t0 , u0 ) of the scalar system (3.12) is

AC

defined in the interval [t0 , ∞).

If V (t0 , y(t; t0 , x0 )) 6 u0 , then we have V (s, y(t; s, x)) 6 r(t; s, t0 , u0 ),

t0 6 s 6 t.

Specifically, if s = t, then V (t, x(t; t0 , x0 )) 6 r˜(t; t0 , u0 ), where r˜(t; t0 , u0 ) = r(t; t, t0 , u0 ).

11

(3.13)

ACCEPTED MANUSCRIPT

Proof. Let x0 ∈ Rn and x(t) = x(t; t0 , x0 ) be any solution of system (2.2) such that ||x0 || < ρ.

Let y(t) = y(t; s, x(s)) be any solution of system (2.3) with the initial value (s, x(s)) in the interval (t0 , t] and V (t0 , y(t; t0 , x0 )) 6 u0 . Set m(t, s) = V (s, y(t; s, x(s))). For k = 1, 2, · · · , s 6= tk , and for h > 0 small enough, we have the following result.

Since for each (t, s), V (t, x) and ||y(t; s, x)|| are locally Lipschitzian in x with Lipschitz

constants L and M , respectively. Then, for any fixed t, we have n X (−1)r+1 αCr m(t, s − rh)

CR IP T

m(t, s) −

r=1

= V (s, y(t; s, x)) −

n X (−1)r+1 αCr V (s − rh, y(t; s − rh, S(x, h, r, α))) r=1

n X 6 V (s, y(t; s, x)) − (−1)r+1 αCr V (s − rh, y(t; s − rh, x − hα F (s, x)))

+ LM

AN US

r=1

n X

αCr (hα ),

r=1

where L > 0, M > 0, S(x, h, r, α) = x(s) − hα F (s, x) − (hα ) with

(hα ) hα

details about the form of S(x, h, r, α) can be seen in [25].

→ 0 as h → 0. The

M

The proof of the inequality above is not a new one, which is similar to that of Theorem 4.2 in [25], so we omit it here.

α D+ m(t, s)

n

X 1 (−1)r+1 αCr m(t, s − rh)) = lim sup α (m(t, s) − h→0+ h r=1

1 6 lim sup α V (s, y(t; s, x)) h→0+ h n X
− (−1)r+1 αCr V (s − rh, y(t; s − rh, x − hα F (s, x)))

AC

CE

PT

C

ED

Dividing through hα by two sides and taking limits as h → 0+ , we get

r=1

+ LM lim sup h→0+

n
1 X αCr (hα ) . α h r=1

The right side does approach to 0 as h → 0+ and hence C

α α D+ m(t, s) 6 C D+ V (s, y(t; s, x))

6 g(t, s, V (s, y(t; s, x)), σk (V (tk , y(t; tk , x(tk ))))) = g(t, s, m(t, s), σk (m(t, tk ))), where s 6= tk , k = 1, 2, · · · .

To prove (3.13), firstly, we should prove that V (s, y(t; s, x)) 6 r0 (t; s, t0 , u+ 0 ), 12

t0 6 s 6 t,

s ∈ (t0 , t1 ],

(3.14)

ACCEPTED MANUSCRIPT

where r0 (t; s, t0 , u+ 0 ) is the maximal solution of the equation

C α t0 Dt u(t)

= g(t, s, u, σ0 (u+ 0 )) with

u(t0 ) = u0 = u+ 0 in the interval (t0 , t1 ]. Then on (t0 , t1 ], it follows from (3.14) that C

α D+ m(t, s) 6 g(t, s, V (s, y(t; s, x)), σ0 (V (t0 , y(t; t0 , x0 ))))

= g(t, s, m(t, s), σ0 (m(t, t0 ))).

CR IP T

Since m(t, t0 ) = V (t0 , y(t; t0 , x0 )) 6 u+ 0 , due to the monotonic character of function σ0 in u and g(t, s, u, v) in v, by applying Lemma 3.1 with appropriate modifications, we obtain m(t, s) 6 r0 (t; s, t0 , u+ 0 ),

s 6 t,

s ∈ (t0 , t1 ].

V (s, y(t; s, x)) 6 r0 (t; s, t0 , u+ 0 ),

t0 6 s 6 t.

AN US

That is

Specially, we have m(t, t1 ) 6 r0 (t; t1 , t0 , u+ 0 ) with s = t1 . Then, applying Condition (2) and the fact that ψk (u) is nondecreasing in u, we obtain that + + + m(t, t+ 1 ) = V (t1 , y(t; t1 , x(t1 )))

M

+ = V (t+ 1 , y(t; t1 , x(t1 ) + I1 (x(t1 ))))

6 ψ1 (V (t1 , y(t; t1 , x(t1 )))) = ψ1 (m(t, t1 ))

ED

+ + + 6 ψ1 (r0 (t; t1 , t0 , u+ 0 )) = r0 (t; t1 , t0 , u0 ) = u1 .

PT

Then on (t1 , t2 ], by (3.14), we have

  C Dα m(t, s) 6 g(t, s, m(t, s), σ1 (m(t, t1 ))),

CE

+



(3.15)

+ m(t, t+ 1 ) 6 u1 .

It also implies due to the monotonic character of σ1 in u and g(t, s, u, v) in v, by Lemma 3.1

AC

with appropriate modifications, that m(t, s) 6 r1 (t; s, t1 , u+ 1 ),

s 6 t,

s ∈ (t1 , t2 ],

V (s, y(t; s, x)) 6 r1 (t; s, t1 , u+ 1 ),

t0 6 s 6 t,

and

where r1 (t; s, t1 , u+ 1 ) is the maximal solution of the equation

C α t0 Dt u(t)

= g(t, s, u, σ1 (u+ 1 )) with

+ + + u+ 1 = ψ1 (r0 (t; t1 , t0 , u0 )) = r0 (t; t1 , t0 , u0 ) in the interval (t1 , t2 ].

