# On the Asymptotical and Practical Stability of Stochastic Control Systems

Authors: Fakhreddin Abedi; Wah June Leong; Sarkhosh Seddighi Chaharborj
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The asymptotical and practical stability in probability of
stochastic control systems by means of feedback laws is provided.
The main results of this work enable us to derive the sufficient
conditions for the existence of control Lyapunov function that
play a leading role in the existence of stabilizing feedback laws.
Particularly, the sufficient conditions for practical stability in
probability are established and numerical examples are also given
to illustrate the usefulness of our results.


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## Body

## 1. Introduction

The stabilization of various types of linear and nonlinear systems has been widely studied in the past years (see, for instance, Karafyllis and Tsinias [1], Phat et al. [2], Thuan et al. [3], Bay et al. [4], and Trinh and Fernando [5]). In these papers, the authors derived Lyapunov-Krasovskii functions and established the necessary and sufficient conditions for robust global asymptotic stability and robust H∞ stability at the equilibrium state of linear and nonlinear systems.Stabilization of stochastic control systems (SCSs) by means of state feedback laws is important in the control theory. The stochastic version of the Lyapunov theorem has been used to derive necessary and sufficient conditions for stabilization of SCSs at their equilibrium state. In recent years, the stabilizability of various types of SCSs has been studied for different concepts of stochastic stability (see, for instance, [6–13]).Florchinger [6] established the necessary and sufficient conditions for the asymptotic stability in probability of the SCSs at their equilibrium state. Under these conditions, Deng and Krstić [7] designed an inverse optimal control law for strict-feedback systems which guarantees global asymptotic stability in probability (GASP). Moreover, Deng et al. [8] also developed the notion of uniform in time GASP to the problem of feedback stabilization for a class of SCSs. On the other hand, the concept of design and analysis of controller for SCS has been introduced by Xie and Tian [9], Pan and Başar [10], Lin et al. [11], and Abedi et al. [13]. Necessary and sufficient conditions are derived by Abedi et al. [12] for non-uniform in time GASP for a class of SCSs.Thus, our aim in this paper is to explore further the asymptotical stabilization in probability problem for a larger class of SCSs than that described in [6, 13]. This class of SCSs can be characterized in terms of computable control Lyapunov functions (CLFs) which depend on the system's coefficients. In addition, the paper is also intended to fill the gap of the previous works by establishing sufficient conditions for practical stability in probability for this broader class of SCSs. Our main result, Theorem 14, which is an extension of Theorem 2.1 established in Tsinias [14], asserts that this broader class of SCSs is asymptotically stable in probability (ASP), if it admits a CLF at the origin. In addition, we obtain a computable stabilizing feedback law under Propositions 15 and 16 and Theorems 18 and 19 that are extension of Propositions 1.5 and 2.3 and Theorems 2.4 and 2.6, respectively, proved by Tsinias [14] for deterministic control systems to the larger class of SCSs driven by a Wiener process. For some backgrounds related to this paper, readers may refer to the paper by Florchinger [6] that gives necessary and sufficient conditions for ASP of a special case of our SCSs. Both of the results and the methods used in this paper, however, are different from those in the references. The main tools used in this paper are, indeed, the stochastic versions of converse Lyapunov theorems established in Kushner [15].The paper is organized as follows. In Section2, we introduce the class of stochastic systems, some basic definitions, and results that we are dealing with in this paper. Section 3 describes a broader class of SCSs and is focused on the properties of stochastic CLF which play an important role in the stabilization theory. In Section 4, we state and prove the main results of the paper on the asymptotical and practical stabilization of the larger class of SCSs. Finally in Section 5 we provide some numerical examples to validate our results.


