Generalized Lyapunov approach for functional differential inclusions

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Generalized Lyapunov approach for functional differential inclusions✩ ∗

Zuowei Cai a,b , , Lihong Huang c a

Department of Information Technology, Hunan Women’s University, Changsha 410004, China School of Mathematics and Statistics, Central South University, Changsha 410083, China c School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China b

article

info

Article history: Received 11 August 2019 Received in revised form 4 November 2019 Accepted 11 November 2019 Available online xxxx

a b s t r a c t This paper is concerned with the generalized Lyapunov approach for functional differential inclusions (FDI). At first, we prove a class of generalized Halanay’s inequalities. Then, by applying the generalization of Halanay’s inequalities, we investigate the uniformly ultimate boundedness and uniformly asymptotic stability of FDI. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Functional differential inclusions Filippov solution Generalized Halanay inequalities Uniformly ultimate boundedness (UUB) Uniformly asymptotic stability (UAS)

1. Introduction Discontinuity is widely discovered in a variety of science and engineering fields, such as system structure variation caused by mode transition, impacting machines, switching in electronic circuits, Coulomb friction (Cortés, 2008; di Bernardo, Budd, Champneys, & Kowalczyk, 2008; Filippov, 1988; Forti, Grazzini, Nistri, & Pancioni, 2006). Moreover, time-delays are often inevitable in most physical, biological, neural, chemical and other natural systems due to the energy propagating with a finite speed, processing and reaction time of signals (Hale, 1977; Liu & Sun, 2016). In the past few decades, there has been growing interests and needs for modeling and analysis of discontinuous dynamical systems with time delays (Benchohra & Ntouyas, 2001; Cai & Huang, 2018; Haddad, 1981a, 1981b; Lupulescu, 2004; Surkov, 2007; Wang & Michel, 1996). A particular case is represented by time-delayed nonlinear dynamical systems whose discontinuity depends on the state variable. This type of discontinuous time-delayed dynamical system is usually described by a delayed ✩ This work was supported by National Natural Science Foundation of China (11701172, 11771059). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor André L. Tits. Part of this work was conducted when the first author was visiting the Central South University, and the authors would like to thank Xianhua Tang ([email protected]. edu.cn) for his some constructive comments. ∗ Corresponding author at: Department of Information Technology, Hunan Women’s University, Changsha 410004, China. E-mail addresses: [email protected] (Z. Cai), [email protected] (L. Huang).

differential equation (DDE) possessing discontinuous right-hand side. In this case, the existence of a continuously differentiable solution (i.e., classical solution) for DDE is not guaranteed since the given vector field is discontinuous. In 1964, Filippov proposed a solution to discontinuous differential equation that only time and state variables in the right-hand side are required to be Lebesgue measureable (Filippov, 1988). Since then, the functional differential inclusion (FDI) has been developed and further investigated. As pointed out by Aubin and Cellina (1984), FDI expresses velocity well depending not only on the state of the system at every instant, but also on the history of the trajectory until this instant. In 1981, Haddad gave a systematic introduction for the solution sets of FDI (see Haddad, 1981b). In Benchohra and Ntouyas (2001) and Lupulescu (2004), the existence of the solutions for convex and nonconvex FDI has been presented. Actually, the solution of DDE could be transformed into a solution of FDI by constructing a Filippov set-valued map, which is called the Filippov regularization. Such a concept in the sense of Filippov has been universally accepted as a good method for the switching or discontinuous nonlinear dynamical systems with time delays. Generally speaking, FDI is considered as a generalization of DDE. Furthermore, in the process of simplification of many practical problems, FDI with Filippov-framework can maximally relax certain restrictions, but does not affect the essence of the problems (Cai & Huang, 2014). Nonetheless, it is worth noting that the issues of the qualitative analysis and stability analysis for FDI is still incomplete and lacks new and efficacious methods as well. It must be emphasized that stability analysis is one of the most significant and meaningful research topic for FDI or time-delayed dynamical systems with discontinuous property. The standard

