On the Asymptotical and Practical Stability of Stochastic Control Systems

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.

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 Krstic [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 Basar [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 Section 2, 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’minskii [17], and also the paper by Abedi et al. [13].

Let (Ω,𝐹,𝑃) be a complete probability space, and denote by (𝑤_𝑡)_𝑡≥0 a standard 𝑅^𝑚-valued Wiener process defined on this space.

Consider the stochastic process solution 𝑥_𝑡∈𝑅^𝑛 of the Ito stochastic differential system as 𝑥_𝑡=𝑥₀+^𝑡₀𝑓𝑥_𝑠𝑑𝑠+^𝑚_𝑘=1^𝑡₀ℎ_𝑘𝑥_𝑠𝑑𝑤^𝑘_𝑠, (1)

where(i)𝑥₀∈𝑅^𝑛 is given,(ii)𝑓∶𝑅^𝑛→𝑅^𝑛 and ℎ_𝑘∶𝑅^𝑛→𝑅^𝑛×𝑚,1≤𝑘≤𝑚, are locally Lipschitz functionals mapping with 𝑓(0)=0,ℎ_𝑘(0)=0 and there exists a constant 𝐾≥0 such that for any 𝑥∈𝑅^𝑛, the following linear growth condition holds: 𝑓(𝑥)+^𝑚_𝑘=1ℎ_𝑘(𝑥)≤𝐾(1+|𝑥|). (2)

Notations. Throughout this paper we adopt the following notation.(i)For any 𝑠≥0 and 𝑥∈𝑅^𝑛,??𝑥_𝑡^𝑠,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 any 𝑥∈𝑅^𝑛, 𝑥^𝑇 denotes its transpose.

Definition 1. The equilibrium 𝑥_𝑡=0 of the system (1) is(i)stable in probability, if for every 𝑠≥0 and 𝜀>0, lim_𝑥→0𝑃sup_𝑠≤𝑡𝑥_𝑡^𝑠,𝑥>𝜀=0, (3)(ii)ASP, if it is stable in probability and lim_𝑥→0𝑃lim_𝑡→+∞𝑥_𝑡^𝑠,𝑥=0=1. (4)

Definition 2. A function 𝛾∶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 if 𝛾(𝑟)>0 for all 𝑟>0, and 𝛾(0)=0.

Definition 3. The equilibrium 𝑥_𝑡=0 of the system (1) is(i)globally stable in probability, if for any 𝜀>0, there exists a class 𝐾-function 𝛾(⋅) such that 𝑃𝑥_𝑡<𝛾𝑥₀≥1−𝜀,∀𝑡≥0,∀𝑥₀∈𝑅^𝑛⧵{0}, (5)(ii)GASP, if it is globally stable in probability and 𝑃lim_𝑡→∞𝑥_𝑡=0=1,∀𝑥₀∈𝑅^𝑛. (6)

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 Section 4. 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 (𝑤_𝑡)_𝑡≥0 a standard 𝑅^𝑚-valued Wiener process defined on this space.

Consider the following stochastic process solution 𝑥_𝑡∈𝑅^𝑛 of the SCS: 𝑥_𝑡=𝑥₀+^𝑡₀𝑓𝑥_𝑠+^𝑝_𝑧=1𝑔_𝑧𝑥_𝑠𝑢^𝑧+𝑑𝑠_{𝑚𝑘=1𝑡0}ℎ_𝑘𝑥_𝑠+^𝑝_𝑧=1𝑞_𝑘,𝑧𝑥_𝑠𝑢^𝑧𝑑𝑤^𝑘_𝑠, (7) where(i)𝑥₀∈𝑅^𝑛 is given,(ii)𝑢 is an 𝑅^𝑝-valued control law,(iii)𝑓 and ℎ_𝑘 are functionals mapping satisfying the hypothesis given in Section 2, (iv)𝑔_𝑧∶𝑅^𝑛→𝑅^𝑛×𝑝,1≤𝑧≤𝑝, and 𝑞_𝑘,𝑧∶𝑅^𝑛→𝑅^{𝑛×𝑚×𝑝},1≤𝑘≤𝑚,1≤𝑧≤𝑝, are 𝐶^∞ functionals mapping with 𝑔_𝑧(0)=0,𝑞_𝑘,𝑧(0)=0 and there exists a constant 𝐾≥0 such that for any 𝑥∈𝑅^𝑛, the following linear growth condition holds: ^𝑝_𝑧=1𝑔_𝑧(𝑥)+^𝑚_𝑘=1𝑞_𝑘,𝑧(𝑥)≤𝐾(1+|𝑥|). (8)

