Stability Analysis for Markovian Jump Neutral Systems with Mixed Delays and Partially Known Transition Rates

and Applied Analysis 3 known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here. Moreover, in contrast with the recent research on uncertain transition probabilities, our proposed concept of the partly unknown transition probabilities does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties. In addition, the relationship between the stability criteria currently obtained for the usual MJLS and switched linear system under arbitrary switching is exposed by our proposed systems. Furthermore, the number of matrix inequalities conditions obtained in this paper is much more than some existing results due to the introduced free matrices based on the system itself and the information of transition probabilities in this paper, which may increase the complexity of computation. However, it would decrease the conservativeness for the delay-dependent stability conditions. Finally, two numerical examples are provided to illustrate the effectiveness of our results. 2. Problem Statement and Preliminaries Consider the following neutral system with markovian jump parameters: ẋ t − C rt ẋ t − τ2 rt A rt x t B rt x t − τ1 rt , x t0 θ φ θ , ∀θ ∈ −τ, 0 , 2.1 where {rt}, t ≥ 0 is a right-continuous Markov process on the probability space taking values in a finite state space, ℘ {1, 2, . . . ,N}with generator I λij , i, j ∈ ℘ given by Pr { rt Δ j | rt i } { λijΔ o Δ , j / i, 1 λiiΔ o Δ , j i, 2.2 whereΔ > 0, limΔ→ 0 o Δ /Δ 0, λij ≥ 0, for j / i, is the transition rate frommode i at time t to mode j at time t Δ, λii − ∑N j 1 λij .A rt , B rt , and C rt are known matrix functions of the markovian process, x t ∈ R is the state vector, and φ θ is the initial condition function. τ1 rt and τ2 rt are mode-dependent delays, when rt i ∈ ℘, τ1 rt τ1i, τ2 rt τ2i, and τ max τ1i, τ2i . Since the transition probability depends on the transition rates for the continuous-time MJSs, the transition rates of the jumping process are considered to be partly accessible in this paper. For instance, the transition rate matrix I with N operation modes may be expressed as I ⎛ ⎜⎜⎜ ⎝ λ11 ? λ13 · · · ? ? ? ? ? λ2N .. .. .. . . . .. ? λN2 λN3 · · · λNN ⎞ ⎟⎟ ⎠ , 2.3 where ? represents the unknown transition rate. For notational clarity, for all i ∈ ℘, the set U denotes U U k ⋃ U uk with U i k {j : λij is known for j ∈ ℘}, U uk {j : λij is unknown for j ∈ ℘}, and λ k j∈Ui k λij . 4 Abstract and Applied Analysis Moreover, ifU k / ∅, it is further described as U k { k 1, k i 2, . . . , k i m, } , 2.4 where m is a nonnegation integer with 1 ≤ m ≤ N, and k j ∈ Z , 1 ≤ k j ≤ N j 1, 2, . . . , m represent the jth known element of the setU k in the ith row of the transition rate matrix I. Remark 2.1. It is worthwhile to note that if U k ∅, U U uk which means that any information between the ith mode and the otherN−1modes is not accessible, thenMJSs with N modes can be regarded as ones withN − 1 modes. It is clear that when U uk ∅, U U k, the system 2.1 becomes the usual assumption case. For the sake of simplicity, the solution x t, φ θ , r0 with r0 ∈ ℘ is denoted by x t . It is known from 38 that {x t , t} is a Markov process with an initial state {φ θ , r0}, and its weak infinitesimal generator, acting on function V , is defined in 39 : LV x t , t, i lim Δ→ 0 1 Δ ε V x t Δ , t Δ, rt Δ | x t , rt i − V x t , t, i . 2.5 Throughout this paper, the following definition is necessary. More details refer to 23 . Definition 2.2 see, 32 . The system 2.1 is said to be stochastically stable if the following holds:


Introduction
A switched system is a dynamic system consisted of a number of subsystems and a rule that manages the switching between these subsystems.In the past, a large number of excellent papers and monographs on the stability of switched systems have been published such as 1-7 and the references cited therein.Among the results for switched systems, the stabilization problem of switched neutral systems has also been explored by some researchers 8-22 , and mainly two kinds of switching rule are designed in these articles.Some state-dependent switching rules are obtained assuming the convex combination of the systems matrix, see, for example, 8, 10, 20 .To reduce the conservative, the authors in 11 have investigated the stabilization for switched neutral systems without the assumption that the restraint of known or completely unknown transition probabilities, which can be viewed as two special cases of the ones tackled here.Moreover, in contrast with the recent research on uncertain transition probabilities, our proposed concept of the partly unknown transition probabilities does not require any knowledge of the unknown elements, such as the bounds or structures of uncertainties.In addition, the relationship between the stability criteria currently obtained for the usual MJLS and switched linear system under arbitrary switching is exposed by our proposed systems.Furthermore, the number of matrix inequalities conditions obtained in this paper is much more than some existing results due to the introduced free matrices based on the system itself and the information of transition probabilities in this paper, which may increase the complexity of computation.However, it would decrease the conservativeness for the delay-dependent stability conditions.Finally, two numerical examples are provided to illustrate the effectiveness of our results.

Problem Statement and Preliminaries
Consider the following neutral system with markovian jump parameters: ẋ t − C r t ẋ t − τ 2 r t A r t x t B r t x t − τ 1 r t , x t 0 θ ϕ θ , ∀θ ∈ −τ, 0 , 2.1 where {r t }, t ≥ 0 is a right-continuous Markov process on the probability space taking values in a finite state space, ℘ {1, 2, . . ., N} with generator I λ ij , i, j ∈ ℘ given by Pr r t Δ j | r t i λ ij Δ o Δ , j / i, where Δ > 0, lim Δ → 0 o Δ /Δ 0, λ ij ≥ 0, for j / i, is the transition rate from mode i at time t to mode j at time t Δ, λ ii − N j 1 λ ij .A r t , B r t , and C r t are known matrix functions of the markovian process, x t ∈ R n is the state vector, and ϕ θ is the initial condition function.τ 1 r t and τ 2 r t are mode-dependent delays, when r t i ∈ ℘, τ 1 r t τ 1i , τ 2 r t τ 2i , and τ max τ 1i , τ 2i .
Since the transition probability depends on the transition rates for the continuous-time MJSs, the transition rates of the jumping process are considered to be partly accessible in this paper.For instance, the transition rate matrix I with N operation modes may be expressed as where ?represents the unknown transition rate.For notational clarity, for all i ∈ ℘, the set Moreover, if U i k / ∅, it is further described as where m is a nonnegation integer with 1 ≤ m ≤ N, and k i j ∈ Z , 1 ≤ k i j ≤ N j 1, 2, . . ., m represent the jth known element of the set U i k in the ith row of the transition rate matrix I.
uk which means that any information between the ith mode and the other N −1 modes is not accessible, then MJSs with N modes can be regarded as ones with N − 1 modes.It is clear that when U i uk ∅, U i U i k , the system 2.1 becomes the usual assumption case.
For the sake of simplicity, the solution x t, ϕ θ , r 0 with r 0 ∈ ℘ is denoted by x t .It is known from 38 that {x t , t} is a Markov process with an initial state {ϕ θ , r 0 }, and its weak infinitesimal generator, acting on function V , is defined in 39 : Throughout this paper, the following definition is necessary.More details refer to 23 .
Definition 2.2 see, 32 .The system 2.1 is said to be stochastically stable if the following holds: for every initial condition ϕ ∈ R n and r 0 ∈ ℘.

