The Stability and Stabilization of Stochastic Delay-Time Systems

Abstract

The aim of this paper is to investigate the stability and the stabilizability of stochastic time-delay deference system. To do this, we use mainly two methods to give a list of the necessary and sufficient conditions for the stability and stabilizability of the stochastic time-delay deference system. One way is in term of the operator spectrum and 𝐻-representation; the other is by Lyapunov equation approach. In addition, we introduce the notion of unremovable spectrum of stochastic time-delay deference system, describe the PBH criterion of the unremovable spectrum of time-delay system, and investigate the relation between the unremovable spectrum and the stabilizability of stochastic time-delay deference system.

1. Introduction

The stochastic time-delay system is one of the fundamental research branches in the theory of control systems, which is usually applied in the fields of electronics, machinery, chemicals, life sciences, economics, and so on. As is well known, the stability is an essential concept in linear system theory, which is relative to the system matrix root-clustering in subregions of the complex plane, and also the spectral operator approach is effective in the study of the eigenvalue placement of a matrix (see [1]). Since two classic books [2, 3] appeared, stochastic stability and stabilization of Itô differential systems have been investigated by many researchers for several decades; we refer the reader to [4–6] and the references therein. More specifically, for linear time-invariant stochastic (LTIS) systems, most work is concentrated on the investigation of mean square stabilization, which has important applications in system analysis and design. Some necessary and sufficient conditions for the mean square stabilization of LTIS systems have been obtained in terms of generalized algebraic Riccati equation (GARE) or linear matrix inequality (LMI) in [7–14] or spectra of some operators in [5, 9, 15]. For the stochastic delay-time systems, the present results were mainly obtained by Lyapunov functional approach. We concentrate our attention upon the stability and stabilization of stochastic systems by the operator spectrum.

The structure of this paper is as follows. In Section 2, with the aid of the operator spectrum, 𝐻-representation, and Lyapunov equation approach, some necessary and sufficient conditions are given for the stability and the stabilizability of stochastic delay-time systems. In Section 3, the unremovable spectrum of stochastic delay-time systems is introduced, and PBH criterion of the stabilizability of stochastic delay-time systems is presented.

For convenience, we adopt the following traditional notations. 𝑆^𝑛: the set of all symmetric matrices, whose components may be complex; 𝑁={0,1,2,…}; 𝐴^′(Ker(𝐴)): the transpose (kernel space) of a matrix 𝐴; 𝐴≥0  (𝐴>0) is a positive semidefinite (positive-definite) symmetric matrix 𝐴; 𝐼: identity matrix; 𝜎(𝐿): spectral set of the operator or matrix 𝐿; 𝐶⁻(𝐶⁻⁰): the open left (closed left) hand side complex plane. 𝐷(0,𝛼)={𝜆∣‖𝜆‖<𝛼}; ‖⋅‖ is the 𝑙₂-norm; 𝐿²_ℱ𝑡(𝑅⁺,𝑅^𝑛𝑥): space of nonanticipative stochastic processes 𝑥(𝑡)∈𝑅^𝑛𝑥 with respect to an increasing 𝜎-algebra {ℱ_𝑡}_𝑡≥0 satisfying 𝐸‖𝑥(𝑡)‖²<∞. Finally, we make the assumption throughout this paper that all systems have real coefficients.

2. The Stability of Stochastic Delay-Time Systems

In this section, we will investigate the stability and stabilizability of the stochastic time-delay deference system using the spectrum of operator and Lyapunov equation approach. At first, we introduce a Lyapunove operator. Consider the following linear difference system with constant delays: 𝑥(𝑡+1)=𝐹₀𝑥(𝑡)+𝐺₀𝑥(𝑡)𝑤(𝑡)+∑^𝑚_𝑗=1𝐹_𝑗𝑥𝑡−𝑗+𝐺_𝑗𝑥𝑡−𝑗𝑤(𝑡),𝑡∈𝑁, (1) with the initial condition 𝑥(𝑘)=𝜑(𝑘),𝑘=0,−1,−2,…,−𝑚. (2) Here, 𝑥∈𝑅^𝑛 is a column vector, 𝐹_𝑗,𝐺_𝑗∈𝑅^𝑛×𝑛, 𝑗=0,1,…,𝑚, are constant coefficient matrices, 𝜑(𝑘)∈𝑅^𝑛 is a deterministic initial condition, {𝑤(𝑡)∈𝑅,𝑡∈𝑁} is a sequence of real random variables defined on a complete probability space {Ω,ℱ,ℱ_𝑡,𝜇} which is a wide sense stationary, second-order process with 𝐸(𝑤(𝑡))=0 and 𝐸(𝑤(𝑡)𝑤(𝑠))=𝛿_𝑠,𝑡, where 𝛿_𝑠,𝑡 is the Kronecker delta with ℱ_𝑡=𝜎{𝑤(𝑠)∶0≤𝑠≤𝑡}.