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and effective analysis methods are often based on Lyapunov methods (Liu, Sun, Liu, & Teel, 2016; Surkov, 2007; Wang & Michel, 1996). In Surkov (2007), Surkov investigated the stability of FDI by using Lyapunov functions. In Wang and Michel (1996), a new argument has been shown for the stability analysis of FDI. To the best of our knowledge, in order to study the stability problems of FDI as well as ordinary differential inclusion, most of methods in the literature require that the Lyapunov functions are differential or smooth. Afterwards, the methods based on regular (C-regular) Lyapunov functions which are allowed to be local Lipschitz continuous or non-smooth have been widely developed (see Baciotti & Ceragioli, 1999; Forti et al., 2006; Guo & Huang, 2009). However, in the existing literature, there is still quite little research employing non-smooth Lyapunov function to handle the stability, uniformly ultimate boundedness, uniformly asymptotic stability for FDI. On the other hand, Halanay inequality and its generalization is also an effective approach to dealing with stability and dissipativity of time-delayed system. There are some results concerning generalizations of the Halanay inequality and some suitable applications. For example, Wang and Ding (2012) and Wen, Yu, and Wang (2008) gave the generalizations of the Halanay inequality and its application in handling the dissipativity of Volterra equations. Liu, Lu, and Chen (2011) proposed some variants of generalized Halanay inequalities to discuss the dissipativity and stability of timedelayed neural networks. In Hien, Phat, and Trinh (2015), new generalized Halanay-type functional differential inequalities were studied. However, these Halanay-type inequalities are only effective for handling DDE with continuous right-hand side. Inspired by the above discussions, this paper will propose more generalized Halanay-type inequality which is valid to deal with DDE with discontinuous right-hand sides or FDI. Such a class of Halanay inequality does not request that the Lyapunov function V (t) is differentiable with respect to t for everywhere. Meanwhile, based on the framework of FDI, the stability problems of discontinuous DDE will be studied by using generalized Halanay-type inequality. Notation. Let R (R+ ) denote the set of (nonnegative) real numn bers, 2R denote the family of all nonempty subsets of Rn . Given n x ∈ R , ∥x∥ denotes vector norm of x. Given a real number τ ≥ 0 and t0 ∈ R+ , C ([t0 − τ , t0 ], Rn ) denotes the Banach space of continuous functions φ mapping the interval [t0 − τ , t0 ] into Rn with the norm ∥φ∥C = supt0 −τ ≤s≤t0 ∥φ (s)∥. ϕ −1 denotes the inverse function of ϕ . A non-degenerate interval I = [ℓ1 , ℓ2 ] (or (ℓ1 , ℓ2 ) or [ℓ1 , ℓ2 ) or (ℓ1 , ℓ2 ]) means that ℓ1 < ℓ2 . 2. Preliminaries

dt

= f (t , x(t), x(t − τ (t))), a.e. t ≥ 0

(1)

where t denotes time; x(t) = (x1 (t), x2 (t), . . . , xn (t))T denotes state vector; x(t − τ (t)) = (x1 (t − τ (t)), x2 (t − τ (t)), . . . , xn (t − τ (t)))T represents time-varying delayed state vector and the timedelay τ (t) is a continuous function; dx/dt denotes the time derivative of x and f : R × Rn × Rn → Rn is measurable and essentially locally bounded. In this case, the DDE (1) is allowed to possess discontinuous right-hand side. n Construct the set-valued map F : R × Rn × Rn → 2R : F t , x(t), x(t − τ (t)) =

(

)

⋂

⋂

fxc ,

(2)

ρ1 >0,ρ2 >0 meas(N)=0,meas(M)=0

where = co f t , B(x, ρ1 )\N, B(x(t − τ (t)), ρ2 )\M ; meas(N) (meas(M)) is the Lebesgue measure of set N(M); intersection is taken over all sets N (M) of measure zero and over all ρ1 > 0 fxc

[ (

)]

the closure of the convex hull of some set E. Definition 1. The function x(t) defined on a non-degenerate interval I ⊆ R is called a Filippov solution for DDE (1), if it is absolutely continuous on any compact subinterval [t1 , t2 ] of I, and for a.e. t ∈ I, x(t) satisfies the following FDI dx dt

∈ F (t , x(t), x(t − τ (t))).

