Practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion

Practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion Tom´as Caraballoa∗Faten Ezzineb Mohamed Ali Hammamib aUniversidad de Sevilla, Depto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, c/ Tarfia s/n, 41008-Sevilla (Spain) bUniversity of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia E.mail: [email protected]u.tn Abstract In this paper, practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion (G-SDEs) is studied. The analysis of the global practical uniform pth moment exponential stability, as well as the global practical uniform exponential stability with respect to a part of the variables of G-SDEs are investigated by means of the G-Lyapunov functions. An illustrative example to show the usefulness of the practical stability with respect to a part of the variables notion is also provided. Keywords: G-Stochastic differential equations, G-Lyapunov techniques, G-Itˆo formula, G- Brownian motion, Nontrivial solution, Practical stability with respect to a part of the variables. Mathematics Subject Classification Primary 93E03 Secondary 60H10. ∗The research of Tom´as Caraballo has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andaluc´ıa (Consejer´ıa de Econom´ıa y Conocimiento) and FEDER under projects US-1254251 and P18-FR-4509. 1 1 Introduction Stochastic differential equations are widely used for modeling physical, technical, ecological, economical dynamical systems, wherein significant uncertainty is present. With the rapid development of stochastic analysis theory, the stability of SDEs has become a very prevalent theme of recent research in Mathematics and its applications. Some stochastic systems do not fulfill stability properties with respect to all the unknown variables of the system. Nevertheless, it is very crucial to investigate if it is still possible to prove some stability criteria with respect to some of the variables in the problem, which is called ”Stability with respect to a part of the variables.” In 1892, Lyapunov introduced the theory of stability for deterministic dynamical systems and firstly established the so-called Lyapunov exponent method. Since then, the method of Lyapunov functions has become one of the most powerful tools to study the stability of stochastic dynamical systems, we would like to mention here the references [5, 6, 19], among others. With the development of the second method of Lyapunov, Ignatyev [2], Peiffer and Rouche [23], Rouche et al. [27], Rumyantsev [28], Rumyantsev and Oziraner [29], Savchenko and Ignatyev [30], Vorotnikov [31], Vorotnikov and Rumyantsev [32] investigated the concept of stability with respect to a part of the variables especially is supposed to be an equilibrium point. The stability with respect to a part of the variables has been used in analyzing the qualitative properties of equilibria and boundedness properties of motions of dynamical systems determined by ordinary differential equations, difference equations, functional differential equations, stochastic differential equations, etc. It involves a notion of stability with respect to only a prespecified subset of the state variables describing the motions of the system under investigation. Several intrinsic variants to Lyapunov’s original concepts of practical stability were proposed in [2, 4]. In the case that the origin is not an equilibrium point, we can investigate the stability of the SDEs in a small neighborhood of the origin in terms of convergence of solution in probability to a small ball. This property is