Stability of Nonlinear Stochastic Discrete-Time Systems

Abstract

This paper studies the stability for nonlinear stochastic discrete-time systems. First of all, several definitions on stability are introduced, such as stability, asymptotical stability, and pth moment exponential stability. Moreover, using the method of the Lyapunov functionals, some efficient criteria for stochastic stability are obtained. Some examples are presented to illustrate the effectiveness of the proposed theoretical results.

1. Introduction

Stability is the first of all the considered problems in the system analysis and synthesis of modern control theory, which plays an essential role in dealing with infinite-horizon linear-quadratic regulator, 𝐻₂/𝐻_∞ robust optimal control, and other control problems; see [1–5]. In 1892, Lyapunov introduced the concept of stability of dynamic systems and created a very powerful tool known as the Lyapunov method in the study of stability. It can be found that the Lyapunov method has been developed and applied to investigate stochastic stability of the Itô-type systems, and many important classical results on deterministic differential equations have been generalized to the stochastic Itô systems; we refer the reader to Arnold [6], Friedman [7], Has’minskii [8], Kushner [9], Kolmanovskii and Myshkis [10], Ladde and Lakshmikantham [11], Mohammed [12], and Mao [13].

Compared with the plenty of fruits of the continuous-time Itô systems, few results have been obtained on the stability of discrete-time nonlinear stochastic systems: 𝑥(𝑡+1)=𝑓(𝑥(𝑡),𝑤(𝑡),𝑡). (1)

In [14], the mean square stability of the discrete-time time-varying Markov jump system 𝑥(𝑡+1)=𝐴₀𝑡,𝜂_𝑡+^𝑟_𝑘=1𝐴_𝑘𝑡,𝜂_𝑡𝑤_𝑘(𝑡)𝑥(𝑡) (2) was studied. Nevertheless [15], based on the exact observability assumption, extensively researched the mean square stability of the following linear discrete-time time-invariant system with multiplicative noise: 𝑥(𝑡+1)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡)𝑤(𝑡), (3) where the Classical Lyapunov Theorem was extended. Reference [16] considered the 𝑝th mean stability of the following difference equations 𝑥()()𝑥(𝑡)𝑥()()𝑥(𝑡)((𝑡𝑡+1=𝐴𝑡,𝜔,𝑡=0,1,2,…,𝑡+1=𝐴𝑡,𝜔+𝑓𝑡,𝑥)),𝑡=0,1,2,…, (4) with random coefficients. For the nonlinear stochastic difference equation 𝑥(𝑡+1)=^𝑡+ℎ_𝑗=0𝑎_𝑗𝑥𝑡−𝑗+^𝑡+ℎ_𝑗=0𝜎_𝑗𝑥𝜉(𝑡)(()(𝑡,𝑡−𝑗+𝑔𝑡,𝑥−ℎ,…,𝑥)) (5) its stability in probability was investigated in [17]. It is not difficult to find that, different from the continuous-time Itô systems, up to now, there lacks the systematic theory on stability of nonlinear discrete-time stochastic systems. The aim of this paper is to develop a parallel theory for stability of general nonlinear stochastic discrete-time systems, and some sufficient criteria for various stabilities are given.

Different from the Itô systems, most sufficient criteria are presented via ℒ𝑉(𝑥)≤0 or ℒ𝑉(𝑥)<0 together with other assumptions on the Lyapunov function 𝑉(𝑥), where ℒ is the so-called infinitesimal generator associated with the given Itô system. In discrete-time stochastic systems, most stability criteria are given via 𝐸[Δ𝑉(𝑥(𝑡))]≤0 or 𝐸[Δ𝑉(𝑥(𝑡))]<0, where 𝐸 represents the mathematical expectation. So general discrete stochastic stability is more difficult to be tested due to the appearance of the mathematical expectation 𝐸.

The organization of this paper is as follows. Section 2 presents some stability definitions. Section 3 is devoted to developing some efficient criteria for various stabilities. Section 4 contains three examples provided to show the efficiency of the proposed results. Finally, we end this paper by Section 5 with a brief conclusion.

For convenience, we adopt the following notations: 𝐴^𝑇: the transpose of the matrix 𝐴; 𝐴≥0 (𝐴>0): 𝐴 is a positive semidefinite (positive definite) matrix; 𝐷_𝑟∶={𝑥∈𝑅^𝑛∶|𝑥|<𝑟} for 𝑟>0; 𝐶²(𝑈): the class of functions 𝑉(𝑥) twice continuously differential with respect to 𝑥∈𝑈; 𝑍⁺∶={0,1,2,…}; 𝑃{𝐵}: the probability of event 𝐵; a.s.: almost surely, or with probability 1; 𝐼_𝐵: the indicator function of a set 𝐵; that is, 𝐼_𝐵(𝑥)=1 if 𝑥∈𝐵 or otherwise 0; 𝑎∧𝑏: the minimum of 𝑎 and 𝑏.

