A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality

A new result on stabilization analysis for stochastic nonlinear affine systems via Gamidov’s inequality Tom´as Caraballoa∗Faten Ezzineb Mohamed Ali Hammamic aUniversidad de Sevilla, Depto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, c/ Tarfia s/n, 41012-Sevilla (Spain) {b,c}University of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia b: E.mail: [email protected] c: E.mail: [email protected]u.tn Abstract The Lyapunov approach is one of the most effective and efficient methods for the investigation of the stability of stochastic systems. Several authors analyzed the stability and stabilization of stochastic differential equations via Lyapunov techniques. Nevertheless, few results are concerned with the stability of stochastic systems based on the knowledge of the solution of the system explicitly. The originality of our work is to investigate the problem of stabilization of stochastic perturbed control–bilinear systems based on the explicit solution of the system by using the integral inequalities of the Gronwall type in particular Gamidov’s inequality. Namely, under some restrictions on the perturbed term, and based on the method of integral inequalities, we prove that the stochastic system can be stabilized by constant feedback. Further, we study the problem of stabilization of stochastic perturbed control affine systems based on the use of bilinear approximation. Different examples are provided to verify the effectiveness of the proposed results. Keywords: Affine systems, Bilinear systems, Stochastic systems, Integral inequalities, Gamidov inequalities, Stabilization, Practical stability. 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 Gronwall’s inequality was first proposed and proved as its differential form in 1911 by the Swedish mathematician Thomas Hacon Gronwall [19]. Later on, the integral form was proved in 1943 by the American mathematician Bellmen [8]. Since then, many authors introduced several generalizations of this inequality, see [6, 7, 15, 20, 22]. Gronwall Lemmas are very useful in many analysis problems. In particular, it is an essential tool in the analysis of the problem of boundedness, uniqueness, and other aspects of the qualitative behavior of solutions of differential and stochastic equations. As it is well known, environmental noise exists and cannot neglect it in many dynamical systems. Indeed, it is essential to analyze whether the presence of some random terms in the equations of the models may produce a very different behavior of their solutions. Although there exists a wide literature on this topic, see [1, 21], [23]-[25]. The result is related to the relation between a perturbed stochastic system and the associated unperturbed one. Given two solutions to the perturbed stochastic system and the associated unperturbed one with initial conditions that are close at the same value of time, these solutions will remain close over the entire time interval and not just at the initial time. Different intrinsic variants to Lyapunov’s original concepts of practical stability were proposed in [4, 16]. 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 [2, 5, 9, 17, 18, 26]. 