## 2. Stochastic Stability

In the following we introduce the class of stochastic systems and recall some definitions of ASP and GASP that we are dealing with in the rest of the paper.A detailed exposition on the subject can be found in the books of Speyer and Chung [16], Has’minskiĭ [17], and also the paper by Abedi et al. [13].Let(Ω,F,P) be a complete probability space, and denote by (wt)t≥0 a standard Rm-valued Wiener process defined on this space.Consider the stochastic process solutionxt∈Rn of the Ito stochastic differential system as
(1)xt=x0+∫0tf(xs)ds+∑k=1m∫0thk(xs)dwsk,where(i)
x0∈Rn is given,(ii)
f:Rn→Rn and hk:Rn→Rn×m,1≤k≤m, are locally Lipschitz functionals mapping with f(0)=0,hk(0)=0 and there exists a constant K≥0 such that for any x∈Rn, the following linear growth condition holds:
(2)|f(x)|+∑k=1m‍|hk(x)|≤K(1+|x|).Notations. Throughout this paper we adopt the following notation.(i)
For anys≥0 and x∈Rn,  xts,x, where s≤t, denotes the solution of the Ito stochastic differential system (1) starting from the state at time s.(ii)
∥·∥ is the usual Euclidean norm.(iii)
For anyx∈Rn, xT denotes its transpose.Definition 1.
The equilibriumxt=0 of the system (1) is(i)
stable in probability, if for everys≥0 and ε>0,
(3)limx→0P{sups≤t|xts,x|>ε}=0,(ii)
ASP, if it is stable in probability and(4)limx→0P{limt→+∞|xts,x|=0}=1.Definition 2.
A functionγ:R+→R+ is a(i)
K-function if it is continuous, strictly increasing, and γ(0)=0,(ii)
K∞-function if it is a K-function and γ(r)→∞ as r→∞,(iii)
positive definite function ifγ(r)>0 for all r>0, and γ(0)=0.Definition 3.
The equilibriumxt=0 of the system (1) is(i)
globally stable in probability, if for anyε>0, there exists a class K-function γ(·) such that
(5)P{|xt|<γ(|x0|)}≥1-ε,∀t≥0,∀x0∈Rn∖{0},(ii)
GASP, if it is globally stable in probability and(6)P{limt→∞|xt|=0}=1,∀x0∈Rn.