Definition 1. The trivial stationary solution 𝑥(𝑡)=0 of the system (1) is called mean square stable if, for any arbitrarily small number 𝜀>0, there exists a number 𝛿>0, when ‖𝜑‖<𝛿, such that 𝐸‖𝑥(𝑡)‖²<𝜀, (3) for a solution 𝑥(𝑡)=𝑥(𝑡,𝜑) of (1).

Definition 2. The trivial stationary solution 𝑥(𝑡)=0 of the system (1) is called asymptotically mean square stable if it is stable in the sense of Definition 1 and, moreover, any solution 𝑥(𝑡)=𝑥(𝑡,𝜑) of (1) satisfies lim_𝑡→+∞𝐸‖𝑥(𝑡)‖²=0. (4)

We consider the problem of finding criteria and sufficient conditions for the mean square asymptotic stability of the trivial stationary solution 𝑥(𝑡)=0 by operator spectra. Since the stochastic system (1) is a system with time-delays, it seems impossible to construct the operator directly for (1) like the operator in [15]. So, first of all, we introduce the following column vector 𝑥(𝑡) of new variables of dimension 𝑛(𝑚+1): 𝑥(𝑡)=𝑥^′(𝑡),𝑥^′(𝑡−1),…,𝑥^′(𝑡−𝑚)^′. (5)

The stochastic system (1) with time-delays can now be written in the form of an equivalent stochastic system of dimension 𝑛(𝑚+1) without delay; namely, 𝑥(𝑡+1)=[𝐹+𝐺𝜔(𝑡)]𝑥(𝑡), (6) where 𝐹 and 𝐺 denote the following 𝑛(𝑚+1)×𝑛(𝑚+1) matrices: 𝐹𝐹=⁰𝐹₁⋯𝐹_𝑚−1𝐹_𝑚,𝐺𝐼0⋯00⋮⋮⋮⋮⋮00⋯𝐼0𝐺=⁰𝐺₁⋯𝐺_𝑚−1𝐺_𝑚.00⋯00⋮⋮⋮⋮⋮00⋯00 (7)

If we set 𝑋(𝑡)=𝐸𝑥(𝑡)𝑥^′(𝑡), 𝑋(𝑡) satisfies the following difference equation: 𝑋(𝑡+1)=𝐹𝑋(𝑡)𝐹^′+𝐺𝑋(𝑡)𝐺^′. (8) Motivated by (8), we introduce the following linear Lyapunov operator: ℒ_𝐹,𝐺∶𝑋∈𝑆^𝑛(𝑚+1)⟼𝐹𝑋(𝑡)𝐹^′+𝐺𝑋(𝑡)𝐺^′∈𝑆^𝑛(𝑚+1). (9) With the use of the Kronecker matrix product, the matrix equation (8) can be rewritten in the vector matrix form as follows: 𝑋^→(𝑡+1)=𝐹𝑋^→, (10) where 𝑋^→(𝑡) denotes the 𝑛²(𝑚+1)²-dimensional column vector 𝑋^→(𝑡)=𝑋_1,1(𝑡),…,𝑋_1,𝑛(𝑡)𝑋,…,_{1,𝑛(𝑚+1)}(𝑡),…,𝑋_{𝑛(𝑚+1),𝑛(𝑚+1)}(𝑡)^′ (11) and 𝐹∈𝑅^{𝑛2(𝑚+1)2×𝑛2(𝑚+1)2} has the form 𝐹=𝐹⊗𝐹+𝐺⊗𝐺.

Now, we are in a position to give a spectral description for the stability of system (1) by 𝐻-representation in [14].

Lemma 3. Let 𝐻_𝑛(𝑚+1) be a 𝑛²(𝑚+1)²×(𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2) matrix and rank(𝐻_𝑛(𝑚+1))=(𝑛(𝑚+1)[𝑛(𝑚+1)+1])/2; then 𝐻^′_𝑛(𝑚+1)𝐻_𝑛(𝑚+1) is invertible.

Theorem 4. The trivial solution 𝑥(𝑡)=0 of system (1) is asymptotically mean square stable if and only if 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1).

