Mean square stability for controlled hybrid neutral stochastic differential equations with in nite delay

Mean square stability for controlled hybrid neutral stochastic differential equations with infinite delay Tom´as Caraballo Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla (Spain). [email protected] Lassaad Mchiri Department of Mathematics, Faculty of sciences of Sfax, Tunisia. lmc[email protected] BELFEKI Mohsen Department of Mathematics, Faculty of sciences of Sfax, Tunisia. [email protected] Mohamed Rhaima University of Tunis El Manar, Faculty of Sciences of Tunis, Department of Mathematics, Tunisia. [email protected]u.tn Abstract In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by the control function, the infinite time delay, and the highly nonlinear coefficients of the systems itself, which makes our problem more sophisticated. Different from the classical Khasminskii-types conditions in the literature, we provide some sufficient conditions for 1 the mean square and almost sure exponential stability for controlled HNSFDEswID by using the M-matrix technique and a suitable Lyapunov functional. In this sense, the main contributions of our results, compared with those in the literature, are the infinite time delay, the neutral term and the new Lyapunov functions. Finally, we illustrate our results by a numerical example. 1 Introduction The applications of dynamical systems are found in many branches of technologies, sciences, economy, physics, biology amongst others. Many researchers have devoted much more attention to the investigation and study of different types of deterministic and stochastic dynamical systems (see [1, 2, 5, 6, 9, 12, 13, 31, 33]). One important class of stochastic dynamical systems is given by stochastic functional differential equations (SFDEs), which are very useful to characterize equations with states depending on the present and on the past (see [11]-[19], [23] and [24]-[29]). One of the most important class of SFDEs are the hybrid SFDEs (HSFDEs) or SFDES with Markovian switching, which contain, in particular, the cases modeled by stochastic differential equations with delay (SDEswD) and Markovian switching, which have been proved very appropriate to describe many phenomena in the real world. Stability theory is an essential topic in the analysis of HSFDEs (see [10] and [20]-[22]). On the one hand, many published papers in the literature impose that the coefficients in the equations must satisfy some linear growth condition (see [4] and [27]). But, on the other hand, in practical situations, many SFDEs do not often satisfy this type of linear growth condition (i.e. SFDEs can be highly nonlinear). Therefore, there exist many papers in which the authors studied the asymptotic properties and establish some stability criteria of the HSFDEs which are highly nonlinear (see [7]-[10], [22] and [26]). In many