Partial stability analysis of stochastic differential equations with a general decay rate

J Eng Math (2021) 130:4 https://doi.org/10.1007/s10665-021-10164-w Partial stability analysis of stochastic differential equations with a general decay rate Tomás Caraballo ·Faten Ezzine · Mohamed Ali Hammami Received: 2 November 2020 / Accepted: 17 June 2021 / Published online: 28 August 2021 © The Author(s), under exclusive licence to Springer Nature B.V. 2021 Abstract This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results. Keywords Decay function ·Itô formula ·Lyapunov construction ·Partial almost sure practical stability ·Stochastic systems Mathematics Subject Classification Primary 93E03 ·Secondary 60H10 1 Introduction Stochastic models are useful for modeling physical, biological, technical, as well as dynamical structures in engineering and mechanics in which notable uncertainty is present either by the intrinsic nature of the model or by the influence of external sources of noise. The theory of stochastic differential equations in finite and infinite-dimensional spaces is already an established area of research. The corresponding study of the stability properties of solutions has received much attention during the last three decades. The reader is referred to [1,2], for more details. In our present paper, we will analyze a quite general model of stochastic differential system which can cover a wide class of real models subjected to noise effects, and we emphasize that our results can be applied to many The research of Tomás Caraballo has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PGC2018-096540-B-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) and FEDER under Projects US-1254251 and P18-FR-4509. T. Caraballo (B) Depto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain e-mail: [email protected] F. Ezzine ·M. A. Hammami Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia e-mail: [email protected] 123 4Page 2 of 17 T. Caraballo et al. interesting examples related to the analysis of stochastic stability in different kinds of structures (see, [3]formore details about the use of Lyapunov’s technique to study this problem). In many situations, the stochastic model considered may not fulfill sufficient conditions ensuring the stability of solutions. However, it is still possible that the system is still stable with respect to a part of the unknowns. It is crucial to investigate whether it is still possible to ascertain some stability properties to some of the variables in the problem which is called “Partial Stability.” Partial stability has proved to have powerful applications in many branches of biotechnology, electro-magnetics, combustion systems, vibrations in rotating machinery, inertial navigation systems, spacecraft stabilization via gimballed gyroscopes, and/or flywheels. The reader is referred to [4–9], for more details. The method of Lyapunov functions is one of the most powerful tools to investigate the stability of stochastic systems. Later on, by the development of the second method of Lyapunov, different authors have been working on partial stability with stochastic differential equations, see [6,8,10–15]. When the origin is not a