2. Definitions of Stability

We will investigate various types of stabilities in probability for the 𝑛-dimensional stochastic discrete-time system 𝑥(𝑡+1)=𝑓(𝑥(𝑡),𝑤(𝑡),𝑡),𝑥𝑡₀=𝑥₀, (6) where 𝑥₀∈𝑅^𝑛 is a constant vector. For any given initial value 𝑥(𝑡₀)=𝑥₀∈𝑅^𝑛, (6) has a unique solution that is denoted by 𝑥(𝑡;𝑡₀,𝑥₀) or 𝑥(𝑡;𝑡₀,𝑥₀)=𝑥(𝑡) simply. 𝑤(𝑡) is a one-dimensional stochastic process defined on the complete probability space (Ω,𝐹,𝑃). We assume that 𝑓(0,𝑤(𝑡),𝑡)≡0 for all 𝑡∈𝐼∶={𝑡₀+𝑘∶𝑘∈𝑍⁺}, so (6) has the solution 𝑥(𝑡)≡0 corresponding to the initial value 𝑥(𝑡₀)=0. This solution is called the trivial solution or the equilibrium position.

Definition 1. The trivial solution of (6) is said to be stochastically stable or stable in probability if, for every 𝜀>0 and ℎ>0, there exists 𝛿=𝛿(𝜀,ℎ,𝑡₀)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,𝑡≥𝑡₀, (7) when |𝑥₀|<𝛿. Otherwise, it is said to be stochastically unstable.
If the previous 𝛿 is independent of 𝑡₀, that is, 𝛿=𝛿(𝜀,ℎ)>0, then the trivial solution of (6) is said to be stochastically uniformly stable in probability.

Definition 2. The trivial solution of (6) is said to be stochastically asymptotically stable in probability if it is stochastically stable, and for every 𝜀>0, there exists 𝛿=𝛿(𝜀,𝑡₀)>0, such that 𝑃lim_𝑡→∞𝑥(𝑡)=0≥1−𝜀, (8) when |𝑥₀|<𝛿.

Definition 3. The trivial solution of (6) is said to be stochastically uniformly asymptotically stable in probability if it is stochastically uniformly stable in probability, and for every 𝜀>0, ℎ>0, there exist 𝛿=𝛿(𝜀,ℎ)>0 and a 𝑇(𝜀)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀+𝑇(𝜀),when𝑥₀<𝛿. (9)

Definition 4. The trivial solution of (6) is said to be stochastically asymptotically stable in the large in probability if it is stochastically stable, and for all 𝑥₀∈𝑅^𝑛, 𝑃lim_𝑡→∞𝑥(𝑡)=0=1. (10)

Definition 5. The trivial solution of (6) is said to be uniformly bounded if, for every 𝛼>0 and 𝑡₀∈𝑍⁺, there exists 𝛽=𝛽(𝛼)>0, such that |𝑥(𝑡)|<𝛽,a.s., (11) when |𝑥₀|<𝛼 and 𝑡≥𝑡₀.

Definition 6. The trivial solution of (6) is said to be stochastically uniformly asymptotically stable in the large in probability if the following are satisfied:(i)it is stochastically uniformly stable;(ii)it is uniformly bounded;(iii)for any 𝛼>0, ℎ>0, and 𝜀>0, there exists 𝑇(𝜀,𝛼)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀+𝑇(𝜀,𝛼),𝑥₀<𝛼. (12)

Definition 7. The trivial solution of (6) is said to be 𝑝th moment exponentially stable if there exist positive constants 𝜆 and 𝐶, such that 𝐸|𝑥(𝑡)|^𝑝≤𝐶𝑥₀^𝑝𝑒^{−𝜆(𝑡−𝑡0)}, (13) where 𝑝>0, 𝑡≥𝑡₀, and 𝑥₀∈𝑅^𝑛. When 𝑝=2, it is usually said to be exponentially stable in mean square.
Below, we consider such a continuous function 𝑉(𝑥)∶𝑅^𝑛⟶𝑅, (14) with 𝑉(0)=0, and write Δ𝑉(𝑥(𝑡))=𝑉(𝑥(𝑡+1))−𝑉(𝑥(𝑡)). (15)

Definition 8 (see [13]). A continuous function 𝜑∶[0,+∞)→[0,+∞) is said to belong to class 𝐾 if it is strictly increasing and 𝜑(0)=0.

Definition 9 (see [13]). A continuous function 𝑉(𝑥) defined on 𝐷_𝑟 is said to be positive definite (in the sense of Lyapunov) if 𝑉(0)=0 and, for some 𝜑∈𝐾, 𝑉(𝑥)≥𝜑(|𝑥|). (16)
A continuous function 𝑉(𝑥) defined on 𝐷_𝑟 is said to be negative definite (in the sense of Lyapunov) if −𝑉(𝑥) is positive definite.