Then SCS (7) is said to be ASP at the origin if there exists a neighborhood 𝐷 of the origin in 𝑅^𝑛 and a function 𝑢=𝑘(𝑥_𝑠) mapping 𝐷 into 𝑅^𝑝, vanishing at the origin, such that(i)for every 𝑥∈𝐷, the solution 𝑥_𝑡 of the resulting closed-loop system 𝑥=𝑥₀+^𝑡₀𝑓𝑥_𝑠+^𝑝_𝑧=1𝑔_𝑧𝑥_𝑠𝑘𝑥_𝑠^𝑧+𝑑𝑠_{𝑚𝑘=1𝑡0}ℎ_𝑘𝑥_𝑠+^𝑝_𝑧=1𝑞_𝑘,𝑧𝑥_𝑠𝑘𝑥_𝑠^𝑧𝑑𝑤^𝑘_𝑠 (9)is uniquely defined,(ii)the equilibrium solution 𝑥_𝑡≡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 {𝑘_𝑟,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 𝑥_𝑟(𝑡,𝑥₀) of the resulting closed-loop system 𝑥=𝑥₀+^𝑡₀𝑓𝑥_𝑠+^𝑝_𝑧=1𝑔_𝑧𝑥_𝑠𝑘_𝑟𝑥_𝑠^𝑧+𝑑𝑠_{𝑚𝑘=1𝑡0}ℎ_𝑘𝑥_𝑠+^𝑝_𝑧=1𝑞_𝑘,𝑧𝑥_𝑠𝑘_𝑟𝑥_𝑠^𝑧𝑑𝑤^𝑘_𝑠 (10)

enters 𝑆(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 by 𝐃 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(𝑅^𝑛,𝑅) by 𝐃Φ(𝑥)=^𝑛_𝑖=1𝑓^𝑖(𝑥)𝜕Φ(𝑥)𝜕𝑥_𝑖+12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘ℎ^𝑗_𝑘𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗. (11)

For any 𝑧∈(1,…,𝑝), let 𝐃_𝑧 be the second order differential operator defined for any function Φ in 𝐶²(𝑅^𝑛,𝑅) and is given by 𝐃_𝑧Φ(𝑥)=^𝑛_𝑖=1𝑔^𝑖_𝑧(𝑥)𝜕Φ(𝑥)𝜕𝑥_𝑖+^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘𝑞^𝑗_𝑘,𝑧𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗, (12)and, for any 𝑧,𝑟∈(1,…,𝑝), 𝐃_𝑧,𝑟 denotes the second order differential operator defined for any function Φ in 𝐶²(𝑅^𝑛,𝑅) by 𝐃_𝑧,𝑟Φ(𝑥)=12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}𝑞^𝑖_𝑘,𝑧(𝑥)𝑞^𝑗_𝑘,𝑟(𝑥)𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗. (13)

We also denote 𝐃₀ 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(𝑅^𝑛,𝑅) given by 𝐃₀Φ(𝑥)=𝐃Φ(𝑥)+^𝑝_𝑧=1𝐃_𝑧Φ(𝑥)𝑘(𝑥)^𝑧+^𝑝_𝑧,𝑟=1𝐃_𝑧Φ(𝑥)𝐃_𝑧,𝑟Φ(𝑥)𝑘(𝑥)^𝑧2. (14)

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 𝑅^𝑛 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: 𝐃_𝑧Φ(𝑥)𝐃=0,_𝑧,𝑟Φ(𝑥)=0⇒𝐃Φ(𝑥)≤−𝜌(𝑥)<0. (15)