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 asymptotic 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. The Lyapunov method is one of the most effective ones for the investigation of the stability of stochastic systems, without knowing the explicit solution form of the system. The Lyapunov stability has attracted the attention of many authors. There exists a huge amount of work on this subject, see [10, 11, 12, 13, 14]. However, the construction of a suitable Lyapunov function is still a difficult task. The novelty of our work is to develop the problem of stability and stabilization of stochastic perturbed control-bilinear systems based on the explicit solution formed through integral inequalities of the Gronwall type, in particular Gamidov’s inequality. The qualitative behavior of the solutions of perturbed stochastic systems is usually studied 2 by considering a Lyapunov function candidate for the unperturbed system and using it as an appropriate Lyapunov function candidate for the stochastic perturbed system. Nevertheless, unlike the linear case, the construction of a suitable Lyapunov function is still a difficult task for nonlinear stochastic differential equations. This motivates us to investigate the problem of stability of stochastic perturbed systems by using integral inequalities of Gronwall type under some restrictions on the perturbation term. The usual property of the solutions that can be deduced for such systems is ultimate boundedness. That means that the solutions remain in some neighborhoods of the origin after a sufficiently large time, (see [3], [10]-[13]). In different cases, the linearized system is independent of the control. Therefore, we can study the stabilization problem for such systems via a bilinear system. The organization of this paper is as follows: In Section 2, we investigate the stability of linear time–invariant stochastic perturbed systems via non linear–integral inequalities. In Section 3, we analyze the problem of stabilization of stochastic perturbed control–bilinear systems under some restrictions on the bound of perturbations. In Section 4, we prove that the problem of stabilization of stochastic perturbed affine system can be performed by considering a bilinear approximation. Further, we display some illustrative examples to exhibit the applicability of our abstract theory. In Section 5, some conclusions are included. 2 Linear time–invariant stochastic perturbed systems Consider the following linear time–invariant system: dx(t) = Ax(t)dt, (2.1) where x∈Rn,Ais a constant matrix (n×n) . Assume that some parameters are excited or perturbed by Brownian motion, and the linear time-invariant stochastic perturbed system is expressed by the following form: dx(t) = Ax(t)dt +φ(t, x(t))dBt,(2.2) where φ:R+×Rn−→ Rn×m, Bt= (B1(t), ..., Bm(t))Tis an m-dimensional Brownian motion defined on a complete probability space (Ω,F,P).In the sequel, ||·||represents the Euclidean norm. We will study the asymptotic behaviors of the solutions of the stochastic perturbed system (2.2) in the sense that all state trajectories are bounded and approach a sufficiently small neighborhood of the origin. In