The Stability and Stabilization of Stochastic Delay-Time Systems

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 andH-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.


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][5][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][8][9][10][11][12][13][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, -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.

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 2 Mathematical Problems in Engineering the spectrum of operator and Lyapunov equation approach.
Remark 5.In Theorem 4, a necessary and sufficient condition for the asymptotically mean square stability of system (1) via the spectrum of L , is presented, which can be called "spectral criterion." Theorem 6.The trivial solution  = 0 of system (1) is asymptotically mean square stable if and only if, for any  ∈  (+1) with  > 0, there exists a  ∈  (+1) such that  > 0 and  is a solution of the following Lyapunov equation: Proof.We introduce an  2 ( + 1) 2 -parameter stochastic Lyapunov function as a quadratic form: The role of parameters is played by  2 ( + 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() = 0 of system ( 6) is asymptotically mean square stable if and only if for any  > 0, the Lyapunov equation ( 15) has a solution  > 0. By the proof of Theorem 4, the trivial solution  = 0 of system (1) is asymptotically mean square stable if and only if the trivial solution x() = 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.
Proof.By Theorems 4 and 6, (L , ) ⊂ (0, 1) holds if and only if there is a matrix  ∈  (+1) such that  > 0 and  is a solution of the following Lyapunov equation: for any  > 0. So, there exists a  ∈  ∈  (+1) such that  > 0 and  is a solution of the following Lyapunov equation: which is equivalent to () ⊂ (0, 1); that is, the system 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  = 0 of the system (1) is mean square stable, then (L , ) ⊂ (0, 1).Theorem 10. (L , ) ⊂ (0, 1) if and only if one of the following conditions holds.
(1) For any  > 0 and  > 0, the following Lyapunov equation has a positive-definite solution .
Theorem 13.The trivial solution  = 0 of system (1) is mean square stable if and only if, for any  ≥ 0, there exists a  ∈  n(+1) such that  > 0 and  is a solution of the following Lyapunov equation: Proof.We introduce an  2 ( + 1) 2 -parameter stochastic Lyapunov function as a quadratic form: 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 representation.
For a state feedback control law () = (), we introduce a linear operator L  associated with the closedloop system: where  ∈   is a column vector,   ,   ∈  × ,  = 0, 1, . . .,  are constant coefficient matrices, () is a deterministic initial condition, and () ∈   is a control input.
Definition 15.The trivial stationary solution () = 0 of the system (30) is called mean square stabilization if there exists an input feedback  such that, for any arbitrarily small number  > 0, one can find a number  > 0, when ‖‖ < , satisfying for a solution () = (, ) of (30).

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: Definition 24.We say that  is an unremovable spectrum of system (50) with state feedback if there exists  ̸ = 0 ∈  (+1) , such that,for any  ∈  (+1)×(+1) , L *  () = ( + )   ( + ) holds.
Remark 25.It is easy to see that the operator L *  is the adjoint operator of the operator L  with the inner product ⟨, ⟩ = trace( * , ) for any ,  ∈  (+1) .As we restrict the coefficients to real matrices, (L *  ) = (L  ).By Corollaries 17-23, we know that any one of them can characterize the stabilizability of system (50).
Obviously, if  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.