Definition 10 (see [13]). A function 𝑉(𝑥) defined on 𝐷_𝑟 is said to be radially unbounded if lim_|𝑥|→∞inf𝑉(𝑥)=∞. (17)

Definition 11. A function 𝑉(𝑥) defined on 𝐷_𝑟 is said to have infinite small upper bound if there exists 𝜑∈𝐾 such that |𝑉(𝑥)|≤𝜑(|𝑥|). (18)

3. Main Results

In this section, we state our main results in this paper. By using the method of the Lyapunov functionals, some efficient criteria for the stability are obtained.

Theorem 12. If there exists a positive definite function 𝑉(𝑥)∈𝐶²(𝐷_𝑟), such that 𝐸[Δ𝑉(𝑥(𝑡))]≤0, (19) for all 𝑥(𝑡)∈𝐷_𝑟, then the trivial solution of (6) is stochastically stable in probability.

Proof. By the definition of 𝑉(𝑥), we obtain that 𝑉(0)=0 and that there exists a function 𝜑∈𝐾, such that 𝑉(𝑥)≥𝜑(|𝑥|),∀𝑥∈𝐷_𝑟. (20)
For any 𝜀∈(0,1) and ℎ>0, without loss of generality, we assume that ℎ<𝑟. Because 𝑉(𝑥) is continuous, we can find that 𝛿=𝛿(𝜀,ℎ,𝑡₀)>0, such that 𝑉(𝑥)≤𝜀𝜑(ℎ),∀𝑥∈𝐷_𝛿. (21)
It is obvious that 𝛿<ℎ. We fix the initial value 𝑥₀∈𝐷_𝛿 arbitrarily. Let 𝜇 be the first exit time of 𝑥(𝑡) from 𝐷_ℎ; that is, 𝜇=inf𝑡≥𝑡₀∶𝑥(𝑡)∉𝐷_ℎ. (22)
Let 𝜏=𝜇∧𝑡, for any 𝑡≥𝑡₀, we have 𝑉𝑥𝜇∧𝑡−𝑉𝑥₀(𝑥(𝜏(𝑥((𝑥((𝑥(𝑥𝑡=𝑉))−𝑉𝜏−1))+𝑉𝜏−1))−𝑉𝜏−2))+⋯+𝑉₀+1−𝑉𝑥₀=^𝜏−1_𝑡=𝑡0Δ𝑉(𝑥(𝑡)). (23)
Taking the expectation on both sides, it is easy to see that 𝐸𝑉𝑥𝜇∧𝑡≤𝑉𝑥₀. (24)
If 𝜇≤𝑡 and we note that |𝑥(𝜇∧𝑡)|=|𝑥(𝜇)|=ℎ, then 𝜑(ℎ)𝑃𝜇≤𝑡≤𝐸𝐼_{{𝜇≤𝑡}}𝑉𝑥𝜇≤𝐸𝑉𝑥𝜇∧𝑡. (25)
From (21) and (24), we achieve that 𝑃𝜇≤𝑡≤𝜀. (26)
Letting 𝑡→+∞, then 𝑃{𝜇<∞}≤𝜀; that is, 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,𝑡≥𝑡₀. (27)
Therefore, the trivial solution of (6) is stochastically stable.

Theorem 13. If there exists a positive definite and infinite small upper bounded function 𝑉(𝑥), such that 𝐸[Δ𝑉(𝑥(𝑡))]≤0, (28) then the trivial solution of (6) is stochastically uniformly stable in probability.

Proof. By the assumptions, there exist 𝜑∈𝐾 and 𝜓∈𝐾, such that 𝜑(|𝑥|)≤𝑉(𝑥)≤𝜓(|𝑥|),∀𝑥∈𝐷_𝑟. (29)
Let 𝜀∈(0,1) and ℎ>0 be arbitrary. Without loss of generality, we may assume that ℎ<𝑟. We define 𝜓(𝛿)=𝜑_ℎ(𝜀). (30)
Because of 𝜑∈𝐾, we can obtain 𝛿=𝜓⁻¹(𝜑_ℎ(𝜀)), and it has nothing to do with 𝑡₀.
Similar to the proof of Theorem 12, Theorem 13 is established.

Remark 14. We note that 𝐸[Δ𝑉(𝑥(𝑡))]≤0 in Theorems 12-13 corresponds to ℒ𝑉(𝑥)≤0 in the Itô systems. In Theorem 13, 𝑉(⋅) is not only a positive function, but it is also an infinite small upper bounded function; this is because Theorem 13 is stronger than Theorem 12.