this objective, we recall the following definitions, see [10, 11, 12, 13]. Let’s consider the following stochastic system : dx(t) = F(t, x(t))dt +G(t, x(t))dBt, t ≥0,(2.3) 3 where x∈Rn, F :R+×Rn→Rn, G :R+×Rn→Rn×m,and Btis an m-dimensional Brownian motion. We assume that both functions Fand Gsatisfy the following conditions: ||F(t, x)||2+||G(t, x)||2≤α11 + ||x||2,for all t≥0, x ∈Rn, ||F(t, x)−F(t, y)||∨||G(t, x)−G(t, y)|| ≤ α2||x−y||,for all t≥0, x, y ∈Rn, where α1and α2are given positive real constant, then under the precedent assumptions, there exists a unique global solution x(t) corresponding to the initial condition x0∈Rn(see Mao [25]). We assume that there exists tsuch that F(t, 0) 6= 0 or G(t, 0) 6= 0, i.e., the stochastic differential equation (2.3) does not have the trivial solution x≡0. The study of the exponential stability of the solutions of the stochastic system (2.3) leads to analyze the stability behavior of a ball centered at the origin: Br:= {x∈Rn:||x|| ≤ r}, r > 0. Definition 2.1. i) The ball Bris said to be almost surely globally uniformly exponentially stable, if there exists a pair of positive constants µ1and µ2, such that for all x0∈Rn, the following inequalities are satisfied: ||x(t)|| ≤ µ1||x0||e−µ2t+r, a.s., ∀t≥0.(2.4) ii) The stochastic system (2.3) is said to be almost surely globally practically uniformly exponentially stable, if there exists r > 0 such that Bris almost surely globally uniformly exponentially stable. Eq.(2.4) implies that x(t) will be bounded by a small bound r > 0, that is ||x(t)|| will be small for sufficiently large t. It means that the solution given in (2.4) will be uniformly ultimately bounded for sufficiently large t. This means that solution given in (2.4) will be uniformly ultimately bounded for sufficiently large t. The factor µ2in Eq.(2.4) is called the convergence speed, whereas the factor µ1is called the transient estimate. It is also worth noticing that, in the previous definition, if we take r= 0, then we recover the standard concept of the global exponential stability of the origin considered as an equilibrium point. Further, we out to investigate the asymptotic behavior of a small ball centered at the origin for 0 ≤ ||x(t)||−r, ∀t≥0, so that the initial conditions are taken outside the ball Br. If ris small enough, the trajectories tend to the origin when tgoes to infinity. 4 Our main objective in this section is to state sufficient conditions ensuring the almost sure practical uniform exponential stability of the linear time–invariant stochastic system (2.2). If we suppose that the perturbation term is bounded, then the origin is not necessarily an equilibrium point of the stochastic perturbed system (2.2). That’s why, we will study the convergence of the solutions toward a neighborhood of origin. Remark 2.1. Different authors tackle the problem of practical stability of stochastic differential equations via Lyapunov functions, see [10, 11, 12, 13, 14]. The construction of appropriate Lyapunov functions is not always possible, which motivates us to look for another method. Our approach in this paper is to analyze the stability and stabilization of stochastic perturbed control bilinear systems by