trivial solution, we investigate the partial stability of the SDEs with respect to a small neighborhood of the origin. In this sense, we can analyze the ultimate boundedness of the solutions of the stochastic system or the possibility of proving the partial convergence of the solutions towards a small ball centered at the origin, this is the so-called “Partial Practical Stability.” Several results on the stability of the nontrivial solution of stochastic systems are proposed in [16–18]. In the analysis of the asymptotic behavior of solutions to stochastic differential systems, one can find that a solution can be asymptotically stable but may not be exponential. Further, in the nonlinear and/or nonautonomous situations, it may happen that the stability cannot always be exponential but can even be sub or super-exponential, see [19]. For this reason, this paper deals with the analysis of almost sure partial practical stability of stochastic systems with general decay rate. Our main objective in this paper is to generalize the results in [18], which investigated the partial practical exponential stability of SDEs, to the partial practical stability with a general decay rate (including polynomial, logarithmic, sub-exponential and super-exponential, as well as exponential decay). We establish sufficient conditions ensuring the almost sure partial practical stability by using Lyapunov method. Further, one of our main purpose of this paper is to construct suitable Lyapunov functions to prove the partial practical stability with general decay rate. Moreover, we will illustrate the theory with an application example. The arrangement of the paper is presented as follows. In Sect. 2, we evoke some essential preliminaries and results. In Sect. 3, some sufficient conditions ensuring almost sure partial practical stability with general decay rate are proven. In Sect. 4, we establish some sufficient conditions based on the construction of appropriate Lyapunov functions. In Sect. 5, we present an example to illustrate the theoretical findings. Finally, in Sect. 6, some conclusions and prospects are discussed. 2 Preliminaries We consider the nonlinear stochastic differential equation in the following form: dx(t)=f(t,x(t))dt+g(t,x(t))dBt,t≥0,(2.1) where f:R+×Rd−→ Rd,g:R+×Rd−→ Rd×m,x=(x1,...,xd)T, and Bt=(B1(t), . . . , Bm(t))Tis an m-dimensional Brownian motion on a probability space (, F,P). For the well-posedness of system (2.1), we impose that fand gsatisfy the following assumptions: ||f(t,x)||2+||g(t,x)||2≤C1(t)1+||x||2,∀t≥0,x∈Rd,(2.2) ||f(t,x)−f(t,x)||∨||g(t,x)−g(t,x)|| ≤ C2(t)||x−x||,∀t≥0,x,x∈Rd,(2.3) with C1(·)and C2(·)being nonnegative functions. In the sequel, the same symbol is used to express the norm in Rdor in Rd×m. 123 Partial stability analysis of stochastic differential equations... Page 3 of 17 4 Let x=(x1,x2)T, where x1=(x1.1,...,x1.d1)T∈Rd1,x2=(x2.1,...,x2.d2)T∈Rd2,d1>0,d2≥0,d1+d2= d; ||x1|| = x1.12+···+x1.d1 2,||x2|| = x2.12+···+x2.d2 2,||x|| = ||x1||2+||x2||2. Then, based on the previous conditions, there exists a unique global solution with initial value x0∈Rddefined in an interval [t0,T):x(t,t0,x0)=(x1(t,t0,x0), x2(t,t0,x0)) or simply x(t)=(x1(t), x2(t)) to express