## 3. Preliminary Results

In this section, we introduce a broader class of SCSs and focus on the properties of CLF and control Lyapunov family which play an important role in asymptotical and practical stability in probability, respectively, in Section4. A detailed exposition on the subject can be found in the paper of Abedi et al. [13].Let(Ω,F,P) be a complete probability space, and denote by (wt)t≥0 a standard Rm-valued Wiener process defined on this space.Consider the following stochastic process solutionxt∈Rn of the SCS:
(7)xt=x0+∫0t‍(f(xs)+∑z=1p‍gz(xs)uz)ds+∑k=1m‍∫0t‍(hk(xs)+∑z=1p‍qk,z(xs)uz)dwsk,
where(i)
x0∈Rn is given,(ii)
u is an Rp-valued control law,(iii)
f and hk are functionals mapping satisfying the hypothesis given in Section 2,(iv)
gz:Rn→Rn×p,1≤z≤p, and qk,z:Rn→Rn×m×p,1≤k≤m,1≤z≤p, are C∞ functionals mapping with gz(0)=0,qk,z(0)=0 and there exists a constant K≥0 such that for any x∈Rn, the following linear growth condition holds:
(8)∑z=1p‍(|gz(x)|+∑k=1m‍|qk,z(x)|)≤K(1+|x|).Then SCS (7) is said to be ASP at the origin if there exists a neighborhood D of the origin in Rn and a function u=k(xs) mapping D into Rp, vanishing at the origin, such that(i)
for everyx∈D, the solution xt of the resulting closed-loop system
(9)x=x0+∫0t‍(f(xs)+∑z=1p‍gz(xs)k(xs)z)ds+∑k=1m‍∫0t‍(hk(xs)+∑z=1p‍qk,z(xs)k(xs)z)dwskis uniquely defined,(ii)
the equilibrium solutionxt≡0 of the closed-loop system (9) is ASP.Moreover, SCS (7) is said to be practically stabilizable in probability at the origin by means of the family of feedback laws {kr,r>0}, if for any sufficiently small r and for any x0 near the open sphere S(0,r) of radius r around origin, the corresponding trajectory xr(t,x0) of the resulting closed-loop system
(10)x=x0+∫0t‍(f(xs)+∑z=1p‍gz(xs)kr(xs)z)ds+∑k=1m‍∫0t‍(hk(xs)+∑z=1p‍qk,z(xs)kr(xs)z)dwskentersS(0,r) after some time t=T<+∞ and it stays in this region thereafter. The concept of practical stability is first introduced by La Salle and Lefschetz [18].Denote byD the infinitesimal generator of the stochastic process solution of the uncontrolled part of SCS (7); that is, D is the second order differential operator defined for any function Φ in C2(Rn,R) by
(11)DΦ(x)=∑i=1n‍fi(x)∂Φ(x)∂xi+12∑i,j=1n‍∑k=1m‍hkihkj∂2Φ(x)∂xi∂xj.For anyz∈(1,…,p), let Dz be the second order differential operator defined for any function Φ in C2(Rn,R) and is given by
(12)DzΦ(x)=∑i=1n‍gzi(x)∂Φ(x)∂xi+∑i,j=1n‍∑k=1m‍hkiqk,zj∂2Φ(x)∂xi∂xj,and, for any z,r∈(1,…,p), Dz,r denotes the second order differential operator defined for any function Φ in C2(Rn,R) by
(13)Dz,rΦ(x)=12∑i,j=1n‍∑k=1m‍qk,zi(x)qk,rj(x)∂2Φ(x)∂xi∂xj.We also denoteD0 as the infinitesimal generator for the stochastic process solution of the closed-loop system (9); that is, D0 is the differential operator defined for any function Φ in C2(Rn,R) given by
(14)D0Φ(x)=DΦ(x)+∑z=1p‍DzΦ(x)k(x)z+∑z,r=1p‍DzΦ(x)Dz,rΦ(x)(k(x)z)2.Using these notations, we recall some notions of CLF as follows.Definition 4.
The SCS (7) is said to satisfy a stochastic Lyapunov condition at the origin if there exists a neighborhood D of the origin in Rn and a C2 positive definite function Φ:D→R+, and a positive definite function ρ, such that for all x∈D-{0} the following condition holds:
(15)DzΦ(x)=0,Dz,rΦ(x)=0⇒DΦ(x)≤-ρ(x)<0.Definition 5.
A continuously differentiable real functionΦ is called a CLF, if it is positive definite and satisfies condition (15).Definition 6.
The CLF “Φ” is said to satisfy the bounded control property, if there exists a positive real function c:D→R such that c is bounded on D and for every x∈D-{0} there exists a control u∈Rp satisfying the following inequalities:
(16)∥u∥<c(x),(17)DΦ(x)+∑z=1p‍DzΦ(x)uz+∑z,r=1p‍Dz,rΦ(x)(uz)2≤-ρ(x)<0.
If in additionlimx→0c(x)=0, then the Φ is said to satisfy the small control property.A useful tool to study the stabilizability of SCS is the stochastic version of Artstein's theorem [19], established by Florchinger [6], and is given in the following theorem.Theorem 7.
(i) The SCS (7) satisfies a stochastic Lyapunov condition at the origin, if and only if it is asymptotically stabilizable by means of a feedback law u=k(x), which is smooth in a neighborhood of the origin except possibly at x=0.