science, engineering and economic fields, many phenomena are influenced by random factors and time delay even in the terms containing derivatives, which motivates and justify to model such systems by hybrid neutral stochastic functional differential equations (HNSFDEs). HNSFDEs are important extensions of HSFDEs (see [3], [25] and [30]). HNSFDEs have been used when a neutral stochastic functional differential equation experiences sudden changes in its coefficients structures due to some environmental phenomena. The exponential stabilization of HSFDEs by means of state feedback controllers has been widely discussed (see [21] and [22]). In [22], the authors constructed a delay control based on discrete-time state observations to guarantee the stability of HSFDEs (which are highly nonlinear) and they assumed that the time lag 2 generated by the controller in each discrete observation should be different. To the best of our knowledge, there is no existing result on the exponential stability and the mean square stability of highly nonlinear HNSFDEswID. In this sense, our paper extends the work in [22] to the neutral case with infinite delay. Different from the previous works in the literature, the main highlights of our paper are as follows: (1) Our system (2.1) is infinite-dimensional and its coefficients are highly nonlinear. (2) The controlled equation (2.6) includes not only discrete modes and continuous states but also new discrete states. (3) Comparing with [22], our model contains a neutral term and infinite delays which make the system more complicated to deal with. (4) We investigate existence and uniqueness of solution to system (2.6) by using the Lyapunov techniques. (5) Different from the Khasminskii-type conditions in the literature, we study the mean square exponential stability and the almost sure exponential stability via the M-matrix method. The paper is arranged as follows. In Section 2, we recall some preliminaries and fundamental concepts for our analysis. In Section 3, we study the existence and uniqueness of the solution of HNSFDEswID. In Section 4, we investigate the mean square and the exponential stability of HNSFDEswID. Eventually, in Section 5, we illustrate our results by a numerical example. 2 Preliminaries and basic notions Denote by C([ρ, 0],Rc) the family of continuous functions from [ρ, 0] into Rcand W(ϑ) = (W1(ϑ), . . . , Wp(ϑ))Tap-dimensional Brownian motion defined on a complete probability space (Ω,F,P). Let |s|denote the Euclidean norm of s∈Rcand |L|=ptrace(LTL) the trace norm, for a matrix L. Let BC((−∞,0],Rc) be the set of bounded continuous functions µ: (−∞,0] →Rc equipped with the norm ||µ|| = supζ≤0|µ(ζ)|. For q > 0, denote by Lq((−∞,0],Rc) the set of measurable functions ψ: (−∞,0] →Rcsuch that R0 −∞ |ψ(ϑ)|qdϑ < ∞. Let N((−∞,0),R+) be the family of non-negative continuous bounded functions ν(·) satisfying R0 −∞ ν(ϑ)dϑ = 1. For each ε > 0, let N(ε)((−∞,0),R+) be the set of continuous bounded non-negative functions ν(·) such that 0 ≤M(ε)=R0 −∞ e−εϑν(ϑ)dϑ < ∞. Denote by {l(ϑ), ϑ ∈[0,+∞)}a Markov chain (which is right-continuous) on {Ω,F,(Fϑ)ϑ≥0,P}, taking values in Θ = {1,2, . . . , N}, whose generator Σ = (σbd)N×Nis given by P(l(ϑ+ ∆) = d|l(ϑ) = b) =    σbd∆ + o(∆),if b6=d 1 + σbb∆ + o(∆),if b=d, 3 where ∆ >0. Here σbd ≥0 is the transition rate from bto d, if b6=d, while σbb =−X d6=b σbd. We say that the matrix O(or the vector t > 0) is positive if all its elements are positive. We say that the square matrix O= (obd)N×Nis a Z-matrix if it has non-positive off-diagonal entries (namely obd ≤0, ∀b6=d) and all positive diagonal entries. Definition 2.1. The square matrix O= (obd)N×Nis said to be a nonsingular M-matrix if Ocan take the form O=sI −Tsuch that s>ρ(T), all elements of Tare nonnegative, Iis the identity matrix and ρ(T) is the spectral radius of T. Lemma 2.1. If Ois a Z-matrix, the following assertions are equivalent: (i) Ois a nonsingular M-matrix. (ii) Ois semi-positive. (iii) O−1exists and its elements are all nonnegative. Suppose that l(·) and W(·) are independent. Consider the following HNSFDEswID:    d(τ(ϑ)−F(ϑ, τϑ)) = h1(τ(ϑ), τϑ, ϑ, l(ϑ)) dϑ +h2(τ(ϑ), τϑ, ϑ, l(ϑ)) dW (ϑ), ϑ ≥0 τ0=χ, (2.1) where the initial function χ∈ BC((−∞,0],Rc)∩Lq((−∞,0],Rc)) l(0) = b0∈Θ, τ(ϑ) = (τ1(ϑ), τ2(ϑ), . . . , τi(ϑ))Tand τϑ: (−∞,0] →Rcis the segment of the solution defined as τϑ(θ) = τ(ϑ+θ) for θ∈(−∞,0]. Consequently, the initial condition τ0=χmeans τ(θ) = χ(θ) for all θ≤0. Assume that h1:Rc×BC((−∞,0],Rc)×R+×Θ→Rc, h2:Rc×BC((−∞,0],Rc)×R+×Θ→Rc×p, F :R+×BC((−∞,0],Rc)→Rc. Assume that F(ϑ, 0) = h1(0,0, ϑ, r) = h2(0,0, ϑ, r) = 0 for all (ϑ, r)∈R+×Θ. Let C1,2(Rc×R+×Θ,R+) be the set of all non-negative functions V(τ, ϑ, b) on Rc×R+×Θ, which are twice continuously differentiable with respect to τand once continuously differentiable with respect to ϑ. 4 Let LV:Rc×BC((−∞,0],Rc)×R+×Θ→Rbe the operator satisfying (see [20]) LV(τ, φ, ϑ, b) = Vϑ(τ−F(ϑ, φ), ϑ, b) + Vτ(τ−F(ϑ, φ), ϑ, b)h1(φ, ϑ, b) +1 2trace hT 2(φ, ϑ, b)Vττ (τ−F(ϑ, φ), ϑ, b)h2(φ, ϑ, b) + N X k=1 γbkV(τ−F(ϑ, φ), ϑ, k), where Vϑ(τ, ϑ, b) = ∂V (τ, ϑ, b) ∂ϑ , Vτ(τ, ϑ, b) = ∂V (τ, ϑ, b) ∂τ1 ,...,∂V (τ, ϑ, b) ∂τc, Vττ (τ, ϑ, b) = ∂2V(τ, ϑ, b) ∂τm∂τnc×c . Lemma 2.2. Let q > 1,ε > 0and (i1, i2)∈R2. Then, |i1+i2|q≤1 + ε1 q−1q−1|i1|q+|i2|q ε. Proof. See ([18]). Remark 2.3. Let q > 1 and (i1, i2)∈R2. By taking ε= 1 in Lemma 2.2, we obtain |i1+i2|q≤2q−1|i1|q+|i2|q. Assumption 2.4. For each ε > 0, there exists e Kε>0such that |h1(φ1(0), φ1, ϑ, b)−h1(φ2(0), φ2, ϑ, b)|∨|h2(φ1(0), φ1, ϑ, b)−h2(φ2(0), φ2, ϑ, b)| ≤ e Kε||φ1−φ2|| (2.2) ∀φ1, φ2∈ BC((−∞,0],Rc)verifying ||φ1||∨||φ2|| ≤ εand ∀(ϑ, b)∈R+×Θ. Assumption 2.5. Suppose that there is a constant γ∈(0,1) such that ∀ϑ≥0and ∀φ∈ BC((−∞,0],Rc), we have |F(φ)| ≤ γ|φ(0)|.