Proof. If we set 𝑋(𝑡)=𝐸𝑥(𝑡)𝑥^′(𝑡), 𝑋(𝑡) satisfies 𝑋(𝑡+1)=𝐹𝑋(𝑡)𝐹^′+𝐺𝑋(𝑡)𝐺^′,𝑋(𝑘)=𝑥(𝑘)𝑥^′(𝑘)∈𝑆^𝑛(𝑚+1),𝑘=0,−1,−2,…,−𝑚,𝑡∈𝑁. (12) Since 𝑋(⋅) is real symmetric, (12) is a linear matrix equation with 𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2 different variables; that is, it is in fact an 𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2 th-order linear system. We define a map ℒ from 𝑆^𝑛(𝑚+1) to 𝐶^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} as follows.
For any 𝑌=(𝑌_𝑖𝑗)_{𝑛(𝑚+1)×𝑛(𝑚+1)}∈𝑆^𝑛(𝑚+1), set 𝑌=ℒ(𝑌)=𝑌₁₁,…,𝑌_{1,𝑛(𝑚+1)},…,𝑌_{𝑛(𝑚+1)−1,𝑛(𝑚+1)−1,}𝑌_{𝑛(𝑚+1)−1,𝑛(𝑚+1)},…,𝑌_{𝑛(𝑚+1),𝑛(𝑚+1)}^′; (13) then there exists an unique matrix 𝜃(𝐻_𝑛(𝑚+1))∈𝑅^{(𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2)×(𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2)}, by 𝐻-representation of [14], such that (12) is equivalent to 𝑋(𝑡+1)=ℒℒ_𝐹,𝐺(𝑋)=𝜃𝐻_𝑛(𝑚+1)𝑋(𝑡),𝑋(𝑘)=𝐻^′_𝑛()𝑚+1𝐻_𝑛(𝑚+1)⁻¹𝐻^′_𝑛()𝑚+1×𝑋^→(𝑘)∈𝑅^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2},𝑘=0,−1,−2,…,−𝑚,𝑡∈𝑁, (14) where 𝜃(𝐻_𝑛(𝑚+1))=[𝐻^′_𝑛(𝑚+1)𝐻_𝑛(𝑚+1)]⁻¹𝐻^′_𝑛(𝑚+1)[𝐹⊗𝐹+𝐺⊗𝐺]𝐻_𝑛(𝑚+1), 𝑋∈𝑅^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2}. Obviously, since the system of (14) for moments is deterministic, the proof of the theorem is carried out by the standard method for deterministic difference equations. Seeking the general solution of system (14) in the exponential form 𝑋(𝑡)=𝑐𝜆^𝑡, where 𝑐,𝜆 = const, we arrive at the characteristic equation det(𝜆𝐼−𝜃(𝐻_𝑛(𝑚+1)))=0. That is, lim_𝑡→+∞𝐸𝑥(𝑡)𝑥^′(𝑡)=lim_𝑡→+∞𝑋(𝑡)=0⇔lim_𝑡→+∞𝑋(𝑡)=0⇔𝜎(𝜃(𝐻_𝑛(𝑚+1)))⊂𝐷(0,1).
By (9) and (14), for any eigenvalue 𝜆 and its corresponding eigenvector 𝑌=(𝑌_𝑖𝑗)_𝑛×𝑛∈𝑆^𝑛 of ℒ_𝐹,𝐺, from ℒ_𝐹,𝐺(𝑌)=𝜆𝑌, we have ℒ(ℒ_𝐹,𝐺(𝑌))=𝜃(𝐻_𝑛(𝑚+1))𝑌=𝜆𝑌, which yields 𝜎(ℒ_𝐹,𝐺)=𝜎(𝜃(𝐻_𝑛(𝑚+1))). The above discussion concludes the proof of Theorem 4. The proof of Theorem 4 is complete.

Remark 5. In Theorem 4, a necessary and sufficient condition for the asymptotically mean square stability of system (1) via the spectrum of ℒ_𝐹,𝐺 is presented, which can be called “spectral criterion.”

Theorem 6. The trivial solution 𝑥=0 of system (1) is asymptotically mean square stable if and only if, for any 𝑄∈𝑆^𝑛(𝑚+1) with 𝑄>0, there exists a 𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−ℒ_𝐹,𝐺(𝑃)=𝑄. (15)

Proof. We introduce an 𝑛²(𝑚+1)²-parameter stochastic Lyapunov function as a quadratic form: 𝑉𝑋^→(𝑡)=𝑋^→′𝑃𝑋^→,𝑃∈𝑆^𝑛(𝑚+1). (16)
The role of parameters is played by 𝑛²(𝑚+1)² elements of the positive-definite matrix, which should be determined. The statement of the theorem can be established in a way that is standard for the method of Lyapunov functions for stochastic difference equations. So the trivial solution 𝑥(𝑡)=0 of system (6) is asymptotically mean square stable if and only if for any 𝑄>0, the Lyapunov equation (15) has a solution 𝑃>0. By the proof of Theorem 4, the trivial solution 𝑥=0 of system (1) is asymptotically mean square stable if and only if the trivial solution 𝑥(𝑡)=0 of system (6) is asymptotically mean square stable. The proof of Theorem 6 is complete.