(3)

The given initial value is given as (t0 , φ ) ∈ R+ × C ([t0 − τ , t0 ], Rn ). Definition 2. If for any t ∈ R, 0 ∈ F (t , 0, 0), then x = 0 is called a zero solution of the FDI (3) or DDE (1). Definition 3. Let C = C ([t0 − τ , t0 ], Rn ). The solutions of system (1) (or FDI (3)) are said to be uniformly ultimately bounded if there exists a M > 0 such that, for any δ > 0, there is a constant T = T (δ ) > 0 such that ∥x(t0 , φ )(t)∥ ≤ M for t ≥ t0 + T and for all t0 ∈ R+ , φ ∈ C , ∥φ∥C ≤ δ . Definition 4. A function ϕ : R+ → R+ is said to be a K -function which is denoted by ϕ ∈ K if ϕ is continuous and strictly increasing with ϕ (0) = 0. A function ϕ : R+ → R+ is said to be a K∞ -function which is denoted by ϕ ∈ K∞ if it is a K -function and also satisfies limr →+∞ ϕ (r) = +∞. 3. Main results First of all, we state and prove a class of important inequality named generalized Halanay’s inequality as follows. 3.1. Generalized Halanay’s inequality Theorem 1 (Generalized Halanay Inequality). Suppose that V (t) is absolutely continuous and nonnegative for t ∈ (−∞, +∞). Assume further that dV (t) dt

≤γ (t) + ξ (t)V (t) + η(t) ·

sup

V (s),

t −τ (t)≤s≤t

for a.e. t ≥ t0 , and V (t) = |ψ (t)|, for t ≤ t0 .

(4)

Here, the function ψ (t) is bounded and continuous for all t ≤ t0 . The continuous bounded functions γ (t) ≥ 0, η(t) ≥ 0 and ξ (t) ≤ 0 for all t ∈ [t0 , +∞), τ (t) ≥ 0. Moreover, there exists a σ > 0 such that

ξ (t) + η(t) ≤ −σ , for t ≥ t0 .

Consider the non-autonomous DDE of the vector form: dx

(ρ2 > 0); B(x, ρ1 ) is the ball of center x and radius ρ1 ; B(x(t − τ (t)), ρ2 ) is the ball of center x(t − τ (t)) and radius ρ2 ; co[E] is

(5)

Then for all t ≥ t0 ,

γ∗ ∗ + Ψ e−µ (t −t0 ) , (6) σ where γ ∗ = supt0 ≤t ≤∞ γ (t), Ψ = sup−∞≤s≤t0 |ψ (s)|, µ∗ = inft ≥t0 {µ(t) : µ(t) + ξ (t) + η(t)eµ(t)τ (t) = 0}. V (t) ≤

Proof. Firstly, we prove the following statement is true V (t) ≤

γ∗ + Ψ , for all t ≥ t0 . σ

(7)

We only need to prove (7) holds for the case of γ ∗ > 0. To do so, γ∗ we denote W (t) = V (t) − σ . If Ψ > 0, then for any ε > 1, we can obtain from (4) that W (t) < V (t) < ε Ψ for t ≤ t0 ; from this we will deduce that W (t) < ε Ψ , for all t ≥ t0 .

(8)

Please cite this article as: Z. Cai and L. Huang, Generalized Lyapunov approach for functional differential inclusions. Automatica (2019) 108740, https://doi.org/10.1016/j.automatica.2019.108740.

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For this purpose, we will prove that (8) holds by way of contraγ∗ diction. Since W (t) = V (t) − σ is a continuous function with respect to t, if (8) is not true, then there exists t1 > t0 such that W (t) < ε Ψ (−∞ < t < t1 ) and W (t1 ) = ε Ψ , which yields that there exist ε1 (1 < ε1 < ε ) and t2 (t0 < t2 < t1 ) such that W (t) < ε1 Ψ (−∞ < t < t2 ) and W (t2 ) = ε1 Ψ . On the other hand, it follows from (4) and (5) that dW (t) dt