(2.3) Assumption 2.6. Let q≥2and ν∈ N ((−∞,0),R+). Suppose that for each m∈Θthere exist constants Ξ3<0,Ξ1,Ξ2and Ξ4≥0such that Ξ2≥Ξ1≥0and |φ(0) −F(ϑ, φ)|q−2(φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, b) + q−1 2|h2(φ(0), φ, ϑ, b)|2 ≤Ξ4|φ(0) −F(ϑ, φ)|q−2+ Ξ3|φ(0) −F(ϑ, φ)|q+ Ξ1Z0 −∞ |φ(θ)|qν(θ)dθ −Ξ2|φ(0)|q.(2.4) 5 Let π > π ≥0, while we shall assume πa∈[π, π] and π+π > π. Hence, set a bounded function v:R+→[π, π +π) defined by v(ϑ) = ϑ−aπ for ϑ∈[aπ +πa,(a+ 1)π+πa+1), a = 0,1,2, ... (2.5) If the HNSFDEswID (2.1) is unstable, we need to construct a delay feedback controller u(τ(ϑ−v(ϑ)), ϑ, l(ϑ)) to guarantee that equation (2.1) becomes stable. Consequently, we will study the controlled HNSFDEswID, for ϑ≥0 d(τ(ϑ)−F(ϑ, τϑ)) = (h1(τ(ϑ), τϑ, ϑ, l(ϑ))+u(τ(ϑ−v(ϑ)), ϑ, l(ϑ)))dϑ+h2(τ(ϑ), τϑ, ϑ, l(ϑ)) dW (ϑ), (2.6) with the initial condition bτπ0=bχ∈ BC((−∞,0],Rc)∩Lq((−∞,0],Rc), l(0) = b0∈Θ,(2.7) and the Borel measurable control function u:Rc×R+×Θ−→ Rc. Assumption 2.7. Suppose that there exists a positive constant ρsatisfying |u(z, ϑ, b)−u(z, ϑ, b)| ≤ ρ|z−z|,(2.8) ∀(ϑ, b)∈R+×Θand ∀z, z ∈Rc. Moreover, suppose that u(0, ϑ, b)≡0, for all (ϑ, b)∈R+×Θ. Clearly this hypothesis implies ∀(z, ϑ, b)∈Rc×R+×Θ |u(z, ϑ, b)| ≤ ρ|z|.(2.9) Definition 2.2. (i) System (2.1) is said to be asymptotically bounded in q-th moment if any of its solutions τ(·) satisfies lim sup ϑ→+∞ E|τ(ϑ)|q< M, (2.10) with Mis a positive constant. (ii) System (2.1) is called exponentially stable in q-th moment if lim ϑ→+∞ 1 ϑlog (E|τ(ϑ)|q)<0, for any solution τ(·). (iii) System (2.1) is called almost surely exponentially stable if lim ϑ→+∞sup 1 ϑlog |τ(ϑ)|<0,P−a.s., for any solution τ(·). 6 3 Existence, Uniqueness and Boundedness In this section, we will investigate the existence and uniqueness of the solution and the mean square exponential stability of system (2.6). Theorem 3.1. Under Assumptions (2.4)-(2.6), for any initial condition satisfying (2.7), there is a unique global solution τ(ϑ)to equation (2.6). Proof. We will split the proof into two steps. Step 1: Let V:Rc×R+×Θ→R+be the Lyapunov function: V(τ, ϑ, b) := V(τ) = |τ|q.(3.1) We then have LV (φ(0), φ, ϑ, b) = q|φ(0) −F(ϑ, φ)|q−2(φ(0) −F(ϑ, φ))T[h1(φ(0), φ, ϑ, b) + u(ϑ, φ(−v(ϑ)), b)] +1 2q|φ(0) −F(ϑ, φ)|q−2|h2(φ(0), φ, ϑ, b)|2 +1 2q(q−2)|φ(0) −F(ϑ, φ)|q−4|(φ(0) −F(ϑ, φ))Th2(φ(0), φ, ϑ, b)|2.(3.2) Using the inequality |(φ(0) −F(ϑ, φ))Th2(φ(0), φ, ϑ, b)|2≤ |φ(0) −F(ϑ, φ)|2|h2(φ(0), φ, ϑ, b)|2,(3.3) we deduce LV (φ(0), φ, ϑ, b) =q|φ(0) −F(ϑ, φ)|q−2h(φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, b) + q−1 2|h2(φ(0), φ, ϑ, b)|2 + (φ(0) −F(ϑ, φ))Tu(ϑ, φ(−v(ϑ)), b)i ≤qΞ4|φ(0) −F(ϑ, φ)|q−2+qΞ3|φ(0) −F(ϑ, φ)|q+qγ|φ(0) −F(ϑ, φ)|q−1|φ(−v(ϑ))| +qΞ1Z0 −∞ |φ(θ)|qν(θ)dθ −qΞ2|φ(0)|q.(3.4) Applying Young’s inequality, for ε > 0 arbitrary, we can derive that qγ|φ(0) −F(ϑ, φ)|q−1|φ(−v(ϑ))|= (qγ)q q−1|φ(0) −F(ϑ, φ)|q (εq)1 q−1!q−1 q (εq|φ(−v(ϑ))|q)1 q ≤(qγ)q q−1 ε1 q−1|φ(0) −F(ϑ, φ)|q+ε|φ(−v(ϑ))|q.