From the proof of Theorem 4 and the method of Lyapunov functions for difference equations, we immediately get the following result.

Theorem 7. The trivial solution 𝑥=0 of system (1) is asymptotically mean square stable if and only if, for any 𝑄∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} with 𝑄>0, there exists a 𝑃∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝐹^′𝑃𝐹=𝑄. (17)

Corollary 8. If 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1), then 𝜎(𝐹)⊂𝐷(0,1).

Proof. By Theorems 4 and 6, 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1) holds if and only if there is a matrix 𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−ℒ_𝐹,𝐺(𝑃)=𝑄, (18) for any 𝑄>0. So, there exists a 𝑃∈𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝐹^′𝑃𝐹>0, (19) which is equivalent to 𝜎(𝐹)⊂𝐷(0,1); that is, the system 𝑥(𝑡+1)=𝐹₀𝑥(𝑡)+^𝑗=𝑚₁𝐹_𝑗𝑥,𝑥(𝑘)(𝑘)𝑡−𝑗=𝜑,𝑘=0,−1,−2,…,−𝑚,𝑡∈𝑁, (20) is asymptotically Lyapunov stable. The proof of Corollary 8 is complete.

Now, we present some results about mean square stability of system (1). From the process of Theorems 4–7, we easily obtain the following Theorems 9–10, so we omit their proofs.

Theorem 9. If the trivial stationary solution 𝑥=0 of the system (1) is mean square stable, then 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1).

Theorem 10. 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1) if and only if one of the following conditions holds.(1)For any 𝜀>0 and 𝑄>0, the following Lyapunov equation 𝑃−ℒ_{𝑒−𝜀𝐹,𝑒−𝜀𝐺}(𝑃)=𝑄 (21) has a positive-definite solution 𝑃.(2)𝜎(ℒ_{𝑒−𝜀𝐹,𝑒−𝜀𝐺})⊂𝐷(0,1).

Corollary 11. If 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1), then 𝜎(𝐹)⊂𝐷(0,1).

Proof. Since 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1), we have, for any sufficient small 𝜀>0, 𝜎(ℒ_{𝑒−𝜀𝐹,𝑒−𝜀𝐺})⊂𝐷(0,1). By Theorem 4, (𝑒^−𝜀𝐹,𝑒^−𝜀𝐺) is mean square stable, which implies 𝜎(ℒ_{𝑒−𝜀𝐹})⊂𝐷(0,1) by Corollary 8. Let 𝜀→0, we have 𝜎(𝐹)⊂𝐷(0,1) by continuity of spectrum [16]. The proof of Corollary 11 is complete.

Remark 12. 𝜎(𝐹)⊂𝐷(0,1) does not imply 𝜎(ℒ_𝐹,𝐺)⊂𝐷(0,1), which is one of the essential differences between stochastic system and deterministic system.

Theorem 13. The trivial solution 𝑥=0 of system (1) is mean square stable if and only if, for any 𝑄≥0, there exists a 𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−ℒ_𝐹,𝐺(𝑃)=𝑄. (22)

Proof. We introduce an 𝑛²(𝑚+1)²-parameter stochastic Lyapunov function as a quadratic form: 𝑉𝑋^→(𝑡)=𝑋^→′𝑃𝑋^→,𝑃∈𝑅^{𝑛(𝑚+1)×𝑛(𝑚+1)}. (23) By the method of Lyapunov functions for stochastic difference equations, we can get the result. The proof of Theorem 13 is complete.

Now, we give an example to show how to solve the spectrum of stochastic time-delay deference system by 𝐻-representation.

