Practical stability with respect to a part of variables of stochastic differential equations

Practical stability with respect to a part of variables of stochastic differential equations Tom´as Caraballoa∗Faten Ezzine b Mohamed Ali HammamibLassaad Mchiric aUniversidad de Sevilla, Depto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, 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 nonlinear stochastic differential equations (SDEs) are studied. We investigate, the global practical uniform asymptotic, 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 based on Lyapunov techniques. We provide also some illustrative examples to show the usefulness of the notion of stability with respect to a part of variables for SDEs. Keywords: Stochastic systems, Lyapunov techniques, Itˆo formula, Brownian motion, nontrivial solution, practical stability with respect to a part of the variables . 1 Introduction Stability of stochastic differential equations (SDEs) has become a very prevalent theme of recent research in mathematics and its applications. Stochastic model can be used to solve the problem ∗The research of Tom´as Caraballo has been partially supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2011-22411 1 which evinces by accident, noise, etc. This paper aims mainly to establish some criteria for the global practical uniform asymptotic, 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 (SDEs) of the form: dx(t) = f(x(t), t)dt +g(x(t), t)dB(t). In this way, considerable attention has been paid to the concept of stability with respect to a part of the system’s states. Such concept arises from the study of electro-magnetics [23], inertial navigation systems [22], spacecraft stabilization via gimballed gyroscopes and/or flywheels [24], combustion systems [3], vibrations in rotating machinery [20], and biocenology [21]. The method of Lyapunov functions is one of the most powerful tool to study the stability of stochastic dynamical system, with the emergence of the second method of Lyapunov Peiffer and Rouche (1969), Rouche et al. (1977), Rumyantsev (1957), Rumyantsev and Oziraner (1987), Savchenko and Ignatyev (1989), Sontag and Wang (2001), Vorotnikov (1998), Vorotnikov and Rumyantsev (2001), involved the notion of stability with respect to a part of the variables especially when the origin considered as an equilibrium point. We would like to mention here the references [7],[12],[19],[24], among others. Stability with respect to a part of the variables, has been used in investigating 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 characterizing the motions of the system under investigation. Several interesting and important variants to Lyapunov’s original concepts of practical stability were proposed in [1]-[4]-[10]. When the origin is not necessarily an equilibrium point, we can study the stability with respect to a part of the variables of the SDE in a small neighborhood of the origin in terms of convergence of solution in probability with respect to a part of the variables to a small ball. Our main objective in this paper is to highlight situations in which the results in [10] cannot be applied to obtain the global practical stability in all variables but at least we can prove for some of the variables. The remainder of this paper is organized as follows. In section 2,we give necessary preliminaries and results. In sections 3,4,and 5,we give sufficient conditions to ensure the global practical uniform asymptotic, the global practical uniform pth moment exponential stability, and the global practical uniform exponential stability with respect to a part of the variables of SDEs. Moreover, we exhibit some illustrative examples to show the applicability of the main results. 