defined as ”Practical stability”. The practical stability, in the sense introduced in [3, 9, 20, 21]. In fact, it is very important and very useful for analyzing the stability or for designing practical controllers of dynamical systems since controlling system to an idealized point are either expensive or impossible in the presence of disruptions and the best which we can hope in such situations is to use practical stability. In practice, we may only need to stabilize a system into the region of phase space in which the implementation is still acceptable. It is well known that asymptotic stability is more important than stability. Also, the desired system may oscillate near the origin. Thus the notion of practical stability is more suitable in several situations than Lyapunov stability. In this case, all state trajectories are bounded and approach a sufficiently small neighborhood of the origin. One also desires that the state approaches the origin (or some sufficiently small neighborhood of it) in a sufficiently fast manner especially in presence of perturbations. In general, we know some information on the upper bound of the term of perturbation whose size influences the size of the ball. Recently, in [7, 8], the practical stability of stochastic stability with respect to a part of the variables is investigated. 2 For the dormant applications in uncertainty problems, risk measures, and superhedging in finance, considerable attention has been paid to the theory of nonlinear expectation. Notably, Peng [24] built the fundamental basics of time-consistent G-expectation as well as G-conditional expectation where G is the infinitesimal generator of a nonlinear heat equation. Under the G- framework, Peng [22, 24] introduced the notion of G-normal distribution, G-Brownian motion, and he also established the corresponding stochastic calculus of Itˆo’s type. Since then, much researches have been carried out on the stochastic analysis concerning the G-Brownian motion. On that basis, Gao [11] and Peng [22] studied the existence and uniqueness of the solution of stochastic differential equations driven by G-Brownian motion (G-SDEs, in short) under a standard Lipschitz condition. Moreover, Lin [17] obtained the existence and uniqueness of the solution of G-SDEs with reflecting boundary. The G-Brownian motion has a very rich and interesting new structure which non-trivially, for a recent account and development of this theory, we refer the reader to see [1, 15, 16]. To the best of our knowledge, no work has been published about the practical stability of stochastic differential equations driven by G-Brownian motion. Our main objective in this paper is to extend the results in [7], which investigated the practical stability with respect to a part of the variables of stochastic differential equations driven by the standard Brownian motion, to the case of G-Brownian motion. Based on the method of Lyapunov and recently developed Itˆo calculus for SDE driven by G-Brownian motion, we establish some criteria for the global practical uniform pth moment exponential stability, and the global practical uniform exponential stability with respect to a part of the variables of a class of nonlinear stochastic differential equations driven by G-Brownian motion. This paper is structured in the following way: In Section 2,we recall some necessary preliminaries and results. In Sections 3 and 4,we establish sufficient conditions to ensure the global practical uniform pth moment exponential stability, and the global practical uniform exponential stability with respect to a part of the variables of G-SDEs by using G-Lyapunov techniques and Itˆo’s calculus for G-SDEs. Moreover, we exhibit an illustrative example to show the applicability of our abstract theory. Eventually, in Section 6, some conclusions are included. 