Example 14. Consider the following stochastic system: 𝑥(𝑡+1)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡−1)+𝑐𝑥(𝑡)𝜔(𝑡)+𝑑𝑥(𝑡−1)𝜔(𝑡). (24) Letting 𝑥(𝑡)=(𝑥(𝑡),𝑥(𝑡−1))^′, 𝑥(𝑡+1)=𝑎𝑏10𝑥(𝑡)+𝑐𝑑00𝑥(𝑡)𝜔(𝑡). (25) Letting 𝑋(𝑡)=𝐸𝑥(𝑡)𝑥^′(𝑡), 𝑋(𝑡+1)=𝑋(𝑡)+𝑋(𝑡).𝑎𝑏10𝑎1𝑏0𝑐𝑑00𝑐0𝑑0 (26) Letting 𝐹=^𝑎𝑏₁₀, 𝐺=^𝑐𝑑₀₀, 𝑋^→(𝑡)=(𝑥₁₁,𝑥₁₂,𝑥₁₂,𝑥₂₂)^′, 𝑋^→(𝑡+1)=(𝐹⊗𝐹+𝐺⊗𝐺)𝑋^→(𝑡)=𝑎²+𝑐²𝑎𝑏+𝑐𝑑𝑎𝑏+𝑐𝑑𝑏²+𝑑²𝑎0𝑏0𝑎𝑏001000^→𝑋(𝑡). (27) Choose 𝐻₂=.100010010001 (28) Let 𝑋(𝑡)=(𝑥₁₁,𝑥₁₂,𝑥₂₂)^′; then 𝑋^→(𝑡)=𝐻₂𝑋(𝑡) and 𝑋(𝑡+1)=𝐻^′₂𝐻₂⁻¹𝐻^′₂(𝐹⊗𝐹+𝐺⊗𝐺)𝐻₂𝑋(𝑡). (29) So 𝜎((𝐻^′₂𝐻₂)⁻¹𝐻^′₂(𝐹⊗𝐹+𝐺⊗𝐺)𝐻₂)=𝜎(ℒ) if we choose 𝑎=1, 𝑏=0, 𝑐=0, 𝑑=1/4, and 𝜎(ℒ)={0,1/2,1/2}.
For a state feedback control law 𝑢(𝑡)=𝐾𝑥(𝑡), we introduce a linear operator ℒ_𝐾 associated with the closed-loop system: 𝑥(𝑡+1)=𝐹₀𝑥(𝑡)+𝑀₀𝑢(𝑡)+𝐺₀𝑥(𝑡)+𝑁₀𝑢(𝑡)𝑤(𝑡)+^𝑚_𝑗=1𝐹_𝑗𝑥𝑡−𝑗+𝑀_𝑗𝑢+𝐺𝑡−𝑗_𝑗𝑥𝑡−𝑗+𝑁_𝑗𝑢𝑤(𝑡,𝑥(𝑘)(𝑘)𝑡−𝑗)=𝜑∈𝑅^𝑛,𝑘=0,−1,…,−𝑚,𝑡∈𝑁, (30) where 𝑥∈𝑅^𝑛 is a column vector, 𝐹_𝑗,𝐺_𝑗∈𝑅^𝑛×𝑛, 𝑗=0,1,…,𝑚 are constant coefficient matrices, 𝜑(𝑘) is a deterministic initial condition, and 𝑢(𝑡)∈𝑅^𝑛 is a control input.

Definition 15. The trivial stationary solution 𝑥(𝑡)=0 of the system (30) is called mean square stabilization if there exists an input feedback 𝐾 such that, for any arbitrarily small number 𝜀>0, one can find a number 𝛿>0, when ‖𝜑‖<𝛿, satisfying 𝐸‖𝑥(𝑡)‖²<𝜀, (31) for a solution 𝑥(𝑡)=𝑥(𝑡,𝜑) of (30).

Definition 16. The trivial stationary solution 𝑥(𝑡)=0 of the system (30) is called asymptotically mean square stabilization if it is stable in the sense of Definition 15 and, moreover, lim_𝑡→+∞𝐸‖𝑥(𝑡)‖²=0. (32)

We introduce the following column vectors 𝑥(𝑡) and 𝑢(𝑡) of new variables of dimension 𝑛(𝑚+1): 𝑥(𝑡)=𝑥^′(𝑡),𝑥^′(𝑡−1),…,𝑥^′(𝑡−𝑚)^′,𝑢(𝑡)=𝑢^′(𝑡),𝑢^′(𝑡−1),…,𝑢^′(𝑡−𝑚)^′. (33)