2 2 Auxiliary Facts and Results Consider the following n-dimensional stochastic differential equation (SDE): dx(t) = f(x(t), t)dt +g(x(t), t)dBt,∀t≥0.(2.1) Where f:Rn×R+−→ Rn, g :Rn×R+−→ Rn×m, x = (x1, ...., xn)Tand Bt= (B1(t), ...., Bm(t))T is an m-dimensional Brownian motion defined on a complete probability space (Ω,F,P). We assume that there exists tsuch that f(0, t)6= 0 or g(0, t)6= 0, i.e. the stochastic differential equation (2.1) does not have the trivial solution x= 0. The functions f(x(t), t) and g(x(t), t) verify the following standard assumptions for Itˆo calculus [16]. ZT 0||f(x(s), s)||ds < ∞a.s. ∀T > 0, ZT 0||g(x(s), s)||2ds < ∞a.s. ∀T > 0. We assume that both fand gsatisfy the following conditions: ||f(x, t)||2+||g(x, t)||2≤K11 + ||x||2,for all t≥0, x ∈Rn,(2.2) ||f(x, t)−f(y, t)||∨||g(x, t)−g(y, t)|| ≤ K2||x−y||,for all t≥0, x, y ∈Rn,(2.3) where k1and k2are given positive reals. Denote x= (y, z)Twhere y= (y1, ...., yn1)T∈Rn1, z = (z1, ...., zn2)T∈Rn2, n1+n2=n; ||y|| =qy2 1+... +y2 n1,||z|| =qz2 1+... +z2 n2,||x|| = (||y||2+||z||2)1 2. The SDE (2.1) has a unique global solution x(t, t0, x0) with initial condition x0(Mao, 1997, Oksendal, 2003). In what follows we use x(, t0, x0), or simply x(t) for a solution on some small interval. 3 Global practical uniform asymptotic stability of stochastic differential equations with respect to a part of the variables In this section, we study the asymptotic stability in probability with respect to a part of the variables of stochastic differential systems when the origin is not an equilibrium point. Therefore, we will study the asymptotic stability with respect to x1of the SDE in a small neighborhood of the origin in terms of convergence of solution in probability with respect to a part of the variables to a small ball Br:= {x∈Rn:||x|| ≤ r},r > 0. 3 Definition 3.1. The solution of (2.1) is said to be globally uniformly bounded in probability, 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),a.s. (3.1) Definition 3.2. (i) The ball Bris said to be uniformly stable with respect to yin probability, if for each ε∈]0,1[ and k > r, there exists δ=δ(ε, k)>0 such that P||y(t, t0, x0)|| < k, ∀t≥t0≥0≥1−εfor all ||x0|| < δ. (3.2) (ii) The ball Bris said to be globally uniformly stable with respect to yin probability, if it is uniformly stable with respect to yin probability and the solution of (2.1) is globally uniformly bounded in probability. Definition 3.3. The ball Bris said to be globally uniformly attractive with respect to yin probability, if for each ε∈]0,1[, k > r and c > 0 (independent of t0), there exists T=T(ε, c)>0 such that, for all t0≥0, it holds P||y(t, t0, x0)|| < k, ∀t≥t0+T≥1−εfor all x0∈Rnsuch that ||x0|| < c. (3.3) Definition 3.4. The system (2.1) is said to be globally uniformly practically asymptotically stable with respect to yin probability, if there exists r > 0 such that Bris globally uniformly stable with respect to yin probability and globally uniformly attractive with respect to yin probability. Definition 3.5 (Class