=

dV (t) dt

≤ γ ∗ + ξ (t)V (t) + η(t)

sup t −τ (t)≤s≤t

ξ (t) + η(t) ∗ γ + ξ (t)V (t) + η(t) sup V (s) −σ t −τ (t)≤s≤t ) ( ) ( γ∗ γ∗ + η(t) sup V (s) − = ξ (t) V (t) − σ σ t −τ (t)≤s≤t

(14)

Actually, if (14) is not true, then there exists some t > t0 such that

γ∗ + ε ∗ + Ψ e−µ (t −t0 ) . σ

(15)

For convenience, let

= ξ (t)W (t) + η(t)

γ∗ + ε ∗ + Ψ e−µ (t −t0 ) , for all t ≥ t0 . σ and W(t) = S(t) − V (t). Obviously, W(t) is differentiable for almost all t ≥ t0 because of the almost everywhere differentiability

W (s), for a.e. t ≥ t0 .

sup

S(t) =

(9)

of V (t) and S(t). Moreover, we can obtain

( ∫ ) t Multiplying both sides of (9) by exp − t ξ (s)ds and then inte0 grating both sides of (9) from t2 to t1 , we have

ε Ψ = W (t1 ) ≤ W (t2 ) exp

t1

(∫

t2

∫

t1

+

η (ρ )

t2

≤ ε1 Ψ exp

sup

ρ−τ (ρ )≤s≤ρ

(∫

t1

t1

(−ξ (ρ ))ε Ψ exp

ξ (θ )dθ (∫

t1

ρ

ξ (θ )dθ dρ

dV (t) dt

∫

(∫

−

dV (t) dt

˙ (θ )dθ = W t0

t1

t∗

∫ ≥ t1

ξ (θ )dθ

, for a.e. t ≥ t0 .

t∗

0 ≥ W(t∗ ) − W(t0 ) =

)]

∫

t∗

S˙ (θ ) − V˙ (θ ) dθ

(

)

t0

[ ∗ −Ψ µ∗ e−µ (θ −t0 )

t0

( − γ (θ ) + ξ (θ )V (θ ) + η(θ ) ·

εΨ ,

t2

which implies

ε1 Ψ exp

−

Let t∗ = inf{t > t0 : S(t) − V (t) < 0}. Then we have W(t∗ ) = S(t∗ ) − V (t∗ ) = 0 which implies t∗ > t0 , and W(t) ≥ 0 for all t ∈ [t0 , t∗ ). Therefore, we can obtain

)

(10) t1

dt

)

t2

(∫

dS(t)

= −Ψ µ e

) ξ (θ )dθ dρ ρ t2 (∫ t1 ) [ (∫ ≤ ε1 Ψ exp ξ (θ )dθ + 1 − exp ∫

+

=

dt

∗ −µ∗ (t −t0 )

W (s) exp

ξ (θ )dθ

dW(t)

)

t2

>

∫

t∗

[

)] sup

θ −τ (θ )≤s≤θ

V (s)

dθ

∗ (θ −t ) 0

−Ψ µ∗ e−µ

t0

ξ (θ )dθ

)

≥ εΨ exp

(∫

t2

t1

)

ξ (θ )dθ , i.e., ε1 ≥ ε.

t2

This contradicts the fact 1 < ε1 < ε . Therefore, the inequality (8) must hold. Since ε > 1 is arbitrary, we let ε → 1 and obtain W (t) ≤ Ψ , for t ≥ t0 , i.e., V (t) ≤

γ∗ + Ψ for t ≥ t0 . σ

(11)

If Ψ = 0, applying similar technique as proving (8), we can γ∗ prove that V (t) ≤ σ for all t ≥ t0 . Hence, for the case of γ ∗ > 0, the inequality (7) holds. For the case of γ ∗ = 0, (7) is still true by similar proof. Now, we prove (6) is true. Let us denote H(µ) by µτ (t)

H(µ) = µ + ξ (t) + η(t)e

.