2 Preliminaries In this section, we briefly recall some notations and preliminaries about sublinear expectations and G-Brownian motions. For more details, one can see [14, 22, 24, 25]. Notations on G-stochastic calculus •Rn: the space of n-dimensional real column vectors. • hx, yi: the scalar product of two vectors x, y ∈Rn. •If x∈Rn,|x|denotes its Euclidean norm. 3 •Ωt:= {ω·∧t:ω∈Ω}. • Ft=B(Ωt). • B(Ω) : the Borel σ-algebra of Ω. • Cb,Lip(Rn): the space of all bounded real-valued Lipschitz continuous functions. •L0(Ω) : the space of all B(Ω)-measurable real functions. •L0(Ωt) : the space of all B(Ωt)-measurable real functions. •Bb(Ω) : all bounded elements in L0(Ω), Bb(Ωt) := Bb(Ω) ∩L0(Ωt). •Lp G(Ω) : Banach space under the natural norm ||X||p=b E(|X|p)1 p. •Mp,0 G([0, T]) = {ζ:= ζt(ω) = PN−1 j=0 ζj1[tj,tj+1)(t),∀N > 0, 0 = t0< ... < tN=T, ζj∈Lp G(ωtj), j = 0,1,2, .., N −1}. •Mp G([0, T]) : the completion of Mp,0 Gunder ||η||Mp G=|ZT 0b E(|η(t)|p)dt|1 p. Let Ω be a given set and Hbe a linear space of real valued functions defined on Ω.We further suppose that Hsatisfies a∈ H for each constant aand |X| ∈ H if X∈ H. Definition 2.1. A sublinear expectation b Eon His a functional b E:H → Rsatisfying the following properties: for all X, Y ∈ H, i) Monotonicity: if X≥Y, then b E(X)≥b E(Y). ii) Constant preserving: b E(a) = a, ∀a∈R. iii) Sub-additivity: b E(X+Y)≤b E(X) + b E(Y). iv) Positive homogeneity: b E(λX) = λb E(X), λ ≥0. The triple (Ω,H,b E) is called a sublinear expectation space. X∈ H is called a random variable in (Ω,H,b E). Y = (Y1, ..., Yn),where Yi∈ H is called an n-dimensional random vector in (Ω,H,b E). Definition 2.2. Weakly compact sets are defined by sets which are compact with respect to the weak topology of a Banach space. The representation of a sublinear expectation can be expressed as a supermum of linear expectations. Theorem 2.1. [24] There exists a weakly compact family Pof probability measures defined on (Ω,B(Ω)), such that b E(X) = sup p∈P Ep(X), X ∈L1 G(Ω). 4 Definition 2.3. In a sublinear expectation space (Ω,H,b E), an n-dimensional random vector Y= (Y1, .., Yn)∈ H is said to be independent from an m-dimensional random vector X= (X1, .., Xm)∈ H under the sublinear expectation b E, if for any test function ϕ∈ Cb,Lip(Rm+n) b E(ϕ(X, Y )) = b Eb E(ϕ(x, Y ))|x=X. Definition 2.4. Let X1and X2be two n-dimensional random vectors defined on sublinear expectation spaces (Ω1,H1,b E1) and (Ω2,H2,b E2),respectively. They are called identically distributed, denoted by X1 d =X2,if b E1(ϕ(X1)) = b E2(ϕ(X2)),∀ϕ∈ Cb,Lip(Rn). ¯ Xis said to be an independent copy of Xif ¯ Xd =Xand ¯ Xis independent from X. Definition 2.5. (G-Normal Distribution) A random variable Xon a sublinear expectation space (Ω,H,b E) is called G-normal distributed, denoted by X∼N(0,[σ2,¯σ2]), for a given pair 0≤σ≤¯σ, if for any a, b ≥0, aX +b¯ Xd =√a2+b2X, where ¯ Xis an independent copy of X. Definition 2.6. A stopping time τrelative to the filtration Ftin a map on Ω with values in [0, T], such that for every t≥T, {τ≤t} ∈ Ft. Let Ω be the space of all Rd−valued continuous paths (ωt)t≥0with ω0= 0.We assume moreover that Ω is a metric space equipped with the following distance: ρ(ω1, ω2) := ∞ X N=1 2−Nmax 0≤t≤N(|ω1 t−ω2 t|)∧1, and consider the canonical process Bt(ω) = ωt, t ∈[0,+∞),for ω∈Ω; then for each