Finite-Time Stability and Stabilization of Itô-Type Stochastic Singular Systems

This paper is concerned with the finite-time stability and stabilization problems for linear Itˆo stochastic singular systems. The condition of existence and uniqueness of solution to such class of systems are first given. Then the concept of finite-time stochastic stability is introduced, and a sufficient condition under which an Itˆo stochastic singular system is finite-time stochastic stable is derived.Moreover,thefinite-timestabilizationisinvestigated,andasufficientconditionfortheexistenceofstatefeedbackcontrollerispresentedintermsofmatrixinequalities.Inthesequel,analgorithmisgivenforsolvingthematrixinequalitiesarisingfromfinite-timestochasticstability(stabilization).Finally,twoexamplesareemployedtoillustrateourresults.


Introduction
Stochastic systems, especially for the systems governed by Itô-type stochastic differential equations, have received considerable attention due to its both theoretical and practical importance.Some results for this class of systems have been reported in the monographs and literatures, for example, stochastic stability and stabilization [1][2][3], linear/nonlinear stochastic  ∞ control and filtering [4][5][6][7], and output tracking control for high-order stochastic nonlinear systems [8].Meanwhile, singular systems (descriptor systems, implicit systems, generalized state-space systems, and differential-algebraic systems) have also attracted much attention of researchers and made a rapid progress.Many results have been achieved on different subjects related to such class of systems, for example, stability and impulsive elimination [9,10], linear quadratic optimal control [11], and  ∞ control/filtering and  2 / ∞ control [12][13][14].Consequently, Itô stochastic singular systems have received attention in recent years.Reference [15] is concerned with the problems of stability of Itô singular stochastic systems with Markovian jumping.Reference [16] investigated the  ∞ control/filtering for a class of singular stochastic time-delay systems.To the best of our knowledge, most of the results on stability of Itô stochastic singular systems are concerned with Lyapunov asymptotic stability or exponential stability, which is defined over an infinite-time interval.
In many practical situations, however, we are interested in stability of the system over a fixed finite-time interval.Such kind of stability is called finite-time stability (FTS).The concept of FTS was first introduced in the Russian literature.Later, this concept appeared in the western control literatures.Roughly speaking, a system is said to be finitetime stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval.Compared with infinite-time stability, the FTS can be used in the problem of controlling the trajectory of a space vehicle from an initial point to a final point in a prescribed time interval and all those applications where large values of the states should be attained, for instance, in the presence of saturations.Much effort has been devoted to FTS for its stability analysis and stabilization, for instance, linear continuous-time systems [17], linear discrete-time systems [18], stochastic systems [19][20][21][22], singular systems/Markovian jumping singular systems [23,24], and stochastic singular biological economic systems [25].Nevertheless, the FTS in [17][18][19][20][21][22][23][24][25] only requires that the state trajectory does not exceed a given upper bound during a prespecified time interval.
Recently, [26] gave a new "finite-time stochastic stability" for linear Itô stochastic systems, which quantifies the state trajectory of some complex practical systems over a finitetime interval in more detail.Roughly speaking, a stochastic Itô system is called finite-time stochastically stable if its state trajectories do not exceed an upper bound  2 and are not less than a lower bound  1 ( 1 <  2 ) in the mean square sense during a specific time interval.
In this paper, motivated by [26], we consider finite-time stability and stabilization problems for Itô stochastic singular systems.Because of the special structure of Itô stochastic singular systems, the problems considered are of more complexity than those in [26].By using stochastic analysis technology, the stability criterion and some stabilizing conditions are obtained.The contributions of this paper lie in the following three aspects: (1) the condition for the existence and uniqueness of solution to linear Itô stochastic singular systems is given.Our proof is different from that in [15], which may better reflect the essential characteristics of this class of systems.
(2) The definition of finite-time stochastic stability for linear Itô stochastic singular systems is given.By the generalized Itô formula and mathematical expectation properties, some new stability criteria and the conditions of existence for state feedback controller are obtained as well.(3) A solving algorithm for the matrix inequalities arising from finite-time stochastic stability (stabilization) is given.By adjusting the parameters in this algorithm, the less conservative results can be attained.
The remainder of this paper is organized as follows.The definition of finite-time stochastic stability of linear Itô stochastic singular systems and some preliminaries are presented in Section 2. A sufficient condition to verify finitetime stochastic stability is given in Section 3. Section 4 gives some sufficient conditions for finite-time stochastic stabilization and a solving algorithm for the matrix inequalities arising from finite-time stochastic stability (stabilization).Section 5 employs two examples to illustrate the results.Finally, concluding remarks are made in Section 6.