Definition 5. A continuously differentiable real function Φ 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∈𝑅^𝑝 satisfying the following inequalities: ‖𝑢‖(𝑥),(𝑥)+<𝑐𝐃Φ^𝑝_𝑧=1𝐃_𝑧Φ(𝑥)𝑢^𝑧+^𝑝_𝑧,𝑟=1𝐃_𝑧,𝑟Φ(𝑥)𝑢^𝑧2≤−𝜌(𝑥)<0. (16)(17)
If in addition lim_𝑥→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 ‖𝑘(𝑥)‖<𝑐(𝑥), (18) 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∈𝑅^𝑛, denote by 𝑏(𝑥) the function defined by 𝑏(𝑥)=𝐃_𝑧Φ(𝑥). (19)
Then the feedback law 𝑘(𝑥)𝑏(𝑥)=−Ψ(𝑥), (20) where 𝑏(𝑥) is given by (19) and Ψ(𝑥)=1+sup_{1≤𝑧,𝑟≤𝑝}𝐃_𝑧,𝑟Φ(𝑥)², (21)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 𝐷_𝑟 of the origin in 𝑅^𝑛 such that 𝑆(0,𝑟)⊂𝐷_𝑟, a family of 𝐶² positive definite function Φ_𝑟∶𝑅^𝑛→𝑅, where 𝑟>0, and a positive definite function 𝜌, such that for any sufficiently small 𝑟, condition (15) holds with Φ=Φ_𝑟, and minΦ_𝑟(𝑥),𝑥∈𝜕𝐷_𝑟>maxΦ_𝑟(𝑥),𝑥∈𝑆(0,𝑟), (22) where 𝜕𝐷 is the boundary of 𝐷. Moreover, for any sequences {𝑟_𝑖}⊂𝑅⁺ and {𝑥_𝑟𝑖}⊂𝐷 with lim𝑟_𝑖=0 and lim𝑥_𝑟𝑖=𝑎 as 𝑖→+∞, we have limΦ_𝑟𝑖𝑥_𝑟𝑖=0for𝑎=0,limΦ_𝑟𝑖𝑥_𝑟𝑖>0for𝑎>0. (23)

Definition 10. A continuously differentiable real function Φ_𝑟 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 {Φ_𝑟} 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 𝑚_𝑟, where 𝑟>0 such that 𝑚_𝑟→0 as 𝑟→0 and for any sufficiently small 𝑟, conditions (17), (22), and (23) hold with Φ=Φ_𝑟,𝑥∈𝐷_2𝑟−𝐷_1𝑟, and for some 𝑢∈𝑅¹ where ‖𝑢‖<𝑐(𝑥)+𝑚_𝑟. (24)
If, in addition, 𝑐(𝑥)→0 as 𝑥→0, then the control Lyapunov family {Φ_𝑟} 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 Theorem 7.

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={𝑘_𝑟(𝑥)}.
(ii) If the control Lyapunov family {Φ_𝑟} satisfies the stochastic bounded (small) control property, then for any 𝑟 the corresponding stabilizer 𝑢 satisfies the following inequality: 𝑘_𝑟(𝑥)<𝑐(𝑥)+𝑚_𝑟, (25) where 𝑐 and 𝑚_𝑟 are those given in Definition 11.

Remark 13. A direct result of Definitions 4 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 𝑑𝑥₁𝑥₂=𝑓₁𝑥₁,𝑥₂𝑓₂𝑥₁,𝑥₂0𝑔𝑑𝑡+₂𝑥₁,𝑥₂+ℎ𝑢𝑑𝑡₁𝑥₁ℎ₂𝑥₁,𝑥₂𝑑𝑤_𝑡+0𝑞₂𝑥₁,𝑥₂𝑢𝑑𝑤_𝑡, (26) where 𝑥=^𝑥1_𝑥2∈𝑅^𝑛1×𝑅^𝑛2,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 𝑑𝑥₁𝑥₂=𝑓₁𝑥₁,𝑥₂𝑓₂𝑥₁,𝑥₂0𝑢+ℎ𝑑𝑡+𝑑𝑡₁𝑥₁ℎ₂𝑥₁,𝑥₂𝑑𝑤_𝑡+0𝑢𝑑𝑤_𝑡, (27) where 𝑥=^𝑥1_𝑥2∈𝑅^𝑛1×𝑅^𝑛2 and 𝑢 is an 𝑅^𝑛2-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 (Theorem 14). The formulas for the special case of SCS (7) with unity intensity noise and vanishing 𝑞_𝑘,𝑧(𝑥) without the term ℎ_𝑘(𝑥)𝑑𝑤were given by Abedi et al. [13] and for the system (7) with affinity in the noise and control 𝑢, whereℎ_𝑘(𝑥) may be nonvanishing at the origin, without the term 𝑞_𝑘,𝑧(𝑥)𝑢^𝑧𝑑𝑤were 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 𝑔_𝑧(𝑥) and 𝑞_𝑘,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): 𝑑𝑥₁=𝑓₁𝑥₁,𝜐𝑑𝑡+ℎ₁𝑥₁𝑑𝑤_𝑡,𝜐∈𝑅^𝑛2,𝑑𝑥₂=𝑔₂𝑥₁,𝑥₂𝑢𝑑𝑡+𝑞₂𝑥₁,𝑥₂𝑢𝑑𝑤_𝑡,𝑢∈𝑅^𝑝. (28)(29)