Repeating the procedure above, generally, let s ∈ (tk−1 , tk ], k > 1, V (s, y(t; s, x)) 6 rk−1 (t; s, tk−1 , u+ k−1 ), 13

t0 6 s 6 t,

(3.16)

ACCEPTED MANUSCRIPT

+ C α where rk−1 (t; s, tk−1 , u+ k−1 ) is the maximal solution of the equation t0 Dt u(t) = g(t, s, u, σk−1 (uk−1 )) + + + with u+ k−1 = ψk−1 (rk−2 (t; tk−1 , tk−2 , uk−2 )) = rk−2 (t; tk−1 , tk−2 , uk−2 ) in the interval (tk−1 , tk ].

Next, we should prove that V (s, y(t; s, x)) 6 rk (t; s, tk , u+ k ),

t0 6 s 6 t

with s ∈ (tk , tk+1 ].

CR IP T

By (3.16), when s = tk , we have

m(t, tk ) = V (tk , y(t; tk , x(tk ))) 6 rk−1 (t; tk , tk−1 , u+ k−1 ). It follows from Conditions (2) and (3) that + + + m(t, t+ k ) = V (tk , y(t; tk , x(tk )))

AN US

+ = V (t+ k , y(t; tk , x(tk ) + Ik (x(tk ))))

6 ψk (V (tk , y(t; tk , x(tk )))) = ψk (m(t, tk ))

+ + + 6 ψk (rk−1 (t; tk , tk−1 , u+ k−1 )) = rk−1 (t; tk , tk−1 , uk−1 ) = uk .

+

CE

ED

m(t, s) 6 rk (t; s, tk , u+ k ),

and

s 6 t,

V (s, y(t; s, x)) 6 rk (t; s, tk , u+ k ),

s ∈ (tk , tk+1 ], t0 6 s 6 t,

where rk (t; s, tk , u+ k ) is the maximal solution of the equation

AC

(3.17)

+ m(t, t+ k ) 6 uk .

PT

Similarly, we obtain



M

Then on (tk , tk+1 ], we have   C Dα m(t, s) 6 g(t, s, m(t, s), σk (m(t, tk ))),

C α t0 Dt u(t)

= g(t, s, u, σk (u+ k )) with

+ + + u+ k = ψk (rk−1 (t; tk , tk−1 , uk−1 )) = rk−1 (t; tk , tk−1 , uk−1 ) in the interval (tk−1 , tk ].

Thus,

m(t, s) = V (s, y(t; s, x)) 6 rk (t; s, tk , u+ k ),

k = 0, 1, 2, · · · ,

and V (t, x(t; t0 , x0 )) 6 rk (t; tk , u+ k ), with s = t.

14

k = 0, 1, 2, · · ·

ACCEPTED MANUSCRIPT

(3.18)

CR IP T

Generally, for any s ∈ [t0 , t], we have the following equalities   r0 (t; s, t0 , u+ s ∈ (t0 , t1 ],  0 ),      r1 (t; s, t1 , u+ s ∈ (t1 , t2 ],  1 ),    .. .. r˜(t; s, t0 , u0 ) = . .      rk (t; s, tk , u+ s ∈ (tk , tk+1 ],  k ),     .. ..  . .

+ α where r˜(t; s, t0 , u0 ) is the maximal solution of equation C t0 Dt u(t) = g(t, s, u, σk (uk )) in the inter+ + + val (tk , tk+1 ], k = 0, 1, 2, · · · for which u+ k = ψk (rk−1 (t; tk , tk−1 , uk−1 )) = rk−1 (t; tk , tk−1 , uk−1 ),

k = 1, 2, · · · and u+ 0 = u0 . Thus, we can get that

t0 6 s 6 t.

AN US

m(t, s) = V (s, y(t; s, x)) 6 r(t; s, t0 , u0 ), Specifically, we have

V (t, x(t)) 6 r˜(t; t0 , u0 ),



with s = t, where r˜(t; t0 , u0 ) = r(t; t, t0 , u0 ).

M

Remark 3.4. In Theorem 3.3, the variational Lyapunov function V (t, x) plays the role of connecting the solutions of systems (2.2), (2.3) and (3.12). Based on this, the stability properties

ED

of system (2.2) can be obtained by the corresponding stability properties of system (2.3). Remark 3.5. In Theorem 3.3, if Conditions (1) and (2) are replaced with C

α V (s, y(t; s, x)) > g(t, s, V (s, y(t; s, x)), σ (V (t , y(t; t , x(t ))))), D+ k k k k

PT

(1)

(2) V (s, y(t; s, x + Ik (x(s)))) > ψk (V (s, y(t; s, x))).

CE

and r(t; s, t0 , u0 ) represents the minimal solution with initial value (t0 , u0 ) of the scalar system (3.12) defined in the interval [t0 , ∞), while other conditions remain unchanged, then, we can

obtain that if V (t0 , y(t; t0 , x0 )) > u0 , then for t0 6 s 6 t, we have V (s, y(t; s, x)) > r(t; s, t0 , u0 ).