using the explicit solution form and it is based on integral inequalities of the Gronwall type. We suppose that the matrix Ais asymptotically stable. A basic result in systems theory is that σ(A)⊂C−, where σ(A) denotes the set of eigenvalues of a the matrix A. With this condition, we have Reλ(A)<0,where Reλ(A) denotes the real parts of the eigenvalues of matrix A. A simple result for the asymptotic stability is that the eigenvalues of the matrix (all of which are real) remain strictly in the left-half complex plane: Reλ(A)<0. First, we suppose the following assumption required for the stability purposes. (H1) We assume that, Reλ(A)<0. Note that, the assumption (H1) implies that ||etA|| ≤ ke−γt,∀t≥0, for a certain k > 0 and γ≤min 1≤i≤n|Reλi(A)|. (H2) There exists a continuous nonnegative known function ξ(t), such that ||φ(t, x)|| ≤ ξ(t),∀x∈Rn,∀t≥0, where lim t→+∞ξ(t) = 0. Theorem 2.2. Under assumptions (H1)and (H2)the linear time–invariant stochastic perturbed system (2.2) is almost surely globally practically uniformly exponentially stable. In order to prove Theorem 2.2, we need to recall the following lemma. Lemma 2.3. [25] Let G= (G1, ...., Gn)∈L2(R+,Rn),T, a, b be any positive numbers. Then, Psup 0≤t≤TZt 0 G(s)dBs−a 2Zt 0||G(s)||2ds> b≤e−ab. 5 Proof of Theorem 2.2. The solution with initial condition x0of the linear time–invariant stochastic perturbed system is the following: x(t) = etA x0+Zt 0 e(t−s)Aφ(s, x(s))dBs. Assign ε∈]0,1[ arbitrarily, and let n= 1,2,··· .Then, using Lemma 2.3, we obtain Psup 0≤t≤nZt 0 e(t−s)Aφ(s, x(s))dBs−ε 2Zt 0||e(t−s)A||2||φ(s, x(s))||2ds>2 ε ln(n) n≤e1 n n2. By an application of the well known Borel-Cantelli lemma, we see that for almost all ω∈Ω, there exists an integer n0=n0(ω), such that if n≥n0, it yields that Zt 0 e(t−s)Aφ(s, x(s))dBs≤2 ε ln(n) n+ε 2Zt 0||e(t−s)A||2||φ(s, x(s))||2ds, for all 0 ≤t≤n. (2.5) Based on inequality (2.5), one obtains ||x(t)|| ≤ ||etA||||x0||+ε 2Zt 0||e(t−s)A||2||φ(s, x(s))||2ds +2 ε ln(n) n,for all 0 ≤t≤n, n ≥n0. Taking into account assumption (H1) and inequality (3.6), one deduce ||x(t)|| ≤ k||x0||e−γt +ε 2Zt 0 k2e−2γ(t−s)||φ(s, x(s))||2ds +2 ε ln(n) n,for all 0 ≤t≤n, n ≥n0. Using assumption (A2), one obtains ||x(t)|| ≤ k||x0||e−γt +ε 2k2e−2γt Zt 0 e2γsξ2(s)ds +2 ε ln(n) n,for all 0 ≤t≤n, n ≥n0. Since ξ(·)→0 as t→+∞, then there exists ¯ ξ > 0, such that ξ(t)≤¯ ξ, ∀t≥0. ||x(t)|| ≤ k||x0||e−γt +ε 2k2e−2γt Zt 0 e2γs ¯ ξ2ds +2 ε ln(n) n ≤k||x0||e−γt +ε 2k2e−2γt Zt 0 e2γs ¯ ξ2ds +2 ε ln(n) n ≤k||x0||e−γt +ε 2 ¯ ξ2 2γk2e−2γt(e2γt −1) + 2 ε ln(n) n ≤k||x0||e−γt +ε 2 ¯ ξ2 2γk2+2 ε ln(n) nfor all 0 ≤t≤n, n ≥n0. 6 Further, sine ln(n) n→0 as n→+∞, then there exists α > 0, such that ln(n) n≤α, ∀n≥n0.(2.6) Then, one obtains ||x(t)|| ≤ k||x0||e−γt +¯ ξ2 4γεk2+2 εα, for all 0 ≤t≤n, n ≥n0. Letting ε→1, we deduce that ||x(t)|| ≤ k||x0||e−γt +¯ ξ2k2 4γ+ 2α, a.s. Thus, the ball Brwith r=¯ ξ2k2 4γ+ 2αis almost surely globally uniformly exponentially stable. Then, the stochastic perturbed system (2.2) is almost sure globally practically uniformly exponentially stable. 