(12)

For any given fixed t ≥ t0 , it is easy to see that H(0) = ξ (t) + η(t) ≤ −σ < 0,

dµ

γ∗ + ε ∗ + Ψ e−µ (t −t0 ) , for all t ≥ t0 . σ

V (t) ≤

≤

t −τ (t)≤s≤t

and dH(µ)

do so, we first prove that the following inequality holds for any given ε > 0

V (t) >

V (s)

3

lim H(µ) = +∞

µ→+∞

= 1 + τ (t)η(t)eµτ (t) > 0.

Hence, for any given fixed t ≥ t0 , there is a unique positive µ such that

µ + ξ (t) + η(t)eµτ (t) = 0.

(13)

This means that (13) defines an implicit function µ(t) for t ≥ t0 . From this definition, we can obtain that µ∗ ≥ 0. Obviously, for the case of µ∗ = 0, it follows straightway from (7) that (6) holds. Next, we will prove that (6) still holds for the case of µ∗ > 0. To

( )] ∗ − γ + ε + ξ (θ )V (θ ) + η(θ ) · sup V (s) dθ θ −τ (θ )≤s≤θ [ ∗ −µ∗ (ι−t0 ) = (t∗ − t0 ) · −Ψ µ e ( )] − γ ∗ + ε + ξ (ι)V (ι) + η(ι) · sup V (s) , ι−τ (ι)≤s≤ι

for some ι ∈ [t0 , t∗ ].

(16)

If ι − τ (ι) ≥ t0 , it is not difficult to deduce from (16) that 0 ≥ W(t∗ ) − W(t0 )

[ ∗ > (t∗ − t0 ) · −Ψ µ∗ e−µ (ι−t0 ) ( ( ∗ ) γ +ε ∗ − γ ∗ + ε + ξ (ι) + Ψ e−µ (ι−t0 ) σ ( ∗ ))] γ +ε −µ∗ (ι−τ (ι)−t0 ) +η(ι) + Ψe σ [ ∗ γ +ε = − (t∗ − t0 ) · (σ + ξ (ι) + η(ι)) σ ( )] ∗ ∗ +Ψ e−µ (ι−t0 ) µ∗ + ξ (ι) + η(ι)eµ τ (ι) .

(17)

From the definition of function µ(t), it is easy to see that

µ∗ + ξ (ι) + η(ι)eµ

∗ τ (ι)

= µ∗ + ξ (ι) + η(ι)eµ τ (ι) − µ(ι) − ξ (ι) − η(ι)eµ(ι)τ (ι) ( ∗ ) ( ) = µ∗ − µ(ι) + η(ι) eµ τ (ι) − eµ(ι)τ (ι) ≤ 0. ∗

(18)

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Noting (5), we derive from (17) that 0 ≥ W(t∗ ) − W(t0 ) > 0.

(19)

This is a contradiction. If ι − τ (ι) < t0 , it follows from (16) that

Theorem 2. Assume that there exists a C-regular and locally Lipschitz continuous function V : Rn → R+ and functions ϕ1 , ϕ2 ∈ K∞ such that (i) ϕ1 (∥x∥) ≤ V (x) ≤ ϕ2 (∥x∥), ∀x ∈ Rn ; (ii) for almost all t ≥ t0 ≥ 0, the following inequality holds dV (x(t0 , φ )(t))

0 ≥ W(t∗ ) − W(t0 )

[

∗ −µ∗ (ι−t0 )

> (t∗ − t0 ) · −Ψ µ e ( ( ∗ ) γ +ε ∗ −µ∗ (ι−t0 ) − γ + ε + ξ (ι ) + Ψe σ { })] +η(ι) max sup V (s), sup V (s) s≤t t0 ≤s≤ι [ 0 ∗ −µ∗ (ι−t0 ) ≥ (t∗ − t0 ) · −Ψ µ e ) ( ( ∗ γ +ε ∗ + Ψ e−µ (ι−t0 ) − γ ∗ + ε + ξ (ι ) σ ( ))] γ∗ + ε + η (ι ) Ψ + σ [ ∗ γ +ε = − (t∗ − t0 ) · (σ + ξ (ι) + η(ι)) σ ( )] ∗ ∗ +Ψ e−µ (ι−t0 ) µ∗ + ξ (ι) + η(ι)eµ (ι−t0 ) [ ∗ γ +ε ≥ − (t∗ − t0 ) · (σ + ξ (ι) + η(ι)) σ ( )] ∗ ∗ +Ψ e−µ (ι−t0 ) µ∗ + ξ (ι) + η(ι)eµ τ (ι) .