(ii) The corresponding CLF,Φ, satisfies the bounded control property, if and only if there exists a stabilizer u=k(x), which is smooth in a neighborhood D of the origin, except possibly at zero and satisfies
(18)∥k(x)∥<c(x),
where c is defined in (16). This implies that the function k is bounded in a neighborhood of the origin and if, in addition, Φ satisfies the small control property then k(x)→0 as x→0.In the following we introduce the stochastic version of Florchinger control law [20] established by Abedi et al. [12], which gives the property of GASP for the resulting closed-loop system (9).Theorem 8.
Consider the SCS (7) and its corresponding closed-loop system (7). Let Φ be a CLF associated with the system, and for any x∈Rn, denote by b(x) the function defined by
(19)b(x)=DzΦ(x).
Then the feedback law(20)k(x)=-b(x)Ψ(x),
where b(x) is given by (19) and
(21)Ψ(x)=1+(sup1≤z,r≤pDz,rΦ(x))2,guarantees that the resulting closed-loop system (9) is GASP.Under a slight change of assumption, that is on the property of small control family, introduced by Tsinias [14] we give the definitions on small control family, in the notion of stochastic, as follows.Definition 9.
The SCS (7) is said to satisfy a stochastic practical Lyapunov condition at the origin if there exist compact neighborhoods D and Dr of the origin in Rn such that S(0,r)⊂Dr, a family of C2 positive definite function Φr:Rn→R, where r>0, and a positive definite function ρ, such that for any sufficiently small r, condition (15) holds with Φ=Φr, and
(22)min{Φr(x),x∈∂Dr}>max{Φr(x),x∈S(0,r)},
where ∂D is the boundary of D. Moreover, for any sequences {ri}⊂R+ and {xri}⊂D with limri=0 and limxri=a as i→+∞, we have
(23)lim¯Φri(xri)=0fora=0,lim_Φri(xri)>0fora>0.Definition 10.
A continuously differentiable real functionΦr is regarded as a member of control Lyapunov family, if it is positive definite and is satisfying the conditions given in Definition 9.Definition 11.
The control Lyapunov family{Φr} is said to satisfy the stochastic bounded control property, if there exists a positive real function c:D→R such that c is bounded on D, and a family of nonnegative real numbers mr, where r>0 such that mr→0 as r→0 and for any sufficiently small r, conditions (17), (22), and (23) hold with Φ=Φr,x∈D2r-D1r, and for some u∈R1 where
(24)∥u∥<c(x)+mr.
If, in addition,c(x)→0 as x→0, then the control Lyapunov family {Φr} is said to satisfy the stochastic small control property.The following theorem presents sufficient conditions for practical stabilization. The proof of this theorem is similar to the proof of Theorem7.Theorem 12.
(i) If the SCS (7) satisfies the stochastic practical Lyapunov condition, then it is practically stabilizable in probability at the origin by means of a family of smooth feedback laws u={kr(x)}.
(ii) If the control Lyapunov family{Φr} satisfies the stochastic bounded (small) control property, then for any r the corresponding stabilizer u satisfies the following inequality:
(25)∥kr(x)∥<c(x)+mr,
where c and mr are those given in Definition 11.Remark 13.
A direct result of Definitions4 and 9 is that if the SCS (7) satisfies the stochastic Lyapunov condition, it will also satisfy the stochastic practical Lyapunov condition.In the methodology used in this paper, we will consider the SCS (7) in the differential form as
(26)d(x1x2)=(f1(x1,x2)f2(x1,x2))dt+(0g2(x1,x2))udt+(h1(x1)h2(x1,x2))dwt+(0q2(x1,x2))udwt,
where x=(x1x2)∈Rn1×Rn2,n=n1+n2, f1,f2,g2,h1,h2, and q2 are C∞ functionals vanishing at the origin, and we will derive the sufficient conditions for the existence of CLF guaranteeing stabilization (Theorem 14) in the form of (26). We extend the similar decomposition used by Tsinias [14] in deterministic case to the stochastic case. Thus, the study of asymptotical and practical stability in probability at the equilibrium state of the SCS (7) can be replaced by the study of asymptotical and practical stability in probability at the equilibrium state of the SCS (26). Special emphasis is given to asymptotical and practical stabilization in probability for SCS (26), whose control terms g2 and q2 are constants; that is, the system has the form
(27)d(x1x2)=(f1(x1,x2)f2(x1,x2))dt+(0u)dt+(h1(x1)h2(x1,x2))dwt+(0u)dwt,
where x=(x1x2)∈Rn1×Rn2 and u is an Rn2-valued control law.