fixed T∈[0,+∞),we have L0 ip(ΩT) := ϕ(Bt1, Bt2, ..., Btn) : n≥1,0≤t1≤... ≤tn≤T, ϕ ∈Cb,lip(Rd×n). Definition 2.7. On the sublinear expectation space (Ω, L0 ip(ΩT),b E),the canonical process (Bt)t≥0 is called a G-Brownian motion, if the following properties are satisfied: (i) B0= 0. (ii) for t, s ≥0, the increment Bt+s−Bt d =√sX, where Xis G-normal distributed. (iii) for t, s ≥0, the increment Bt+s−Btis independent from (Bt1, Bt2, ..., Btn) for each n∈N, and 0 ≤t1≤t2≤... ≤tn≤t. Moreover, the sublinear expectation b E(·) is called G-expectation. 5 Let (Bt)t≥0be a 1-dimensional G-Brownian motion. The letter Gdenotes the function G(a) := 1 2b E(aB2 1) = 1 2σ2a+−σ2a−, a ∈R, with σ2:= −b E(−B2 1)≤b E(B2 1) := σ2,0≤σ≤σ < ∞.Recall that a+= max{0, a}and a−=−min{0, a}. Definition 2.8. For p≥1 and T∈R+fixed, we first consider the following type of simple processes: Mb,0([0, T]) = {η:= ηt(ω) = N−1 X j=0 ζj1[tj,tj+1)(t),∀N > 0, 0 = t0< ... < tN=T, ζj∈Bb(Ωtj), j = 0,1,2, .., N −1}. For each p≥1, we denote by Mp ?([0, T]) the completion of Mb,0([0, T]) under the following norm: ||η||Mp([0,T ]) =b EZT 0|ηt|pdt1 p . We have Mp ?([0, T]) ⊃Mq ?([0, T]) for p≤q. Definition 2.9. We define C1.2(R+×Rn,R) by the family of all real-valued functions V(t, x) defined on R+×Rnwhich are once continuously differentiable in tand twice in x. Now, we introduce the following G-Itˆo formula that can be found in [15]. Lemma 2.2. Let V∈C1.2(R+,Rn),and x(t) = x(0) + Zt 0eϕ(s)ds +Zt 0eχ(s)dhBis+Zt 0e ψ(s)dBs, where eϕ, eχ∈M1 ?([0, T]),e ψ∈M2 ?([0, T]).Then, for t > 0, we have V(t, x(t)) −V(0, x(0)) Zt 0 Vx(s, x(s)) e ψ(s)dBs+Zt 0 (Vt(s, x(s)) + Vx(s, x(s))eϕ(s)) ds Zt 0Vx(s, x(s))eχ(s) + 1 2e ψT(s)Vxx(s, x(s)) e ψ(s)dhBis, where Vt(t, x) = ∂V ∂t (t, x);Vx(t, x)=(∂V ∂x1 (t, x), .., ∂V ∂xn (t, x)) ;Vxx(t, x) = ∂2V ∂xi∂xj (t, x)n×n. 6 Now, we introduce the natural Choquet capacity. Definition 2.10. Let B(Ω) the Borel σ−algebra and Pbe a weakly compact collection of probability measures Pdefined on (Ω,B(Ω)), then the capacity ˆ C(·) associated to Pis defined by ˆ C(A) := sup P∈P P(A), A ∈ B(Ω). Definition 2.11. A set A⊂ B(Ω) is polar if ˆ C(A) = 0.A property holds ”quasi-surely” (q.s.), if it holds outside a polar set. Lemma 2.3. [10] Let {Ak}⊂B(Ω), such that ∞ X k=1 ˆ C(Ak)<∞. Then, lim k→∞ sup Akis polar. 3 Global practical uniform pth moment exponential stability of G-SDEs with respect to a part of the variables Several authors handled the problem of stability with respect to a part of variables of stochastic differential equations driven by standard Brownian motion when the origin is an equilibrium point in the sense of Lyapunov, see [7, 12, 13, 31]. The innovation in our work is to tackle the problem of stability with respect to a part of variables of stochastic differential equations driven by G-Brownian motion when the origin is not a trivial solution by using the method of G-Lyapunov functions and recently developed Itˆo’s calculus for SDEs driven by G-Brownian motion. Consider the following SDE driven by m-dimensional G-Brownian motion: dx(t) = f(t, x(t))dt +h(t, x(t))dhBit+g(t, x(t))dBt,∀x∈Rn,∀t≥0,(3.1) where, Bt= (B1(t), ...., Bm(t))Tis an m-dimensional G-Brownian motion, and (hBi)t≥0is the quadratic variation process of B. Assume that f, h, and g∈Mp ?([0, T],Rn), p ≥2,satisfy the following condition: |f(t, x)−f(t, ex)|2+|h(t, x)−h(t, ex)|2+|g(t, x)−g(t, ex)|2≤K2|x−ex|2,for all t≥0, x, ex∈Rn, where K2is a given positive real constant. 7 Denote x= (y, z)Twhere y= (y1, ..., yn1)T∈Rn1, z = (z1, ..., zn2)T∈Rn2, n1>0, n2≥ 0; n1+n2=n; |y|=qy2 1+... +y2 n1,|z|=qz2 1+... +z2 n2,|x|=|y|2+|z|21 2. Under the precedent assumptions, there exists a unique global solution x(t, t0, x0)∈Mp ?