Theorem 15. If there exist a function 𝜑∈𝐾 and a positive definite function 𝑉(𝑥)∈𝐶²(𝐷_𝑟), such that 𝐸[Δ𝑉(𝑥(𝑡)]≤−𝐸𝜑(|𝑥(𝑡)|) for all 𝑥(𝑡)∈𝐷_𝑟, then the trivial solution of (6) is stochastically asymptotically stable in probability.

Proof. From Theorem 12, we have that the trivial solution of (6) is stochastically stable. Fix 𝜀∈(0,1) arbitrarily; then there is 𝛿₀=𝛿₀(𝜀,𝑡₀)>0, such that 𝑃𝑥(𝑡<𝑟|)|2𝜀≥1−4, (31) when 𝑥₀∈𝐷_𝛿0.
Fix 𝑥₀∈𝐷_𝛿0 arbitrarily. By the assumptions on function 𝑉(𝑥), we know that 𝑉(0)=0 and that there exist two functions 𝜑₁, 𝜑∈𝐾, such that 𝜑₁𝑥(𝑥),𝐸[(𝑥(𝑡𝑥(𝑡(||)≤𝑉Δ𝑉))]≤−𝐸𝜑(|)|),∀𝑥∈𝐷_𝑟. (32)
Let 0<𝛽<|𝑥₀| arbitrarily, and choose 0<𝛼<𝛽, 0<𝜂<𝛼 sufficiently small; because of 𝑉(𝑥) being continuous, we can find that  0<𝛿=𝛿(𝜀,𝑡₀)<𝛿₀, such that 𝑉(𝑥)≤𝜀4𝜑𝜂,∀𝑥∈𝐷_𝛿. (33)
Define the stopping times 𝜇_𝛼=inf𝑡≥𝑡₀∶|𝑥(𝑡,𝜇)|≤𝛼_𝑟=inf𝑡≥𝑡₀∶|𝑥(𝑡≥𝑟)|2. (34)
Choose 𝜃 sufficiently large, such that 𝑃𝜇_𝛼𝜀<𝜃≥1−2. (35)
Let 𝜏=𝜇_𝛼∧𝜇_𝑟∧𝑡, for any 𝑡≥𝑡₀, we have 0≤𝑉(𝑥(𝜏))−𝑉𝑥₀(𝑥(𝜏(𝑥((𝑥((𝑥(𝑥𝑡=𝑉))−𝑉𝜏−1))+𝑉𝜏−1))−𝑉𝜏−2))+⋯+𝑉₀+1−𝑉𝑥₀=^𝜏−1_𝑡=𝑡0Δ𝑉(𝑥(𝑡)). (36)
Taking the expectation on both sides, we can derive that 0≤𝐸𝑉(𝑥(𝜏))≤𝑉𝑥₀−𝜑(𝛼)𝜏−𝑡₀. (37)
Hence, 𝑉𝑥₀𝜑(𝛼)≥𝐸𝜇_𝛼∧𝜇_𝑟∧𝑡−𝑡₀=𝐸𝜏−𝑡₀≥𝑡−𝑡₀𝑃𝜇_𝛼∧𝜇_𝑟≥𝑡. (38)
This means that 𝑃𝜇_𝛼∧𝜇_𝑟<∞=1. (39) By (31), 𝑃{𝜇_𝑟<∞}≤𝜀/4. So 𝑃𝜇_𝛼+𝜀<∞4≥𝑃𝜇_𝛼<∞+𝑃𝜇_𝑟𝜇<∞≤𝑃_𝛼∧𝜇_𝑟<∞=1, (40) which implies that 𝜀1−4≤𝑃𝜇_𝛼<∞. (41)
Hence, 𝑃𝜇_𝛼<𝜇_𝑟∧𝜃≥𝑃𝜇_𝛼<𝜃∩𝜇_𝑟𝜇=∞≥𝑃_𝛼<𝜃−𝑃𝜇_𝑟3<∞≥1−4𝜀. (42)
Define the two stopping times 𝜇𝜎=^𝛼if𝜇_𝛼<𝜇_𝑟𝜇∧𝜃,∞otherwise,_𝛽=inf𝑡>𝜎∶|𝑥(𝑡)|≥𝛽. (43)
Similar to the proof of (24), we can show that, for 𝑡≥𝜃, 𝐸𝑉(𝑥(𝜎∧𝑡))≥𝐸𝑉𝑥𝜇_𝛽∧𝑡. (44)
If 𝜇_𝛼≥𝜇_𝑟∧𝜃, then we note that |𝑥(𝜇_𝛽∧𝑡)|=|𝑥(𝜎∧𝑡)|=|𝑥(𝑡)|=𝜂, then 𝐸𝐼_{{𝜇𝛼<𝜇𝑟∧𝜃}}𝑉𝑥𝜇_𝛼≥𝐸𝐼_{{𝜇𝛼<𝜇𝑟∧𝜃}}𝑉𝑥𝜇_𝛽∧𝑡. (45)
By (31) and {𝜇_𝛼<𝜇_𝑟∧𝜃}⊃{𝜇_𝛽≤𝑡}, we have 𝜑₁𝜂𝑃𝜇_𝛽≤𝑡≤𝐸𝐼_{{𝜇𝛽∧𝑡}}𝑉𝑥𝜇_𝛽𝑥𝜇∧𝑡≤𝐸𝑉_𝛽∧𝑡. (46) Together with (33), we get 𝜀4≥𝑃𝜇_𝛽≤𝑡. (47)
Letting 𝑡→∞, we obtain 𝜀4≥𝑃𝜇_𝛽<∞. (48)
By (42), it follows that 1−𝜀≤𝑃𝜇_𝛼<𝜇_ℎ∧𝜃−𝑃𝜇_𝛽<∞≤𝑃𝜎<∞,𝜇_𝛽=∞. (49)
This means that 𝑃lim_𝑡→∞sup|𝑥(𝑡)|≤𝛽≥1−𝜀. (50)
Since 𝛽 is arbitrary, then we have 𝑃lim_𝑡→+∞𝑥(𝑡)=0≥1−𝜀. (51)