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 neighborhoods 𝐷 and 𝐷₁ of the origin in 𝑅^𝑛 and 𝑅^𝑛1, respectively, and mappings 𝑠∶𝐷→𝑅^𝑝,𝜙∶𝐷₁→𝑅^𝑛2, and 𝑊∶𝐷→𝑅 such that 𝜙(0)=0 where 𝜙 is continuous, and 𝑊 is continuously differentiable, and let us denote 𝑁=𝑥∈𝐷∶𝑥₂=𝜙𝑥₁,𝑁₁=𝑥∈𝐷∶∇^𝛼𝑊(𝑥),𝑁=0,𝛼=0,1,2₂=𝑥∈𝐷∶𝜕𝑊𝜕𝑥₂𝑔₂𝑠(𝑥)=0,𝜕𝑊𝜕𝑥₂𝑞₂𝑠(𝑥)=0. (30)(31)(i)Assume that for any 𝑥 in a neighborhood of the origin in 𝑅^𝑛,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)Let Φ 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 𝜕𝑊𝜕𝑥₂𝑔₂𝑠(𝑥)≤−𝑑𝑥₂−𝜙𝑥₁^𝑏, (32)for 𝑥 in a neighborhood of the origin in 𝑅^𝑛 then the function Φ satisfies the small control property.

Proof. Part (i): since the equilibrium solution 𝑥₁≡0 of the closed-loop system 𝑑𝑥₁=𝑓₁𝑥₁,𝜙𝑥₁𝑑𝑡+ℎ₁𝑥₁𝑑𝑤_𝑡 (33) is asymptotically stable, then the converse Lyapunov theorem of Kushner [15] asserts that there exists a smooth Lyapunov function 𝑉∶𝑅^𝑛1→𝑅 such that 𝑉(0)=0,𝑉(𝑥₁)>0 and 𝐇𝑉𝑥₁<0 (34)
for 𝑥₁≠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 Φ(𝑥)=𝑉𝑥₁+𝑊(𝑥) (35)
is positive definite, and for any 𝑥∈(𝐷×𝑅^𝑛2)∩𝐷, we have 𝐆_𝑧Φ(𝑥)=^𝑛_𝑖=1𝑔^𝑖_𝑧(𝑥)𝜕Φ(𝑥)𝜕𝑥_𝑖+^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘𝑞^𝑗_𝑘,𝑧𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑔₂𝜕𝑊(𝑥)𝜕𝑥₂+ℎ₁𝑞₂𝜕²𝑊𝜕𝑥₁𝜕𝑥₂+ℎ₂𝑞₂𝜕²𝑊(𝑥)𝜕𝑥²₂,𝐆_𝑧,𝑟Φ(𝑥)=12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}𝑞^𝑖_𝑘,𝑧𝑞^𝑗_𝑘,𝑟𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=12𝑞₂(𝑥)𝜕²𝑊(𝑥)𝜕𝑥²₂. (36)
For any 𝑥≠0 such that (𝜕𝑊(𝑥)/𝜕𝑥₂)𝑔₂(𝑥)=(𝜕𝑊(𝑥)/𝜕𝑥₂)𝑞₂(𝑥)=0,it follows by the assumption that 𝑥₂=𝜙(𝑥₁). Hence, ∇𝑊𝑥₁,𝜙𝑥₁=0,∇²𝑊𝑥₁,𝜙𝑥₁=0, (37)
and so 𝐆Φ(𝑥)=^𝑛_𝑖=1𝑓^𝑖𝜕Φ(𝑡,𝑥)𝜕𝑥_𝑖+12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘ℎ^𝑗_𝑘𝜕²Φ(𝑡,𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑓₁𝜕𝑉𝜕𝑥₁+𝑓₁𝜕𝑊𝜕𝑥₁+𝑓₂𝜕𝑊𝜕𝑥₂+12ℎ²₁𝜕²𝑉𝜕𝑥²₁+𝜕²𝑊𝜕𝑥²₁+12ℎ₁ℎ₂𝜕²𝑊𝜕𝑥₁𝜕𝑥₂+12ℎ₂ℎ₁𝜕²𝑊𝜕𝑥₂𝜕𝑥₁+12ℎ²₂𝜕²𝑊𝜕𝑥²₂=𝐇𝑉𝑥₁<0, (38) 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 that Φ(𝑥) is the corresponding CLF and that (32) holds, with 𝑏≤1. Define 𝜌(𝑥)=−𝐇𝑉𝑥₁,𝜙𝑥₁+𝑥₂−𝜙𝑥₁². (39)
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 ∇𝑓(𝑥)≤𝑑₁,‖∇𝑊(𝑥)‖≤𝑑₂𝑥₂−𝜙𝑥₁,∇²𝑊(𝑥)≤𝑑₃𝑥₂−𝜙𝑥₁ (40)(41)(42)
for 𝑥 in a neighborhood of the equilibrium. Let 𝑎(𝑥)=𝑑₁∇𝑉𝑥₁+𝑑₂𝑓(𝑥+𝑑)³2ℎℎ^𝑇(𝑥). (43)
Then, by taking into account (32)–(42), we have 𝐆Φ+𝜌(𝑥)≤𝑑₁∇𝑉𝑥₁𝑥₂−𝜙𝑥₁+𝑑₂𝑓(𝑥)𝑥₂−𝜙𝑥₁+𝑑₃2ℎℎ^𝑇(𝑥)𝑥₂−𝜙𝑥₁≤𝑎(𝑥)𝑥₂−𝜙𝑥₁𝑎(𝑥)≤−𝑑𝜕𝑊𝜕𝑥₂𝑔₂𝑠(𝑥), (44)and it follows that (𝑥(𝑥)𝐆Φ+𝜌)≤−𝑠𝑎𝑑𝐆_𝑧Φ(𝑥). (45)
Therefore, 𝐆Φ(𝑥)+𝐆_𝑧Φ(𝑥)𝑠𝑎𝑑≤−𝜌(𝑥)<0, (46)for all 𝑥≠0 near zero. Inequality (46) in conjunction with condition ∇²𝑊(𝑥₁,𝜙(𝑥₁))=0 gives 𝐆Φ(𝑥)+𝐆_𝑧Φ(𝑥)𝑠𝑎𝑑+𝐆_𝑧,𝑟Φ(𝑥)𝑠𝑎𝑑²≤−𝜌(𝑥)<0. (47)
Since 𝑠 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={𝑘_𝑟(𝑥)}.