dt

|(3) ≤γ (t) + ξ (t)V (x(t0 , φ )(t)) + η(t) ·

sup

V (x(t0 , φ )(s)),

t −τ (t)≤s≤t

where the continuous functions γ (t) ≥ 0, η(t) ≥ 0 and ξ (t) ≤ 0 for all t ∈ [t0 , +∞), 0 ≤ τ (t) ≤ τ < +∞ and τ (t) is a nonnegative continuous function; (iii) there exists σ > 0 such that ξ (t) + η(t) ≤ −σ , for t ≥ t0 . Then the solutions of the FDI (3) are uniformly ultimately bounded. Especially, if γ ∗ = supt0 ≤t ≤∞ γ (t) = 0 and 0 ∈ F (t , 0, 0) for any t ∈ R, then we can further obtain that the zero solution of the FDI (3) is globally uniformly asymptotically stable. Proof. According to the generalized Halanay’s inequality given in Theorem 1, conditions (ii) and (iii) imply that the following estimation holds

γ∗ ∗ + Ψ e−µ (t −t0 ) , for all t ≥ t0 , (21) σ where γ ∗ = supt0 ≤t ≤∞ γ (t), µ∗ = inft ≥t0 {µ(t) : µ(t) + ξ (t) + η(t)eµ(t)τ (t) = 0}, Ψ = supt0 −τ ≤s≤t0 V (x(t0 , φ )(s)). Since x(t0 , φ )(s) = φ (s) for all t0 − τ ≤ s ≤ t0 , we can obtain from V (t) ≤

(20)

condition (i) that

Therefore, from (5), (18) and (20), we can also obtain (19) which is a contradiction. By now we have proved that the inequality (14) holds. Since ε > 0 is arbitrary, by allowing ε → 0 we have (6) holds for all t ≥ t0 . This completes the proof.

V (x(t0 , φ )(s)) = V (φ (s)) ≤ ϕ2 (∥φ (s)∥)

Remark 1. In Liu et al. (2011), Mohamad and Gopalsamy (2000) and Wen et al. (2008), Mohamad, Wen, Liu, et al. have also given the proofs of generalized Halanay inequalities similar to those of Theorem 1 in this paper. However, our results do not request that V (t) is differentiable with respect to t for everywhere. Said another way, we only request V (t) is absolutely continuous with respect to t ∈ R (or continuous and differentiable for almost all t ∈ R). Thus, our results are more general and more practical.

Ψ =

Remark 2. In Hien et al. (2015), Hien et al. have proved some new generalizations of the Halanay inequalities for nonlinear non-autonomous time-delay systems by the use of upper righthand Dini derivatives. As one of the innovations of this paper, our Halanay-type inequality does not required to calculate the derivative for everywhere (but just calculate the derivative for a.e. t ≥ t0 ). However, the Halanay-type inequality of Hien et al. (2015) required to calculate the upper right-hand Dini derivatives for everywhere. So the main difference is that our Halanay-type inequality is relaxed to calculate derivatives for ‘‘a.e’’. Remark 3. If the function V (t) is defined on the interval [t0 − τ , +∞) for some t0 ∈ R+ and τ (t) ≤ τ < +∞, there are similar results corresponding to Theorem 1. This only need to replace Ψ = sup−∞≤s≤t0 |ψ (s)| with Ψ = supt0 −τ ≤s≤t0 |ψ (s)|. 3.2. UUB and UAS of FDI In this section, x(t0 , φ )(t) denotes the Filippov solution of FDI (3) with given initial condition (t0 , φ ) ∈ R+ × C ([t0 − τ , t0 ], Rn ).

≤ ϕ2 (∥φ∥C ), for s ∈ [t0 − τ , t0 ],

(22)

V (x(t0 , φ )(s)) ≤ ϕ2 (∥φ∥C ).