Theorem 16. If there exist a function 𝜑∈𝐾 and a positive definite, infinite small upper bounded function 𝑉(𝑥), such that 𝐸[Δ𝑉(𝑥(𝑡)]≤−𝐸𝜑(|𝑥(𝑡)|), for all 𝑥(𝑡)∈𝐷_𝑟, then the trivial solution of (6) is stochastically uniformly asymptotically stable in probability.

Proof. By the assumptions, there exist 𝜑₁, 𝜑₂, and 𝜑∈𝐾, such that 𝜑₁(|𝑥|)≤𝑉(𝑥)≤𝜑₂𝑥,𝐸[(𝑥(𝑡𝑥(𝑡.(||)Δ𝑉))]≤−𝐸𝜑(|)|) (52)
From Theorem 13, we know that the trivial solution of (6) is stochastically uniformly stable. Therefore, for every 𝜀>0 and ℎ>0, there exists 𝛿=𝛿(𝜀,ℎ)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀,𝑥₀<𝛿. (53)
According to Definition 3 we only need to show that, for every 𝜀>0 and ℎ>0, there exist 𝛿=𝛿(𝜀,ℎ)>0 and 𝑇(𝜀)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀+𝑇(𝜀),whenever𝑥₀<𝛿. (54)
We use a contradiction argument; take 𝑇(𝜀)=𝜑₂(𝛿₀)/𝜑(𝛿(𝜀,ℎ)); suppose, for any 𝜏, that 𝑡₀≤𝜏≤𝑡₀+𝑇(𝜀), such that |𝑥(𝜏)|≥𝛿(𝜀,ℎ). By 𝐸[Δ𝑉(𝑥(𝑡)]≤−𝜑(|𝑥(𝑡)|), we can show that 𝐸[Δ𝑉(𝑥(𝑡))]≤−𝜑(|𝑥(𝑡)|)≤−𝜑(𝛿(𝜀,ℎ)). (55)
So ^𝜏−1_𝑡=𝑡0𝐸[Δ𝑉(𝑥(𝑡))]≤^𝜏−1_𝑡=𝑡0(𝛿((𝛿(−𝜑𝜀,ℎ))=−𝜑𝜀,𝑟))𝜏−𝑡₀. (56) That is, 𝐸𝑉(𝑥(𝜏))−𝑉𝑥₀≤−𝜑(𝛿(𝜀,ℎ))𝜏−𝑡₀. (57)
Thus, 𝐸𝑉(𝑥(𝜏))≤𝑉𝑥₀−𝜑(𝛿(𝜀,ℎ))𝜏−𝑡₀≤𝜑₂𝑥₀−𝜑(𝛿(𝜀,ℎ))𝜏−𝑡₀<𝜑₂𝛿₀−𝜑(𝛿(𝜀,ℎ))𝜏−𝑡₀, (58) whenever 𝑡₀≤𝜏≤𝑡₀+𝑇(𝜀).
Especially, if 𝜏=𝑡₀+𝑇(𝜀), it follows that 𝐸𝑉𝑥𝑡₀+𝑇(𝜀)<𝜑₂𝛿₀−𝜑(𝛿(𝜀,ℎ))𝑇(𝜀)=0. (59)
This contradicts the positive definite property of 𝑉(𝑥). Then, we can prove that there exists 𝑡₁∈[𝑡₀,𝑡₀+𝑇(𝜀)], such that 𝑥𝑡₁<𝛿(𝜀,ℎ). (60)
According to Definition 3, we have 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,𝑡≥𝑡₁. (61)
Therefore, 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,𝑡≥𝑡₀+𝑇(𝜀). (62)
The proof is complete.