2 A simple extension can be done, if we replace the assumption (H2) by the following assumption on the perturbed term. (H0 2) There exist a continuous positive functions λ1(t) and λ2(t), such that ||φ(t, x)||2≤e−µ2t(λ1(t)||x||q+λ2(t)) ,0< q < 1,∀x∈Rn, t ≥0, where λ1satisfies λ1(t)≤m, ∀t≥0,(2.7) and the function λ2satisfies the following Z+∞ 0 eµ2sλ2(s)ds =¯ λ2<+∞.(2.8) Theorem 2.4. Under assumptions (H1)and (H0 2),the stochastic perturbed system (2.2) is almost surely globally practically uniformly exponentially stable. The proof of Theorem 2.4 is based on the generalized integral inequality of the Gamidov type [20]. Lemma 2.5. If V(t)≤(t) + CZt 0 Φ(s)Vq(s)ds, 7 where all functions are continuous and nonnegative on [0, T),0< q < 1and T, C > 0. Then, there exists a constant ρ > 0, such that V(t)≤(t) + CρqZt 0 Φ1 1−q(s)ds1−q . Taking into account the above integral inequality, we can prove the almost sure global practical uniform exponential stability of the stochastic perturbed system (2.2). Proof of Theorem 2.4 The solution with initial condition x0of the stochastic perturbed system (2.2) is: x(t) = etA x0+Zt 0 e(t−s)Aφ(s, x(s))dBs.(2.9) Using assumption (H1) and Lemma 2.3, it yields that ||x(t)|| ≤ k||x0||e−γt +ε 2Zt 0 k2e−2γ(t−s)||φ(s, x(s))||2ds +2 ε ln(n) n, for all 0 ≤t≤n, n ≥n0. Based on (H0 2), we obtain ||x(t)|| ≤ k||x0||e−γt +ε 2Zt 0 k2e−2γ(t−s)e−γs (λ1(s)||x(s)||q+λ2(s)) ds +2 ε ln(n) n, for all 0 ≤t≤n, n ≥n0. Thus, one obtains ||x(t)|| ≤ k||x0||e−γt +ε 2k2e−2γt Zt 0 eγs (λ1(s)||x(s)||q+λ2(s)) ds +2 ε ln(n) n, for all 0 ≤t≤n, n ≥n0. Multiplying both sides by eγt, it yields that ||x(t)||eγt ≤k||x0||+ε 2k2e−γt Zt 0 eγs (λ1(s)||x(s)||q+λ2(s)) ds +2 ε ln(n) neγt ≤k||x0||+ε 2k2Zt 0 eγs (λ1(s)||x(s)||q+λ2(s)) ds +2 ε ln(n) neγt, for all 0 ≤t≤n, n ≥n0. 8 Using (2.6) and (2.7), we obtain ||x(t)||eγt ≤k||x0||+ε 2k2e−γt Zt 0 eγs (m||x(s)||q+λ2(s)) ds +2 εαeγt ≤k||x0||+ε 2k2mZt 0 eγs||x(s)||qds +ε 2k2Z+∞ 0 eγsλ2(s)ds +2 εαeγt, for all 0 ≤t≤n, n ≥n0. Let V(t) = eγt||x(t)||, and using (4.10), we have V(t)≤k||x0||+2 εαeγt +ε 2k2¯ λ2+ε 2k2mZt 0 e(1−q)γsVq(s)ds, for all 0 ≤t≤n, n ≥n0. Thus, V(t)≤(t) + CZt 0 e(1−q)γsVq(s)ds, where (t) = k||x0||+2 εαeγt +ε 2k2¯ λ2, C =ε 2k2m. By the application of the Gamidov inequality (Lemma 2.5), it follows that V(t)≤(t) + CρqZt 0 exp 1−q 1−qγsds1−q ≤(t) + CρqZt 0 eγsds1−q =(t) + Cρq1 γ1−q eγ(1−q)t. for all 0 ≤t≤n, n ≥n0. Consequently, we obtain ||x(t)|| ≤ k||x0||e−γt +2 εα+ε 2k2¯ λ2+ε 2k2mρq1 γ1−q , for all 0 ≤t≤n, n ≥n0. Letting ε→1, we deduce that ||x(t)|| ≤ k||x0||e−γt + 2α+1 2k2¯ λ2+1 2k2mρq1 γ1−q . 9 Lemma 3.4. [15] Let a(t), b(t), c(t), u(t)be continuous functions for t≥0, and b(t)be nonnegative for t≥0, suppose that u(t)≤a(t) + Zt 0 [b(s)u(s) + c(s)] ds. Then, u(t)≤a(t) + Zt 0 [a(s)b(s) + c(s)] exp Zt s b(τ)dτds. Corollary 3.5. [15] For a(t)≡a, we have u(t)≤aexp Zt 0 b(τ)dτ+Zt 0 c(s) exp Zt s b(τ)dτds. Proof of Theorem 3.3. The solution with initial condition x0of the closed–loop stochastic system (3.3) expressed by the following: x(t) = et¯ Ax0+Zt 0 e(t−s)¯ AG(s, x(s))dBs.