AC

Specifically, if s = t, then V (t, x(t; t0 , x0 )) > r˜(t; t0 , u0 ), where r˜(t; t0 , u0 ) = r(t; t, t0 , u0 ). The proof of this case is similar to that of Theorem 3.3 by setting m(t, s) = −V (s, y(t; s, x(s))) and using Lemma 3.1 under the condition that the inequalities are all >. Corollary 3.6. In Theorem 3.3, if for all k, F (t, y) = 0, then V (t, x(t; t0 , x0 )) 6 r˜(t; t0 , y(t; t0 , x0 )) with V (t0 , x0 ) 6 u0 . In fact, in this case, y(t0 ; t0 , x0 ) = x0 , the definition of the Caputo fractional Dini derivative reduces to C

α D+ V (s, x) = lim sup h→0+

1 [V (s, x) − V (s − h, x − hα F (s, x))]. hα 15

(3.19)

ACCEPTED MANUSCRIPT

Remark 3.7. Note that Corollary 3.6 is the Theorem 3.1 of [32] when we set s = t in equation (3.19). So, the result of [32] can be regarded as a special case of Theorem 3.3. For some special cases of the function g, the following corollaries are derived from Theorem 3.3. The proofs are straightforward, so we omit them here.

u0 = V (t0 , y(t; t0 , x0 )) and for all k, ψk (uk ) = u, we have

CR IP T

Corollary 3.8. In Theorem 3.3 and due to Remark 3.5, in the case g(t, s, u, σk (uk )) = 0,

V (t, x(t; t0 , x0 )) 6 (>)V (t0 , y(t; t0 , x0 )). If V (t, x) = ||x||, then

||x(t; t0 , x0 )|| 6 (>)||y(t; t0 , x0 )||.

AN US

Corollary 3.9. In Theorem 3.3, in the case g(t, s, u, σk (uk )) = βu, u0 = V (t0 , y(t; t0 , x0 )) and for all k, ψk (uk ) = u, we have

V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 ))Eα (β(t − t0 )α ),

t ∈ [t0 , ∞).

Corollary 3.10. Assume that assumption (H) holds. In Theorem 3.3, suppose that C

α V (s, y(t; s, x)) 6 −c(h (s, y(t; s, x)))λ(s, σ ), where λ(t, σ ) : R × R → R is inteD+ 1 + + k k

M

(1)

grable and c ∈ K, h1 ∈ Γ0 , s 6= tk ;

PT

Then for t > t0 , we have

ED

+ + (2) V (t+ k , y(t; tk , x(tk ))) 6 V (tk , y(t; tk , xk )), k = 1, 2, · · · .

(1) if k = 0, t ∈ (t0 , t1 ], then

AC

CE

V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 )) Z t 1 − (t − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ, Γ(α) t0

(2) if k = 1, 2, · · · , t ∈ (tk , tk+1 ], then V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 )) Z ti k X 1 − (ti − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σi−1 )dτ Γ(α) ti−1 i=1 Z t 1 − (t − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σk )dτ. Γ(α) tk

16

ACCEPTED MANUSCRIPT

Proof. Let t0 < s 6 t1 , s 6 t. Set W (s, y(t; s, x)) 1 = V (s, y(t; s, x)) + Γ(α) = V (s, y(t; s, x)) +

Z

s

t0

(s − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ

−α t0 Ds c(h1 (s, y(t; s, x)))λ(s, σ0 ).

C

CR IP T

By using Lemma 2.4, we get α D+ W (s, y(t; s, x))

−α α α = C D+ V (s, y(t; s, x)) + C t0 Ds t0 Ds c(h1 (s, y(t; s, x)))λ(s, σ0 )

6 −c(h1 (s, y(t, s, x)))λ(s, σ0 ) + c(h1 (s, y(t; s, x)))λ(s, σ0 ) 6 0.

AN US

On the other hand,

W (t0 , y(t; t0 , x0 )) = V (t0 , y(t; t0 , x0 )) = u0 and + + W (t+ 1 , y(t; t1 , x(t1 )))

Z t1 1 + =V (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ Γ(α) t0 Z t1 1 6 V (t1 , y(t; t1 , x1 )) + (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ Γ(α) t0

ED

M

+ + (t+ 1 , y(t; t1 , x(t1 )))

= W (t1 , y(t; t1 , x(t1 ))). By Corollary 3.6, we have

PT

W (s, y(t; s, x)) 6 W (t0 , y(t; t0 , x0 )),

AC

CE

which implies, by definition of W , V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 )) Z t 1 − (t − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ Γ(α) t0

with s = t.

If s ∈ (t1 , t2 ], s 6 t, set W (s, y(t; s, x))

Z t1 1 (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ = V (s, y(t; s, x)) + Γ(α) t0 Z s 1 + (s − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ1 )dτ. Γ(α) t1 17

ACCEPTED MANUSCRIPT

Similarly, we get C

α D+ W (s, y(t; s, x)) 6 0,

and + + W (t+ 1 , y(t; t1 , x(t1 )))

Z t1 1 (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ =V + Γ(α) t0 Z t1 1 6 V (t1 , y(t; t1 , x1 )) + (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ Γ(α) t0 = W (t1 , y(t; t1 , x(t1 ))) 6 r(t; t1 , t0 , u0 ) = u+ 1 = u0 = W (t0 , y(t; t0 , x0 )),

CR IP T

+ + (t+ 1 , y(t; t1 , x(t1 )))

AN US

where r(t; t1 , t0 , u0 ) is the maximal solution of the equation (3.12) with g(t, s, u, σk (uk )) = 0 and initial condition u0 = V (t0 , y(t; t0 , x0 )) in the interval (t1 , t2 ]. Also, we can prove that

+ + W (t+ 2 , y(t; t2 , x(t2 ))) 6 W (t2 , y(t; t2 , x(t2 ))).

By the similar proof of Theorem 3.3, based on Lemma 3.1, Corollary 3.8 and the definition of

M

W , when s = t, we have

PT

ED

V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 )) Z t1 1 − (t1 − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ0 )dτ Γ(α) t0 Z t 1 − (t − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ1 )dτ. Γ(α) t1

Stability in terms of two measures

AC

4



CE

If t ∈ (tk , tk+1 ], by repeating the procedure above, the result can be obtained.