Remark 17. By comparing Theorems 12–15, we know that 𝐸[Δ𝑉(𝑥(𝑡))]≤−𝐸𝜑(|𝑥(𝑡)|) guarantees the system to be stochastically asymptotically stable. The difference between Theorems 15 and 16 is that 𝑉(𝑥) is additionally required to have an infinite small upper bound in Theorem 16, which ensures the trivial solution of (6) to be stochastically uniformly asymptotically stable in probability.

Theorem 18. If there exist a function 𝜑∈𝐾 and a positive definite radially unbounded function 𝑉(𝑥)∈𝐶²(𝐷_𝑟), such that 𝐸[Δ𝑉(𝑥(𝑡)]≤−𝐸𝜑(|𝑥(𝑡)|), for all 𝑥(𝑡)∈𝐷_𝑟, then the trivial solution of (6) is stochastically asymptotically stable in the large.

Proof. By Theorem 12, we know that the trivial solution of (6) is stochastically stable.
Let 𝜀∈(0,1) be arbitrary, and fix any 𝑥₀. Since 𝑉(𝑥) is radially unbounded, then we can choose 𝑟>|𝑥₀| sufficiently large, such that inf_{|𝑥|≥𝑟,𝑡≥𝑡0}𝑉(𝑥)≥4𝑉𝑥₀𝜀. (63)
Define the stopping time 𝜇_𝑟=inf𝑡≥𝑡₀∶|𝑥(𝑡)|≥𝑟. (64)
Similar to the proof of (24), we can obtain that, for any 𝑡≥𝑡₀, 𝑉𝑥₀≥𝐸𝑉𝑥𝜇_𝑟∧𝑡. (65)
From (63), we have 𝑉𝑥₀≥𝐸𝑉𝑥𝜇_𝑟≥𝑥∧𝑡4𝑉₀𝜀𝑃𝜇_𝑟≤𝑡. (66)
Together with (65), it yields that 𝑃𝜇_𝑟≤𝜀≤𝑡4. (67)
Let 𝑡→∞; we have 𝑃{𝜇_𝑟<∞}≤𝜀/4. That is to say, 𝑃𝑥(𝑡}𝜀{|)|≤𝑟≥1−4,∀𝑡≥𝑡₀. (68)
In the same way as that of the proof of Theorem 15, we can show that 𝑃lim_𝑡→+∞𝑥(𝑡)=0≥1−𝜀. (69)
This immediately implies that 𝑃{lim_𝑡→+∞𝑥(𝑡)=0}=1. The proof is complete.

Theorem 19. If there exist a function 𝜑∈𝐾 and a positive definite, infinite small upper bound and radially unbounded function 𝑉(𝑥), such that 𝐸[Δ𝑉(𝑥(𝑡)]≤−𝐸𝜑(|𝑥(𝑡)|), for all 𝑥(𝑡)∈𝐷_𝑟, then the trivial solution of (6) is stochastically uniformly asymptotically stable in the large in probability.

Proof. Under the conditions of Theorem 19, there exist 𝜑₁, 𝜑₂, and 𝜑∈𝐾, such that 𝜑₁(|𝑥|)≤𝑉(𝑥)≤𝜑₂𝑥,𝐸[(𝑥(𝑡𝑥(𝑡.(||)Δ𝑉))]≤−𝐸𝜑(|)|) (70)
By Theorem 13, we know that the trivial solution of (6) is stochastically uniformly stable.
In the following, we first verify that the trivial solution of (6) is uniformly bounded. Actually, for any 𝛼>0,  𝑡₀≥0, due to 𝜑₁, 𝜑₂∈𝐾, there exists 𝛽=𝛽(𝛼), such that 𝜑₁𝛽=𝜑₂(𝛼),thatis,𝛽=𝜑₁⁻¹𝜑₂(𝛼). (71)
It is easy to show that 𝐸𝑉(𝑥(𝑡))−𝑉𝑥₀=^𝑡−1_𝑡=𝑡0𝐸[Δ𝑉(𝑥(𝑡))]≤0,𝑡≥𝑡₀. (72)
When |𝑥₀|<𝛼, we have 𝜑₁(|𝑥(𝑡)|)≤𝐸𝑉(𝑥(𝑡))≤𝑉𝑥₀≤𝜑₂𝑥₀<𝜑₂(𝛼)=𝜑₁𝛽. (73)
Because of 𝜑₁being strictly increasing, so |𝑥(𝑡)|<𝛽, a.s., 𝑡≥𝑡₀. This implies that the trivial solution of (6) is uniformly bounded.
We further show that, for every 𝛼>0, 𝜀>0, and ℎ>0, there exists 𝑇(𝜀,𝛼)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀+𝑇(𝜀,𝛼),whenever𝑥₀<𝛼. (74)
As previously stated the trivial solution of (6) is stochastically uniformly stable. Therefore, for every 𝜀>0 and ℎ>0, there exists 𝛿=𝛿(𝜀,ℎ)>0, such that 𝑃{|𝑥(𝑡)|<ℎ}≥1−𝜀,∀𝑡≥𝑡₀,𝑥₀<𝛿. (75) The rest is similar to the proof of Theorem 16 and is thus omitted.