Kfunction).A continuous function α: [0, a)→[0,+∞) is said to belong to class K, if it is strictly increasing and α(0) = 0. It is said to belong to class K∞if a= +∞ and α(r)→+∞as r→+∞. Definition 3.6. (see [24]) A function V:R+×Rn→Ris said to be positive definite with respect to y(or y-positive definite) if there exist α1∈ K such that V(t, x)≥α1(||y||). Definition 3.7. (see [24]) A function V:R+×Rn→Ris said to have an infinitesimal upper limit in y if there exists α2∈ K such that V(t, x)≤α2(||y||). We will define now C2,1(Rn×R+,R) the family of all real-valued functions V(x, t) defined on Rn×R+which are twice continuously differentiable in xand once in t. If V∈C2,1(Rn×R+,R), we set Vt(x, t) = ∂V ∂t (x, t) ; Vx(x, t) = (∂V ∂y (x, t),∂V ∂z (x, t)); Vxx(x, t) = ∂2V ∂xi∂xj (x, t)n×n. Define the differential operator Las associated with equation (2.1) by LV (x, t) = Vt(x, t) + Vx(x, t)f(x, t) + 1 2trace[g(x, t)TVxx(x, t)g(x, t)]. 4 By Itˆo’s formula, it follows dV (x(t), t) = LV (x(t), t)dt +Vx(x(t), t)g(x(t), t)dBt. Our first main result in this section reads as follow. Theorem 3.1. Assume that there exist V∈C2,1(Rn×R+,R+),α1, α2∈ K∞, with α1:R∗ +→R∗ +and α3∈ K,M > 0, such that for all t≥0, and all x∈Rn, α1(||y||)≤V(x, t)≤α2(||y||),(3.4) LV (x, t)≤ −α3(||y||) + ϕ(t),(3.5) where ϕ(t)is a continuous nonnegative function with lim t→+∞ϕ(t)=0,(3.6) and Z+∞ 0 ϕ(t)dt ≤M < +∞.(3.7) Furthermore, we suppose that for all t≥t0≥0,||z(t, t0, x0)|| ≤ h, a.s. where h is a positive constant independent of x0and t0. Then, system (2.1) is globally uniformly practically asymptotically stable with respect to yin probability. Remark 3.2. If ϕ(t) is uniformly continuous nonnegative function, then By Barbalat’s Lemma (Khalil, 2002) (3.7) implies (3.6) . Proof. Let us start prove the uniform stability of Brwith respect to yin probability. Let ε∈]0,1[, and take a small positive real number r, and k > r. Assume that there exists δ=δ(ε, k)∈]0, k[ such that, for x0∈Sδ,Sδ={x∈Rn/||x|| < δ}and α1(k)>M ε, we deduce V(x0,0) ε≤α1(k)−M ε. Define the stopping time τk= inf{t≥0; ||y(t;t0, x0)|| ≥ k}. By Itˆo’s formula, for any t≥0, V(x(τk∧t), τk∧t) = V(x0,0) + Zτk∧t 0 LV (x(s), s)ds +Zτk∧t 0 Vx(x(s), s)g(x(s), s)dBs. 5 Taking the expectation on both sides, we obtain that 0≤EV(x(τk∧t), τk∧t)=V(x0,0) + EZτk∧t 0 LV (x(s), s)ds. This, together with (3.5) and (3.7), implies EV(x(τk∧t), τk∧t)≤V(x0,0) + EZτk∧t 0 ϕ(s)−α3(||y(s)||)ds ≤V(x0,0) + EZτk∧t 0 ϕ(s)ds ≤V(x0,0) + EZ+∞ 0 ϕ(s)ds ≤V(x0,0) + M ≤εα1(k). Note that, ||y(τk∧t;t0, x0)|| =||y(τk;t0, x0)|| =kif τk≤t. Hence, EV(x(τk∧t), τk∧t)≥E1{τk≤t}V(x(τk), τk) ≥E1{τk≤t}α1(||y(τk;t0, x0)||) ≥E1{τk≤t}α1(k) =α1(k)E1{τk≤t} =α1(k)Pτk≤t. Consequently, α1(k)Pτk≤t≤εα1(k). This implies immediately that, Pτk≤t≤ε. Letting, t→+∞, we get Pτk<+∞≤ε. That is, for all ε∈]0,1[, r≥0 and k > r, there exists δ=δ(ε, k)∈]0, k[, such that P||y(t;t0, x0)|| < k, ∀t≥t0≥0≥1−ε, for all ||x0|| < δ. (3.8) Let us now prove the global uniform attractivity of Brwith respect to yin probability. It is sufficient to prove that, for each ε∈]0,1[, λ > r and c > 0,there exists T=T(ε, c)>0 (independent of t0), such that for all t≥t0+T, we have P||y(t;t0, x0)|| < λ≥1−ε, for all ||x0|| < c. (3.9) Let ε∈]0,1[, h>rlarge enough and c > 0 (independent of t0), such that ||x0|| < c and α1(λ)>M ε, we deduce α1(h)−M ε≥V(x0,0) ε. 6 Suppose now that there exists T=T(ε, c)>0 such that, for all