The concepts of stability in terms of two measures can unify many known stability notions

such as Lyapunov stability, eventual stability, and partial stability in a single setup, and offer a general framework for investigation of stability theory. The significance of the study of the stability theory in terms of two measures is demonstrated in many studies [28, 37–39]. The aim of this section is to discuss the stability properties in terms of two measures of impulsive hybrid systems with fractional order by applying the fractional variational comparison principle obtained in Section 3. Several stability and instability criteria for system (2.2) are derived. Let ρ > 0 and h ∈ Γ0 . Define S(h, ρ) = {(t, x) ∈ R+ × Rn : h(t, x) < ρ}. Throughout this

section, h(t, x) and V (t, x) stand for h(t, x(t)) and V (t, x(t)), respectively. 18

ACCEPTED MANUSCRIPT

In the following theorems, we present the results of stability, uniform stability, and unstability properties in terms of two measures for system (2.2). Theorem 4.1. Assume that (1) h0 , h ∈ Γ0 and h0 is finer than h; 0

C

α D+ V (s, y(t; s, x)) 6 0,

where s 6= tk ;

CR IP T

(2) V ∈ ν0 , V (t, x) and y(t; s, x) are locally Lipschitzian in x for each (t, x);

(3) V (s, y(t; s, x + Ik (x(s)))) 6 V (s, y(t; s, x)), s = tk , k = 1, 2, · · · ;

AN US

(4) V (t, x) is h-positive definite on S(h, ρ) and weakly h0 -decrescent, where (t, x) ∈ S(h, ρ), ρ > 0;

(5) There exists a ρ0 , 0 < ρ0 < ρ such that h(tk , x) < ρ0 implies h(t+ k , x + Ik (x)) < ρ. ˜ 0, h ˜ 0 )-stability of system (2.3) implies the corresponding (h0 , h)-stability of system Then, the (h (2.2).

M

Proof. Let 0 < ε < ρ0 and t0 ∈ R+ be given, and u0 = V (t0 , y(t; t0 , x0 )). Since h0 is finer than

ED

h, there exist a δ1 > 0 and a function φ ∈ P CK such that h(t, x) 6 φ(t, h0 (t, x)),

(t, x) ∈ S(h0 , δ1 ).

(4.1)

PT

whenever h0 (t, x) < δ1 with φ(t, δ1 ) 6 ρ for t > t0 .

CE

Since V (t, x) is h-positive definite on S(h, ρ), there exists a function b ∈ K such that b(h(t, x)) 6 V (t, x),

(t, x) ∈ S(h, ρ).

(4.2)

AC

Because V (t, x) is weakly h0 -decrescent, there exist a δ0 > 0 and a function a ∈ P CK such that V (t, x) 6 a(t, h0 (t, x)),

(t, x) ∈ S(h0 , δ0 ).

(4.3)

Due to the property of function a, choose η = η(t0 , ε) < min{ρ, δ0 , δ1 } such that for t > t0 , a(t0 , h0 (t0 , y(t))) < b(ε),

(4.4)

whenever h0 (t0 , y(t)) < η. ˜ 0, h ˜ 0 )-stable. Then, for the η chosen above, there exists a Assume that system (2.3) is (h δ = δ(t0 , η) > 0 (δ < η) such that h0 (t0 , x0 ) < δ implies h0 (t0 , y(t)) < η, 19

t > t0 ,

(4.5)

ACCEPTED MANUSCRIPT

where y(t) = y(t; t0 , x0 ) is any solution of system (2.3). Suppose that x(t) = x(t; t0 , x0 ) is any solution of system (2.2) with h0 (t0 , x0 ) < δ. In views of (4.1)-(4.5) and the choice of δ, note that b(h(t0 , x0 )) 6 V (t0 , x0 ) 6 a(t0 , h0 (t0 , x0 )) < b(ε), which implies h(t0 , x0 ) < ε.

h(t, x) < ε,

t > t0 ,

with h0 (t0 , x0 ) < δ.

CR IP T

We claim that with this δ, system (2.2) is (h0 , h)-stable. That is

If it does not hold, then there would exist a solution x(t) = x(t; t0 , x0 ) of (2.2) with h0 (t0 , x0 ) < δ and a t∗ > t0 such that tk < t∗ 6 tk+1 for some k, satisfying t ∈ [t0 , tk ].

AN US

h(t∗ , x(t∗ )) > ε and h(t, x(t)) < ε, Since 0 < ε < ρ0 , by Condition (5), we have

+ h(t+ k , x(tk )) < ρ. 0

Hence, we can find a t ∈ (tk , t∗ ) such that 0

0

M

ε 6 h(t , x(t )) < ρ and h(t, x(t)) < ρ,

0

t ∈ [t0 , t ].

ED

It follows from Condition (1), Condition (2) and Corollary 3.8, that V (t, x(t)) 6 V (t0 , y(t)),

0

t ∈ [t0 , t ].

PT

By using Condition (4), and (4.2)-(4.5), we have 0

0

0

0

0

0

b(ε) 6 b(h(t , x(t ))) 6 V (t , x(t )) 6 V (t0 , y(t ; t0 , x0 )) 6 a(t0 , h0 (t0 , y(t ))) < b(ε),

CE

which is a contradiction. Thus, system (2.2) is (h0 , h)-stable. The proof is completed.



Theorem 4.2. In Theorem 4.1, if Conditions (1) and (4) are replaced with

AC

(1) h0 is uniformly finer than h; (2) V (t, x) is h-positive definite on S(h, ρ) and h0 -decrescent, where ρ > 0.