([0, T],Rn), with x(t, t0, x0) = (y(t, t0, x0), z(t, t0, x0)) corresponding to the initial condition x0= (y0, z0)∈Rn (see [18] for more details). In what follows, we use x(t, t0, x0) = (y(t, t0, x0), z(t, t0, x0)), or simply x(t)=(y(t), z(t)) to denote a solution of our system. Suppose that there exists tsuch that f(t, 0) 6= 0 or h(t, 0) 6= 0 or g(t, 0) 6= 0, i.e., the G-stochastic differential equation (3.1) does not have the trivial solution x≡0.Therefore, we will study the pth moment exponential stability with respect to yof the G-SDE in a small neighborhood of the origin. The study of the practical stability with respect to a part of the variables of G-SDE lead back to the study of the stability with respect to a part of the variables of a small ball Br:= {x∈Rn:|x| ≤ r},r > 0. Definition 3.1. The solutions of (3.1) are said to be quasi surely globally uniformly bounded, if for each α > 0, there exists c=c(α)>0 (independent of t0), such that for every t0≥0,and all x0∈Rnwith |x0| ≤ α, sup t≥t0|x(t, t0, x0)| ≤ c(α),q.s. (3.2) Definition 3.2. i) Let p > 0, the ball Bris said to be globally uniformly pth moment exponentially stable with respect to y, if there exist λ1>0 and λ2>0, such that b E(|y(t, t0, x0)|p)≤λ1|x0|pe−λ2(t−t0)+r, ∀t≥t0,∀x0∈Rn.(3.3) ii) The system (3.1) is said to be globally practically uniformly pth moment exponentially stable with respect to yif there exists r > 0, such that Bris globally uniformly pth moment exponentially stable with respect to y. Definition 3.3. For each V∈C1.2(R+×Rn,R), we define an operator L(called as G-Lyapunov function) from R+×Rn→Ras follows: LV (t, x) := Vt(t, x) + Vxf(t, x) + G(hVx(t, x),2h(t, x)i+hVxx(t, x)g(t, x), g(t, x)i). Now, we aim to prove the global practical pth moment exponential stability of stochastic differential equations driven by G-Brownian motion with respect to a part of the variables, via the G-Lyapunov functions. Theorem 3.1. Assume that there exist V∈C1.2(R+×Rn,R+)and nonnegative constants ci(i= 1,2,3), p and ρ, such that for all t≥0, and all x= (y, z)∈Rn, c1|y|p≤V(t, x)≤c2|y|p,(3.4) 8 LV (t, x)≤ −c3|y|p+ρ. (3.5) Moreover, we suppose that z(t, t0, x0)is quasi surely globally uniformly bounded. Then, system (3.1) is globally practically uniformly pth moment exponentially stable with respect to y, with r=ρc2 c1c3 . In order to prove this theorem, we need to introduce the Gronwall lemma proved in [26]. Lemma 3.2. Let ϕ: [0,+∞)−→ [0,+∞)be a continuous function, εis a positive real number and λis a strictly positive real number. Assume that for all t∈[0,+∞)and 0≤v≤t, we have ϕ(t)−ϕ(v)≤Zt v (−λϕ(s) + ε)ds. Then, ϕ(t)≤ε λ+ϕ(0) exp(−λt). Proof of Theorem 3.3. Applying the G-Itˆo’s formula 2.2 on Vfor any t≥v≥0, leads to V(t, x(t)) = V(v, x(v)) + Zt v LV (s, x(s))ds +Zt v Vx(s, x(s))g(s, x(s))dBs. Taking the G-expectation on both sides and taking into consideration conditions (3.4) and (3.5), we obtain 0≤b E(V(t, x(t))) −b E(V(v, x(v))) ≤Zt vb E(LV (s, x(s)))ds ≤Zt v−c3 c2b E(V(s, x(s))) + ρds. In order to use Lemma 3.2, we need to prove that t7→ ˆ E(V(t, x(t))) is continuous on R+. In fact, we denote by F(t, ω) = V(t, x(t)). Then, from the representation of the sublinear expectation (2.1), it follows that, b E(F(t, ω)) = sup p∈P Ep(F(t, ω)) , where Ep(F(t, ω)) = ZΩ F(t, ω)dP(ω),for each P∈ P. Thus, our target now is to prove that t7→ ZΩ F(t, ω)dP(ω) is continuous on R+. 