a solution to our system and, as we will be interested in analyzing the asymptotic behavior of solutions, we assume T=+∞. Definition 2.1 [20] Denote by C1.2(R+×Rd,R)the family of all real-valued functions V(t,x)defined on R+×Rd which are once continuously differentiable in tand twice in x. Let V∈C1.2(R+×Rd,R),weset Vt(t,x)=∂V ∂t(t,x);Vx(t,x)=∂V ∂x1 (t,x), ∂V ∂x2 (t,x);Vxx(t,x)=∂2V ∂xi∂xj (t,x)d×d. We define operators Land Hassociated with Eq. (2.1)asfollows:forx∈Rd,t∈R+, LV(t,x)=Vt(t,x)+Vx(t,x)f(t,x)+1 2trace[gT(t,x)Vxx(t,x)g(t,x)], and HV(t,x)=||Vx(t,x)g(t,x)||2. Applying the well-known Itô formula [20], it follows for x(·), a solution to (2.1), that dV(t,x(t)) =LV(t,x(t))dt+Vx(t,x(t))g(t,x(t))dBt. Now, we recall the following technical lemmas, which will be useful in our analysis. Lemma 2.1 [20]For all x0∈Rd,such that x0= 0, we have Px(t;t0,x0)= 0,∀t≥t0=1. This means that almost all sample paths of any solution beginning from a nonzero state will never arrive at the origin. The following lemma is known as the exponential martingale inequality and will play an important role in the analysis of some results in this paper. Lemma 2.2 [20]Let g =(g1,...,gd)∈L2(R+,Rd), and τ,μ,ηare positive constants. Then, for t0≥0, Psup t0≤t≤τt t0 g(s)dBs−μ 2t t0||g(s)||2ds>η ≤exp(−μη). 3 Partial practical stability of stochastic differential equations with general decay rate We state the definition of partial convergence towards a small ball centered at the origin with a general decay function λ(t).Mao[21] was the first who introduced the concept of stability with the polynomial decay rate. After that, this concept was generalized to the stability with general decay rate, see [17,19,22]. Definition 3.1 Let λ(t)be a positive function defined for sufficiently large t>0, such that λ(t)→+∞as t→ +∞. A solution x(t)=(x1(t), x2(t)) to system (2.1) is said to tend to the ball Br:= {x∈Rd:||x|| ≤ r},r>0 with respect to x1with decay function λ(t)and order at least γ>0, if its generalized Lyapunov exponent is less than or equal to −γwith probability one, i.e., lim t→+∞sup ln(||x1(t,t0,x0)||−r) ln λ(t)≤−γ, a.s. (3.1) 123 4Page 4 of 17 T. Caraballo et al. If in addition, 0 is a solution to system (2.1), the zero solution is said to be almost surely practically asymptotically stable with respect to x1with decay function λ(t)and order at least γ, if every solution to system (2.1) tends to the boundary of the ball Brwith respect to x1with decay function λ(t)and order at least γ, for all r>0 sufficiently small. Remark 3.1 Clearly, replacing in the above definition the decay function λ(t)by O(et)leads to the partial practical exponential stability, which is investigated in [18]. Remark 3.2 We should mention that the authors in [18] have established sufficient conditions for partial practical exponential stability while here we will prove the convergence to a small ball by using Lyapunov functions. Definition 3.2 [18] The solutions of (2.1) are said to be globally uniformly bounded with probability one, if for each β>0, there exists c=c(β) > 0 (independent of t0), such that for every t0≥0,and all x0∈Rdwith ||x0|| ≤ β, sup t≥t0||x(t,t0,x0)|| ≤ c(β), a.s. (3.2) Now, we state and prove one of our main