Proof. The proof of this proposition is a direct consequence of Theorems 7(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 Theorem 14 for a particular case of SCS (27). Assume, without loss of generality that 𝑓₂=0. In this case, the SCS (27) becomes 𝑑𝑥₁𝑥₂=𝑓₁𝑥₁,𝑥₂00𝑢+ℎ𝑑𝑡+𝑑𝑡₁𝑥₁ℎ₂𝑥₁,𝑥₂𝑑𝑤_𝑡+0𝑢𝑑𝑤_𝑡, (48) where 𝑥=^𝑥1_𝑥2∈𝑅^𝑛1×𝑅^𝑛2, 𝑢 is an 𝑅^𝑛2-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 solution 𝑥₁≡0 for the closed-loop system 𝑑𝑥₁=𝑓₁𝑥₁,𝜙𝑥₁𝑑𝑡+ℎ₁𝑥₁𝑑𝑤_𝑡 (49) is asymptotically stable, then the converse Lyapunov theorem of Kushner [15] asserts that there exists a smooth Lyapunov function 𝑉∶𝑅^𝑛1→𝑅 defined in a neighborhood 𝐷 of the origin in 𝑅^𝑛1 such that 𝐇𝑉𝑥₁<0 (50)
for 𝑥₁≠0 near zero. Let 𝑊(𝑥)=12𝑥₂−𝜙𝑥₁². (51)
Obviously, 𝑊 is nonnegative definite and (𝜕𝑊/𝜕𝑥₂) is continuously differentiable in a neighborhood of the origin. If we define the functional 𝑟 on 𝑅^𝑛 by the following: 𝑟_𝑖(𝑥)=−sgn𝑥₂−𝜙𝑥_1𝑖, (52) then 𝜕𝑊𝜕𝑥₂𝑔₂𝑟(𝑥)=𝜕𝑊𝜕𝑥₂𝑟(𝑥)≤−𝑥₂−𝜙𝑥₁, (53) where 𝑟 is uniformly bounded on 𝑅^𝑛 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. Let 𝜙∶𝑅^𝑛→𝑅^𝑚 be a real mapping that satisfies the Lipschitz condition on a compact neighborhood 𝑈 of the origin in 𝑅^𝑛, and 𝜙(0)=0. Then there exist a neighborhood 𝑈₁ of the origin in 𝑅^𝑛, a constant 𝐾>0, and a family of smooth mappings 𝜙_𝜖∶𝑈₁→𝑅^𝑚,𝜖>0 such that for every sufficiently small 𝜖 the following conditions hold: 𝜙(𝑥)−𝜙_𝜖(𝑥)<𝜖,∀𝑥∈𝑈₁(),𝜙−𝑆0,𝜖_𝜖(0)=0,∇𝜙_𝜖(𝑥)<𝐾,∀𝑥∈𝑈₁. (54)(55)(56)
Moreover, for any sequence {𝜖_𝑖}⊂𝑅⁺ and {𝑥_𝑖}⊂𝑈₁ such that 𝜖_𝑖→0 and 𝑥_𝑖→𝑥∈𝑈₁, as 𝑖→+∞, there exist subsequences {𝜖^′_𝑖} and {𝑥^′_𝑖} of {𝜖_𝑖} and {𝑥_𝑖}, respectively, and a positive constant 0≤𝛼≤1 satisfying 𝜙_𝜖′𝑖𝑥^′₁→𝛼𝜙(𝑥). (57)