9 Theorem 4.3. Assume that there exist a function V∈C1.2(R+×Rn,R+)and constants p∈N∗, β1≥1,γ≥β1, ζ ≥0and β2∈R,β3≥0,such that for all t≥t0≥0, and x= (y, z)∈Rn, 1. β1|y|p≤V(t, x), 2. Vt(t, x) + Vx(t, x)f(t, x)≤β2V(t, x), 3. Vx(t, x)h(t, x)≤γ, 4. |Vx(t, x)g(t, x)|2≥β3V2(t, x) + ζ. Moreover, we suppose that z(t, t0, x0)is quasi surely globally uniformly bounded. Then, lim t→+∞sup 1 tln |y(t, t0, x0)|−γ β11 p!≤ −[σ2β3−2 (β2+ ¯σ2)] 2q.s. In particular, if σ2β3>2(β2+ ¯σ2), then system (3.1) is said to be quasi surely globally practically uniformly exponentially stable with respect to y, with r=γ β11 p . Proof. Fix x06= 0 in Rn. By Lemma 4.1, x(t)6= 0, for all t≥0 quasi surely. Suppose that β1|y(t)|p> γ, then we have β1|y(t)|p−γ=β1|y(t)|p−γ β1 =β1 |y(t)|p− γ β11 p!p!. Based on the following inequality, ap−bp= (a−b)(ap−1+ap−2b+ap−3b2+... +a0bp−1), we deduce β1|y(t)|p−γ=β1|y(t)|−(γ β1 )1 p |y(t)|p−1+|y(t)|p−2γ β11 p +... +γ β1p−1 p! ≥β1 |y(t)|−γ β11 p!γ β1p−1 p . Since β1≥1 and γ≥β1,we obtain V(t, x(t)) ≥β1|y(t)|p≥β1|y(t)|p−γ≥β1 |y(t)|−γ β11 p!γ β1p−1 p ≥ |y(t)|−γ β11 p!. 16 Thus, |y(t)|−γ β11 p!≤V(t, x(t)), and ln |y(t)|−γ β11 p!≤ln V(t, x(t))),∀t≥0. The application of G-Itˆo’s formula 2.2 leads to, for all t≥0, dln(V(t, x(t)))=Vt(t, x(t)) + Vx(t, x(t))f(t, x(t)) V(t, x(t)) dt +Vx(t, x(t))g(t, x(t)) V(t, x(t)) dBt +Vx(t, x(t))h(t, x(t)) V(t, x(t)) dhBit−1 2|Vx(t, x(t))g(t, x(t))|2 V2(t, x(t)) dhBit. That is, ln(V(t, x(t))) = ln(V(0, x(0))) + Zt 0 Vs(s, x(s)) + Vx(s, x(s))f(s, x(s)) V(s, x(s)) ds+ Zt 0 Vx(s, x(s))g(s, x(s)) V(s, x(s)) dBs+Zt 0 Vx(s, x(s))h(s, x(s)) V(s, x(s)) dhBis−1 2Zt 0 |Vx(s, x(s))g(s, x(s))|2 V2(s, x(s)) dhBis. (4.3) Assign ε∈]0,1[ arbitrarily and let n= 1,2, ... It follows from Lemma 4.2 that, b Csup 0≤t≤nZt 0 Vx(s, x(s))g(s, x(s)) V(s, x(s)) dBs−ε 2Zt 0 (Vx(x(s), s)g(x(s), s))2 V2(x(s), s)dhBis>2 εln(n)≤1 n2. Using the Borel-Cantelli lemma (Lemma 2.3) for capacity, we deduce that there exists n0∈N, such that Zt 0 Vx(s, x(s))g(s, x(s)) V(s, x(s)) dBs≤2 εln(n)+ε 2Zt 0 (Vx(x(s), s)g(x(s), s))2 V2(x(s), s)dhBis,for all n≥n0,0≤t≤n. Combining this result together with (4.3), we obtain ln(V(t, x(t))) ≤ln(V(0, x(0))) + Zt 0 Vs(s, x(s)) + Vx(s, x(s))f(s, x(s)) V(s, x(s)) ds +Zt 0 Vx(s, x(s))h(s, x(s)) V(s, x(s)) dhBis+2 εln(n) +ε−1 2Zt 0 (Vx(s, x(s))g(s, x(s)))2 V2(s, x(s)) dhBis,0≤t≤n, n ≥n0,q.s. 17 By Peng [22, Chapter III], we have for each 0 ≤s≤t≤T, σ2(t−s)≤ hBit−hBis≤¯σ2(t−s). Based on this fact, we obtain ln(V(t, x(t))) ≤ln(V(0, x(0))) + Zt 0 Vs(s, x(s)) + Vx(s, x(s))f(s, x(s)) V(s, x(s)) ds + ¯σ2Zt 0 Vx(s, x(s))h(s, x(s)) V(s, x(s)) ds +2 εln(n) +ε−1 2σ2Zt 0 (Vx(s, x(s))g(s, x(s)))2 V2(s, x(s)) ds, 0≤t≤n, n ≥n0,q.s. Taking into account the assumptions, it follows that ln(V(t, x(t))) ≤ln(V(0, x(0))) + Zt 0 β2V(s, x(s)) + ¯σ2γ V(s, x(s)) ds +2 εln(n) +ε−1 2σ2Zt 0 β3V2(s, x(s)) + ζ V2(s, x(s)) ds ≤ln(V(0, x(0))) + β2t+Zt 0 ¯σ2γ β1|y(s)|pds +2 εln(n) −1−ε 2σ2β3t−1−ε 2σ2Zt 0 ζ V2(s, x(s))ds ≤ln(V(0, x(0))) + (β2+ ¯σ2)t+2 εln(n)−1−ε 2σ2β3t−1−ε 2σ2Zt 0 ζ V2(s, x(s))ds, for all 0 ≤t≤n, n ≥n0,q.s. Since 0 <ε<1,the following inequality holds −1−ε 2σ2Zt 0 ζ V2(s, x(s))ds ≤0. As a consequence, we obtain ln(V(t, x(t))) ≤ln(V(0, x(0))) + (β2+ ¯σ2)t+2 εln(n)−1−ε 2σ2β3t, 0≤t≤n, n ≥n0,q.s. Then, it follows that 1 tln(V(t, x(t))) ≤ −1 2(1 −ε)σ2β3−2(β2+ ¯σ2)+ln(V(0, x(0))) + 2 εln(n) n−1. 18 The inequality above implies that, lim t→+∞sup 1 tln(V(t, x(t)) ≤ −1 2(1 −ε)σ2β3−2(β2+ ¯σ2),q.s. Putting ε−→ 0 gives, lim t→+∞sup 1 tln |y(t)|−γ β11 p!≤ −[σ2β3−2(β2+ ¯σ2)] 2,q.s. If the inequality σ2β3>2(β2+¯σ2) is satisfied, then system (3.1) is quasi surely globally practically uniformly exponentially stable with respect to y. 2 5 Example Consider the following G-stochastic system:            dx1(t) = −2x1(t)dt +1 2sin2(x2)e−2tx1dhBit+ (1 + e−t)1 2dBt dx2(t) = −2x2(t)dt +1 2cos2(x1)e−tx2dhBit+1 pch(t)dBt dx3(t) = Ax3(t)dt, (5.1) where x= (x1, x2, x3)∈R4, x3= (z1, z2)∈R2,Btis a one-dimensional G-Brownian motion and Bt∼N(0,[1 2,1]), A2=0−1 1 0 , with initial value x0= (x10, x20, x30),and x30= (z10, z20). By simple computations, we obtain x3(t) = z10(t, w) cos(t)−z20(t, w) sin(t) z10(t, w) sin(t) + z20(t, w) cos(t). Using the Euclidean norm leads to |x3(t)|=|x30|. It is clear that x3(t) is quasi surely globally uniformly bounded. In fact, for all α0>0 (independent of t0), all t≥t0≥0,and all x30∈R2with |x30| ≤ α0,we have |x3(t)| ≤ α0,q.s. Let consider the following G-Lyapunov-like function: V(t, x) = x2 1+x2 2. 19 It is easy to verify that, LV (t, x) = −4x2 1−4x2 2+G2 sin2(x2)e−2tx2 1+ 2 cos2(x1)e−tx2 2+ 2(1 + e−t)+2 1 ch(t). By the sub-additivity of the function G, we obtain LV (t, x)≤ −4x2 1−4x2 2+G2 sin2(x2)e−2tx2 1+G2 cos2(x1)e−tx2 2+G2(1 + e−t)+G21 ch(t) ≤ −4x2 1−4x2 2+x2 1sin2(x2)e−2t+ cos2(x1)e−tx2 2+ (1 + e−t) + 1 ch(t) ≤ −3x2 1−3x2 2+ 3. That is, LV (t, x)≤ −4x2 1−4x2 2+x2 1+x2 2+ 3. Thus, the constants in Theorem 3.1 become c1=c2= 1, c3=ρ= 3, p = 2, and all conditions of Theorem 3.1 are satisfied, hence system (5.1) is globally practically uniformly pth moment exponentially stable with respect to (x1, x2), with r= 1. 6 Conclusion In this paper, practical stability with respect to a part of the variables procedure for certain classes of nonlinear stochastic differential equations driven by G-Brownian motion was developed. Some stability criteria are established. Then, using G-Lyapunov techniques, the notion of practical stability with respect to a part of the variables is studied for G-SDEs. In this context, some sufficient conditions are presented to ensure the desired stability property. An example has been introduced to validate the developed methods. Acknowledgement The authors would like to thank the editor and the anonymous reviewer for valuable comments and suggestions, which allowed us to improve the paper. References [1] X. Bai, Y. Lin, On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients, Acta Mathematicae Applicatae Sinica, English Series, 30 (2010), 589-610. [2] A. BenAbdallah, I. Ellouze, M. A. Hammami, Practical stability of nonlinear time-varying cascade systems, Journal of Dynamical and Control Systems, 15 (2009) 45-62. 20 [3] B. Ben Hamed, I. Ellouze, and M. A. Hammami, Practical uniform stability of nonlinear differential delay equations, Mediterranean Journal of Mathematics, 8 (2011), 603-616. [4] T. Caraballo, M. A. Hammami, L. Mchiri, On the practical global uniform asymptotic stability of stochastic differential equations, Stochastics An International Journal of Probability and Stochastic Processes, 88 (2016), 45-56. [5] T. Caraballo, M. J. Garrido-Atienza, J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stochastic Analysis and Applications, 21 (2003), 301327. [6] T. Caraballo, M. J. Garrido-Atienza, J. Real, Stochastic stabilization of differential systems with general decay rate, Systems and Control Letters, 48 (2003), 397-406. [7] T. Caraballo, F. Ezzine, M. Hammami, L. Mchiri, Practical stability with respect to a part of variables of stochastic differential equations, Stochastics An International Journal of Probability and Stochastic Processes, 6 (2020), 1-18. [8] T. Caraballo, F. Ezzine, M. Hammami, Partial stability analysis of stochastic differential equations with a general decay rate, Journal of Engineering Mathematics, 1 (2021). [9] M. Corless, Guaranteed rates of exponential convergence for uncertain systems, Journal of Optimization Theory and Applications, 64 (1990), 481-494. [10] L. Denis, M. Hu, S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion path, Potential Analysis, 34 (2010), 139-161. [11] F. Gao, Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion, Stochastic Processes and their Applications, 119 (2009), 3356- 3382. [12] O. Ignatyev, Partial asymptotic stability in probability of stochastic differential equations, Statistics and Probability Letters, 79 (2009), 597-601. [13] O. Ignatyev, New criterion of partial asymptotic stability in probability of stochastic differential equations, Applied Mathematics and Computation, 219 (2013), 10961-10966. [14] P. Luo, F. Wang, Stochastic differential equations driven by G-Brownian motion and ordinary differential equations, Stochastic Processes and their applications, 124 (2014), 3869-3885. [15] X. Li, S. Peng, Stopping times and related G-Itˆocalculus with G-Brownian motion, Stochastic Processes and their Applications, 121 (2009), 1492-1508. [16] Q. Lin, Some properties of stochastic differential equations driven by G-Brownian motion, Acta Mathematica Sinica, 29 (2013), 923-942. 21 [17] Y. Lin, Stochastic differential eqations driven by G-Brownian motion with reflecting boundary, Electronic Journal of Probability, 18 (2013), 1-23. [18] X. Li, X. Lin, Y. Lin, Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion, Mathematical Analysis and Application, 439 (2016), 235-255. [19] X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, 1997. [20] E. Sontag, Input to State Stability: Basic Concepts and Results, DOI: 10.1007/978 −3− 540 −77653 −63.In book: Nonlinear and optimal control theory. [21] E. Sontag, Y. Wang, on characterizations of input-to-state stability with respect to compact sets, IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995. [22] S. Peng, Nonlinear expectations and stochastic calculus under uncertaintly-with robust central limit theorem and G-Brownian motion, Springer-Verlag Berlin Heidelberg, 2010. [23] K. Peiffer, N. Rouche, Liapunov’s second method applied to partial stability, Journal of Mecanique, 2 (1969), 323-334. [24] S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itˆo’s type, arXiv:math.PR/0601035v1, 2006. [25] S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G- expectation, Stochastic Processes and their Applications, 118 (2008), 223-225. [26] Q. C. Pham, N. Tabareau, J. E. Slotine, A contraction theory approach to stochastic incremental stability, IEEE transactions on Automatic Control, 54 (2009), 1285-1290. [27] N. Rouche, P. Habets, M. Lalog, Stability Theory by Liapunovs Direct Method, Springer- Verlag, New York, 1977. [28] V. V. Rymanstev, On the stability of motions with respect to part of variables, Moscow University Mathematics Bulletin, 4 (1957), 9-16. [29] V. V. Rumyantsev, A. S. Oziraner, Partial Stability and Stabilization of Motion, Nauka, Moscow (in Russian), 1987. [30] A. Ya. Savchenko, O. Ignatyev, Some Problems of Stability Theory, Naukova Dumka, Kiev (in Russian), 1989. [31] V. I. Vorotnikov, Partial Stability and Control, Birkhuser Boston, 1998. [32] V. I. Vorotnikov, V. V. Rumyantsev, Stability and Control with Respect to a Part of the Phase Coordinates of Dynamic Systems: Theory, Methods, and Applications, Scientific World, Moscow (in Russian), 2001. 22 [33] D. Zhang, Z.Chen, Exponential stability for stochastic differential equation driven by G- Brownian motion, Applied Mathematics Letters, 25 (2012), 1906-1910. 23