Stability Analysis for Neutral Markovian Jump Systems
The purpose of this section is to state the stability analysis for neutral markovian jump systems with partly unknown transition rates.Throughout the paper, the matrix C r t is assumed to be ρ C r t < 1.Before giving the stability result of systems 2.1 with a partly unknown transition rate matrix 2.3 , the stability of neutral markovian jump systems 2.1 with all transition probabilities known is firstly investigated.With the introduced free matrices and the novel analysis technique of matrix, the stability conditions are presented in this section.
Theorem 3.1.The system 2.1 with a fully known transition rate matrix is stochastically stable if there exist matrices P i > 0, Q 1i > 0, Q 2i > 0, R 1 > 0, and R 2 > 0 and any matrices N 1 , N 2 , N 3 , and N 4 with appropriate dimensions satisfying the following linear matrix inequalities:

3.4
Proof.Construct a stochastic Lyapunov functional candidate as , and the weak infinitesimal operator L of the stochastic process x t along the evolution of V k x t , t, i k 1, . . ., 7 are given as λ ij P j x t .

3.7
According to the definition of the weak infinitesimal operator L and the expression 2.2 , it can be shown that

3.8
Similar to the above, we can obtain 3.9 Moreover, there exist matrices N k k 1, . . ., 4 with appropriate dimensions, such that the following equality holds according to 2.1 : where

3.11
From 3.7 -3.10 and with 3.2 -3.3 , one can obtain that where ϕ i are defined in this theorem.Therefore, which means that systems 2.1 are stochastic stability.The proof is completed.
Based on the result of Theorem 3.1, the next theorem will relate to the stability condition of systems 2.1 with partially known transition probabilities.
Theorem 3.2.The system 2.1 with a partly unknown transition rate matrix 2.4 is stochastically stable if there exist matrices i with appropriate dimensions satisfying the following linear matrix inequalities: 15 3.17 3.19 Abstract and Applied Analysis

3.22
and λ i d is a given lower bound for the unknown diagonal element.
Proof.For the case of the systems 2.1 with partly unknown transition probabilities, and taking into account the situation that the information of transition probabilities is not accessible completely, due to N j 1 λ ij 0, the following zero equation holds for arbitrary matrices W i W T i is satisfied: and the inequality 3.12 can be rewritten as where ξ t has already been defined on the above with the elements are the same as those in ϕ i , except for and note that λ ii − j∈U i k λ ij and λ ij ≥ 0 for all j / i, namely, λ ii < 0 for all i ∈ ℘.Therefore, it follows from easy computation that if i ∈ U i k , inequalities 3.20 and the formula 3.25 less than 0 imply that LV x t , t, i < 0. 3.27 On the other hand, for the same reason, if i ∈ U i uk , inequalities 3.20 -3.21 and the formula 3.25 less than 0 also imply that inequality 3.27 holds.Therefore, which means that systems 2.1 with partly unknown transition probabilities are stochastically stable.Note that the formula 3.25 less than 0 can be represented as follows: where λ ij is an unknown element in transition probabilities matrix, and

3.31
One can note that 3.29 can be separated into two cases, i ∈ U i k and i ∈ U i uk .
Case I i ∈ U i k : it should be first noted that in this case one has λ i k < 0. In fact, we only need to consider λ i k < 0 because λ i k 0 means all the elements in the i throw of the transition rate matrix.Now the formula 3.29 can be rewritten as

3.32
It follows from 0 Similar to the above proof, 3.2 and 3.3 can be rewritten as 3.16 and 3.18 , respectively, for this case.Accordingly, for 0 ≤ λ ij ≤ λ i k , ϕ i < 0 is equivalent to 3.14 which is satisfied for all j ∈ U i uk , which also implies that, in the presence of unknown elements λ ij , the system stability is ensured if 3.14 , 3.16 , 3.18 , and 3.20 hold.
Case II i ∈ U i uk : for the sake of simple expression, let Ψ j diag 0, 0, τ 1j Q 1i , τ 2j Q 2i .In this case, λ ii is unknown, λ i k ≥ 0, and λ ii ≤ 0, and following the same analysis of the above case, we just consider λ ii < −λ i k .And now the formula 3.29 can be rewritten as