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)If 𝜙 is continuous at the origin, then SCS (48) satisfies the practical Lyapunov condition.(ii)If 𝜙 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 𝑅^𝑛1 and a positive definite and strictly increasing continuous function 𝛾∶𝑅⁺→𝑅⁺ such that 𝛾(0)=0 and 𝐇𝑉𝑥₁,𝜙𝑥₁≤−𝛾𝑥₁ (58)
for any 𝑥₁∈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 𝑅^𝑛2. Consider the positive constants 𝑐₁ and 𝑐₂ such that ‖𝜙(𝑥₁)‖<𝑐₁ for 𝑥₁∈𝑈₁ and ‖∇𝑓₁(𝑥)‖<𝑐₂ for every 𝑥 belonging to 𝐷=𝑥∈𝑅^𝑛∶𝑥₁∈𝑈₁,𝑥₂<𝑐₁. (59)
For any 𝑟>0, such that the sphere 𝑆(0,𝑟) is contained to 𝐷, consider a positive 𝑙=𝑙(𝑟)<1 where 𝐷_𝑙=𝑥∈𝑅^𝑛∶𝑥₁<𝑙,𝑥₂≤2𝐾𝑙, (60) is contained to 𝑆(0,𝑟). Since 𝑓(0)=0 and ℎ(0)=0, there exist nonnegative constants 𝑐₃ and 𝑐₄ such that 𝑓(𝑥)≤𝑐₃,ℎℎ^𝑇≤𝑐₄, (61) for all 𝑥 in a neighborhood of the origin in 𝑅^𝑛. Let 𝜖=𝜖(𝑟) with 𝜖<𝑙 and 𝜖<min𝑐₃⁻¹‖∇𝑉‖⁻¹+2𝑐₄⁻¹∇²𝑉⁻¹×𝛾𝑥₁∶𝑥₁∈𝑈₁,𝑥₁≥𝑙. (62)
Then, by (54)–(56) and (58)–(61), we have for any 𝑥₁∈𝑈₁ and ‖𝑥₁‖>𝑙𝐇𝑉𝑥₁,𝜙_𝜖𝑥₁≤𝐇𝑉𝑥₁,𝜙_𝜖𝑥₁−𝐇𝑉𝑥₁,𝜙𝑥₁+𝐇𝑉𝑥₁,𝜙𝑥₁≤𝑐₃‖‖+1∇𝑉2𝑐₄∇²𝑉𝜙𝑥₁−𝜙_𝜖𝑥₁−𝛾𝑥₁ (63)
Now, for 𝜖=𝜖(𝑟), we can define Φ_𝑟(𝑥)=𝑉𝑥₁+12𝑥₂−𝜙_𝜖𝑥₁². (64)
Obviously, Φ_𝑟(𝑥) is continuously differentiable and positive definite for 𝑥∈𝐷. Let 𝛽_𝑟∶𝐷→𝑅^𝑛2 be a real mapping where 𝛽_𝑟(𝑥)=45ℎ₁𝜕𝜙_𝜖𝑥₁𝜕𝑥₁−ℎ₂−𝑥₂−𝜙_𝜖𝑥₁. (65)
Denoting 𝐃₀ 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 𝛽_𝑟(𝑥) given by (65). This yields 𝐃₀Φ(𝑥)=𝐃Φ(𝑥)+^𝑝_𝑧=1𝐃_𝑧Φ(𝑥)𝛽_𝑟(𝑥)+^𝑝_𝑧,𝑟=1𝐃_𝑧Φ(𝑥)𝐃_𝑧,𝑟Φ(𝑥)𝛽_𝑟(𝑥)²=𝑓₁𝜕𝑉𝑥₁𝜕𝑥₁+𝜕𝜙_𝜖𝑥₁𝜕𝑥₁𝑥₂−𝜙_𝜖𝑥₁+45ℎ₁𝜕𝜙_𝜖𝑥₁𝜕𝑥₁−ℎ₂−𝑥₂−𝜙_𝜖𝑥₁²+8ℎ25¹𝜕𝜙_𝜖𝑥₁𝜕𝑥₁−ℎ₂−𝑥₂−𝜙_𝜖𝑥₁³<0, (66) for every 𝑥∈𝐷−𝐷_𝑙. Note that both 𝑥∈𝐷−𝐷_𝑙 and ‖𝑥₁‖<𝑙 give ‖𝑥₂‖>2𝐾𝑙 and by (56) that ‖𝜙_𝜖(𝑥₁)‖<𝐾𝑙. Therefore,𝑥₂−𝜙_𝜖𝑥₁≥𝑥₂−𝐾𝑥₁>𝐾𝑙≠0. (67)