(23)

which yields sup t0 −τ ≤s≤t0

Again using condition (i), we can get from inequality (21) and (23) that

ϕ1 (∥x(t0 , φ )(t)∥) ≤ V (t) γ∗ ∗ ≤ + ϕ2 (∥φ∥C )e−µ (t −t0 ) , ∀t ≥ t0 , σ

(24)

which implies

∥x(t0 , φ )(t)∥ ≤ ϕ1−1

(

) γ∗ ∗ + ϕ2 (∥φ∥C )e−µ (t −t0 ) , ∀t ≥ t0 . σ

(25)

The above inequality means that the solutions of the FDI (3) are uniformly ultimately bounded. Obviously, if γ ∗ = 0, then the zero solution of the FDI (3) is uniformly asymptotically stable. The proof is complete. 4. Numerical example Example 1. Consider the following DDE with discontinuous switching property. dx(t) dt

= −d(t)x(t) + a(t)x(t − τ (t))sign(x(t)), a.e. t ≥ 0,

(26)

where sign(x) denotes sign function, d(t) = 5, τ (t) = 1, a(t) = 3 − 0.8 cos t + D(t), { D(t) denotes the Dirichlet function but re1, if t is irrational, versed, i.e., D(t) = Consider the Lyapunov 0, if t is rational. function V (t) = |x(t)|. Calculating derivative of V (t) along the

Please cite this article as: Z. Cai and L. Huang, Generalized Lyapunov approach for functional differential inclusions. Automatica (2019) 108740, https://doi.org/10.1016/j.automatica.2019.108740.

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5

References

Fig. 1. Time-domain behavior of the state variable x(t) of DDE (26).

right half trajectory of DDE (26), we have dV (t) dt

|(26) ≤ ξ (t)V (t) + η(t) ·

sup

V (s), a.e.t ≥ 0,

t −τ (t)≤s≤t

where ξ (t) = −5 ≤ 0, η(t) = 4 − 0.8 cos t > 0 and ξ (t) + η(t) ≤ −0.2. This means that all conditions of Theorem 2 are satisfied. Thus, the zero solution of discontinuous DDE (26) is uniformly asymptotically stable. As shown in Fig. 1, under given 10 random initial values x(t) = φ (s) for s ∈ [−1, 0], the solution trajectories of (26) eventually tend to zero solution . Remark 4. The DDE (26) has discontinuous property due to the switching term sign(x(t)). Moreover, because D(t) is the Dirichlet function, the DDE (26) is differentiable for almost everywhere (a.e.). The Example 1 demonstrates that the generalized Halanaytype inequality of this paper holds for a.e. and is valid to deal with the stability of DDE with discontinuous right-hand sides. However, the existing Halanay-type inequalities of Liu et al. (2011), Wang and Ding (2012) and Wen et al. (2008) are only effective for DDE with continuous right-hand sides. In addition, another advantage of this class of Halanay-type inequality is that it does not need to construct complex Lyapunov–Krasovskii functionals in the study of stability for DDE with discontinuous right-hand sides or FDI. 5. Conclusion In this paper, we have generalized a class of inequalities: generalized Halanay’s inequalities. Such a class of new inequalities will be very effective to deal with the stability, dissipativity and synchronization for DDE with discontinuous right-hand sides or FDI. Then, we have discussed the uniformly ultimate boundedness and uniformly asymptotic stability of Filippov solution for FDI via generalized Lyapunov approach. Finally, a simple numerical example was provided. We think it would be interesting to apply the methods established in this paper to investigating other classes of discontinuous DDE or FDI such as impulsive semilinear FDI. The Lyapunov–Krasovskii functionals and Razumikhin method for time-delayed system with discontinuity should also be developed (Cai, Huang, & Huang, 2017; Ning, He, Wu, & Shen, 2014; Pepe & Ito, 2012; Zhang, Shen, & Jiao, 2009). These issues will be considered further.

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Please cite this article as: Z. Cai and L. Huang, Generalized Lyapunov approach for functional differential inclusions. Automatica (2019) 108740, https://doi.org/10.1016/j.automatica.2019.108740.