## 4. Main Results

The aim of this section is to derive sufficient conditions for the existence of CLFs that play a leading role in the existence of stabilizing feedback laws that are smooth, except possibly at the equilibrium state of the system (Theorem14). The formulas for the special case of SCS (7) with unity intensity noise and vanishing qk,z(x) without the term hk(x)dwwere given by Abedi et al. [13] and for the system (7) with affinity in the noise and control u, wherehk(x) may be nonvanishing at the origin, without the term qk,z(x)uzdwwere given by Deng et al. [8]. In [12], Abedi et al. derived the necessary and sufficient conditions for nonuniform in time GASP for the system (7) in the case where gz(x) and qk,z(x) may be nonvanishing at the origin. Here, we extend Propositions 1.5 and 2.3 and Theorems 2.4 and 2.6 of Tsinias [14] in deterministic case and obtain a computable stabilizing feedback law as Propositions 15 and 16 and Theorems 18 and 19, respectively, for a broader class of SCS (26).Consider the following lower-dimensional subsystems of the SCS (26):
(28)dx1=f1(x1,υ)dt+h1(x1)dwt,υ∈Rn2,(29)dx2=g2(x1,x2)udt+q2(x1,x2)udwt,u∈Rp.Then, we can present the following result, where the stochastic Lyapunov condition for the SCS (26) is characterized in terms of some appropriate positive functions for the SCS (28) and (29).Theorem 14.
Suppose that there exist neighborhoodsD and D1 of the origin in Rn and Rn1, respectively, and mappings s:D→Rp,ϕ:D1→Rn2, and W:D→R such that ϕ(0)=0 where ϕ is continuous, and W is continuously differentiable, and let us denote
(30)N={x∈D:x2=ϕ(x1)},N1={x∈D:∇αW(x)=0,α=0,1,2},(31)N2={x∈D:(∂W∂x2g2s)(x)=0,u(∂W∂x2q2s)(x)=0}.(i)
Assume that for anyx in a neighborhood of the origin in Rn,W(x)≥0 and N2⊂N1=N and the SCS (28) is ASP by means of the feedback law υ=ϕ(x1). Then, the SCS (26) satisfies a stochastic Lyapunov condition at the origin.(ii)
LetΦ be the corresponding CLF. If we further assume that s is bounded, ϕ and (∂W/∂x2) are continuously differentiable, and there exist positive constants d>0 and b≤1 such that
(32)(∂W∂x2g2s)(x)≤-d∥x2-ϕ(x1)∥b,for x in a neighborhood of the origin in Rn then the function Φ satisfies the small control property.Proof.
Part (i): since the equilibrium solutionx1≡0 of the closed-loop system
(33)dx1=f1(x1,ϕ(x1))dt+h1(x1)dwt
is asymptotically stable, then the converse Lyapunov theorem of Kushner [15] asserts that there exists a smooth Lyapunov function V:Rn1→R such that V(0)=0,V(x1)>0 and
(34)HV(x1)<0
forx1≠0 near zero, where H denotes the infinitesimal generator for the stochastic process solution of the closed-loop system (33). Obviously, by the assumption of the theorem, the function
(35)Φ(x)=V(x1)+W(x)
is positive definite, and for anyx∈(D×Rn2)∩D, we have
(36)GzΦ(x)=∑i=1n‍gzi(x)∂Φ(x)∂xi+∑i,j=1n‍∑k=1m‍hkiqk,zj∂2Φ(x)∂xi∂xj=g2∂W(x)∂x2+h1q2∂2W∂x1∂x2+h2q2∂2W(x)∂x22,Gz,rΦ(x)=12∑i,j=1n‍∑k=1m‍qk,ziqk,rj∂2Φ(x)∂xi∂xj=12q2(x)∂2W(x)∂x22.
For anyx≠0 such that (∂W(x)/∂x2)g2(x)=(∂W(x)/∂x2)q2(x)=0,it follows by the assumption that x2=ϕ(x1). Hence,
(37)∇W(x1,ϕ(x1))=0,∇2W(x1,ϕ(x1))=0,
and so(38)GΦ(x)=∑i=1n‍fi∂Φ(t,x)∂xi+12∑i,j=1n‍∑k=1m‍hkihkj∂2Φ(t,x)∂xi∂xj=f1∂V∂x1+f1∂W∂x1+f2∂W∂x2+12h12(∂2V∂x12+∂2W∂x12)+12h1h2∂2W∂x1∂x2+12h2h1∂2W∂x2∂x1+12h22∂2W∂x22=HV(x1)<0,
where G is the infinitesimal generator for the stochastic process solution of the uncontrolled part of the SCS (26). Therefore, the SCS (26) satisfies a stochastic Lyapunov condition at the origin, and the function Φ(x)=V(x1)+W(x) is the associated CLF.
Part (ii): let us assume thatΦ(x) is the corresponding CLF and that (32) holds, with b≤1. Define
(39)ρ(x)=-HV(x1,ϕ(x1))+∥x2-ϕ(x1)∥2.
Clearly,ρ is positive definite. Since f is continuously differentiable, W is twice differentiable, ∇W(x1,ϕ(x1))=0, and ∇2W(x1,ϕ(x1))=0, then there are positive constants d1,d2, and d3 such that
(40)∥∇f(x)∥≤d1,(41)∥∇W(x)∥≤d2∥x2-ϕ(x1)∥,(42)∥∇2W(x)∥≤d3∥x2-ϕ(x1)∥
forx in a neighborhood of the equilibrium. Let
(43)a(x)=d1∥∇V(x1)∥+d2∥f(x)∥+d32∥(hhT)(x)∥.
Then, by taking into account (32)–(42), we have
(44)|(GΦ+ρ)(x)|≤d1∥∇V(x1)∥∥x2-ϕ(x1)∥+d2∥f(x)∥∥x2-ϕ(x1)∥+d32∥(hhT)(x)∥∥x2-ϕ(x1)∥≤a(x)∥x2-ϕ(x1)∥≤-a(x)d(∂W∂x2g2s)(x),and it follows that