(3.5) 7 Hence, LV (φ(0), φ, ϑ, j)≤ −e K|φ(0) −F(ϑ, φ)|q+qΞ4|φ(0) −F(ϑ, φ)|q−2+ε|φ(−v(ϑ))|q +qΞ1Z0 −∞ |φ(θ)|qν(θ)dθ −|φ(0)|q.(3.6) Choose ε > (qγ)q (−qΞ3)q−1such that e K=−qΞ3−(qγ)q q−1 ε1 q−1 >0, and note that V(t) = qΞ4tq−2−e Ktq has a finite supremum value over [0,+∞) denoted by ϕ4= sup t∈[0,∞)V(t). Hence, LV (φ(0), φ, ϑ, b)≤ϕ4+qΞ1Z0 −∞ |φ(θ)|qν(θ)dθ −|φ(0)|q+ε|φ(−v(ϑ))|q.(3.7) Step 2: As the coefficients of equation (2.6) are continuous and locally Lipschitz, for any given initial condition (2.7), by the standard truncation method, there is a unique maximal local strong solution of equation (2.6) on (−∞, νe), with νethe explosion time (see, e.g., [19, Theorem 3.2.2, p.95] and [28, Theorem 3.3]). Let c0>0 be sufficiently large satisfying ||bχ|| < c0. For each c∈N∗such that c≥c0, let νc be the stopping time defined by νc= inf{ϑ∈[0, νe); |τ(ϑ)| ≥ c}. It is clear that νcis increasing. Set ν∞= lim c→∞ νc. It is obvious that ν∞≤νea.s. We will prove that ν∞=∞a.s, then νe=∞a.s. Set eτ(ϑ) = τ(ϑ)−F(ϑ, τϑ) and eτ(π0) = bτ(π0)−F(ϑ, bτπ0). Applying Itˆo’s formula and (3.7), EV(eτ(ϑ∧νc)) = EV(eτ(π0)) + EZϑ∧νc π0 LV (τ(s), τs, s, l(s)) ds ≤EV(eτ(π0)) + ϕ4ϑ+εEZϑ∧νc π0|τ(s−v(s))|qds +qΞ1EZϑ∧νc π0Z0 −∞ |τ(s+θ)|qν(θ)dθ −|τ(s)|qds.(3.8) 8 Applying now Fubini’s theorem, EZϑ∧νc π0Z0 −∞ |τ(s+θ)|qν(θ)dθ −|τ(s)|qds =EZ0 −∞ Zϑ∧νc π0|τ(s+θ)|qdsν(θ)dθ−EZϑ∧νc π0|τ(s)|qds ≤EZ0 −∞ Zϑ∧νc −∞ |τ(s)|qdsν(θ)dθ−EZϑ∧νc π0|τ(s)|qds =EZπ0 −∞ |bχ(s)|qds. (3.9) Besides, for all s≥π0, we have 0 ≤s−v(s)≤s. This yields E|τ(s−v(s))|q1[π0,νc](s)≤sup 0≤w≤s E|τ(w∧νc)|q. Therefore, by (2.3), we obtain EV(eτ(ϑ∧νc)) ≤EV(eτ(π0)) + ϕ4ϑ+qΞ1EZπ0 −∞ |bχ(s)|qds +εZϑ π0 E|τ(s−v(s))|q1[π0,νc](s)ds ≤EV(eτ(π0)) + ϕ4ϑ+qΞ1EZπ0 −∞ |bχ(s)|qds +εZϑ π0 sup 0≤w≤s E|τ(w∧νc)|qds ≤E|eτ(π0)|q+ϕ4ϑ+qΞ1EZπ0 −∞ |bχ(s)|qds +εZϑ π0 sup 0≤w≤s E|τ(w∧νc)|qds ≤2q−1E|bτπ0|q+ 2q−1E|F(π0,bτπ0)|q+ϕ4ϑ+qΞ1EZπ0 −∞ |bχ(s)|qds +εZϑ π0 sup 0≤w≤s E|τ(w∧νc)|qds ≤2q−1E||bχ||q(1 + κq) + ϕ4ϑ+qΞ1EZπ0 −∞ |bχ(s)|qds +εZϑ π0 sup 0≤w≤s E|τ(w∧νc)|qds. (3.10) 9 are non singular M-matrix and define (η1, η2, ..., ηN)T= Λ−1 1(1,1, ..., 1)T,(bη1,bη2, ..., bηN)T= Λ−1 2(1,1, ..., 1)T,(4.4) where {ηb}1≤b≤Nand {bηb}1≤b≤Nare positive constants. Theorem 4.4. Let Assumptions (2.4)-(2.7) and (4.3) hold. Suppose that there exist positive constants k,δ1,δ2,δ3,p≥q≥2,αj, j = 1,...,5, such that 0< α1<1, a nonnegative constant λand a function W(·)∈C(Rc,R+)such that α3e φ(0) p+q−2≤ W(e φ(0)) ≤α4+α5e φ(0) p+q−2 ,(4.5) and LU(φ, ϑ, b) + δ1|h1(φ(0), φ, ϑ, b)|2+δ2|h2(φ(0), φ, ϑ, b)|2+5 2δ32ηbe φ(0)+qbηbe φ(0) q−12 ≤ −ke φ(0)2+|φ(0)|2−α1Z0 −∞ |φ(θ)|2ν(θ)dθ −λW(φ(0)) −W(e φ(0)) + α2Z0 −∞ W(φ(θ))ν(θ)dθ, (4.6) ∀(ϑ, φ, b)∈R+×BC((−∞,0],Rc)×Θ. Suppose also π∗is sufficiently small such that π∗<pkδ3(1 −α1) 2ρ2and π∗≤√δ1δ3 √2ρ∧δ2δ3 ρ2∧1 2√10ρ.(4.7) Therefore, for any initial condition (2.7), the solution of equation (2.6) satisfies lim ϑ→+∞ 1 ϑlog E|τ(ϑ)|2<0. Proof. We split this proof into four steps. Step 1: Using Assumption (2.7), (4.2) and (4.3), we can derive that LU(τ(ϑ), τϑ, ϑ, l(ϑ)) ≤LU(τϑ, ϑ, l(ϑ)) + 5 2δ32ηl(ϑ)|eτ(ϑ)|+qbηl(ϑ)|eτ(ϑ)|q−12 +ρ2 10δ3|τ(ϑ)−τ(ϑ−v(ϑ))|2. 16 Thus, b U(τϑ, ϑ,blϑ)≤b U(τπ0, π0,blπ0) + Zϑ π0 Lb U(τs, s,bls)ds +c M(ϑ),(4.8) where, Lb U(τϑ, ϑ,blϑ) = LU(τϑ, ϑ, l(ϑ)) + 5 2δ32ηl(ϑ)|eτ(ϑ)|+qbηl(ϑ)|eτ(ϑ)|q−12 +ρ2 10δ3|τ(ϑ)−τ(ϑ−v(ϑ))|2+ωπ∗Φ(τϑ, ϑ, l(ϑ)) −ωZϑ ϑ−π∗ Φ(τs, s, l(s))ds. (4.9) On the other hand, by Assumptions (2.4), (2.7) and Theorem 3.2, we deduce sup π0≤ϑ<∞ ELb U(τϑ, ϑ,blϑ)<∞.(4.10) Step 2: Let ω=ρ2 δ3 . Using (4.8) and (4.10), we obtain for any ϑ≥π0 eεϑEb U(τϑ, ϑ,blϑ)≤eεπ0b U(τπ0, π0,blπ0) + EZϑ π0 eεs εb U(τs, s,bls) + Lb U(τs, s,bls)ds. (4.11) By condition (4.7), we can see that 2 (π∗)2ρ2 δ3≤δ1and π∗ρ2 δ3≤δ2. Then, by elementary inequality and (2.9), we obtain ωπ∗Φ(τs, s, l(s)) ≤2 (π∗)2ρ2 δ3|h1(s, τs, l(s))|2+π∗ρ2 δ3|h2(s, τs, l(s))|2 +2 (π∗)2ρ2 δ3|u(τ(s−v(s)), s, l(s))|2 ≤δ1|h1(s, τs, l(s))|2+δ2|h2(s, τs, l(s))|2+2 (π∗)2ρ4 δ3|τ(s−v(s))|2. (4.12) 17 Plugging (4.12) into (4.9) and using (4.6), we have Lb U(τs, s,bls)≤LU(τs, s, l(s)) + δ1|h1(s, τs, l(s))|2+δ2|h2(s, τs, l(s))|2 +5 2δ32ηl(s)|eτ(s)|+qbηl(s)|eτ(s)|q−12+2 (π∗)2ρ4 δ3|τ(s−v(s))|2 +ρ2 10δ3|τ(ϑ)−τ(ϑ−v(ϑ))|2−ρ2 δ3Zs s−π∗ Φ(τw, w, l(w))dw ≤ −k|eτ(s)|2+|τ(s)|2−α1Z0 −∞ |τ(s+θ)|2ν(θ)dθ−λW(τ(s)) −W(eτ(s)) +α2Z0 −∞ W(τ(s+θ))ν(θ)dθ +2 (π∗)2ρ4 δ3|τ(s−v(s))|2 +ρ2 10δ3|τ(s)−τ(s−v(s))|2−ρ2 δ3Zs s−π∗ Φ(τw, w, l(w))dw. (4.13) Using (4.7), we can derive that ρπ∗≤1 2√10 and 2 (π∗)2ρ4 δ3|τ(s−v(s))|2≤(4π∗)2ρ4 δ3|τ(s)|2+ρ2 10δ3|τ(s)−τ(s−v(s))|2. Hence, Lb U(τs, s,bls)≤ −k|eτ(s)|2+kα1Z0 −∞ |τ(s+θ)|2ν(θ)dθ −λW(τ(s)) −W(eτ(s)) +α2Z0 −∞ W(τ(s+θ))ν(θ)dθ +ρ2 5δ3|τ(s)−τ(s−v(s))|2 − k−4 (π∗)2ρ4 δ3!|τ(s)|2−ρ2 δ3Zs s−π∗ Φ(τw, w, l(w))dw. (4.14) Plugging (4.14) into (4.11), we have eεϑEb U(τϑ, ϑ,blϑ)≤eεπ0b U(τπ0, π0,blπ0) + EZϑ π0 εeεs b U(τs, s,bls)ds +I1+I2+I3−I4,(4.15) 18 where I1=EZϑ π0 eεs −k|eτ(s)|2− k−4 (π∗)2ρ4 δ3!|τ(s)|2+kα1Z0 −∞ |τ(s+θ)|2ν(θ)dθ!ds I2=EZϑ π0 eεs −λW(τ(s)) −W(eτ(s)) + α2Z0 −∞ W(τ(s+θ))ν(θ)dθds I3=ρ2 5δ3 EZϑ π0 eεs |τ(s)−τ(s−v(s))|2ds I4=ρ2 δ3 EZϑ π0 eεs Zs s−π∗ Φ(τw, w, l(w))dwds. Step 3: Using the substitution method, we obtain Zϑ π0Z0 −∞ eεs |τ(s+θ)|2ν(θ)dθds =Z0 −∞ e−εθν(θ)Zϑ π0 eε(s+θ)|τ(s+θ)|2dsdθ =Z0 −∞ e−εθν(θ)Zϑ+θ π0+θ eεs |τ(s)|2dsdθ ≤M(ε)Zϑ −∞ eεs |τ(s)|2ds ≤M(ε)eεπ0Zπ0 −∞ |bχ(s)|2ds +M(ε)Zϑ π0 eεs |τ(s)|2ds. Therefore, I1≤kα1eεπ0M(ε)Zπ0 −∞ |bχ(s)|2ds − k−4 (π∗)2ρ4 δ3−kα1M(ε)!EZϑ π0 eεs |τ(s)|2ds −kEZϑ π0 eεs |eτ(s)|2ds. (4.16) By the same method, we have I2≤α2eεπ0M(ε)Zπ0 −∞ |W(bχ(s))|ds +M(ε)α2−λEZϑ π0 eεsW(τ(s))ds −EZϑ π0 eεsW(eτ(s))ds. (4.17) Applying the Fubini theorem, we can derive that ρ2 5δ3 EZϑ π0 eεs |τ(s)−τ(s−v(s))|2ds =ρ2 5δ3Zϑ π0 eεsE|τ(s)−τ(s−v(s))|2ds. 19 By the Itˆo isometry and the H¨older inequality, E|τ(s)−τ(s−v(s))|2 ≤3E|F(s, τs)−F(s, τs−v(s))|2+ 3EZs s−v(s)π∗|h1(τ(w), τw, w, l(w)) −u(τ(w−v(w)), w, l(w))|2 +|h2(τ(w), τw, w, l(w))|2dw ≤3E|F(s, τs)−F(s, τs−v(s))|2+ 3EZs s−π∗π∗|h1(τ(w), τw, w, l(w)) −u(τ(w−v(w)), w, l(w))|2 +|h2(τ(w), τw, w, l(w))|2dw. Then, by Theorem 3.2, we have I3≤3 5I4+ 3γ2(E|τ(s)|2+E|τ(s−v(s))|2) ≤3 5I4+ 6γ2C4.(4.18) Substituting (4.16)–(4.18) into (4.15), eεϑEb U(τϑ, ϑ,blϑ)≤C5+EZϑ π0 εeεs b U(τs, s,bls)ds − k−4 (π∗)2ρ4 δ3−kα1M(ε)!EZϑ π0 eεs |τ(s)|2ds −kEZϑ π0 eεs |eτ(s)|2ds −λ−M(ε)α2EZϑ π0 eεsW(τ(s))ds −EZϑ π0 eεsW(eτ(s))ds −2 5I4, where C5=eεπ0b U(τπ0, π0,blπ0) + kα1eεπ0M(ε)Zπ0 −∞ |bχ(s)|2ds +α2eεπ0M(ε)Zπ0 −∞ |W(bχ(s))|ds + 6γ2C4. Step 4: By an elementary inequality and (4.5), we obtain |τ|q≤ |τ|2+|τ|p+q−2≤ |τ|2+W(τ) α3 . 20 Using now the definition of b U, ω1eεϑE|eτ(ϑ)|2≤eεϑEb U(τϑ, ϑ,blϑ) ≤C5−1−εω3 α3EZϑ π0 eεsW(eτ(s))ds −(k−εω2−εω3)EZϑ π0 eεs |eτ(s)|2ds − k−4 (π∗)2ρ4 δ3−kα1M(ε)!EZϑ π0 eεs |τ(s)|2ds −λ−M(ε)α2EZϑ π0 eεsW(τ(s))ds −2 5I4+I5,(4.19) where ω1= min i∈Θηi, ω2= max i∈Θηi, ω3= max i∈Θbηi and I5=ερ2 δ3 EZϑ π0 eεs Z0 −π∗Zs s+v Φ(τw, w, l(w))dwdvds. It is straightforward to show that I5≤ερ2 δ3 EZϑ π0 eεs π∗Zs s−π∗ Φ(τw, w, l(w))dwds =επ∗I4. We may choose ε > 0 to be sufficiently small and satisfying επ∗≤2 5, εω3 α3≤1, εω2+εω3≤k kα1M(ε)+4(π∗)2ρ4 δ3≤kand M(ε)α2≤λ. Substituting these into (4.19), we obtain E|eτ(ϑ)|2≤C5 ω1 e−εϑ,∀ϑ≥π0. We know that |eτ(ϑ)|2≥(1 1 + ε−γ2 ε)|τ(ϑ)|2.(4.20) We have lim ε→∞ ε 1 + ε−γ2= 1 −γ2>0. Then, there exists ε > 0, such that 1 1 + ε−γ2 ε=1 ε(ε 1 + ε−γ2)>0. 21 Hence, E|τ(ϑ)|2≤1 1 1+ε−γ2 ε C5 ω1 e−εϑ,∀ϑ≥π0, as desired. Theorem 4.5. Suppose that all assumptions of Theorem 4.4 hold. Then, for any initial condition satisfying (2.7), we have (i) for any bq∈[2, q), the solution of equation (2.6) satisfies lim ϑ→+∞ 1 ϑlog E|τ(ϑ)| bq<0. (ii) The solution of equation (2.6) is almost surely exponentially stable. Proof. (i) By the H¨older inequality and Theorem 3.2, we can derive that, for any 2 ≤bq < q, E|τ(ϑ)| bq≤E|τ(ϑ)|2q−bq q−2(E|τ(ϑ)|q) bq−2 q−2 ≤ 1 1 1+ε−γ2 ε C5 ω1!q−bq q−2 C bq−2 q−2 4e−εϑ q−bq q−2. (ii) Using Itˆo’s formula and proceeding as (4.8) and (4.11), we obtain eεϑ |eτ(ϑ)|2≤C5 ω1 +f M(ϑ),(4.21) where f M(ϑ) is a local continuous martingale with initial value f M(π0) = 0. Using the non-negative semi-martingale convergence theorem, one can derive that lim ϑ→∞ sup eεϑ |eτ(ϑ)|2<∞, a.s. (4.22) Then, there exists a finite positive random variable νsatisfying sup π0≤ϑ<∞ eεϑ |eτ(ϑ)|2≤ν, a.s. (4.23) Proceeding as in (4.20) and the proof of Theorem 2 in [3], we have sup π0≤ϑ<∞ eεϑ |τ(ϑ)|2≤C6,(4.24) where C6=1 1 1+ε−γ2 ε ν. This yields that, for all ϑ≥π0, |τ(ϑ)|2≤C6e−εϑ.(4.25) Therefore, lim ϑ→∞ sup ln |τ(ϑ)| ϑ≤ −ε 2,(4.26) as desired. 