Lemma 1.
If there is a pair of nonsingular matrices  ∈ R × ,  ∈ R × or  1 ∈ R × ,  1 ∈ R × for the triplet (, , ) such that (at least) one of the following conditions is satisfied: where  is nilpotent matrix with nilpotent index ℎ, then (1) has a unique solution.
Proof.Let () =  −1 () = [  1 ()   2 ()]  ,   1 () ∈ R  and   2 () ∈ R − , then under the conditions of (I), the system (1) is equivalent to which are called the slow and fast subsystems, respectively.Note that the slow subsystem ( 4) is nothing more than an Itô stochastic differential equation.Applying the existence and uniqueness theorem of stochastic differential equations [27], the solution of (4) exists and is unique.
We note that the fast subsystem ( 5) is actually an ordinary differential equation.Taking the Laplace transforms on both sides of ( 5) and letting ℓ[ 2 ()] = Γ(), we have From this equation, we obtain Taking the inverse Laplace transform on both sides of (7) gives which implies that (5) has a unique solution.So (1) has a unique solution.
The proof is different from that in [15], which may better reflect the essential characteristics of this class of systems, such as, impulse behaviors.It is obviously observed from the proof of Lemma 1 that the response of system (1) may contain impulse terms.For convenience, we introduce the following definition.
Definition 2. If the state response of an Itô stochastic singular system, starting from an arbitrary initial value, does not contain impulse terms, then the system is called impulse-free.
Referring to some results on impulse-free of singular systems in [28], the following result is obtained.Proposition 3. The following statements are equivalent under the conditions of Lemma 1: Next, we extend the finite-time stochastic stability in [26] to Itô stochastic singular systems.Definition 4. Given some positive scalars  1 ,  2 ,  3 ,  4 ,  with 0 <  1 <  3 <  4 <  2 and a positive definite matrix , system (1) is said to be finite-time stochastically stable with respect to ( 1 ,  2 ,  3 ,  4 , , ), if Definition 4 can be described as follows: system (1) is said to be finite-time stochastically stable if, given a bound on the initial condition and a fixed time interval, its state trajectories are required to remain in a certain domain of ellipsoidal shape in the mean square sense during this time interval.
A 2-dimension case of Definition 4 is illustrated by Figure 1.A point  lies in the shaped area.The trajectory starting from point  cannot escape the disc from 0 to  2 during the time interval [0, ] in the mean square sense.
Remark 5.In [17][18][19][20][21][22][23][24][25], the finite-time stability only requires the state trajectory not to exceed a given upper bound.A 2dimension case of this finite-time stability can be illustrated by Figure 2. Nevertheless, the current finite-time stability requires the state trajectory not only not to exceed a given upper bound but also not to be less than a given lower bound.
In the following, we give a proposition equivalent to Definition 4. Proposition 6. System (1) is finite-time stochastically stable with respect to ( 1 ,  2 ,  3 ,  4 , , ) if and only if where () > 0 is the solution to Proof.Letting () = E[()()  ], we easily obtain Applying Itô's formula to E[()    ()], we obtain Under the conditions of Lemma 1, ( 14) has unique solution ().So the proof is completed.By Kronecker's product theory, ( 14) can be rewritten as where ⃗  denotes the vector formed by stacking the rows of  = (  ) × into one long vector; that is, and ⊗ represents the Kronecker product of two matrices.
Remark 7. Proposition 6 is actually to solve a set of ordinary differential equations and avoids solving a stochastic differential equation (1), which provides an easier method to test finite-time stochastic stability of system (1).
Based on Definition 4, we define the finite-time stochastic stabilization as follows.
Definition 9.The following Itô stochastic singular controlled system is said to be finite-time stochastically stabilizable, if there exists a state feedback control law () = (), such that is finite-time stochastically stable.
Before proceeding further, we give some lemmas which will be used in the next sections.
(iii) If  is a nonsingular matrix,  and Φ are two symmetric positive definite matrices,  and  satisfy (25),  is a diagonal matrix from (27), and the following equality holds: Then the symmetric positive definite matrix  =  −1/2    −1  −1/2 is a solution of (28).
Proof.We split the proof of Theorem 13 into three steps as follows.
By (34) and (41), we obtain Integrating both sides of (48) from 0 to  with  ∈ [0, ] and then taking the expectation, it yields that By Lemma 11, we conclude that According to condition (30), it follows that From ( 32), (51), we obtain So, the proof is completed.
Theorem 13 provides a criterion for finite-time stochastic stability of system (1).To design finite-time controller conveniently, the following corollary is given.Corollary 14.Under the conditions of Lemma 1, if there exist positive matrices  > 0, nonsingular matrix , and two scalars  ≥ 0,  ≥ 0 satisfying then the system (1) is impulse-free and finite-time stochastically stable with respect to ( 1 ,  2 ,  3 ,  4 , , ).

Finite-Time Stochastic Stabilization
In this section, we aim to design a finite-time stabilizing controller for system (18).To this aim, the following result is obtained.
Proof.If the state feedback controller is taken into account, then the state equation of system (18) becomes Therefore, we can replace  by  +  in Corollary 14.As a result, condition (57) and (58) turn to By setting  =   , (62) and ( 63) become ( 59) and (60), respectively.This completes the proof of Theorem 16.
Remark 18.It is important to notice that once we have fixed the values for  and , the feasibility of the conditions stated in Theorem 17 can be turned into the following LMIs based feasibility problem.The algorithm of how to choose  and  for Theorem 17 is given in the following.
Algorithm 19.Consider the following steps.

Examples
In this section, we will present two examples to illustrate the obtained results.

Conclusion
In this paper, we have dealt with finite-time stability and stabilization problems for linear Itô stochastic singular systems and also established a condition of the existence and uniqueness of solution of linear Itô stochastic singular systems.A new sufficient condition has been provided to guarantee that the linear Itô stochastic singular system is impulse-free and finite-time stochastic stable.Based on the obtained result, we have also derived the corresponding stabilization criteria.Moreover, the finite-time stochastic stabilization has been studied via state feedback, and some new sufficient conditions have been given.Two examples are presented to illustrate the effectiveness of the proposed results.In addition, we can refer to [29][30][31] and extend the results of this paper to Markovian jump systems, networked systems, and linear parameter varying systems.

Figure 1 :Figure 2 :
Figure 1: Illustration of finite-time stochastic stability in this paper.

Figure 4 :Figure 5 :
Figure 4: A region by  and .