(45)|(GΦ+ρ)(x)|≤-sa(x)dGzΦ(x).
Therefore,(46)GΦ(x)+GzΦ(x)(sad)≤-ρ(x)<0,for all x≠0 near zero. Inequality (46) in conjunction with condition ∇2W(x1,ϕ(x1))=0 gives
(47)GΦ(x)+GzΦ(x)(sad)+Gz,rΦ(x)(sad)2≤-ρ(x)<0.
Sinces is bounded and a is continuous with a(0)=0, it follows that Φ satisfies the small control property.Proposition 15.
If the SCS (7) is asymptotically stabilizable in probability by means of the feedback u=k(x), which is continuous in a neighborhood of origin, then it is practically stabilizable in probability by means of a family of smooth feedback u={kr(x)}.Proof.
The proof of this proposition is a direct consequence of Theorems7(i) and 12 and Remark 13. Indeed, suppose that the SCS (7) is asymptotically stabilizable in probability, then according to Theorem 7(i) the SCS (7) satisfies the stochastic Lyapunov condition at the origin. Therefore, by Remark 13, the SCS (7) satisfies the stochastic practical Lyapunov condition at the origin. Hence, the desired practically stability in probability for the SCS (7) is assured by Theorem 12.In the following, we will customize Theorem14 for a particular case of SCS (27). Assume, without loss of generality that f2=0. In this case, the SCS (27) becomes
(48)d(x1x2)=(f1(x1,x2)0)dt+(0u)dt+(h1(x1)h2(x1,x2))dwt+(0u)dwt,
where x=(x1x2)∈Rn1×Rn2, u is an Rn2-valued control law, and g and q are constant function. The following proposition, which is an immediate consequence of Theorem 14 and is also an extension of Proposition 2.3 established in [14], provides a CLF for the SCS (48) that satisfies the small control property. In the proof of this proposition we use the stochastic version of converse Lyapunov theorem established in [15].Proposition 16.
Consider the SCS (48) and suppose that (28) is asymptotically stabilizable in probability by means of the feedback law v=ϕ(x1),ϕ(0)=0, which is continuously differentiable in a neighborhood of the origin. Then (48) satisfies the stochastic Lyapunov condition and the corresponding CLF satisfies the small control property.Proof.
Since the equilibrium solutionx1≡0 for the closed-loop system
(49)dx1=f1(x1,ϕ(x1))dt+h1(x1)dwt
is asymptotically stable, then the converse Lyapunov theorem of Kushner [15] asserts that there exists a smooth Lyapunov function V:Rn1→R defined in a neighborhood D of the origin in Rn1 such that
(50)HV(x1)<0
forx1≠0 near zero. Let
(51)W(x)=12∥x2-ϕ(x1)∥2.
Obviously,W is nonnegative definite and (∂W/∂x2) is continuously differentiable in a neighborhood of the origin. If we define the functional r on Rn by the following:
(52)ri(x)=-sgn(x2-ϕ(x1))i,
then
(53)(∂W∂x2g2r)(x)=(∂W∂x2r)(x)≤-∥x2-ϕ(x1)∥,
where r is uniformly bounded on Rn and such that W satisfies the properties in Theorem 14 (i). In particular, (32) holds with b=1. Therefore, the function Φ=V+W satisfies the small control property.As in deterministic case (see Tsinias [14]) we can easily establish the following elementary result that is a useful tool in proving some of our results (Theorem 18).Lemma 17.
Letϕ:Rn→Rm be a real mapping that satisfies the Lipschitz condition on a compact neighborhood U of the origin in Rn, and ϕ(0)=0. Then there exist a neighborhood U1 of the origin in Rn, a constant K>0, and a family of smooth mappings ϕϵ:U1→Rm,ϵ>0 such that for every sufficiently small ϵ the following conditions hold:
(54)∥ϕ(x)-ϕϵ(x)∥<ϵ,∀x∈U1-S(0,ϵ),(55)ϕϵ(0)=0,(56)∥∇ϕϵ(x)∥<K,∀x∈U1.
Moreover, for any sequence{ϵi}⊂R+ and {xi}⊂U1 such that ϵi→0 and xi→x∈U1, as i→+∞, there exist subsequences {ϵi′} and {xi′} of {ϵi} and {xi}, respectively, and a positive constant 0≤α≤1 satisfying
(57)ϕϵi′(x1′)→αϕ(x).The following theorem is an extension of Theorem 2.4 established in [14] and provides a control Lyapunov family for the SCS (48) that depends directly on the dynamics of the SCS (33). The proof of this theorem is the stochastic analogue of the deterministic case of Tsinias [14] where the author used the converse Lyapunov theorem given in Massera [21, 22] to establish his result. In the proof of the following theorem we use the stochastic version of converse Lyapunov theorem established in Kushner [15].Theorem 18.