results in this paper. Theorem 3.3 Assume that there exist a function V∈C1.2(R+×Rd,R+), three continuous functions ψ1(t)∈ R,ψ 2(t)≥0,ρ(t)>0, some constants q ∈N∗,m≥0,α 1≥0,α 2≥0,α 3∈R,and a small constant ξ≥0, such that for all t ≥t0and all x =(x1,x2)∈Rd, (H1)λ(t)m||x1||q≤V(t,x), (H2)LV(t,x)≤ψ1(t)V(t,x)+ρ(t), (H3)HV(t,x)≥ψ2(t)V2(t,x)+ξ, (H4)lim t→+∞sup t Tψ1(s)ds ln λ(t)≤α3,∀T≥t0, lim t→+∞inf t Tψ2(s)ds ln λ(t)≥2α1,∀T≥t0, lim t→+∞sup ln(t) ln λ(t)≤α2 2, (H5)lim t→+∞sup t ln λ(t)=C≥0,lim t→+∞ ρ(t) λ(t)m=ζ>0. Let x0∈Rd,x0= 0, such that the corresponding solution x(t,t0,x0)=(x1(t,t0,x0), x2(t,t0,x0)) satisfies: (i) ||x1(t,t0,x0)|| >ρ(t) λ(t)m1 q,∀t≥t0, (ii) x2(t,t0,x0)is globally uniformly bounded with probability one. Then, for every β∈(0,1), it follows lim t→+∞sup ln ||x1(t,t0,x0)||−ρ(t) λ(t)m1 q ln λ(t)≤− m−1 βα2+α3−α1(1−β) +C,a.s. Proof For x0= 0, from Lemma 2.1 we deduce x(t)=(x1(t), x2(t)) = 0, ∀t≥t0almost surely. Observe that λ(t)m||x1(t)||q−ρ(t)=λ(t)m||x1(t)||q−ρ(t) λ(t)m=λ(t)m||x1(t)||q−ρ(t) λ(t)m1 qq. Based upon the following inequality: aq−bq=(a−b)(aq−1+aq−2b+aq−3b2+···+a0bq−1), 123 Partial stability analysis of stochastic differential equations... Page 5 of 17 4 we deduce λ(t)m||x1(t)||q−ρ(t)=λ(t)m||x1(t)||−ρ(t) λ(t)m1 qq  k=1||x1(t)||q−kρ(t) λ(t)mk−1 q.(3.3) Notice that, from condition (H5), we have limt→+∞ ρ(t)/λ(t)m=ζ>0.Thus, for 0 <ζ 0<ζ, there exists T≥t0such that ρ(t)/λ(t)m≥ζ0,∀t≥T. Then, as we are also assuming that ||x1(t)|| >(ρ(t)/λ(t)m) 1 q,∀t≥0, it yields q  k=1||x1(t)||q−kρ(t) λ(t)mk−1 q=||x1(t)||q−1+||x1(t)||q−2ρ(t) λ(t)m1 q+···+ρ(t) λ(t)mq−1 q ≥ζ=qζ(q−1)/q 0,∀t≥T. It follows that that there exists t0such that λ(t)m||x1(t)||q−ρ(t)≥λ(t)m||x1(t)||−ρ(t) λ(t)m1 qζ,∀t≥T. Hence, V(t,x(t)) ≥λ(t)m||x1(t)||q≥λ(t)m||x1(t)||q−ρ(t)≥λ(t)m||x1(t)||−ρ(t) λ(t)m1 qζ. That is, ζλ(t)m||x1(t)||−ρ(t) λ(t)m1 q≤V(t,x(t)), and ln(ζ )+ln λ(t)m||x1(t)||−ρ(t) λ(t)m1 q≤ln V(t,x(t)). Thus, we obtain ln(ζ )+mln λ(t)+ln ||x1(t)||−ρ(t) λ(t)m1 q≤ln V(t,x(t)),∀t≥T. Invoking Itô’s formula for V(·)along the trajectory x(·)of the stochastic system (2.1), we obtain ∀t≥T, dln(V(t,x(t)))=LV(t,x(t)) V(t,x(t)) dt+Vx(t,x(t))g(t,x(t)) V(t,x(t)) dBt−1 2 HV(t,x(t)) V2(t,x(t)) dt. That is, t T d(ln(V(s,x(s))))ds=t T LV(s,x(s)) V(s,x(s)) ds+t T Vx(s,x(s))g(s,x(s)) V(s,x(s)) dBs−1 2t T HV(s,x(s)) V2(s,x(s)) ds. 123 4Page 6 of 17 T. Caraballo et al. Hence, we obtain ln(V(t,x(t))) =ln(V(T,x(T))) +t T LV(s,x(s)) V(s,x(s)) ds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds,(3.4) where M(t)=t T Vx(s,x(s))g(s,x(s)) V(s,x(s)) dBs is a continuous martingale with M(T)=0. Using (H1)and (H2), it follows that, ∀t≥T, ln(V(t,x(t))) ≤ln(V(T,x(T))) +t T ψ1(s)ds+t T ρ(s) λ(s)m||x1(s)||qds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds.(3.5) Consequently, we obtain ln(V(t,x(t))) ≤ln(V(T,x(T))) +t T ψ1(s)ds+(t−T)+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds.(3.6) Thanks to the exponential martingale inequality (2.2), we have Pω:sup T≤t≤τM(t)−μ 2t T HV(s,x(s)) V2(s,x(s)) ds>η ≤e−μη, for any positive constants μ, η, and τ. In particular, take 0 <β<1, and set μ=β, η =2 βln(k−1), τ =k∈N,k>T+1. We then apply the well-known Borel–Cantelli lemma to obtain that, for almost all ω∈,∃k0=k(ε, ω) > 0, with M(t)≤2 βln(k−1)+β 2t T HV(s,x(s)) V2(s,x(s)) ds,∀T≤t≤k,k≥k0(ε, ω). Substituting this into Eq. (3.6) and taking into account condition (H3), we obtain ln(V(t,x(t))) ≤ln(V(T,x(T))) +2 βln(k−1)+t T ψ1(s)ds+(t−T)−1−β 2t T ψ2(s)ds,(3.7) for T≤t≤k,k≥k0(ε, ω). From condition (H4),forε>0, ∃˜ T≥T>T0, such that ∀t>˜ T,wehave t T ψ1(s)ds≤(α3+ε) ln λ(t), t T ψ2(s)ds≥(2α1−ε) ln λ(t), ln(t)≤1 2(α2+ε) ln λ(t). Then, inequality (3.7) becomes ln(V(t,x(t))) ≤ln(V(T,x(T))) +α2+ε βln λ(t)+(α3+ε) ln λ(t)−1−β 2(2α1−ε) ln λ(t)+(t−T), for k−1≤t≤k,k≥k0(ε, ω), which yields, by using condition (H5), that lim t→+∞sup ln(V(t,x(t))) ln λ(t)≤α2+ε β+(α3+ε) −1−β 2(2α1−ε) +C,a.s. 123 Partial stability analysis of stochastic differential equations... Page 7 of 17 4 Recall that, for t≥Tand q∈N,wehave ln ||x1(t)||−ρ(t) λ(t)m1 q≤ln(V(t,x(t))) −mln λ(t)−ln(ζ ). Taking into account the fact that ε>0 is arbitrary, we deduce that lim t→+∞sup ln ||x1(t)||−ρ(t) λ(t)m1 q ln λ(t)≤− m−1 βα2+α3−α1(1−β) +C,a.s., as required.  Remark 3.4 Note that, in the above theorem, the decaying order of the solution depends on the parameter β.The crucial question imposed concerns the possibility of determining the largest value for γβ. Indeed, in order to find the optimal valued γ=sup0<β<1γβ,we require to find out the minimum value Jfor the following function: J(β) =1 βα2+α3−α1(1−β) +C when the parameter β∈(0,1). Hence, we deduce that the optimal value of γwill hold with γ=(m−J). It is straightforward to check that J=2(α1α2)1 2+α3−α1+Cif 0 ≤α2<α 1, α2+α3+Cif α1≤α2. Finally, we obtain γ=⎧ ⎨ ⎩ m−2(α1α2)1 2+α3−α1+Cif 0 ≤α2<α 1, m−α2+α3+Cif α1≤α2. In the next corollary, we will deduce the partial convergence to a ball with a general decay rate. Corollary 3.5 Assume that there exist a function V∈C1.2(R+×Rd,R+), three continuous functions ψ1(t)∈ R,ψ 2(t)≥0,ρ(t)>0, some constants q ∈N∗,m≥0,α 1≥0,α 2≥0,α 3∈R,and a small constant ξ≥0, such that ∀t≥t0and all x =(x1,x2)∈Rd, assumptions (H1)−(H5)are satisfied. Let x0∈Rd,x0= 0, such that the corresponding solution x(t,t0,x0)=(x1(t,t0,x0), x2(t,t0,x0)) satisfies: (i) ||x1(t,t0,x0)|| >ρ(t)/λ(t)m1 q,∀t≥t0, (ii) x2(t,t0,x0)is globally uniformly bounded with probability one. Then, if additionally there exists  ζ≥ζ>0, such that ||x1(t,t0,x0)|| > ζ,∀t≥t0, it follows lim t→+∞sup ln ||x1(t,t0,x0)||− ζ1 q ln λ(t)≤−γ,a.s., 123 4Page 8 of 17 T. Caraballo et al. where γ=⎧ ⎨ ⎩ m−2(α1α2)1 2+α3−α1+Cif 0≤α2<α 1, m−α2+α3+Cif α1≤α2. In particular, if γ>0, then the solution to system (2.1)tends to the ball Br, with r = ζ1 qalmost surely with respect to x1with decay function λ(t)and order at least γ. Proof Let x0= 0inRd. It immediately follows from Lemma 2.1,x(t)=(x1(t), x2(t)) = 0, ∀t≥0 almost surely. Based upon Theorem 3.3,wehave lim t→+∞sup ln ||x1(t)||−ρ(t) λ(t)m1 q ln λ(t)≤−γ,a.s., where γ=⎧ ⎨ ⎩ m−2(α1α2)1 2+α3−α1+C,if 0 ≤α2<α 1, m−α2+α3+C,if α1≤α2. Since, lim t→+∞ ρ(t) λ(t)m=ζ≤ ζ,then there exists T≥t0such that ρ(t) λ(t)m≤ ζ,∀t≥T. Consequently, lim t→+∞sup ln ||x1(t)||− ζ1 q ln λ(t)≤lim t→+∞sup ln ||x1(t)||−ρ(t) λ(t)m1 q ln λ(t)≤−γa.s. Hence, if γ>0, then the solution to system (2.1) tends to the ball Br, with r= ζ1 qwith respect to x1with decay function λ(t)and order at least γ. Now, we will improve