Since 𝑓₁,ℎ_𝑖, and 𝜙_𝜖 are continuously differentiable, where 𝑓₁(0)=ℎ_𝑖(0)=𝜙_𝜖(0)=0, and because of (54) and (56), there is a positive constant 𝑀 which is independent of 𝑟, satisfying 𝛽_𝑟(𝑥)<𝑀‖𝑥‖+𝑚_𝑟, (68) for every 𝑥∈𝐷. This implies that for every sufficiently small 𝑟, condition (22) holds. In fact, there exist sequences {𝑥_𝑟𝑖}⊂𝜕𝐷 and {𝑧_𝑟𝑖}⊂𝑆(0,𝑟_𝑖) such that 𝑥_𝑟𝑖→𝑥=(𝑥₁,𝑥₂)∈𝜕𝐷 and Φ_𝑟𝑖(𝑥_𝑟𝑖)≤Φ_𝑟𝑖(𝑧_𝑟𝑖). Since 𝜖(𝑟)→0, it follows from (57) that there exist subsequences {𝑥^′_𝑟𝑖}⊂{𝑥_𝑟𝑖} and {𝑧^′_𝑟𝑖}⊂{𝑧_𝑟𝑖}, and real constants 𝛼₁ and 𝛼₂ satisfying 𝑉𝑥₁+12𝑥₂−𝛼₁𝜙𝑥₁²=limΦ_𝑟𝑖𝑥^′_𝑟𝑖≤limΦ_𝑟𝑖𝑧^′_𝑟𝑖(0)+1=𝑉2𝛼²₂𝜙(0)²=0, (69) and so 𝑥₁=𝑥₂=0, which indicates a contradiction. Similarly, we can show (23). It turns out that {Φ_𝑟(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)If 𝜙 is continuous at the origin, then the SCS (48) is practically stabilizable in probability at the origin.(ii)If 𝜙 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)If 𝜙 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 by 𝑥(𝑡)∈𝑅³ the solution of the following SCS: 𝑑𝑥=−3𝑥₁−5𝑥₂−𝑥₃00𝑥𝑑𝑡+₁𝑥₂+𝑢𝑑𝑡−2𝑥₁−3𝑥₂−𝑥₃𝑑𝑤_𝑡+001𝑢𝑑𝑤_𝑡, (70) where (𝑤_𝑡)_𝑡≥0 is a standard real-valued Wiener process and 𝑢 is a real-valued measurable control law.
Consider the following Lyapunov function candidate: Φ(𝑥)=12𝑥²₁+𝑥²₂+𝑥²₃. (71)
After some simple calculations, it gives 𝐃_𝑧Φ(𝑥)=𝑥₁𝑥₂𝑥₃−𝑥₃,𝐃Φ(𝑥)=−𝑥²₁+𝑥²₂+𝑥²₃,𝐃_𝑧,𝑟Φ(𝑥)=12. (72)
From (72), we get 𝐃₀Φ(𝑥)=−𝑥²₁+𝑥²₂+𝑥²₃−45𝑥₁𝑥₂𝑥₃−𝑥₃²+8𝑥25¹𝑥₂𝑥₃−𝑥₃³≤−Φ(𝑥). (73)
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 Theorem 8 to find an explicit formula for a feedback law. Indeed, by (20) and (72) we can obtain the following feedback law: 𝑘(𝑥)4=−5𝑥₁𝑥₂𝑥₃−𝑥₃ (74) that guarantees the equilibrium solution of the resulting closed-loop system is deduced from the SCS (70) which satisfies the ASP property.