(3.10) Using Lemma 2.3, it follows that ||x(t)|| ≤ ||et¯ A||||x0||+ε 2Zt 0||e(t−s)¯ A||2||G(s, x(s))||2ds +2 ε ln(n) n,for all 0 ≤t≤n, n ≥n0. Using assumption (A1),one obtains ||e¯ At|| =||e(A+¯uB)t|| ≤ µ1e−µ2t,(3.11) where µ2≤min 1≤i≤n|Re(λi(A+ ¯uB)) |. Based on (3.11) and assumption (A0 3), one obtains ||x(t)|| ≤ µ1||x0||e−µ2t+ε 2µ2 1e−2µ2tZt 0 eµ2sζ(s)||x(s)||ds +2 ε ln(n) n,for all 0 ≤t≤n, n ≥n0. Multiplying both sides by eµ2t, it comes that ||x(t)||eµ2t≤µ1||x0||+ε 2µ2 1e−µ2tZt 0 eµ2sζ(s)||x(s)||ds +2 ε ln(n) neµ2t, ≤µ1||x0||+ε 2µ2 1Zt 0 eµ2sζ(s)||x(s)||ds +2 ε ln(n) neµ2t,for all 0 ≤t≤n, n ≥n0. 16 Let U(t) = eµ2t||x(t)||, then we see that U(t)≤µ1||x0||+ε 2µ2 1Zt 0 ζ(s)U(s)ds +2 ε ln(n) neµ2t,for all 0 ≤t≤n, n ≥n0. Using (2.6), it yields that U(t)≤µ1||x0||+ε 2µ2 1Zt 0 ζ(s)U(s)ds +2 εαeµ2t, =µ1||x0||+ε 2µ2 1Zt 0 ζ(s)U(s)ds +2 ε α µ2Zt 0 eµ2sds, for all 0 ≤t≤n, n ≥n0. Applying the Gronwall inequality (Corollary 3.5), it yields that U(t)≤µ1||x0||exp Zt 0 ε 2µ2 1ζ(s)ds+Zt 0 2 ε α µ2 eµ2sexp Zt s ε 2µ2 1ζ(τ)dτds ≤µ1||x0||exp Z+∞ 0 ε 2µ2 1ζ(s)ds+Zt 0 2 ε α µ2 eµ2sexp Z+∞ 0 ε 2µ2 1ζ(τ)dτds, for all 0 ≤t≤n, n ≥n0. Taking into account (3.9), it follows that U(t)≤µ1||x0||exp ε 2µ2 1Θ+Zt 0 2 ε α µ2 eµ2sexp ε 2µ2 1Θds ≤µ1||x0||exp ε 2µ2 1Θ+2 ε α µ2 2 eµ2texp ε 2µ2 1Θ. Thus, one obtains eµ2t||x(t)|| ≤ µ1||x0||exp ε 2µ2 1Θ+2 ε α µ2 2 eµ2texp ε 2µ2 1Θ, for all 0 ≤t≤n, n ≥n0. Then, it follows that ||x(t)|| ≤ µ1exp ε 2µ2 1Θe−µ2t||x0||+2 ε α µ2 2 exp ε 2µ2 1Θ, for all 0 ≤t≤n, n ≥n0. Letting ε→1, it yields that ||x(t)|| ≤ µ1exp µ2 1Θ 2e−µ2t||x0||+2α µ2 2 exp µ2 1Θ 2,a.s. 17 Finally, the ball Brwith r=2α µ2 2 exp µ2 1Θ 2, is almost sure globally uniformly exponentially stable, that is the stochastic perturbed bilinear system (3.3) is almost sure globally practically uniformly exponentially stable. 2 Example 3.6. Let consider the following stochastic perturbed bilinear system: dx(t) = (Ax(t) + uBx(t))dt +G(t, x(t))dBt,(3.12) where x= (x1, x2)∈R2. A=2 0 0−1,B=−1 0 0 0 ,G(t, x) = G1(t, x) G2(t, x), with (G1(t, x) = sin(x2)e−ςt G2(t, x) = e−ςt, ς > 0. The stochastic system (3.12) can be regarded as a bilinear perturbed system of: dx(t) = (Ax(t) + uBx(t))dt. (3.13) The unperturbed nominal system is globally exponentially stabilizable by the constant feedback ¯u(x) = σ, σ > 2,since the closed–loop system dx(t)=(A+ ¯uB)xdt satisfies Reλ(e A)<0,as we can see in the following Fig.1, for σ= 3. Figure 1: Time evolution of the states x1(t) and x2(t) of the bilinear system (3.13) 18 On the other side, we have ||G(t, x)||2=G2 1(t, x) + G2 2(t, x). That is, we have ||G(t, x)||2= sin2(x2)e−2ςt +e−2ςt ≤2e−2ςt. It is clear, that assumption (A3)is satisfied with ϕ(t) = √2e−ςt. Hence, all conditions of Theorem (3.1) are satisfied. Thus, the stochastic perturbed bilinear system (3.12) is almost sure globally practically uniformly exponentially stable, as we can see Fig.2and Fig.3, for ς= 1. Figure 2: Time evolution of the state x1(t) of the stochastic perturbed bilinear system (3.12), with five different Brownian motions 19 Figure 3: Time evolution of the state x2(t) of the the stochastic perturbed bilinear system (3.12), with five different Brownian motions 4 Stabilization within a bilinear approximation In this section, we prove the almost sure local uniform exponential stability to a small ball for stochastic perturbed affine system. We will investigate the asymptotic behavior of the solutions in the sense that the trajectories converge to a small ball centered at the origin Br(0, r), r > 0 small enough in such away B(0, r)⊂B(0, η) and for all solutions starting from B(0, η)\B(0, r), will approach exponentially to B(0, r) for tlarge enough. We consider the stochastic perturbed affine system in closed–loop with the constant feedback ¯ufor tlarge enough. Let’s consider the following affine system: dx = (f(x) + ug(x)) dt, (4.1) where x∈Rnis the state vector, u∈Rnis the control input vector, f, g are two smooth functions defined on Rn,with f(0) = 0, g(0) = 0. Assume that some parameters are excited or perturbed by Brownian motion, and the perturbed stochastic system system is given by the following form: dx(t) = (f(x) + ug(x)) dt +G(t, x(t))dBt.(4.2) 20 where G:R+×Rn−→ Rn×m, Bt= (B1(t), ..., Bm(t))Tis an m-dimensional Brownian motion defined on a complete probability space (Ω,F,P).We assume that there exist tsuch that G(t, 0) 6= 0. The associated closed–loop stochastic system with a constant feedback u= ¯uis expressed by the following: dx(t) = (f(x) + ¯ug(x)) dt +G(t, x(t))dBt.(4.3) Since fand gare smooth functions, locally on a certain neighborhood of the origin V(0), then we can write f(x) and g(x) as the following: f(x) = Ax+χ1(x),and g(x) = Bx+χ2(x), with lim x→0||χ1(x)|| ||x|| = 0,lim x→0||χ2(x)|| ||x|| = 0. Then, the closed–loop stochastic system (4.3) can be written as the following: dx(t) = (Ax+ ¯uBx)dt + (χ1(x) + ¯uχ2(x))dt +G(t, x(t))dBt,∀x∈ V(0).(4.4) Let χ(x) = χ1(x) + ¯uχ2(x),then the stochastic system (4.4) can be regarded as follow: dx = (Ax+ ¯uBx+χ(x)) dt +G(t, x(t))dBt.(4.5) (A4) There exist two continuous positive functions ϕ1(t) and ϕ2(t), such that ||G(t, x)||2≤e−µ2t(ϕ1(t)||x||+ϕ2(t)) ,∀x∈Rn, t ≥0, where ϕ1and ϕ2satisfy, the following condition: Z+∞ 0 ϕ1(s)ds = ¯ϕ1<+∞,(4.6) and Z+∞ 0 eµ2sϕ2(s)ds = ¯ϕ2<+∞.(4.7) Theorem 4.1. Under assumptions (A1)and (A4)the stochastic perturbed affine system (4.3) is almost surely practically uniformly exponentially stable. Proof. The solution of the stochastic system (4.5) with initial condition x0is given by the following: x(t) = et¯ Ax0+Zt 0 e(t−s)¯ Aχ(x(s))ds +Zt 0 e(t−s)¯ AG(s, x(s))dBs.(4.8) 21 Similar to the proof of Theorem (3.1), under assumption (A1) and Lemma 2.3, we obtain ||x(t)|| ≤ µ1||x0||e−µ2t+Zt 0 µ1e−µ2(t−s)χ(x(s))ds +ε 2Zt 0 µ2 1e−2µ2(t−s)||G(s, x(s))||2ds +2 ε ln(n) n, for all 0 ≤t≤n, n ≥n0. Since, lim x→0||χ(x)|| ||x|| = 0, then for a given constant δ > 0, there exists η0>0,such that ∀x∈ B(0, η0)⊂B(0, η),for all t≥0,one obtains ||χ(x)|| ≤ δ||x||. For δ=θµ2 µ1 , it yields that ||χ(x)|| ≤ θµ2 µ1 ,0< θ < 1. Taking into account assumption (A4), we have ||x(t)|| ≤ µ1||x0||e−µ2t+µ1Zt 0 e−µ2(t−s)θµ2 µ1||x(s)||ds +ε 2Zt 0 µ2 1e−2µ2(t−s) ×e−µ2s(ϕ1(s)||x(s)||+ϕ2(s)) ds +2 ε ln(n) n =µ1||x0||e−µ2t+θµ2e−µ2tZt 0 eµ2s||x(s)||ds +ε 2µ2 1e−2µ2tZt 0 eµ2s(ϕ1(s)||x(s)||+ϕ2(s)) ds +2 ε ln(n) n, for all 0 ≤t≤n, n ≥n0. Multiplying both sides by eµ2t,one obtains eµ2t||x(t)|| ≤ µ1||x0||+θµ2Zt 0 eµ2s||x(s)||ds +ε 2µ2 1e−µ2tZt 0 eµ2s(ϕ1(s)||x(s)||+ϕ2(s)) ds +2 ε ln(n) neµ2t ≤µ1||x0||+θµ2Zt 0 eµ2s||x(s)||ds +ε 2µ2 1Zt 0 eµ2s(ϕ1(s)||x(s)||+ϕ2(s)) ds +2 ε ln(n) neµ2t, for all 0 ≤t≤n, n ≥n0. Let e V(t) = eµ2t||x(t)||, we see that e V(t)≤µ1||x0||+θµ2Zt 0e V(s)ds +ε 2µ2 1Zt 0 ϕ1(s)e V(s)ds +ε 2µ2 