t0≥0, τλ= inf{t≥t0+T;||y(t;t0, x0)|| ≥ λ} By the Itˆo formula, for any t≥t0+T, we obtain 0≤EV(x(τλ∧t), τλ∧t)=V(x0,0) + EZτλ∧t 0 LV (x(s), s)ds. Taking the expectation on both sides and making use (3.5) and (3.7), we obtain that EV(x(τλ∧t), τλ∧t)≤V(x0,0) + EZτλ∧t 0 (ϕ(s)−α3(||y(s)||)ds ≤V(x0,0) + M ≤εα1(h). Note that, ||y(τλ∧t;t0, x0)|| =||y(τλ;t0, x0)|| =λwhen τλ≤t. Therefore, we can obtain EV(x(τλ∧t), τλ∧t)≥E1{τλ≤t}V(x(τλ), τλ) ≥E1{τλ≤t}α1(||y(τλ;t0, x0)||) ≥E1{τλ≤t}α1(λ) =α1(λ)E1{τλ≤t} =α1(λ)Pτλ≤t. Then, α1(λ)Pτλ≤t≤εα1(λ), which implies that, Pτλ≤t≤ε. Letting t→ ∞ we get Pτλ<+∞≤ε. That is, for any ε∈]0,1[, λ > r and c > 0 (independent of t0), for a fixed T=T(ε, c)>0, we have, for all t≥t0+T,P||y(t;t0, x0)|| < λ;∀t≥t0+T≥1−ε, for all ||x0|| < c. For a small r∈]0, δ[ and ||x0|| < r, we can take k=k(r)>0, such that for all t≥t0≥0, we have ||y(t;t0, x0)|| < k, a.s , for all ||x0|| < r. We also have for all t≥t0≥0, ||z(t;t0, x0)|| < h, a.s. Consequently, there exist m=m(r)such that, sup t≥t0||x(t)|| < m for all ||x0|| < r a.s. Hence, the system (2.1)is globally uniformly practically asymptotically stable with respect to y in probability. 2 We present an example that implement the previous Theorem. 7 Example 3.3. Sorry Mohamed, in my opinion this example is not valid to justify that the results in our paper [10], namely Theorem 2.1, cannot be applied to all the variables and therefore the current result in this paper is meaningful. Let me explain: first, in the example below there are some mistakes, please check it carefully. Second, if we consider V:= x2 1+x2 2+x2 3 2,then the result Theorem 2.1 in ref. [10] holds and therefore we have stability in the three variables. In my opinion, to show the applicability of the theory in the current paper, we need to find an example which is not stable in all the variables and that we can prove stability in some of the variables. If we cannot provide such example, then our results do not have any interest. Can you please think about this? This is the real motivation and importance of the present study in this paper. Consider the following stochastic system                    dx1(t) = (−x1+1 pch(t))dt +√2x1qx4 2+ 1dBt, dx2(t) = (−x2−x2 1x2(x4 2+ 1))dt, dx3(t) = −βx3dt, (3.10) where β > 0, x(t)=(x1(t), x2(t), x3(t))T∈R3and B(t)is one-dimensional Brownian motion. It is clear that x2(t, t0, x0)is stochastically bounded, in fact, x2(t, t0, x0) = x20(t, ω)e−β(t−t0). Denote, V:= x2 1+x2 2 2. Applying the Itˆo formula on Vleads to LV =x1(−x1+1 pch(t))−x2 2−x2 1x2 2(x4 2+ 1) + x2 1x2 2(x4 2+ 1) =−x2 1−x2 2+x1 1 pch(t) ≤ −1 2x2 1−x2 2+1 2 1 ch(t) ≤ −1 2(x2 1+x2 2) + 1 2 1 ch(t). Note that we cannot apply theorem on practical asymptotic stability in probability [10], because all conditions of this Theorem are not fulfilled. But we can apply Theorem (3.1), since Vis positive definite with respect to (x1, x2)and it have an infinitesimal upper limit in (x1, x2). Furthermore, LV (x, t)≤ −1 2(x2 1+x2 2) + ϕ(t), where ϕ(t) = 1 2 1 ch(t).which satisfied both conditions (3.6) and (3.7). 8 Hence, all conditions of Thorem (3.1) are fulfilled and we get that (3.10) is globally uniformly practically asymptotically stable with respect to (x1, x2)in probability. 4 Global practical uniform pth moment exponential stability of stochastic differential equation with respect to a part of the variables In this section, we present sufficient conditions for global practical pth moment exponential stability of SDE with respect to a part of the variables. Definition 4.1. 