The stochastic system (30) with time-delays can now be written in the form of an equivalent stochastic system of dimension 𝑛(𝑚+1) without time-delay; namely, 𝑥(𝑡+1)=𝐹𝑥(𝑡)+𝑀𝑢(𝑡)+𝐺𝑥(𝑡)+𝑁𝑢(𝑡)𝜔(𝑡), (34) where 𝐹𝐹=⁰𝐹₁⋯𝐹_𝑚−1𝐹_𝑚,𝐼0⋯00⋮⋮⋯⋮⋮00⋯𝐼0𝐺𝐺=⁰𝐺₁⋯𝐺_𝑚−1𝐺_𝑚.00⋯00⋮⋮⋮⋮⋮00⋯00 (35)𝑀𝑀=⁰𝑀₁⋯𝑀_𝑚−1𝑀_𝑚,00⋯00⋮⋮⋮⋮⋮00⋮00𝑁𝑁=⁰𝑁₁⋯𝑁_𝑚−1𝑁_𝑚.00⋯00⋮⋮⋮⋮⋮00⋯00 (36) Take a control input 𝑢(𝑡)=𝐾𝑥(𝑡) with 𝐾=,𝐾0⋯000𝐾⋯00⋮⋮⋮⋮⋮00⋯0𝐾 (37) and set 𝑋(𝑡)=𝐸𝑥(𝑡)𝑥^′(𝑡); 𝑋(𝑡) satisfies the following difference equation: 𝑋(𝑡+1)=𝐹+𝑀𝐾)𝑋(𝑡)(𝐹+𝑀𝐾^′+𝐺+𝑁𝐾𝑋(𝑡)𝐺+𝑁𝐾^′. (38) Motivated by (38), we introduce the following linear Lyapunov operator: ℒ_𝐾∶𝑋∈𝑆^𝑛(𝑚+1)⟼𝐹+𝑀𝐾𝑋(𝑡)𝐹+𝑀𝐾^′+𝐺+𝑁𝐾𝑋(𝑡)𝐺+𝑁𝐾^′∈𝑆^𝑛(𝑚+1). (39) With the use of the Kronecker matrix product, the matrix equation (38) can be rewritten in the vector matrix form as follows: 𝑋^→(𝑡+1)=𝐴𝑋^→, (40) where 𝑋^→(𝑡) denotes the 𝑛²(𝑚+1)²-dimensional column vector 𝑋^→(𝑡)=𝑋_1,1(𝑡),…,𝑋_1,𝑛(𝑡)𝑋,…,_{1,𝑛(𝑚+1)}(𝑡),…,𝑋_{𝑛(𝑚+1),𝑛(𝑚+1)}(𝑡)^′ (41) and 𝐴=(𝐹+𝑀𝐾)⊗(𝐹+𝑀𝐾)+(𝐺+𝑁𝐾)⊗(𝐺+𝑁𝐾)∈𝑆^{𝑛2(𝑚+1)2}.

From Theorems 4–13, we can easily obtain the following results.

Corollary 17. The trivial solution 𝑥=0 of system (30) is asymptotical mean square stabilizaton if and only if 𝜎(ℒ_𝐾)⊂𝐷(0,1).

Corollary 18. The trivial solution 𝑥(𝑡)=0 of system (30) is asymptotically mean square stable if and only if, for any 𝑄∈𝑆^𝑛(𝑚+1) with 𝑄>0, there exists a 𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−ℒ_{𝐹+𝑀𝐾,𝐺+𝑁𝐾}(𝑃)=𝑄. (42)

Corollary 19. The trivial solution 𝑥=0 of system (30) is asymptotically mean square stable if and only if, for any 𝑄∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} with 𝑄>0, there exists a 𝑃∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝜃𝐻^𝐾_𝑛()𝑚+1^′𝑃𝜃𝐻^𝐾_𝑛()𝑚+1=𝑄, (43) where 𝜃𝐻^𝐾_𝑛()𝑚+1=𝐻^′_𝑛()𝑚+1𝐻_𝑛(𝑚+1)⁻¹𝐻^′_𝑛()𝑚+1×𝐹+𝑀𝐾⊗𝐹+𝑀𝐾+𝐺+𝑁𝐾⊗𝐺+𝑁𝐾𝐻_𝑛(𝑚+1). (44)

Corollary 20. The trivial solution 𝑥=0 of system (30) is asymptotically mean square stable if and only if, for any 𝑄∈𝑆^{𝑛2(𝑚+1)2}, 𝑄>0, there exists a 𝑃∈𝑆^{𝑛2(𝑚+1)2} such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝐴^′𝑃𝐴=𝑄. (45)

Corollary 21. The trivial solution 𝑥=0 of system (30) is mean square stable if and only if, for any 𝑄∈𝑆^𝑛(𝑚+1) with 𝑄≥0, there exists a 𝑃∈𝑆^𝑛(𝑚+1) such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−ℒ_{𝐹+𝑀𝐾,𝐺+𝑁𝐾}(𝑃)=𝑄. (46)

Corollary 22. The trivial solution 𝑥=0 of system (1) is mean square stable if and only if, for any 𝑄∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} with 𝑄≥0, there exists a 𝑃∈𝑆^{𝑛(𝑚+1)[𝑛(𝑚+1)+1]/2} such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝜃𝐻^𝐾_𝑛()𝑚+1^′𝑃𝜃𝐻^𝐾_𝑛()𝑚+1=𝑄, (47) where 𝜃𝐻^𝐾_𝑛()𝑚+1=𝐻^′_𝑛()𝑚+1𝐻_𝑛(𝑚+1)⁻¹×𝐻^′_𝑛()𝑚+1[𝐹⊗𝐹+𝐺⊗𝐺]𝐻_𝑛(𝑚+1). (48)

Corollary 23. The trivial solution 𝑥=0 of system (1) is mean square stable if and only if, for any 𝑄≥0, there exists a 𝑃∈𝑆^{𝑛2(𝑚+1)2} such that 𝑃>0 and 𝑃 is a solution of the following Lyapunov equation: 𝑃−𝐴^′𝑃𝐴=𝑄. (49)