Remark 20. Theorems 18 and 19 are stronger versions of Theorems 15 and 16, respectively, where 𝑉(𝑥) is additionally required to be a radially unbounded function that is used to prove the stability in the large.

In what follows, we will discuss the 𝑝th moment exponential stability for (6).

Theorem 21. Suppose that there exist a function 𝑉(𝑥)∈𝐶²(𝐷_𝑟) and positive constants 𝑐₁, 𝑐₂, and 𝑐₃, such that 𝑐₁|𝑥|^𝑝≤𝑉(𝑥)≤𝑐₂|𝑥|^𝑝,𝐸[Δ𝑉(𝑥(𝑡))]≤−𝑐₃𝐸𝑉(𝑥(𝑡)). (76)
Then 𝐸|𝑥(𝑡)|^𝑝≤𝑐₂𝑐₁𝑥₀^𝑝𝑒^{−𝑐3(𝑡−𝑡0)},𝑡≥𝑡₀,∀𝑥₀∈𝑅^𝑛. (77)
That is, the trivial solution of (6) is 𝑝th moment exponentially stable.

Proof. Define the stopping time 𝜇_𝑛=inf𝑡≥𝑡₀∶|𝑥(𝑡)|≥𝑛,𝑛≥𝑥₀. (78)
It is easy to see that 𝜇_𝑛→∞ as 𝑛→∞ almost surely.
By 𝐸[Δ𝑉(𝑥(𝑡))]≤−𝑐₃𝐸𝑉(𝑥(𝑡)), we can derive that 𝐸𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−𝑉𝑥₀=𝐸𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−𝑒^{𝑐3(𝑡∧−1−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−1+𝑒^{𝑐3(𝑡∧𝜇𝑛−1−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−1−𝑒^{𝑐3(𝑡∧𝜏𝑛−2−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−2+⋯+𝑒^{𝑐3(𝑡0+1−𝑡0)}𝑉𝑥𝑡₀+1−𝑉𝑥₀=𝐸𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−𝑉𝑥𝑡∧𝜇_𝑛−1+𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−1−𝑒^{𝑐3(𝑡∧𝜇𝑛−1−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛−1+⋯+𝑒^𝑐3𝑉𝑥𝑡₀+1−𝑉𝑥₀=𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝐸Δ𝑉𝑥𝑡∧𝜇_𝑛−1+𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛1−11−𝑒^𝑐3+⋯+𝑒^𝑐3𝐸Δ𝑉𝑥𝑡₀+𝑒^𝑐3𝑉𝑥₀11−𝑒^𝑐3≤𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}−𝑐₃1+1−𝑒^𝑐3𝐸𝑉𝑥𝑡∧𝜇_𝑛−1+⋯+𝑒^𝑐3−𝑐₃1+1−𝑒^𝑐3𝑉𝑥₀≤0. (79)
By 𝑐₁|𝑥|^𝑝≤𝑉(𝑥)≤𝑐₂|𝑥|^𝑝, we have that 𝑐₁𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝐸𝑥𝑡∧𝜇_𝑛^𝑝≤𝐸𝑒^{𝑐3(𝑡∧𝜇𝑛−𝑡0)}𝑉𝑥𝑡∧𝜇_𝑛≤𝑉𝑥₀≤𝑐₂𝑥₀^𝑝. (80)
Letting 𝑛→∞, then 𝑐₁𝑒^{𝑐3(𝑡−𝑡0)}𝐸|𝑥(𝑡)|^𝑝≤𝑐₂𝑥₀^𝑝, (81) which implies (77).
As a corollary, Theorem 21 yields a sufficient criterion for the exponential stability in mean square sense.

Corollary 22. Suppose that there exist a function 𝑉(𝑥)∈𝐶²(𝐷_𝑟) and positive constants 𝑐₁, 𝑐₂, and 𝑐₃, such that 𝑐₁|𝑥|²≤𝑉(𝑥)≤𝑐₂|𝑥|²,𝐸[Δ𝑉(𝑥(𝑡))]≤−𝑐₃𝐸|𝑥(𝑡)|². (82) Then the trivial solution of (6) is exponentially stable in mean square.