3.34
Similarly, since we have which means that ϕ i < 0 is equivalent to all j ∈ U i uk , j / i, and from the defined λ i d in Theorem 3.2, we have that λ i d ≤ λ ii < −λ i k , which means that λ ii may take any value between λ i d , −λ i k ε for some ε < 0 arbitrarily small.Then, λ ii can be further written as a convex combination where α takes value arbitrarily in 0, 1 .So, 3.37 holds if and only if for all j ∈ U i uk , j / i, simultaneously hold.Since ε is arbitrarily small, 3.39 holds if and only if which is the case in 3.40 when j i for all j ∈ U i uk .Hence, 3.37 is equivalent to 3.15 .Furthermore, following the same line of this proof, 3.2 and 3.3 can be represented as 3.17 and 3.19 , respectively, in this case.
Therefore, from the above discussion, in the presence of unknown elements in the transition probabilities matrix, we can readily draw a conclusion that the system 2.1 with partly known transition rates is stable if the inequalities in Theorem 3.2 are satisfied.It completes this proof.Remark 3.3.In order to obtain the less conservative stability criterion of MJSs with partial information on transition probabilities, similar to 32 , the free-connection weighting matrices are introduced by making use of the relationship of the transition rates among various subsystems, that is, N j 1 λ ij 0 for all i ∈ ℘, which overcomes the conservativeness of using the fixed-connection weighting matrices.However, it is difficult to decrease the conservative using free-connection matrices only based on the above equalities, but not on the systems and the themselves Newton-Leibniz formula.Moreover, this paper is inspired by 30 , and the delay-dependent stability results in this theorem are the extension of 30 to delay systems to some extent.Although the large number of introduced free weighting matrices may increase the complexity of computation, using the technique of free weighting matrices would reduce the conservativeness, which would be reflected in the fifth section.Remark 3.4.It should be noted that the more known elements are there in 2.3 , the lower the conservative of the condition will be.In other word, the more unknown elements are there in 2.3 , the lower the maximum of time delay will be in Theorem 3.2.Actually, if all transition probabilities are unknown, the corresponding system can be viewed as a switched linear system under arbitrary switching.Thus, the conditions obtained in Theorem 3.2 will thereby cover the results for arbitrary switched linear system with mixed delays.In that case, one can see that the stability condition in Theorem 3.2 becomes seriously conservative, for many constraints.Fortunately, we can use the common Lyapunov functional method to analyze the stability for the system under the assumption that all transition probabilities are not known.
For the stability analysis of the neutral markovian jump systems with all transition probabilities is not known, one can construct the following common Lyapunov functional: 3.44

Extension to Uncertain Neutral Markov Jump Systems
In this section, we will consider the uncertain neutral Markov jump systems with partially unknown transition probabilities as follows: 4.8 with and λ i d is a given lower bound for the unknown diagonal element.
Proof.φ i < 0 can be written as where and φ i are defined in Theorem 3.2.According to the approach in 40 with Lemma 2.4 in 41 , there exists a scalar ε such that 4.12 are equivalents to Introducing new variables , W i εW i , and N k εN k k 1, . . ., 4 , and with Schur's complement 42 , yields inequalities 4.3 .Similarly, it concludes 4.4 with the same proof.On the other hand, with the same variables substitution, we note that pre-and postmultiplying, respectively, 3.16 -3.21 by a scalar ε yield 4.5 -4.10 , which completes this proof.Remark 4.2.It should be mentioned that Theorem 4.1 is an extension of 2.1 to uncertain neutral markovian jump systems 4.1 with incomplete transition descriptions.In fact, this technology is frequently adopted in dealing with the robust stability analysis of uncertain systems.