22 5 Illustrative Example In this section, we present the following HNSFDEswID to illustrate our results: d[τ(ϑ)−F(ϑ, τϑ)] = h1(τ(ϑ), τϑ, ϑ, l(ϑ)) dϑ +h2(τ(ϑ), τϑ, ϑ, l(ϑ)) dW(ϑ),(5.1) where W(ϑ) is a one dimensional Brownian motion, the function Fis defined as F(ϑ, φ) = 1 2 φ(0) 1 + e−φ2(−ς(ϑ)) , where ς:R→[0,+∞) is a continuous function which represents the unbounded variable delay. In this sense, when we replace φby τϑ, the expression takes the following form F(ϑ, τϑ) = 1 2 τ(ϑ) 1 + e−τ2(ϑ−ς(ϑ)) . The functions h1and h2have the corresponding expressions h1(τ(ϑ), τϑ, ϑ, 1) = −21 + e−τ2(ϑ−ς(ϑ))2τ(ϑ)−1 4Z0 −∞ |τ(ϑ+θ)|eθdθ, h2(τ(ϑ), τϑ, ϑ, 1) = 1 2Z0 −∞ |τ(ϑ+θ)|eθdθ, h1(τ(ϑ), τϑ, ϑ, 2) = −4 31 + e−τ2(ϑ−ς(ϑ))3τ(ϑ)−1 5Z0 −∞ |τ(ϑ+θ)|eθdθ, h2(τ(ϑ), τϑ, ϑ, 2) = r2 15 Z0 −∞ |τ(ϑ+θ)|eθdθ. Consider the following initial condition bχ=e0.01ϑ−e−1,if ϑ∈(−100,0] 0,if ϑ∈(−∞,−100],(5.2) l(0) = 1. Remark 5.1. Since it is hard to make numerical simulation for the equation with infinite delay, we have considered a special initial condition (5.2) here. Although this is enough to illustrate our previous theoretical results. For the theory of numerical methods of SDEs, see [15], [16] and [32]. 23 Set Θ = {1,2}and the matrix Λ = (λmn)1≤m,n≤2given by −1 1 1−1. Thus, for b= 1 (φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, 1) + 1 2|h2(φ(0), φ, ϑ, 1) |2 ≤ −13 8|φ(0)|2+1 2Z0 −∞ |φ(θ)|eθdθ2 .(5.3) For b= 2 (φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, 2) + 1 2|h2(φ(0), φ, ϑ, 2) |2 ≤ −28 15|φ(0)|2+1 5Z0 −∞ |φ(θ)|eθdθ2 .(5.4) Using the H¨older inequality, we obtain Z0 −∞ |φ(θ)|eθdθ2≤Z0 −∞ |φ(θ)|2eθdθ. (5.5) Furthermore, |φ(0) −F(ϑ, φ)|= 1 2+e−φ2(−ς(ϑ)) 1 + e−φ2(−ς(ϑ)) |φ(0)| ≤ |φ(0)|.(5.6) Substituting (5.5) and (5.6) into (5.3) and (5.4), we have ∀b∈ {1,2} (φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, b) + 1 2|h2(φ(0), φ, ϑ, b)|2 ≤ −13 8|φ(0)|2+1 2Z0 −∞ |φ(θ)|2eθdθ, =−|φ(0)|2−5 8|φ(0)|2+1 2Z0 −∞ |φ(θ)|2eθdθ, ≤ −|φ(0) −F(ϑ, φ)|2−5 8|φ(0)|2+1 2Z0 −∞ |φ(θ)|2eθdθ. (5.7) By the fact that |φ(0) −F(ϑ, φ)|2−2= 1, we can see that ∀b∈ {1,2} (φ(0) −F(ϑ, φ))Th1(φ(0), φ, ϑ, b) + 1 2|h2(φ(0), φ, ϑ, b)|2 ≤ −|φ(0) −F(ϑ, φ)|2+ 1 −5 8|φ(0)|2+1 2Z0 −∞ |φ(θ)|2eθdθ. (5.8) 24 Then, Assumption (2.6) is satisfied with Ξ1=1 2, Ξ2=5 8, Ξ3=−1 and Ξ4= 1. However, letting the initial condition (5.2), from the numerical simulation of the computer based on Euler- Maruyama scheme with time step 10−3, we can see that HNSFDEswID (5.1) is not stable. This result can be clearly illustrated in Figure 1. 0 5 10 15 20 0.5 1 1.5 2 2.5 0 5 10 15 20 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 1: The computer simulation of the sample paths of the Markov chain and the System (5.1) using the EulerMaruyama method with time step 10−3. Now, we will construct a control function uto stabilize the system (5.1). Let the function u:R×R+×S−→ Rdefined by u(τ, ϑ, 1) = 1 2τand u(τ, ϑ, 2) = 4 3τ. (5.9) It is obvious to see that Assumption (2.7) is fulfilled with ρ= 2. By Theorems 3.1 and 3.2, the following system: d(τ(ϑ)−F(ϑ, τϑ)) = (h1(τ(ϑ), τϑ, ϑ, l(ϑ)) + u(τ(ϑ−v(ϑ)), ϑ, l(ϑ))) dϑ+h2(τ(ϑ), τϑ, ϑ, l(ϑ)) dW (ϑ), (5.10) 25