3. Popov-Belevith-Hautus Criterion of the Stabilizability

In this section, we will investigate the properties of unremovable spectrum of time-delay deference system and the relation between unremovable spectrum and the stabilizability of time-delay deference system. Consider the following linear stochastic system with time-delays: 𝑥(𝑡+1)=𝐹₀𝑥(𝑡)+𝑀₀𝑢(𝑡)+𝐺₀𝑥(𝑡)+𝑁₀𝑢(𝑡)𝑤(𝑡)+^𝑚_𝑗=1𝐹_𝑗𝑥𝑡−𝑗+𝑀_𝑗𝑢+𝐺𝑡−𝑗_𝑗𝑥𝑡−𝑗+𝑁_𝑗𝑢𝑤(𝑡,𝑥(𝑘)(𝑘)𝑡−𝑗)=𝜑∈𝑅^𝑛,𝑘=0,−1,−2,…,−𝑚,𝑡∈𝑁. (50)

Definition 24. We say that 𝜆 is an unremovable spectrum of system (50) with state feedback if there exists 𝑍≠0∈𝑆^𝑛(𝑚+1), such that,for any 𝐾∈𝑅^{𝑛(𝑚+1)×𝑛(𝑚+1)}, ℒ^∗_𝐾(𝑍)=(𝐹+𝑀𝐾)^′𝑍𝐹+𝑀𝐾+𝐺+𝑁𝐾^′𝑍𝐺+𝑁𝐾=𝜆𝑍 (51) holds.

Remark 25. It is easy to see that the operator ℒ^∗_𝐾 is the adjoint operator of the operator ℒ_𝐾 with the inner product ⟨𝑍,𝑌⟩=trace(𝑍^∗,𝑌) for any 𝑍,𝑌∈𝑆^𝑛(𝑚+1). As we restrict the coefficients to real matrices, 𝜎(ℒ^∗_𝐾)=𝜎(ℒ_𝐾). By Corollaries 17–23, we know that any one of them can characterize the stabilizability of system (50).

Obviously, if 𝜆 is an unremovable spectrum, then it can be regarded as an uncontrollable mode as in deterministic systems. We give a theorem with respect to the unremovable spectrum below.

Theorem 26. (Stochastic PBH criterion) 𝜆 is an unremovable spectrum of system (50) if and only if there exists 𝑍≠0∈𝑆^𝑛(𝑚+1), such that the following three equalities hold simultaneously: 𝐹^′𝑍𝐹+𝐺^′𝑍𝐺=𝜆𝑍,𝐹^′𝑍𝑀+𝐺^′𝑍𝑁=0,𝑁^′𝑍𝑁=−𝑀^′𝑍𝑀. (52)

Proof. Note that (51) can be written as 𝐹^′𝑍𝐹+𝐾^′𝑀^′𝑍𝑀𝐾+𝐺^′𝑍𝐺^′+𝐾^′𝑁^′𝑍𝑁𝐾+𝐹^′𝑍𝑀+𝐺^′𝑍𝑁𝐾+𝐾^′𝑀^′𝑍𝐹+𝑁^′𝑍𝐺=𝜆𝑍, (53) so if (52) holds, then (53) automatically holds. So the sufficiency is proved.
To prove the necessity, we first take 𝐾=0 in (51), then 𝐹^′𝑍𝐹+𝐺𝑍𝐺^′=𝜆𝑍 (54) holds. Again, from (53), it follows that 𝐾^′𝑀𝑍𝑀^′𝐾+𝐾𝑁𝑍𝑁^′𝐾^′+𝐹^′𝑍𝑀+𝐺^′𝑍𝑁𝐾+𝐾^′𝑀^′𝑍𝐹+𝑁^′𝑍𝐺=0. (55) Let ℱ=𝐹^′𝑍𝑀+𝐺^′𝑍𝑁, ℳ=𝑀𝑍𝑀^′, 𝒩=𝑁𝑍𝑁^′; then (55) becomes ℱ𝐾+𝐾^′ℱ^′=−𝐾^′ℳ𝐾−𝐾^′𝒩𝐾. (56) Since the left-hand side in (56) is linear with respect to 𝐾, we must have ℳ+𝒩=0. In fact, due to the linearity of the following equation 𝐾+𝐾^′(ℳ+𝒩)𝐾+𝐾=4𝐾^′(ℳ+𝒩)𝐾=2𝐾^′(ℳ+𝒩)𝐾, (57)𝐾^′(ℳ+𝒩)𝐾=0. Because of the arbitrariness of 𝐾, it is necessary that ℳ+𝒩=0; that is, 𝑀𝑍𝑀^′=−𝑁𝑍𝑁^′. To prove ℱ=0 or 𝐹^′𝑍𝑀+𝐺^′𝑍𝑁=0, we note that (56) becomes 𝐾^′ℱ^′=−ℱ𝐾. Denote ℱ=(𝑓_𝑖𝑗)_{𝑛(𝑚+1)×𝑛(𝑚+1)}, and take 𝐾=𝐾_𝑖𝑗=𝑘_{𝑙𝑠𝑛(𝑚+1)×𝑛(𝑚+1)}=1,𝑙=𝑖,𝑠=𝑗,0,otherwise. (58) From 𝐾^′ℱ^′=−ℱ𝐾, one knows that 𝑓_𝑖𝑗=0, 𝑖,𝑗=1,2,…,𝑛(𝑚+1); that is, ℱ=0. The proof of Theorem 26 is complete.