Suppose that the SCS (33) is asymptotically stabilizable in probability by means of the feedback law v=ϕ(x1) with ϕ(0)=0.(i)
Ifϕ is continuous at the origin, then SCS (48) satisfies the practical Lyapunov condition.(ii)
Ifϕ is Lipschitz continuous, then the SCS (48) satisfies the practical Lyapunov condition and, additionally, the corresponding control Lyapunov family satisfies the stochastic small control property.Proof.
We prove only statement (ii) of this theorem. The proof of the first part of the theorem can be obtained using the similar argument and is therefore omitted.
Suppose that the SCS (33) is stabilizable in probability by means of the feedback law v=ϕ(x1) and ϕ(0)=0. Then the converse Lyapunov theorem proved by Kushner [15] asserts that there exists a smooth Lyapunov function V defined on a compact neighborhood U of Rn1 and a positive definite and strictly increasing continuous function γ:R+→R+ such that γ(0)=0 and
(58)HV(x1,ϕ(x1))≤-γ(∥x1∥)
for anyx1∈U. Let K>0 and U1⊂U as defined in Lemma 17, where ϕ is the above Lipschitzian stabilizer that is defined on U and it takes values on Rn2. Consider the positive constants c1 and c2 such that ∥ϕ(x1)∥<c1 for x1∈U1 and ∥∇f1(x)∥<c2 for every x belonging to
(59)D={x∈Rn:x1∈U1,∥x2∥<c1}.
For anyr>0, such that the sphere S(0,r) is contained to D, consider a positive l=l(r)<1 where
(60)Dl={x∈Rn:∥x1∥<l,∥x2∥≤2Kl},
is contained to S(0,r). Since f(0)=0 and h(0)=0, there exist nonnegative constants c3 and c4 such that
(61)∥f(x)∥≤c3,∥hhT∥≤c4,
for all x in a neighborhood of the origin in Rn. Let ϵ=ϵ(r) with ϵ<l and
(62)ϵ<min{(c3-1∥∇V∥-1+2c4-1∥∇2V∥-1)×γ(∥x1∥):x1∈U1,∥x1∥≥l(c3-1∥∇V∥-1+2c4-1∥∇2V∥-1)}.
Then, by (54)–(56) and (58)–(61), we have for any x1∈U1 and ∥x1∥>l(63)HV(x1,ϕϵ(x1))≤|HV(x1,ϕϵ(x1))-HV(x1,ϕ(x1))|+HV(x1,ϕ(x1))≤(c3∥∇V∥+12c4∥∇2V∥)∥ϕ(x1)-ϕϵ(x1)∥-γ(∥x1∥)
Now, forϵ=ϵ(r), we can define
(64)Φr(x)=V(x1)+12∥x2-ϕϵ(x1)∥2.
Obviously,Φr(x) is continuously differentiable and positive definite for x∈D. Let βr:D→Rn2 be a real mapping where
(65)βr(x)=45(h1∂ϕϵ(x1)∂x1-h2-(x2-ϕϵ(x1))).
DenotingD0 as the infinitesimal generator of the stochastic process, the solution of the resulting closed-loop system is deduced from the SCS (48) with the function βr(x) given by (65). This yields
(66)D0Φ(x)=DΦ(x)+∑z=1p‍DzΦ(x)βr(x)+∑z,r=1p‍DzΦ(x)Dz,rΦ(x)(βr(x))2=f1[∂V(x1)∂x1+∂ϕϵ(x1)∂x1(x2-ϕϵ(x1))]+45[h1∂ϕϵ(x1)∂x1-h2-(x2-ϕϵ(x1))]2+825[h1∂ϕϵ(x1)∂x1-h2-(x2-ϕϵ(x1))]3<0,
for every x∈D-Dl. Note that both x∈D-Dl and ∥x1∥<l give ∥x2∥>2Kl and by (56) that ∥ϕϵ(x1)∥<Kl. Therefore,(67)∥x2-ϕϵ(x1)∥≥∥x2∥-K∥x1∥>Kl≠0.
Sincef1,hi, and ϕϵ are continuously differentiable, where f1(0)=hi(0)=ϕϵ(0)=0, and because of (54) and (56), there is a positive constant M which is independent of r, satisfying
(68)∥βr(x)∥<M∥x∥+mr,
for every x∈D. This implies that for every sufficiently small r, condition (22) holds. In fact, there exist sequences {xri}⊂∂D and {zri}⊂S(0,ri) such that xri→x=(x1,x2)∈∂D and Φri(xri)≤Φri(zri). Since ϵ(r)→0, it follows from (57) that there exist subsequences {xri′}⊂{xri} and {zri′}⊂{zri}, and real constants α1 and α2 satisfying
(69)V(x1)+12∥x2-α1ϕ(x1)∥2=limΦri(xri′)≤limΦri(zri′)=V(0)+12α22∥ϕ(0)∥2=0,
and so x1=x2=0, which indicates a contradiction. Similarly, we can show (23). It turns out that {Φr(x)} as defined in (64) is a control Lyapunov family which satisfies the stochastic small control property.We will summarize our results in the following theorem.Theorem 19.
Suppose that the SCS (33) is asymptotically stabilizable in probability by means of the feedback law v=ϕ(x1) with ϕ(0)=0.(i)
Ifϕ is continuous at the origin, then the SCS (48) is practically stabilizable in probability at the origin.(ii)
Ifϕ is Lipschitz continuous, then then SCS (48) is practically stabilizable in probability at the origin, where the corresponding control Lyapunov family satisfies the stochastic small control property.(iii)
Ifϕ is continuously differentiable, then the SCS (48) is asymptotically stabilizable in probability at the origin, by means of the feedback law, which is smooth for x≠0 near the origin and continuous at the origin.