the statement of Theorem 3.3, when HV(t,x)is also bounded above. Theorem 3.6 Assume that there exist a function V∈C1.2(R+×Rd,R+), four continuous functions ψ1(t)∈ R,ψ 2(t)≥0,ψ 3(t)≥0,ρ(t)>0some constants q ∈N∗,m≥0,α 1≥0,α 2≥0,α 3∈R,a small constant ξ≥0, such that for all t ≥t0and all x =(x1,x2)∈Rd,(H1), (H2), (H5)hold, and the following assumptions: (H 3)ψ 2(t)V2(t,x)+ξ≤HV(t,x)≤ψ3(t)V2(t,x), (H 4)lim t→+∞sup t Tψ1(s)ds ln λ(t)≤α3,∀T≥t0, lim t→+∞inf t Tψ2(s)ds ln λ(t)≥2α1,∀T≥t0, lim t→+∞sup t Tψ3(s)ds ln λ(t)≤α2,∀T≥t0. Let x0∈Rd,x0= 0, such that the corresponding solution x(t,t0,x0)=(x1(t,t0,x0), x2(t,t0,x0)) satisfies: (i) ||x1(t,t0,x0)|| >ρ(t) λ(t)m1 q,∀t≥t0, (ii) x2(t,t0,x0)is globally uniformly bounded with probability one. 123 Partial stability analysis of stochastic differential equations... Page 9 of 17 4 Hence, lim t→+∞sup ln ||x1(t,t0,x0)||−ρ(t) λ(t)m1 q ln λ(t)≤− m−(α3−α1+C),a.s. Proof Let x0= 0inRd. It follows from Lemma 2.1,x(t)=(x1(t), x2(t)) = 0, ∀t≥0 almost surely. Applying the Itô formula once more implies again Eq. (3.4): ln(V(t,x(t))) =ln(V(T,x(T))) +t T LV(s,x(s)) V(s,x(s)) ds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds, with M(t)=t T Vx(s,x(s))g(s,x(s)) V(s,x(s)) dBs. Taking into account assumptions (H2)and (H 3), we obtain that, ∀t≥T, ln(V(t,x(t))) =ln(V(T,x(T))) +M(t)+t T ψ1(s)ds+t−1 2t T ψ2(s)ds.(3.8) Based upon conditions (H 4)and (H5),itfollowsthat lim t→+∞sup ln(V(t,x(t))) ln λ(t)≤lim t→+∞sup M(t) ln λ(t)+(α3+ε) −1 2(2α1−ε) +C,a.s. Let us denote by M(t)the quadratic variation process associated to M(t). Based on our assumptions we deduce that M(t)is a local martingale vanishing at t=T. Moreover, on account of condition (H 3), we obtain t T ψ2(s)ds+t T ξ V2(s,x(s))ds≤M(t)=t T HV(s,x(s)) V2(s,x(s)) ds≤t T ψ3(s)ds. Since α1>0, it follows that lim t→+∞M(t)=+∞. Thanks to the strong law of large numbers (see, [20]), we obtain lim t→+∞ M(t) M(t)=0,a.s. Note that, for tlarge enough, we have ||M(t)|| ln λ(t)=||M(t)|| M(t)M(t) ln λ(t)≤||M(t)|| M(t)t 0ψ3(s)ds ln λ(t). 123 4Page 16 of 17 T. Caraballo et al. Appendix A: Details of calculus of Eq. (3.3) λ(t)m||x1(t)||q−ρ(t)=λ(t)m||x1(t)||q−ρ(t) λ(t)m1 qq =λ(t)m||x1(t)||−ρ(t) λ(t)m1 q||x1(t)||q−1+||x1(t)||q−2ρ(t) λ(t)m1 q +···+ρ(t) λ(t)mq−1 q =λ(t)m||x1(t)||−ρ(t) λ(t)m1 qq  k=1||x1(t)||q−kρ(t) λ(t)mk−1 q. Appendix B: Details of calculus of Eq. (3.5) ln(V(t,x(t))) ≤ln(V(T,x(T))) +t T ψ1V(s,x(s)) +ρ(s) V(s,x(s)) ds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds ≤ln(V(T,x(T))) +t T ψ1(s)ds+t T ρ(s) V(s,x(s))ds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds ≤ln(V(T,x(T))) +t T ψ1(s)ds+t T ρ(s) λ(s)m||x1(s)||qds+M(t)−1 2t T HV(s,x(s)) V2(s,x(s)) ds. Appendix C: Details of calculus of Eq. (3.7) ln(V(t,x(t))) ≤ln(V(T,x(T))) +t T ψ1(s)ds+(t−T)+2 βln(k−1)+β 2t T HV(s,x(s)) V2(s,x(s)) ds −1 2t T HV(s,x(s)) V2(s,x(s)) ds ≤ln(V(T,x(T))) +t T ψ1(s)ds+(t−T)+2 βln(k−1)−1−β 2t T HV(s,x(s)) V2(s,x(s)) ds ≤ln(V(T,x(T))) +t T ψ1(s)ds+(t−T)+2 βln(k−1)−1−β 2t T ψ2(s)ds −1−β 2t T ξ V2(s,x(s))ds ≤ln(V(T,x(T))) +2 βln(k−1)+t T ψ1(s)ds+(t−T)−1−β 2t T ψ2(s)ds. References 1. Ren Y, Sakthivel R, Sun G (2020) Robust stability and boundedness of stochastic differential equations with delay driven by G-Brownian