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 E(||y(t, t0, x0)||p)≤λ1||x0||pe−λ2(t−t0)+r∀t≥t0,∀x0∈Rn.(4.1) ii) The system (2.1) 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. Within the method of Lyapunov, we present the following result. Theorem 4.1. Assume that there exist V∈C2,1(Rn×R+,R∗ +)and positive constants ci(i= 1,2,3), γ, p, such that for all x6= 0 and all t≥t0≥0, c1||y||p≤V(x, t)≤c2||y||p,(4.2) LV (x, t)≤ −c3||y||p+ψ(t),(4.3) where ψ(t)is a continuous nonnegative function with lim t→+∞ψ(t) = 0,(4.4) and Z+∞ t0 ψ(t)dt ≤ρ < +∞.(4.5) Furthermore, we suppose that for all t≥t0≥0,||z(t, t0, x0)|| ≤ h, a.s. where h is a positive constant independent of x0and t0. Then, system (2.1) is globally practically uniformly pth moment exponentially stable with respect to ywith r=ρ c1. In order to prove this Theorem, we need to recall an important Gronwall-Lemma see Dragomir [11] for other versions. 9 then there exists σ > 0 such that Zt 0 %(s)ds ≤(β3+σ)t. Since 0 <ε<1,the following inequality −1−ε 2Zt 0 ζ V2(x(s), s)ds ≤0. Then inequality (5.6) becomes ln(V(x(t), t)) ≤ln(V(x(0),0) + (β2+ 1)t+2 εln(n)−1−ε 2(β3+σ)t. (5.8) Consequently, for almost all ω∈Ω, if n−1≤t≤nand n≥n0, 1 tln(V(x(t), t)) ≤ −1 2(1 −ε)(β3+σ)−2(β2+ 1)+ln(V(x(0),0)) + 2 εln(n) n−1. This implies , lim t→+∞sup 1 tln(V(x(t), t)) ≤ −1 2(1 −ε)(β3+σ)−2(β2+ 1)a.s. Putting ε−→ 0 gives, lim t→+∞sup 1 tln(||y(t;t0, x0)||−(γ β1 )1 p)≤(−(β3+σ)−2(β2+ 1)) 2a.s. If the inequality (β3+σ)>2(β2+ 1) is satisfied, then the system (2.1) is almost surely globally practically uniformly exponentially stable with respect to y. 2 Now, we provide an illustrative example that implement the previous result. Example 5.4. In this example, the second equation is exponentially stable, so, I guess that my comments in red in the first example are also valid here, I mean, I think that this example is stable in all the variables. Please, we need to find an example in which we cannot prove stability in all the variables, but only in a few of them. Consider the following stochastic system:    dx1(t) = f(x(t))dt +Gx1(t)dBt dx2(t) = αx2(t)dt +βx2(t)dBt, α +β2 2<0,(5.9) 16 where x= (x1, x2)∈R3,and x1= (z1, z2)∈R2, B(t)is one-dimensional Brownian motion, f(x) = (z1+ 1) cos(z2) (z2+ 1) sin(x2),, G =4−0.4 −0.4r. With initial value x0= (x10, x20),and x10 = (z10, z20). It is clear that, for α+β2 2<0, x2(t, t0, x0)is stochastically bounded since x2(t, t0, x0)is geometric Brownian motion (see [16]). Consider the Lyapunov-like function: V(t, x) = x2 1=z2 1+z2 2. It is easy to verify that LV (x, t)=2z1(z1+ 1) cos(z2)+2z2(z2+ 1) sin(x2) + ||Gx1||2 ≤2z1(z1+ 1) + 2z2(z2+ 1) + ||Gx1||2 ≤22.36x2 1+ 2, and ||VxGx1||2=||2x1TGx1||2≥51.84||x1||4+ 0. Thus, the constants in Theorem 5.1 become β1= 1,β2= 22.36,β3= 51.84,p=γ= 2, ζ=σ= 0. Clearly, β3>2(β2+ 1). Note that we cannot apply the result of [10] to deduce the practical uniform exponential stability of the system, because Vis not definite positive,Mohamed, indeed we cannot apply the results in [10] to this precise Lyapunov function Vbut it is possible that we can apply to another V, so it is not fully justified that our problem does not have stability in all the variables.Please, we need an example in which we can ensure that stability in all variables does