˜ 0, h ˜ 0 )-uniform stability of system (2.3) implies (h0 , h)-uniform stability of system Then, the (h (2.2). Proof. Since V (t, x(t)) is h0 -decrescent, then there exist a δ1 > 0 and a function a ∈ K such

that h0 (t, x) < δ1 implies

V (t, x) 6 a(h0 (t, x)). For any given 0 < ε < ρ0 and t0 ∈ R+ , by the property of function a, choose η > 0 such that 20

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a(η) < b(ε) and η < δ1 . ˜ 0, h ˜ 0 )-uniformly stable. Then there exists a δ = δ(η) > 0 such Assume that system (2.3) is (h that for any given t0 ∈ R+ and for η > 0 chosen above, h0 (t0 , x0 ) < δ implies h0 (t0 , y(t; t0 , x0 )) < η,

t > t0 ,

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where y(t; t0 , x0 ) is any solution of system (2.3). Suppose that x(t) = x(t; t0 , x0 ) is any solution of system (2.2) with h0 (t0 , x0 ) < δ. By a similar proof of Theorem 4.1, we can obtain that h(t, x(t)) < ε, t > t0 , where δ is independent of t0 . This completes the proof of (h0 , h)-uniform stability of system (2.2). Theorem 4.3. Let h0 , h ∈ Γ0 . Assume that 0



C

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(1) V ∈ ν0 , V (t, x) and y(t; s, x) are locally Lipschitzian in x for each (t, x); α D+ V (s, y(t; s, x)) > 0,

where s 6= tk ;

M

(2) for all k ∈ Z+ , s = tk and (tk , x) ∈ S(h, ρ),

V (s, y(t; s, x + Ik (x(s)))) > V (s, y(t; s, x));

ED

(3) V (t, x) is h-decrescent on S(h, ρ) and h0 -positive definite. ˜ 0, h ˜ 0 )-unstability of system (2.3) implies (h0 , h)-unstability of system (2.2). Then, the (h

PT

˜ 0, h ˜ 0 )-unstable. Then, there exists a ε0 > 0 such that for Proof. Suppose that system (2.3) is (h any δ ∗ > 0, h0 (t0 , y(t; t0 , x0 )) > ε0 with h0 (t0 , x0 ) < δ ∗ . Thus, for some t∗ > t0 , there exists a

CE

solution y(t; t0 , x0 ) of (2.3) with h0 (t0 , x0 ) < δ ∗ such that h0 (t0 , y(t∗ ; t0 , x0 )) = ε0 . We claim that system (2.2) is (h0 , h)-unstable. If it is not true, then for the above ε0 > 0

AC

and δ ∗ > 0 such that h0 (t0 , x0 ) < δ ∗ implies h(t, x(t)) < a−1 (b(ε0 )), t > t0 . From Condition (1), Condition (2) and Corollary 3.8, we get V (t, x) > V (t0 , y(t; t0 , x0 )), t > t0 .

Since V (t, x(t)) is h-decrescent on S(h, ρ) and h0 -positive definite, there exist a δ1 > 0,

δ1 = max{ε0 , a−1 (b(ε0 ))} < ρ and functions a, b ∈ K such that V (t, x(t)) 6 a(h(t, x)),

(t, x) ∈ S(h, δ1 ),

(4.6)

b(h0 (t, x)) 6 V (t, x(t)),

(t, x) ∈ S(h0 , δ1 ).

(4.7)

and

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Thus, we have b(ε0 ) = a(a−1 (b(ε0 ))) > a(h(t∗ , x(t∗ ))) > V (t∗ , x(t∗ )) > V (t0 , y(t∗ ; t0 , x0 ))

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> b(h0 (t0 , y(t∗ ; t0 , x0 ))) = b(ε0 ), which is a contradiction. Thus, system (2.2) is (h0 , h)-unstable. The proof is completed.



The following theorem presents the relationship between the strict uniform stability of system (2.3) and the uniformly asymptotic stability of system (2.2). The modified concepts of strict stability [40] in terms of two measures promoted in [22] are applied to obtain the sufficient

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condition of asymptotic stability property. Theorem 4.4. Assume that

(1) h0 , h1 , h ∈ Γ0 . h0 is uniformly finer than h1 and h; h1 is uniformly finer than h; 0

(2) V ∈ ν0 and y(t; s, x) are locally Lipschitzian in x for each (t, x);

(4)

C

M

(3) V (t, x) is h-positive definite and h1 -decrescent;

α V (s, y(t; s, x)) 6 −c(h (s, y(t; s, x)))λ(s, σ ), s 6= t , t 6 s 6 t, (s, x) ∈ S(h, ρ), D+ 1 0 k k

ED

where c ∈ K, λ(t, σk ) : R+ × R → R+ is integrable and satisfying: N

1 X Γ(α)

PT

i=1

Z

bi

ai

(bi − t)α−1 λ(t, σi )dt = ∞,

(N → ∞);

CE

(5) V (s, y(t; s, x + Ik (x(s)))) 6 V (s, y(t; s, x)), s = tk , k = 1, 2, · · · ; (6) There exists a ρ0 ∈ (0, ρ) such that h(tk , x) < ρ0 implies h(t+ k , x + Ik (x)) < ρ.

AC

˜ 0, h ˜ 1 )-strict uniform stability of system (2.3) implies (h0 , h)-uniformly asymptotic Then, the (h stability of system (2.2). Proof. Since V (t, x) is h-positive definite, there exist a ρ0 ∈ (0, ρ) and a function b ∈ K such that

b(h(t, x)) 6 V (t, x),

(t, x) ∈ S(h, ρ0 ).