1Zt 0 eµ2sϕ2(s)ds +2 ε ln(n) neµ2t, 22 for all 0 ≤t≤n, n ≥n0. Using (2.6), we obtain ¯ V(t)≤µ1||x0||+Zt 0θµ2+ε 2µ2 1ϕ1(s)e V(s)ds +ε 2µ2 1Zt 0 eµ2sϕ2(s)ds +2 εαeµ2t ≤µ1||x0||+Zt 0θµ2+ε 2µ2 1ϕ1(s)e V(s)ds +ε 2µ2 1Z+∞ 0 eµ2sϕ2(s)ds +2 εαeµ2t, for all 0 ≤t≤n, n ≥n0. Now, using condition (4.7), it yields that V(t)≤µ1||x0||+ε 2µ2 1¯ϕ2+Zt 0θµ2+ε 2µ2 1ϕ1(s)e V(s)ds +2 ε α µ2Zt 0 eµ2sds, for all 0 ≤t≤n, n ≥n0. Applying, the Gronwall lemma 3.5, one obtains ¯ V(t)≤µ1||x0||+ε 2µ2 1¯ϕ2exp Zt 0 θµ2+ε 2µ2 1ϕ1(s)ds +Zt 0 2 ε α µ2 eµ2sexp Zt s θµ2+ 2µ2 1ϕ1(τ)dτds, for all 0 ≤t≤n, n ≥n0. Thus, we have ¯ V(t)≤µ1||x0||+ε 2µ2 1¯ϕ2exp(θµ2t) exp  2µ2 1Z+∞ 0 ϕ1(s)ds +Zt 0 2 ε α µ2 eµ2se(t−s)θµ2exp ε 2µ2 1¯ϕ1ds, for all 0 ≤t≤n, n ≥n0. Based on condition (4.6), it yields that ¯ V(t)≤µ1||x0||+ε 2µ2 1¯ϕ2exp(θµ2t) exp  2µ2 1¯ϕ1 +2 ε α µ2 exp ε 2µ2 1¯ϕ1eθµ2tZt 0 eµ2(1−θ)sds ≤µ1||x0||+ε 2µ2 1¯ϕ2exp(θµ2t) exp  2µ2 1¯ϕ1 +2 ε α µ2 exp ε 2µ2 1¯ϕ11 µ2(1 −θ)eθµ2teµ2(1−θ)t, 23 for all 0 ≤t≤n, n ≥n0. That is, one obtains ¯ V(t)≤µ1||x0||+ε 2µ2 1¯ϕ2exp(θµ2t) exp  2µ2 1¯ϕ1 +2 ε α µ2 exp ε 2µ2 1¯ϕ11 µ2(1 −θ)eµ2t, for all 0 ≤t≤n, n ≥n0. As a consequence, we deduce ||x(t)|| ≤ µ1exp  2µ2 1¯ϕ1||x0||e−µ2(1−θ)t+ε 2µ2 1¯ϕ2exp  2µ2 1¯ϕ1e−µ2(1−θ)t +ε 2 α µ2 1 µ2(1 −θ)exp 2 εµ2 1¯ϕ1, for all 0 ≤t≤n, n ≥n0. That is, ||x(t)|| ≤ µ1exp  2µ2 1¯ϕ1||x0||e−µ2(1−θ)t+ε 2µ2 1¯ϕ2exp  2µ2 1¯ϕ1+2 ε α µ2 1 µ2(1 −θ)exp  2µ2 1¯ϕ1, for all 0 ≤t≤n, n ≥n0. Letting, ε→1, it follows that ||x(t)|| ≤ µ1exp 1 2µ2 1¯ϕ1||x0||e−µ2(1−θ)t+1 2µ2 1¯ϕ2exp 1 2µ2 1¯ϕ1+2α µ2 2(1 −θ)exp 1 2µ2 1¯ϕ1,a.s. Finally, the ball Brwith r=1 2µ2 1¯ϕ2exp 1 2µ2 1¯ϕ1+2α µ2 2(1 −θ)exp 1 2µ2 1¯ϕ1, is almost sure uniformly exponentially stable, thus the stochastic perturbed affine system (4.3) is almost surely practically uniformly exponentially stable. 2 Example 4.2. Let consider the following stochastic perturbed affine system:      dx1(t)=(x1cos(x1)−usin(x1)) dt +e−t/2 √ch(t)p|x1|+e−ϑ/2tdBt dx2(t)=(−x1−sin(x2) cos(x1)) dt, ϑ > 2. (4.9) 24 The previous system can be written as: dx(t)=(f(x(t)) + ug(x(t)))dt +G(t, x(t))dBt,(4.10) with x= (x1, x2)∈R2is the state of the system, u∈Ris the input, f(x) =    x1cos(x1) −x1−sin(x2) cos(x1). , g(x) =    −2 sin(x1) 0. ,G(t, x) =        e−t/2 √ch(t)p|x1|+e−ϑ/2t 0. The stochastic system (4.10) can be regarded as a perturbed affine system of the following: dx(t)=(f(x) + ug(x))dt. (4.11) The bilinear approximation for the associated nominal system of (4.11) is the following: dx(t)=(Ax+uBx)dt, (4.12) where A=1 0 −1−1,B=−1 0 0 0 . Notice that, the previous approximation can be done in a small neighborhood of the origin. It means that there exists η > 0,such that for all x∈B(0, η)the passage from affine system to bilinear system is possible. The objective is to seek a constant r > 0small enough in such away B(0, r)⊂B(0, η)and all solutions starting from B(0, η)\B(0, r),will approach exponentially to B(0, r)for tlarge enough. The unperturbed nominal bilinear system (4.12) is globally uniformly exponentially stabilizable by the constant feedback ¯u(x) = u0, u0>1,since the closed–loop system dx(t)=(A+ ¯uB)xdt satisfies Reλ(e A), as we can see in Fig.4(for u0= 2). That is, one has the following estimation: ||et(A+¯uB)|| ≤ µ1e−µ2t. 25