Example 21. Let 𝑥(𝑡)∈𝑅² be the solution of the SCS as 𝑑𝑥₁𝑥₂=−𝑥₁+𝑥₂𝜙₁𝑥₁,𝑥₂0𝑥𝑑𝑡+₂+𝑥𝑢𝑑𝑡₁𝑥₂𝑑𝑤_𝑡+0𝑥₂+𝜙₂𝑥₁𝑢𝑑𝑤_𝑡, (75) where (𝑤_𝑡)_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 𝑥₁𝜙₂𝑥₁>0, (76) for any 𝑥₁∈𝑅,𝑥₁≠0. Obviously, the SCS (75) has the form (26), and the function Φ, defined on 𝑅^𝑛 as Φ𝑥₁,𝑥₂=12𝑥²₁+14𝑥₂+𝜙₂𝑥₁⁴, (77) is a CLF for SCS (75). Indeed, for any 𝑥≠0 with 𝐆_𝑧Φ(𝑥)=^𝑛_𝑖=1𝑔^𝑖_𝑧(𝑥)𝜕Φ(𝑥)𝜕𝑥_𝑖+^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘𝑞^𝑗_𝑘,𝑧𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑔₂𝜕𝑊(𝑥)𝜕𝑥₂+ℎ₁𝑞₂𝜕²𝑊𝜕𝑥₁𝜕𝑥₂+ℎ₂𝑞₂𝜕²𝑊(𝑥)𝜕𝑥²₂=𝑥₂+3𝜙^′₂𝑥₁+3𝑥₂𝑥₂+𝜙₂𝑥₁³𝐆=0,_𝑧,𝑟Φ(𝑥)=12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}𝑞^𝑖_𝑘,𝑧𝑞^𝑗_𝑘,𝑟𝜕²Φ(𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=12𝑞₂(𝑥)𝜕²𝑊(𝑥)𝜕𝑥²₂=32𝑥₂+𝜙₂𝑥₁³=0, (78)(79) it follows that 𝑥₂=−𝜙₂(𝑥₁), and therefore 𝐆Φ(𝑥)=^𝑛_𝑖=1𝑓^𝑖𝜕Φ(𝑡,𝑥)𝜕𝑥_𝑖+12^𝑛𝑚_{𝑖,𝑗=1𝑘=1}ℎ^𝑖_𝑘ℎ^𝑗_𝑘𝜕²Φ(𝑡,𝑥)𝜕𝑥_𝑖𝜕𝑥_𝑗=𝑓₁𝜕𝑉𝜕𝑥₁+𝑓₁𝜕𝑊𝜕𝑥₁+𝑓₂𝜕𝑊𝜕𝑥₂+12ℎ²₁𝜕²𝑉𝜕𝑥²₁+𝜕²𝑊𝜕𝑥²₁+12ℎ₁ℎ₂𝜕²𝑊𝜕𝑥₁𝜕𝑥₂+12ℎ₂ℎ₁𝜕²𝑊𝜕𝑥₂𝜕𝑥₁+12ℎ²₂𝜕²𝑊𝜕𝑥²₂1=−2𝑥²₁+𝑥₁𝑥₂1=−2𝑥²₁−𝑥₁𝜙₂𝑥₁. (80)
Using condition 𝑥₁𝜙₂(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 𝑑𝑥₁=−𝑥₁+𝜈𝑑𝑡+𝑥₁𝑑𝑤_𝑡, (81)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: 𝑥𝑑𝑥=¹+𝑥³₂001𝑑𝑡+𝑢𝑑𝑡+2𝑥₁𝑥₂𝑑𝑤_𝑡, (82) where (𝑤_𝑡)_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 𝑑𝑥₁=𝑥₁+𝑣³𝑑𝑡+2𝑥₁𝑑𝑤_𝑡 (83)
is stabilized by the continuous law 𝑣=−(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.