(4.8)

Also, since V (t, x) is h0 -decrescent, there exist a δ ∗ > 0 and a function a ∈ K such that V (t, x) 6 a(h1 (t, x)),

22

(t, x) ∈ S(h1 , δ ∗ ).

(4.9)

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From Theorem 4.2, it can be proven that system (2.2) is (h0 , h)-uniformly stable. That is, for any given ε = ρ0 > 0 and t0 ∈ R+ , there exists a δ = δ(ε) = δ(ρ0 ) > 0 such that h(t, x(t)) < ρ0 ,

t > t0 ,

(4.10)

with h0 (t0 , x0 ) < δ. ˜ 0, h ˜ 1 )-strictly uniformly stable. Then, in view of Definition Suppose that system (2.3) is (h

h0 (t0 , y(t; t0 , x0 )) < ε1

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2.15, for any given ε1 > 0 and t0 ∈ R+ , there exists a δ1 = δ1 (ε1 ) > 0 such that for all t > t0 , (4.11)

with h0 (t0 , x0 ) < δ1 . And for every δ2 6 δ1 , whenever h1 (s, x(s)) > δ2 , there exists a ε2 6 δ2 such that

t0 6 s 6 t.

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h1 (s, y(t; s, x(s))) > ε2 ,

(4.12)

From Condition (1), there exists a φ1 ∈ K such that for all t > t0

h1 (t0 , y(t; t0 , x0 )) 6 φ1 (h0 (t0 , y(t; t0 , x0 ))) < φ1 (ε1 ) < δ ∗ ,

(4.13)

with h0 (t0 , y(t; t0 , x0 )) < ε1 . Thus, by the property of function a, we obtain

M

a(h1 (t0 , y(t; t0 , x0 ))) < a(φ1 (ε1 )) < a(δ ∗ ). 0

(4.14)

Take δ = min{δ1 , δ ∗ , δ}. Due to the property of function λ(t, σk ), there exist M > 0 and

ED

T = T (ε) > 0, where γ is a constant, 0 < γ 6 ai − ti and ai < ti+1 , such that T

PT

1 X Γ(α) i=1

Z

αi

ti

(αi − t)α−1 λ(t, σi )dt > M.

0

Next, we should prove that for δ and T = T (ε) above, system (2.2) is (h0 , h)-uniformly

AC

CE

attractive. Choosing η > 0 and η > δ2 , we assume that for any t ∈ [t0 , t0 + T ], the inequality h1 (t, x(t)) > η

(4.15)

0

holds whenever h0 (t0 , x0 ) < δ . Suppose that there exist 2T − 1 impulsive points between t0 and t0 + T . Set ti = t0 + 2i ,

i = 1, 2, · · · , 2T − 1. By Conditions (3)-(6) and Corollary 3.8, we have

V (t, x(t; t0 , x0 )) 6 V (t0 , y(t; t0 , x0 )) k Z 1 X ti − (ti − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σi−1 )dτ Γ(α) t i−1 i=1 Z t 1 − (t − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σk )dτ. Γ(α) tk 23

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Setting t = t0 + T , M =

a(δ ∗ )+1 c(ε2 ) ,

from (4.9), (4.11)-(4.14), we have

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V (t0 + T, x(t0 + T )) 6 V (t0 , y(t0 + T ; t0 , x0 )) 2T −1 Z 1 X ti − (ti − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σi−1 )dτ Γ(α) i=1 ti−1 Z t0 +T 1 − (t0 + T − τ )α−1 c(h1 (τ, y(t; τ, x)))λ(τ, σ2T −1 )dτ Γ(α) t2T −1 2T −1 Z c(ε2 ) X ti 6 a(h1 (t0 , y(t0 + T ))) − (ti − τ )α−1 λ(τ, σi−1 )dτ Γ(α) i=1 ti−1 Z t0 +T c(ε2 ) − (t0 + T − τ )α−1 λ(τ, σ2T −1 )dτ Γ(α) t2T −1 1 T Z 1 c(ε2 ) X t0 +i− 2 6 a(h1 (t0 , y(t0 + T ))) − [(t0 + i − ) − τ ]α−1 λ(τ, σi )dτ Γ(α) 2 t0 +i−1 i=1

∗

< a(δ ) − c(ε2 )M = a(δ ∗ ) − c(ε2 )

a(δ ∗ ) + 1 < 0. c(ε2 )

However, from (4.8), we get 0 < b(h(t0 + T, x(t0 + T ))) 6 V (t0 + T, x(t0 + T )), which shows a

M

contradiction.

0

Thus, in any case, we have h1 (t, x(t)) < η for t > t0 + T whenever h0 (t0 , x0 ) < δ , which

ED

implies that the system (2.2) is (h0 , h1 )-uniformly asymptotically stable. Since h1 is uniformly finer than h, there exist a function φ ∈ K and a ε˜ > 0 such that

PT

h(t, x(t)) 6 φ(h1 (t, x(t))) < φ(η)

0

whenever h0 (t0 , x0 ) < δ with φ(η) < ε˜ for t > t0 . This completes the proof.



CE

In the following theorem, we state a more general stability criteria combining the comparison system (3.12).

AC

Theorem 4.5. Assume that (1) h0 , h ∈ Γ0 ; h0 is finer than h; 0

(2) V ∈ ν0 and y(t; s, x) are locally Lipschitzian in x for each (t, x), C

α V (s, y(t; s, x)) 6 g(t, s, V (s, y(t; s, x)), σk (V (tk , y(t; tk , xk )))), D+

where s 6= tk , k = 1, 2, · · · , g : R2+ ×R+ ×R → R+ is continuous in each of the sets (tk , tk+1 ] in s, k = 0, 1, 2, · · · , for each t ∈ R+ and for all k ∈ N lim

(t,s,u)→(t,t+ k ,w)

g(t, s, u, σk (uk )) = g(t, t+ k , w, σk (wk )) 24

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and for any (t, s, u, v), g(t, s, u, v) is nondecreasing in v, σk (u) ∈ C(R+ , R) and σk (u) is

nondecreasing in u;

(3) V (t, x) is h-positive definite on S(h, ρ) and weakly h0 -decrescent, where (t, x) ∈ S(h, ρ), ρ > 0;

(4) V (s, y(t; s, x + Ik (x(s)))) 6 ψk (V (s, y(t; s, x))), s = tk , k = 1, 2, · · · , where ψk (u) ∈

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C(R+ , R+ ), k = 1, 2, · · · is nondecreasing in u;

(5) There exists a ρ0 ∈ (0, ρ) such that h(tk , x) < ρ implies h(t+ k , x + Ik (x)) < ρ; ˜ 0, h ˜ 0 )-stable. (6) The system (2.3) is (h

Then, for the stability properties of the trivial solution of system (3.12), we can get the corre-

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sponding stability properties of system (2.2)

Proof. Suppose that the initial solution of (3.12) is stable. Let 0 < ε < ρ0 and t0 ∈ R+ be given.

Then given b(ε) > 0, there exists a δ ∗ = δ ∗ (t0 , ε) > 0 such that |u(t; t0 , u0 )| < b(ε),

t > t0 ,

whenever |u0 | < δ ∗ , where u(t; t0 , u0 ) is any solution of (2.3) with initial value (t0 , u0 ) when

M

t = s.

Let V (t0 , y(t; t0 , x0 )) = |u0 |. By using this δ ∗ (δ ∗ < b(ε)) in place of b(ε) in proof of Theorem

ED

4.1, we can find a δ = δ(t0 , ε) > 0 as before. Then, from Conditions (1) and (3), an argument similar to the proof of Theorem 4.1 shows that

PT

b(h(t0 , x0 )) 6 V (t0 , x0 ) 6 a(t0 , h0 (t0 , x0 )) < δ ∗ < b(ε). Thus, we get h(t0 , x0 ) < ε.

CE

We claim that h(t, x(t)) < ε, t > t0 , whenever h0 (t0 , x0 ) < δ. If it is not true, then there exist a solution x(t) = x(t; t0 , x0 ) of system (2.2) and a t1 > t0 , t1 ∈ (tk , tk+1 ] for some k such

AC

that

h(t1 , x(t1 )) > ε and h(t, x(t)) < ε,

t ∈ [t0 , tk ].

+ ˆ ˆ Since 0 < ε < ρ0 , by Condition (5), h(t+ k , x(tk )) < ρ, there exists a t > 0, t ∈ (tk , t1 ) such that

ε 6 h(tˆ, x(tˆ)) < ρ and h(t, x(t)) < ρ,

t ∈ [t0 , tˆ].

By Conditions (2)-(4) and Theorem 3.3, we have V (t, x(t)) 6 r0 (t; t0 , u0 )),

t ∈ [t0 , tˆ],

where r0 (t; t0 , u0 ) is the maximum solution of (3.12) with initial value (t0 , u0 ) when t = s and |u0 | = V (t0 , y(t; t0 , x0 )) 6 a(t0 , h0 (t0 , y(t; t0 , x0 ))) < δ ∗ . 25

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Since V (t, x) is h-positive, then we obtain b(ε) 6 b(h(tˆ, x(tˆ))) 6 V (tˆ, x(tˆ)) 6 r0 (tˆ; t0 , u0 ) < b(ε), which leads to a contradiction. Hence, system (2.2) is (h0 , h)-stable. Next, we prove that the asymptotic stability of the trivial solution of system (3.12) implies (h0 , h)-asymptotic stability of system (2.2).

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Suppose that the trivial solution of (3.12) is asymptotically stable, which implies stability and attractivity. By the proof above, system (2.2) is (h0 , h)-stable. Taking ε = ρ, t0 ∈ R+ , there

exists a δ0 = δ0 (t0 , ε) > 0 such that h(t, x(t)) < ρ, t > t0 with h0 (t0 , x0 ) < δ0 . Corresponding to b(ε), there exist δ0∗ = δ0∗ (t0 , ε) > 0 and T = T (t0 , ε) > 0 such that |u0 | 6 δ0∗ implies that t > t0 + T,

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u(t; t0 , u0 ) < b(ε),

which also means that u(t; t0 , u0 ) → 0, t → ∞ with |u0 | 6 δ0∗ . Taking δ1 = min{δ0∗ , δ0 }, we have

h(t, x(t)) < ρ, with h0 (t0 , x0 ) < δ1 .

t > t0

M

We assert that h(t, x(t)) < ε, t > t0 + T whenever h0 (t0 , x0 ) < δ1 < δ. If it is not true, then there would exist a solution x(t) = x(t; t0 , x0 ) of (2.2) and a divergent sequence {t(m) },

ED

t(m) → ∞(m → ∞), t(m) > t0 + T such that

h(t(m) , x(t(m) )) > ε and h(t, x(t)) < ε,

t ∈ [t0 , t(m) ].

PT

Since h0 (t0 , x0 ) < δ1 and system (2.2) is (h0 , h)-stable, we have ε 6 h(t(m) , x(t(m) )) < ρ and h(t, x(t)) < ε,

t ∈ [t0 , t(m) ].

AC

CE

By Conditions (2)-(4) and Theorem 3.3, we get V (t, x(t) 6 r0 (t; t0 , u0 )),

t ∈ [t0 , t(m) ],

where r0 (t; t0 , u0 ) is the maximum solution of (3.12). By the similar proof as before, we have

|u0 | < δ0∗ . Thus,

r0 (t; t0 , u0 ) → 0,

t → ∞.