motion. Int J Control 93:2886–2895 123 Partial stability analysis of stochastic differential equations... Page 17 of 17 4 2. Ren Y, Sakthivel R, Yin W (2018) Stabilization of stochastic differential equations driven by G-Brownian motion with feedback control based on discrete-time state observation. Automatica 95:146–151 3. Azizi A, Yazdi PG (2019) Computer-based analysis of the stochastic stability of mechanical structures driven by white and colored noise. Springer briefs in applied sciences and technology, Springer, Singapore 4. Awad E, Culick FEC (1986) On the existence and stability of limit cycles for longitunical acoustic modes in a combustion chamber. Combust Sci Technol 46:195–222 5. Lum KY, Bernstein DS, Coppola VT (1995) Global stabilization of the spinning top with mass imbalance. Dyn Stab Syst 10:339–365 6. Rouche N, Habets P, Lalog M (1977) Stability theory by Liapunov’s direct method. Springer, New York 7. Sinitsyn VA (1991) On stability of solution in inertial navigation problem. Certain problems on dynamics of mechanical systems, Moscow, pp 46–50 8. Vorotnikov VI (1988) Partial stability and control. Birkhäuser, Boston 9. Zubov VI (1982) The dynamics of controlled systems. Vysshaya Shkola, Moscow 10. Ignatyev O (2009) Partial asymptotic stability in probability of stochastic differential equations. Stat Probab Lett 79:597–601 11. Peiffer K, Rouche N (1969) Liapunov’s second method applied to partial stability. J Mécanique 2:20–29 12. Rymanstev VV (1957) On the stability of motions with respect to part of variables. Vestnik Moscow University Ser Math Mech 4:9–16 13. Rumyantsev VV, Oziraner AS (1987) Partial stability and stabilization of motion. Nauka, Moscow (in Russian) 14. Savchenko AY, Ignatyev O (1989) Some problems of stability theory. Naukova Dumka, Kiev (in Russian) 15. Vorotnikov VI, Rumyantsev VV (2001) Stability and control with respect to a part of the phase coordinates of dynamic systems: theory, methods, and applications. Scientific World, Moscow (in Russian) 16. Caraballo T, Hammami M, Mchiri L (2016) On the practical global uniform asymptotic stability of stochastic differential equations. Stochastics 88:45–56 17. Caraballo T, Hammami M, Mchiri L (2014) Practical asymptotic stability of nonlinear evolution equations. Stoch Anal Appl 32:77–87 18. Caraballo T, Ezzine F, Hammami M, Mchiri L (2020) Practical stability with respect to a part of variables of stochastic differential equations. Stochastics 6:1–18 19. Caraballo T, Garrido-Atienza MJ, Real J (2003) Asymptotic stability of nonlinear stochastic evolution equations. Stoch Anal Appl 21:301–327 20. Mao X (1997) Stochastic differential equations and applications. Ellis Horwood, Chichester 21. Mao X (1992) Almost sure polynomial stability for a class of stochastic differential equations. Quart J Math Oxford Ser 43:339–348 22. Caraballo T, Garrido-Atienza MJ, Real J (2003) Stochastic stabilization of differential systems with general decay rate. Syst Control Lett 48:397–406 23. Caraballo T, Hammami M, Mchiri L (2017) Practical exponential stability of impulsive stochastic functional differential equations. Syst Control Lett 109:43–48 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123