4. Illustrative Examples

In this section, we present three simple examples to illustrate applications of the stability results developed in this paper. We will let 𝑤(𝑡) be a one-dimensional stochastic process defined on the complete probability space (Ω,𝐹,𝑃), such that 𝐸𝑤(𝑡)=0 and 𝐸[𝑤(𝑡)𝑤(𝑠)]=𝛿_𝑠𝑡, where 𝛿_𝑠𝑡 is the Kronecker delta.

Example 1. Consider the following equation: 𝑥(𝑡+1)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡)𝑤(𝑡),𝑥𝑡₀=𝑥₀, (83) where 𝐴 and 𝐵 are 𝑛×𝑛 matrices. Assume that there is a symmetric positive definite matrix 𝑃, such that 𝐴^𝑇𝑃𝐴+𝐵^𝑇𝑃𝐵−𝑃≤0. (84) Now, define the stochastic Lyapunov function 𝑉(𝑥)=𝑥^𝑇𝑃𝑥. It is obvious that 𝐸[Δ𝑉(𝑥(𝑡))]=𝐸𝑥^𝑇(𝑡)𝐴^𝑇𝑃𝐴+𝐵^𝑇𝑃𝐵−𝑃𝑥(𝑡)≤0. (85) By Theorem 12, we conclude that the trivial solution 𝑥≡0 of (83) is stochastically stable in probability.

Example 2. Consider the following stochastic difference equation: 𝑥()(𝑡)𝑥(𝑡)(𝑡)𝑥(𝑡)𝑤(𝑡)=[𝐴(𝑡)(𝑡)𝑤(𝑡𝑥(𝑡)((𝑡𝑥(𝑡),𝑡+1=𝐴+𝐵+𝐵)]=𝐻𝑡,𝑤)) (86) where 𝐴(𝑡), 𝐵(𝑡), and 𝐻(𝑡,𝑤(𝑡))=𝐴(𝑡)+𝐵(𝑡)𝑤(𝑡)=(ℎ_𝑖,𝑗(𝑡,𝑤(𝑡))) are all 2×2 matrix-valued functions defined on 𝑡=𝑡₀,  𝑡₀+1, and   𝑡₀+2,…,𝑥(𝑡₀)=𝑥₀∈𝑅^𝑛. Assume that max_𝑖=1,2𝐸²_𝑗=1ℎ_𝑖𝑗(𝑡,𝑤(𝑡))²<12, (87) for all 𝑥(𝑡)∈𝑅².
We define the Lyapunov function 𝑉(𝑥)=max_𝑖=1,2{|𝑥_𝑖|²}. It is positive definite and radially unbounded. Moreover, 𝐸𝑉(𝑥(𝑡+1))=max_𝑖=1,2𝐸²_𝑗=1ℎ_𝑖𝑗(𝑡,𝑤(𝑡))𝑥_𝑗(𝑡)²≤max_𝑖=1,2𝐸²_𝑗=1ℎ_𝑖𝑗(𝑡,𝑤(𝑡))²²_𝑗=1𝑥_𝑗(𝑡)²≤max_𝑖=1,2𝐸²_𝑗=1ℎ_𝑖𝑗(𝑡,𝑤(𝑡))²×max_𝑗=1,2𝐸2𝑥_𝑗(𝑡)²<max_𝑗=1,2𝐸𝑥_𝑗(𝑡)²=𝐸𝑉(𝑥(𝑡)). (88) That is, 𝐸[Δ𝑉(𝑥(𝑡))]<0. By Theorem 18, the trivial solution is stochastically asymptotically stable in the large.

Example 3. Consider a one-dimensional linear stochastic difference equation 𝑥(𝑡+1)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡)𝑤(𝑡), (89) where 𝑎, 𝑏 are all constants, and 𝑎²<𝑏²/4. We assume that there exist positive constants 𝑝 and 𝑐<1, such that 5𝑏²<4(1−𝑐)𝑝.
We define the Lyapunov function 𝑉(𝑥)=𝑝𝑥²; then 𝐸[Δ𝑉(𝑥(𝑡))]=5𝑏²4−𝑝𝐸𝑥²(𝑡)<−𝑐𝑝𝐸𝑥²(𝑡). (90)
By Corollary 22, the trivial solution is exponentially stable in mean square.

5. Conclusions

This paper has discussed the stability in probability for stochastic discrete-time systems. Using the method of Lyapunov functionals, some efficient criteria for the stability are obtained. Some results of the stability [13] for stochastic differential equations are generalized to stochastic discrete-time systems. There are some interesting problems such as the almost sure exponential stability and the stochastic nonlinear 𝐻_∞ control that merit further study.

Acknowledgments

This work is supported by NSF of China (Grants nos. 61174078 and 61170054), Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20103718110006), the Research Fund for the Taishan Scholar Project of Shandong Province of China, the SDUST Research Fund (Grant no. 2011KYTD105), and the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources (Grant no. LAPS13018).