Then b(ε) 6 b(h(t(m) , x(t(m) ))) 6 V (t(m) , x(t(m) )) 6 r0 (t(m) ; t0 , u0 ), where r0 (t(m) ; t0 , u0 ) approaches 0 with m → ∞, which leads to a contradiction. Thus, system (2.2) is (h0 , h) attractive. This completes the proof of (h0 , h)-asymptotic stability of system 

(2.2). 26

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Remark 4.6. Note that a ∈ P CK, one can only get nonuniform (h0 , h)-stability properties

for (2.2) even when we assume uniform stability properties of (3.12) as well as (2.3). If a ∈ K,

then uniform (h0 , h)-stability properties can be obtained whenever we assume the corresponding notion for (3.12) and (2.3). Remark 4.7. Theorem 4.5 presents the relationship of stability properties among systems (2.2),

5

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(2.3) and (3.12).

An example

In this section, we discuss the following example to demonstrate our theoretical results obtained in Section 4.

   

4x(tk ) = ck x(tk ), x(t+ 0 ) = x0 ,

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Let 0 < α < 1. Considering the following scalar impulsive fractional hybrid equation  C α  D x(t) = −ax(t) + bk x(tk ), t 6= tk , t > t0 ,    t0 t t = tk ,

k = 1, 2, · · · ,

M

and the fractional differential equation   C Dα y(t) = −ay(t), t0

t

y(t+ 0 ) = x0 ,

(5.17)

ED



(5.16)

where a > 0, −1 < ck 6 0, bk ∈ R, k = 1, 2, · · · , are constants, t0 < t1 < t2 < · · · < tk < tk+1 < · · · are impulsive moments and tk → ∞, k → ∞.

PT

Denote by x(t) = x(t; t0 , x0 ) the solution to system (5.16) and y(t) = y(t; t0 , x0 ) the solution

AC

and

CE

to system (5.17) respectively. It is straightforward to show from (5.17) that y(t; t0 , x0 ) = Eα,1 (−a(t − t0 )α )x0

(5.18)

y(t; s, x(s)) = Eα,1 (−a(t − s)α )x(s),

(5.19)

where Eα,β (z) represents the Mittag-Leffler function with two parameters. For the specific

formulation and properties of the Mittag-Leffler function, we refer the reader to the references ˜ 0, h ˜ 0 )-stable if a > 0 for any t0 ∈ R+ and t > t0 . [4, 34]. An easy induction gives that (5.17) is (h Let V (x(t)) = |x(t)| and h0 (t, x) = h(t, x) = |x(t)| for any t ∈ R+ and x ∈ R. Obviously,

h0 is uniformly finer than h when choosing the function φ as φ(u) = |u|p , where p is a constant

satisfying p > 1 and u ∈ R+ . It is also obvious that V is h-positive and h0 -decrescent. Thus, Conditions (1) and (4) are satisfied in Theorem 4.1. For a given ρ0 , 0 < ρ0 < ρ, h(tk , x(tk )) < ρ

27

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implies h(t+ k , x(tk ) + ck x(tk )) < ρ by −1 < ck 6 0, which shows that Condition (5) holds. By a

direct calculation and the proof of Corollary 4.1 in [41], for t > t0 and s 6= tk , we get C

α α D+ V (s, y(t; s, x)) = C D+ |Eα,1 (−a(t − s)α )x(s)| α = C D+ |y(t; s, x(s))|

α = sgn(y(t; s, x(s)))C D+ y(t; s, x(s))

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α = sgn(y(t; s, x(s)))C t0 Dt y(t; s, x(s))

= sgn(y(t; s, x(s)))(−ay(t; s, x(s))) = −a|y(t; s, x(s))| 6 0. On the other hand, when s = tk , we have

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+ + + α + V (t+ k , y(t; tk , x(tk ))) = |Eα,1 (−a(t − tk ) )x(tk )|

α = |Eα,1 (−a(t − t+ k ) )(1 + ck )x(tk )| α 6 |Eα,1 (−a(t − t+ k ) )x(tk )|

6 |Eα,1 (−a(t − tk )α )x(tk )|

Thus, Conditions (2) and (3) hold.

M

= V (tk , y(t; tk , x(tk ))).

By the previous discussions, all conditions of Theorem 4.1 are satisfied. Consequently, if

Conclusions

PT

6

ED

a > 0 for any t0 ∈ R+ and t > t0 , then it follows from Theorem 4.1 that (5.16) is (h0 , h)-stable.

This paper has employed the promoted variational Lyapunov method to analyze the stability

CE

properties for a class of impulsive hybrid systems with Caputo’s fractional order 0 < α < 1. An extended fractional variational comparison principle was established by using differential inequalities and the Lyapunov-like function in Caputo’s sense. Based on the obtained comparison

AC

theorem, some stability conditions in terms of two different measures for IFHSs was presented, which generalizes the corresponding stability theory. An example was given to illustrate the validity of the theoretical results.

Acknowledgements This work is supported partially by the National Natural Science Foundation of China under Grant Nos. 61573325, 61210011, and the Hubei Provincial Natural Science Foundation of China under Grant 2015CFA010.

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ED

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PT

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AC

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[14] Haddad, W. M., Chellaboina, V. S., Nersesov, S. G., Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, New Jersey: Princeton University Press; 2014. [15] Guan, Z. H., Hill, D. J., Shen, X. M., On hybrid impulsive and switching systems and application to nonlinear control, IEEE Transactiona on Automatic Control 2005;50(7):1058-

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[18] Zavalishchin S. T., Sesekin A. N., Dynamic impulse systems: theory and applications, Springer Science and Business Media; 1997.

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