Theorem 27. If system (50) is asymptotically mean square stabilizable, then all unremovable spectra of system (50) must belong to 𝐷(0,1).

Proof. If there is an unremovable spectrum 𝜇 of (50) with |𝜇|≥1, then, by Theorem 26, there exists 𝑍≠0∈𝑆^𝑛(𝑚+1), such that the following three equalities hold simultaneously: 𝐹^′𝑍𝐹+𝐺^′𝑍𝐺=𝜇𝑍,𝐹^′𝑍𝑀+𝐺^′𝑍𝑁=0,𝑁^′𝑍𝑁=−𝑀^′𝑍𝑀. (59) So, for any feedback gain 𝐾, we obtain ℒ_𝐾(𝑍)=𝜇𝑍,𝜇∉𝐷(0,1). The proof of Theorem 27 is complete. It follows that, for any 𝐾, 𝜇∈𝜎(ℒ_𝐾), which contradicts the asymptotical mean square stabilization of system (50).
By Theorems 26 and 27, the deterministic Popov-Belevith-Hautus Criterion can be stated in another form as follows.

Corollary 28. Assume that 𝐺_𝑗=𝑁_𝑗=0, 𝑗=0,1,…,𝑚; then system (50) is asymptotically stabilizable if and only if all of the unremovable spectra of system (50) belong to 𝐷(0,1); that is, there does not exist a nonzero 𝑍∈𝑆^𝑛(𝑚+1), and 𝜆∉𝐷(0,1) satisfying 𝐹^′𝑍𝐹=𝜆𝑍,𝑍𝑀=0. (60)

Proof. The necessity is obvious. To prove the sufficiency part, we note that if (𝐹₀,𝑀₀) is not stabilizable, then, by Popov-Belevith-Hautus Criterion, there exists a nonzero 𝜉∈𝐶^𝑛(𝑚+1), 𝜆∉𝐷(0,1) satisfying 𝜉^′𝐹=𝜆𝜉^′, 𝜉^′𝑀=0. Take 𝑍=𝜉𝜉^′, then (60) holds, which contradicts the given condition. Corollary 28 is proved.

Remark 29. Corollary 28 indicates that there is no difference between unremovable spectrum and uncontrollable mode for deterministic systems.

Theorem 30. If system (50) is mean square stabilizable, then all the existing unremovable spectra of (50) must belong to 𝐷(0,1).

Proof. If there is an unremovable spectrum 𝜇 of system (50) with 𝜇∉𝐷(0,1), then, by Theorem 26, there exists 𝑍≠0∈𝑆^𝑛(𝑚+1), such that the following three equalities hold simultaneously: 𝐹^′𝑍𝐹+𝐺^′𝑍𝐺=𝜇𝑍,𝐹^′𝑍𝑀+𝐺^′𝑍𝑁=0,𝑁^′𝑍𝑁=−𝑀^′𝑍𝑀. (61) So for any feedback gain 𝐾, we obtain ℒ^∗_𝐾(𝑍)=𝜇𝑍, 𝜇∉𝐷(0,1). It follows that, for any 𝐾, 𝜇∈𝜎(ℒ_𝐾), which contradicts mean square stabilization of system (50). The proof of Theorem 30 is complete.

4. Conclusion

In this paper, we investigate the stability and stabilizability of stochastic delay-time systems. By 𝐻-representation, present the spectral criteria of the stability and stabilizability. By generalized Lyapunov equation approach, the equivalent conditions of mean square stabilizability and asymptotically mean square stabilizability of system (50) are given. We introduce the notion of unremovable spectrum of stochastic time-delay deference system, present the PBH criterion of the unremovable spectrum of stochastic time-delay system, and investigate the relation between the unremovable spectrum and the stabilizability of stochastic time-delay deference system.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant nos. 61174078 and 61201430.