not hold. it is only definite positive with respect to x1.Therefore, all conditions of Theorem (5.1) are satisfied, hence the system (5.9) is almost surely globally practically uniformly exponentially stable with respect to x1 with r=√2. References [1] M. Anabtawi, Practical Stability of Nonlinear Stochastic Hybrid Parabolic Systems of Ito- Type: Vector Lyapunov Functions Approach, Nonlinear Analysis: Real World Applications, Volume 12, Issue 3, (2011) 1386-1400. [2] L. Arnold, Stochastic differential equations: Theory and Applications, John Wiley, New York-London-Sydney, 1974. [3] E. Awad, F.E.C. Culick, On the existence and stability of limit cycles for longitunical acoustic modes in a combustion chamber, Comb. Scien. Tech., Izd. MAI: vol. 46,(1986), 195-222. [4] B. Ben Hamed, I. Ellouze, M.A. Hammami, Practical uniform stability of nonlinear differential delay equations. Mediterranean Journal of Mathematics, 8 (2011), no. 4, 603-616. 17 [5] A. BenAbdallah, M. Dlala, M. A. Hammami, A new Lyapunov function for perturbed non linear systems, Systems and Control letters, 56 (3) (2007) 179-187. [6] A. BenAbdallah, I. Ellouze and M. A. Hammami, Practical stability of nonlinear time-varying cascade systems, Journal of Dynamical and Control Systems, Vol. 15, No. 1, (2009) 4562. [7] A. Ben Makhlouf, Stability with respect to part of the variables of nonlinear Caputo fractional differential equations, Mathematical Communications, Math. Commun. 23(2018), 119-126 [8] T. Caraballo, On the decay rate of solutions of non-autonomous differential systems, Electron. J. Diff. Eqns. 2001, 2001 (05), 1-17. [9] T. Caraballo, Mar´ıa J. Garrido-Atienza, Jos´e Real, Stochastic stabilization of differential systems with general decate rate, Systems and Control Letters, 48 (2003), 397-406. [10] 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, vol. 88, 2016, 45-56. [11] S. S. Dragomir, Some Gronwall type inequalities and applications, (2003). [12] W.M. Haddad, V. Chellaboina, Nonlinear Dynamical Systems and Control, (2007). [13] R. Z.Has’minskii, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, (1980). [14] H. K. Khalil, Nonlinear Systems, Mac-Millan, 2nd edition, (1996). [15] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, Inc.: New York, (1994). [16] X. Mao, Stochastic Differential Equations and Applications, Ellis Horwood, Chichester, U.K, (1997). [17] B. Oksendal, Stochastic Differential Equations: an Introduction with Applications (6th ed.) Springer-Verlag, New York, 2003. [18] U. H. Thygesen, A survey of Lyapunov Techniques for Stochastic Differential Equations, IMM Technical Report n.c, (1997). [19] O.Ignatyev, Partial asymptotic stability in probability of stochastic differential equations, Statistics and Probability Letters, 79 (2009), 597-601. [20] K.-Y. Lum, D.S. Bernstein, V.T. Coppola, Global stabilization of the spinning top with mass imbalance, Dynamics and Stability of Systems, 10 vol. 10, (1995), 339-365. [21] N. Rouche, P. Habets, M. Lalog, Stability Theory by Liapunov’s Direct Method. New York: Springer, 1977. 18 [22] V.A. Sinitsyn, On stability of solution in inertial navigation problem, Certain Problems on Dynamics of Mechanical Systems (in Russian), Izd. MAI: Moscow, (1991), 46-50. [23] V.I. Zubov, The Dynamics of Controlled Systems. Moscow: Vysshaya Shkola, (1982). [24] V. I. Vorotnikov, Partial Stability and Control, (1998). 19