## 5. Applications

In this section, we illustrate our results by giving three numerical examples.Example 20.
Denote byx(t)∈R3 the solution of the following SCS:
(70)dx=(-3x1-5x2-x3)dt+(00x1x2)udt+(-2x1-3x2-x3)dwt+(001)udwt,
where (wt)t≥0 is a standard real-valued Wiener process and u is a real-valued measurable control law.
Consider the following Lyapunov function candidate:(71)Φ(x)=12(x12+x22+x32).
After some simple calculations, it gives(72)DzΦ(x)=x1x2x3-x3,DΦ(x)=-(x12+x22+x32),Dz,rΦ(x)=12.
From (72), we get
(73)D0Φ(x)=-(x12+x22+x32)-45(x1x2x3-x3)2+825(x1x2x3-x3)3≤-Φ(x).
Therefore, condition (15) is satisfied with ρ(s)=s, and, thus, the function Φ(x) is a CLF for the SCS (70). By Theorems 7 and 14(i), the SCS (70) is ASP by means of a feedback law which is smooth in a neighborhood of the origin in R2, except possibly at the origin.Finally, we may invoke Theorem8 to find an explicit formula for a feedback law. Indeed, by (20) and (72) we can obtain the following feedback law:
(74)k(x)=-45(x1x2x3-x3)
that guarantees the equilibrium solution of the resulting closed-loop system is deduced from the SCS (70) which satisfies the ASP property.Example 21.
Letx(t)∈R2 be the solution of the SCS as
(75)d(x1x2)=(-x1+x2ϕ1(x1,x2))dt+(0x2)udt+(x1x2)dwt+(0x2+ϕ2(x1))udwt,
where (wt)t≥0 is a standard real-valued Wiener process, u is a real-valued measurable control law, and ϕ2(x1) is a smooth functional mapping from R2 into R such that
(76)x1ϕ2(x1)>0,
for any x1∈R,x1≠0. Obviously, the SCS (75) has the form (26), and the function Φ, defined on Rn as
(77)Φ(x1,x2)=12x12+14(x2+ϕ2(x1))4,
is a CLF for SCS (75). Indeed, for any x≠0 with
(78)GzΦ(x)=∑i=1n‍gzi(x)∂Φ(x)∂xi+∑i,j=1n‍∑k=1m‍hkiqk,zj∂2Φ(x)∂xi∂xj=g2∂W(x)∂x2+h1q2∂2W∂x1∂x2+h2q2∂2W(x)∂x22=(x2+3ϕ2′(x1)+3x2)(x2+ϕ2(x1))3=0,(79)Gz,rΦ(x)=12∑i,j=1n‍∑k=1m‍qk,ziqk,rj∂2Φ(x)∂xi∂xj=12q2(x)∂2W(x)∂x22=32(x2+ϕ2(x1))3=0,
it follows that x2=-ϕ2(x1), and therefore
(80)GΦ(x)=∑i=1n‍fi∂Φ(t,x)∂xi+12∑i,j=1n‍∑k=1m‍hkihkj∂2Φ(t,x)∂xi∂xj=f1∂V∂x1+f1∂W∂x1+f2∂W∂x2+12h12(∂2V∂x12+∂2W∂x12)+12h1h2∂2W∂x1∂x2+12h2h1∂2W∂x2∂x1+12h22∂2W∂x22=-12x12+x1x2=-12x12-x1ϕ2(x1).
Using conditionx1ϕ2(x1)>0, the latter equality implies GΦ(x)<0. Thus, the function Φ is a CLF for SCS (75). In fact, the feedback law ν=-ϕ2(x1) asymptotically stabilizes
(81)dx1=(-x1+ν)dt+x1dwt,at zero. Therefore, by Theorems 7 and 14(i), system (75) is ASP by means of a feedback law which is smooth in a neighborhood of the origin in R2, except possibly at the origin.Example 22.
Consider the following SCS:(82)dx=(x1+x230)dt+(01)udt+(2x1x2)dwt,
where (wt)t≥0 is a standard real-valued Wiener process and u is a real-valued measurable control law. Obviously, the SCS (82) has the form of (48), and the stochastic system
(83)dx1=(x1+v3)dt+2x1dwt
is stabilized by the continuous lawv=-(2x1)1/3. Therefore, according to Theorem 19(i), the SCS (82) is practically stabilizable in probability at the origin by means of a family of smooth feedback laws.


## 6. Conclusion

In this paper, we have studied the problem of asymptotical and practical stabilization in probability of SCS when both drift and diffusion terms are affine in the control. We have used stochastic version of Artstein theorem and extended the asymptotical and practical stabilization results proved by Tsinias [14] for a larger class of SCSs driven by Wiener process. Moreover, the sufficient conditions for the existence of CLF and control Lyapunov family that play a leading role in the existence of stabilizing feedback laws are derived. Several numerical examples are given to validate our results. Finally, we can make the following summaries on our main results.(1)
The stability results provided in this paper are an extension of deterministic results stated in Tsinias [14]. Indeed, we obtained a computable stabilizing feedback law under Propositions 15 and 16 and Theorems 18 and 19 that are extension of Propositions 1.5 and 2.3 and Theorems 2.4 and 2.6, respectively, proved by Tsinias [14] for deterministic control systems to the larger class of SCSs driven by a Wiener process.(2)
The stabilizability results stated in Florchinger [6, Theorem 3.1] and Abedi et al. [12, Theorem 3.1] are not able to render our previous stabilizing results (Theorem 14 and Proposition 16).(3)
The stabilizability results provided for stochastic control systems in [6, 12, 13] do not permit us to make a conclusion about practical stability in probability (Theorem 18), whereas the results of this paper are still valid.



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