Examples
In order to show the effectiveness of the approaches presented in the above sections, two numerical examples are provided.Solving the inequalities in Theorem 3.2 using LMI toolbox, the maximum of the time delay τ 11 can be computed as τ 11 1.1210.However, in Case II, the maximum of the time delay τ 11 can be computed as τ 11 2.0530 by Theorem 3.1.It is easily seen that the more transition probabilities knowledge we have, the larger the maximum of delay can be obtained for ensuring stability.This shows the trade-off between the cost of obtaining transition probabilities and the system performance.
Furthermore, when the transition probabilities are not fully known, as the delay for one of the subsystems decreases, the maximum of other delays may increase.However, when all transition probabilities are fully known, the conclusion may be on the opposite in some interval.In fact, the above observation is in accordance with the actual.Then, we assume that λ 2 d −0.8, λ 2 d −0.9, and let τ 12 1.1000, τ 14 0.9900, τ 21 τ 11 , τ 22 τ 12 , τ 23 τ 13 , and τ 24 τ 14 , τ 13 be different with τ 11 , and with the same computation in Theorem 3.1, as shown in Table 1.However, just according to the approach of Theorem 3 in 32 , not introducing some other free matrices and some other skills, we cannot find the feasible solutions which contain time delay to verify the stability of the system.Therefore, this example shows that the stability criterion in this paper gives much less conservative delay-dependent stability conditions.This example also shows that the approach presented in this paper is effectiveness.

5.9
For the case of all transition probabilities that are unknown, this example also shows that Corollary 3.5 is less conservative than Theorem 3.2 on the stability analysis for the neutral markovian jump system.

Conclusion
The delay-dependent stability for neutral markovian jump systems with partly known transition probabilities has been investigated.Based on a new class of stochastic Lyapunov-Krasovskii functionals constructed, and combined with the technique of analysis for matrix inequalities, some new stability criteria are obtained.The main contribution of this paper contains the following two-fold: one is the extension of delay-dependent stability conditions for markovian jump delay systems to markovian neutral jump systems; the other is the new method presented to decrease the conservative brought by the markovian jump with partly known transition probabilities.The future work is to investigate the systems with modedependent interval mixed time delays and the systems with unsynchronised control input.Three examples have shown the effectiveness of the conditions presented in this paper.

Nomenclature
R n : n-dimensional real space R m×n : Set of all real m by n matrices x T or A T : Transpose of vector x or matrix A P > 0: P < 0, resp.Matrix P is symmetric positive negative, resp.definite P ≥ 0: P ≤ 0, resp.Matrix P is symmetric positive negative, resp.semidefinite : The elements below the main diagonal of a symmetric block matrix x t θ : x t θ , θ ∈ −τ, 0 .
A r t , B r t are known mode-dependent constant matrices with appropriate dimensions, while ΔA r t , ΔB r t are the time-varying but norm-bounded uncertainties satisfying t , H 1 r t , and H 2 r t are known mode-dependent matrices with appropriate dimensions, and F r t t is the time-varying unknown matrix function with Lebesgue norm measurable elements satisfying F T r t t F r t t ≤ I. Theorem 4.1.The uncertain neutral markovian jump system 4.1 with a partly unknown transition rate matrix 2.3 is stochastically stable if there exist matrices Our purpose here is to check the stability of the above system for the two different cases of transition probabilities.For Case I, it is clear to see that λ 22 and λ 44 are not valued, one can set λ 2

Table 1 :
The maximal allowable delays τ 11 of Example 5.1 for different values of τ 12 in a different case.andletτ 12 1.1000, τ 13 0.7500, τ 14 0.9900, τ 21 τ 11 , τ 22 τ 12 , τ 23 τ 13 , and τ 24 τ 14 .Solving the inequalities in Theorem 4.1 by using LMI toolbox, the maximum of the time delay τ 11 can be computed as τ 11 1.0892.Some of the feasible solutions can be obtained as follows:In a word, this example shows that the robust stability condition of Theorem 4.1 is feasible.It is also approved that the approach provided in this paper is effectiveness.The above matrix 5.8 implies that systems 2.1 with all transition probabilities are not known, which viewed the systems as switched systems with arbitrary switching.Given that τ 11 2.4279, τ 12 1.1000, τ 13 0.95, τ 14 0.9900, τ 21 τ 11 , τ 22 τ 12 , τ 23 τ 13 , and τ 24τ 14 , according to the approach of Theorem 3.2, we cannot find the feasible solutions.However, using the Matlab LMI toolbox, we solve the LMI in Corollary 3.5, and the feasible solutions can be obtained as follows: