# Finite-Time Boundedness for a Class of Delayed Markovian Jumping Neural Networks with Partly Unknown Transition Probabilities

Authors: Li Liang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned with the problem of finite-time boundedness for a class of delayed Markovian jumping neural networks with partly unknown transition probabilities. By introducing the appropriate stochastic Lyapunov-Krasovskii functional and the concept of stochastically finite-time stochastic boundedness for Markovian jumping neural networks, a new method is proposed to guarantee that the state trajectory remains in a bounded region of the state space over a prespecified finite-time interval. Finally, numerical examples are given to illustrate the effectiveness and reduced conservativeness of the proposed results.


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## Body

## 1. Introduction

Over the past decades, delayed neural networks have been successfully applied in the pattern recognition, signal processing, image processing, and pattern recognition problems. However, these successful applications mostly rely on the dynamic behaviors of delayed neural networks and some of these applications are dependent on stability of the equilibria of neural networks. Up to now, there have been a large number of results related to dynamical behaviors of delayed neural networks [1–8].On the one, in the past few decades, Markovian jump systems have gained special research attention. Such class of systems is a special class of stochastic hybrid systems, which may switch from one to another at the different time. Such as component failures, sudden environmental disturbance and abrupt variations of a nonlinear system [9–11]. Moreover, it is shown that such jumping can be decided by a Markovian chain [12]. For the linear Markovian jumping systems, many important issues have been devoted extensively such as stability, stabilization, control synthesis, and filter design [13–16]. In reality, however, it is worth mentioning that most of the gotten results are based on the implicit assumptions that the complete knowledge of transition probabilities is known. It is known that in most situations, the transition probabilities rate of Markovian jump systems and networks is not known; it is difficult to obtain all the transition probabilities. Therefore, it is of great importance to investigate the partly unknown transition probabilities. Very recently, the systems with partially unknown transition probabilities have been fully investigated and many important results have been obtained; for a recent survey on this topic and related questions, one can refer to [17–23]. However, it has been shown that the existing delay-dependent results are conservative.On the other hand, the practical problems which described system stay as not exceeding a given threshold over finite-time interval are considered. Compared with classical Lyapunov stability, finite-time stability was studied to tackle the transient behavior of systems in the finite-time interval. Recently, the concept of finite-time stability has been revisited in the terms of linear matrix inequalities (LMIs); some results have been obtained to guarantee that system is finite-time stable and finite-time bounded [24–39]. To the best of our knowledge, the finite-time stability analysis for Markovian jumping neural networks with mode-dependent time-varying delays and partially known transition rates has not been tackled, and such a situation motivates our present study.The main contribution of this paper lies in proposing a novel method for finite-time boundedness of delayed Markovian jumping neural networks with partly unknown transition probabilities. The considered system is more general than the systems with completely known or completely unknown transition probabilities, which can be regarded as two special cases of the one tackled here. In contrast to study on Markovian jumping neural networks with time delays, the knowledge of the unknown elements is not required in our method. By employing the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are obtained to ensure that the system does not exceed a given threshold in a specified time interval. The finite-time bounded criteria can be tackled in the form of LMIs. Finally, numerical examples are given to demonstrate that the derived results are less conservative and more useful than some existent ones.


## 2. Preliminaries

Given a probability space(Ω,F,P) where Ω, F and P, respectively, represents the sample space, the algebra of events and the probability measure which defined on Ω. In this paper, we consider the following n-neuron Markovian jumping neural network over the space (Ω,F,P) described by
(1)x˙(t)=-Artx(t)+Brtf(x(t))+Crtf(x(t-τrt(t)))+Jx(t)=ϕ(t),t∈[-τ,0),
where x(t)=[x1(t),x2(t),…,xn(t)]⊺ represents the neural state vector of the system, f(x(t))=[f1(x1(t)),f2(x2(t)),…,fn(xn(t))]⊺ is the nonlinear activation function with the initial condition f(0)=0, Art=diag{a1(rt),a2(rt),…,an(rt)} describes the rate with each neuron which would reset its potential to resting state in isolation, Brt=[bij(rt)]n×n and Crt=[cij(rt)] are the connection weight matrices and the delayed connection weight matrices, respectively, and J=[J1,J2,…,Jn]⊺ denotes a constant external input vector. τrt(t) are the time-varying delays which satisfy
(2)0≤τrt(t)≤τrt,0≤τ˙rt(t)≤drt≤1,
where τrt and drt are constant scalars and τ=maxrt{τrt}, d=maxrt{drt}.Remark 1.
This assumption is often employed to investigate the stability of neural networks. It is worth noting that if this assumption is not true, corresponding time-delays are not a continuous function belonging to a given interval; neither the lower nor upper bounds for time-varying delays are available. Therefore, it may lead to more conservativeness.
Let the random form process{rt,t≥0} be the Markovian stochastic process taking values on the finite set 𝒩={1,2,…,N} with transition rate matrix Ω={μij}, i,j∈𝒩; namely, for rt=i, rt+1=j, one has
(3)Pr(rt+h=j∣rt=i)={μijh+o(h),ifj≠i1+μiih+o(h),ifj=i,
where h>0, limh→0(o(h)/h)=0, and μ≥0  (i,j∈𝒩,j≠i), denote switching rate from mode i at time t to mode j at time t+h. For all i∈𝒩, μii=-∑j=1,j≠i‍μij. Moreover, the Markovian process transition matrix Ω is defined as follows:
(4)Ω=[μ11μ12⋯μ1Nμ21μ22⋯μ2N⋮⋮⋱⋮μN1μN2⋯μNN].
Moreover, the transition rates of jumping process in this paper are considered to be partly accessed; that is, some elements in matrixΩ are unknown. Therefore, the transition rates matrix Ω which is Markovian jump system (1) may be as follows:
(5)Ω=[μ11?⋯μ1N?μ22⋯?⋮⋮⋱⋮??⋯μNN],
where ? represents the inaccessible elements. For notational clarity, for all i∈𝒩, we denote 𝒩=𝒩𝒦i+𝒩𝒰𝒦i and we denote that
(6)𝒩𝒦i≡{j:μijisknown},𝒩𝒰𝒦i≡{j:μijisunknown}.
Moreover, if𝒩𝒦i≠∅, 𝒩𝒦i and 𝒩𝒰𝒦i can be further described, respectively, as
(7)𝒩𝒦i={𝒦1i,𝒦2i,…,𝒦mi},𝒩𝒰𝒦i={𝒰𝒦1i,𝒰𝒦2i,…,𝒰𝒦N-mi},
where 𝒩mi∈ℤ+ represents the mth known element with the index 𝒩mi in the ith row of matrix Ω. 𝒰𝒩N-mi∈ℤ+ represents the N-mth unknown element with the index 𝒰𝒩Nmi in the ith row of matrix Ω.
Set𝒩 contains N modes of system (1) and, for rt=i∈𝒩, the system matrices of the ith mode are denoted by Ai, Bi, and Ci, which are considered to be real known with appropriate dimensions.Remark 2.
The Markovian jump process{rt,t≥0} in the literature is always assumed μij ether to be completely known (𝒩𝒦i) or completely unknown (𝒩𝒰𝒦i). Therefore, our transition probabilities matrix considered in this paper is more general than the Markovian jump systems and therefore covers the existing ones.Assumption 3.
The neuron state-based nonlinear functionf(x(t)) considered in Markovian jump system (1) is bounded and satisfies
(8)0≤fs(ς1)-fs(ς2)ς1-ς2≤γs,s=1,2,…,n
for all ς1, ς2∈ℛ, with γs being known real constants with s=1,2,…,n.It should be noted that by using the Brouwer fixed-point theorem, there should exist at least the one equilibrium point for system (1). Assuming that x*=[x1*,x2*,…,xn*]⊺ is the equilibrium point of (1) and using the transformation z(·)=x(·)-x*, system (1) can be converted to the following system:
(9)z˙(t)=-Artz(t)+Brtg(z(t))+Crtg(z(t-τrt(t))),
where z(t)=[z1(t),z2(t),…,zn(t)]⊺, g(z(·))=[g1(z1(x(t))),g2(x(t)),…,gn(x(t))]⊺, and gi(zi(zi(·)))=fi(zi(·)+xi*)-fi(xi*), i=1,2,…,n. According to Assumption 3, one can obtain that
(10)0≤gi(zi(t))zi(t)≤γi,gi(0)=0,i=1,2,…,n.Definition 4 (see [33]).
The nominal time-delayed Markovian jumping neural networks (1) are said to be stochastically finite-time bounded with respect to (c1,c2,T), if
(11)𝔼∥x(t1)∥2≤c1⟹𝔼∥x(t2)∥2≤c2,t1∈[-τ,0],t2∈[0,T].Definition 5 (see [34]).
LetV(xt,rt) be a stochastic positive functional and define its weak infinitesimal operator as
(12)£V(xt,rt=i)=limΔ→01Δ[𝔼{V(xt+Δ,rt+Δ)∣xt,rt=i}-V(xt,rt=i)].


## 3. Finite-TimeH∞ Performance Analysis

In this section, one method would be employed to analyze the finite-time stability of Markovian jump systems with partial information on transition probabilities.Theorem 6.
Given a time constantT>0, the delayed Markovian jumping neural networks (1) are stochastically finite-time bounded with respect to (c1,c2,T), if there exist a positive constant η>0, mode-dependent symmetric positive-definite matrices Pi>0, Q1i>0, Q2i>0, W1>0, W2>0  (i∈𝒩), a set of symmetric matrices Sυ  (υ=1,2,…,N), any appropriately dimensioned matrices Mi, Ni  (i∈𝒩), Γs, and scalars λl  (l=1,2,…,6) such that the following matrix inequalities hold:
(6)∑j∈𝒩𝒦i‍μijQ1j-(1+∑j∈𝒩𝒦i‍μij)W1+∑j∈𝒩𝒦i‍μijQ1i<0,Q1j-W1+Q1i<0,j∈𝒩𝒰𝒦i,j≠i,Q1j-W1+Q1i<0,j∈𝒩𝒰𝒦i,j=i,∑j∈𝒩𝒦i‍μijQ2j-(1+∑j∈𝒩𝒦i‍μij)W2+∑j∈𝒩𝒦i‍μijQ2i<0,Q2j-W2+Q2i<0,j∈𝒩𝒰𝒦i,j≠i,Q2j-W2+Q2i<0,j∈𝒩𝒰𝒦i,j=i,Σi=e1(1+∑j∈𝒩𝒦i‍μij)(-PiAi-Ai⊺Pi)e1⊺+e1∑j∈𝒩𝒦i‍μijPje1⊺+2e1PiBie3⊺+2e1PiCie4+e1Q1ie1⊺-(1-di-∑j∈𝒩𝒦i‍μijτj)e2Q1ie2⊺+e3Q2ie3⊺-(1-di)e4Q2ie4⊺+∑j=1N‍μijτje4Q2ie4⊺+τe1W1e1⊺+τe3W2e3⊺+e1ΓsMiΓse1⊺-e3Mie3⊺+e2ΓsNiΓse2⊺-e4Nie4⊺-e1∑j∈𝒩𝒦i‍μijSυe1⊺<0,e1(-PiAi-Ai⊺Pi+Pj-Sυ)e1⊺+e2τjQ2ie2⊺<0,j∈𝒩𝒰𝒦i,j≠i,e1(-PiAi-Ai⊺Pi+Pj-Sυ)e1⊺+e2τjQ2ie2⊺>0,j∈𝒩𝒰𝒦i,j=i,(14)c1eηT(λ2+τλ3+τγ¯s2λ4+τ2λ5+τ2γ¯s2λ6)<λ1c2,
where
(15)λ1=maxi∈𝒩λmin(Pi),λ2=maxi∈𝒩λmax(Pi),λ3=maxi∈𝒩λmax(Q1i),λ4=maxi∈𝒩λmax(Q2i),λ5=λmax(W1),λ6=λmax(W2),γ¯s=maxs(γs).Proof.
We consider the following the stochastic Lyapunov-Krasovskii functional:(16)V(zt,rt)=∑l=14‍Vl(zt,rt),
where
(17)V1(zt,rt)=z⊺(t)Prtz(t),V2(zt,rt)=∫t-τrt(t)t‍z⊺(s)Q1rtz(s)ds,V3(zt,rt)=∫t-τrt(t)t‍g⊺(z(s))Q2rtg(z(s))ds,V4(zt,rt)=∫-τ0‍∫t+θt‍z⊺(s)W1z(s)dsdθ+∫-τ0‍∫t+θt‍g⊺(z(s))W2g(z(s))dsdθ
with Pi, Q1i, Q2i, (i=1,2,…,N), W1, and W2 being positive definite matrices and
(18)∑j=1N‍μijQ1j<W1,(19)∑j=1N‍μijQ2j<W2.
For notational simplicity, let(20)ξ(t)=[z⊺(t),z⊺(t-τi(t)),g⊺(z(t)),g⊺(z(t-τi(t)))]⊺,es=[0,…,0,︸s-1I,0,…,0︸4-s]⊺,s=1,…,4.
Let£ be the infinitesimal generator of random process {zt,t≥0}; then for each rt=i, i∈𝒩, we can obtain that
(21)£V1(zt,i)=2z⊺(t)Piz˙(t)+z⊺(t)∑j=1N‍μijPjz(t)=ξ⊺(t)e1(-PiAi-Ai⊺Pi+∑j=1N‍μijPj)e1⊺ξ(t)+2ξ⊺(t)e1PiBie3⊺ξ(t)+2ξ⊺(t)e1PiCie4ξ(t),£V2(zt,i)=limΔ→0+1Δ𝔼×{[∫t+Δ-τrt+Δ(t+Δ)t+Δ‍z⊺(s)Q1rt+Δz(s)ds∣rt=i]-∫t-τi(t)t‍z⊺(s)Q1iz(s)ds[∫t+Δ-τrt+Δ(t+Δ)t+Δ‍z⊺(s)Q1rt+Δz(s)ds∣rt=i]}=limΔ→0+1Δ{[Q1i+∑j=1N‍(μijΔ+o(Δ))]∫t+Δ-τi(t+Δ)-∑j=1N‍(μijΔ+o(Δ))τj(t+Δ)t+Δ‍z⊺(s)×[Q1i+∑j=1N‍(μijΔ+o(Δ))]z(s)ds-∫t-τi(t)t‍z⊺(s)Q1iz(s)ds[Q1i+∑j=1N‍(μijΔ+o(Δ))]}=limΔ→0+1Δ{∫t+Δ-τi(t+Δ)-∑j=1N‍(μijΔ+o(Δ))τj(t+Δ)t+Δ‍×z⊺(s)Q1iz(s)ds-∫t-τi(t)t‍z⊺(s)Q1iz(s)ds}+limΔ→0+1Δ∫t+Δ-τi(t+Δ)-∑j=1N‍(μijΔ+o(Δ))τj(t+Δ)t+Δ‍z⊺(s)×∑j=1N‍(μijΔ+o(Δ))×Q1jz(s)ds=limΔ→0+1Δ∫tt+Δ‍z⊺(s)Q1iz(s)ds+limΔ→0+1Δ∫t+Δ-τi(t+Δ)-∑j=1N‍(μijΔ+o(Δ))τj(t+Δ)t+Δ‍×z⊺(s)∑j=1N‍(μijΔ+o(Δ))×Q1jz(s)ds=ξ⊺(t)e1Q1ie1⊺ξ(t)-(1-τ˙i(t)-∑j=1N‍μijτj(t))×ξ⊺(t)e2Q1ie2⊺ξ(t)+∫t-τi(t)t‍z⊺(s)(∑j=1N‍μijQ1j)z(s)ds≤ξ⊺(t)e1Q1ie1⊺ξ(t)-(1-di-∑j=1N‍μijτj(t))×ξ⊺(t)e2Q1ie2⊺ξ(t)+∫t-τi(t)t‍z⊺(s)(∑j=1N‍μijQ1j)z(s)ds.
Similar to the process above, it yields
(22)£V3(zt,i)≤ξ⊺(t)e3Q2ie3⊺ξ(t)-(1-di)ξ⊺(t)e4Q2ie4⊺ξ(t)+∑j=1N‍μijτj(t)ξ⊺(t)e4Q2ie4⊺ξ(t)+∫t-τi(t)t‍g⊺(z(s))(∑j=1N‍μijQ2i)g(z(s))ds,£V4(zt,i)=τξ⊺(t)e1W1e1⊺ξ(t)-∫t-τt‍z⊺(s)W1z(s)ds+τξ⊺(t)e3W2e3⊺ξ(t)-∫t-τt‍g⊺(z(s))W2g(z(s))ds.
From (18) and (19), we obtain that
(23)∫t-τi(t)t‍z⊺(s)(∑j=1N‍μijQ1j)z(s)ds≤∫t-τt‍z⊺(s)(∑j=1N‍μijQ1j)z(s)ds≤∫t-τt‍z⊺(s)W1z(s)ds,∫t-τi(t)t‍g⊺(z(s))(∑j=1N‍μijQ2j)g(z(s))ds≤∫t-τt‍g⊺(z(s))(∑j=1N‍μijQ2j)g(z(s))ds≤∫t-τt‍g⊺(z(s))W2g(z(s))ds.
Also, it results from (10) that for any appropriately dimensioned matrices Mi, Ni, (i=1,2,…,N), one can obtain
(24)0≤ξ⊺(t)e1ΓsMiΓse1⊺ξ(t)-ξ⊺(t)e3Mie3⊺ξ(t),0≤ξ⊺(t)e2ΓsNiΓse2⊺ξ(t)-ξ⊺(t)e4Nie4⊺ξ(t).
From (16)–(24), we have
(25)£V(zt,i)≤ξ⊺(t)Ξiξ(t),
where
(26)Ξi=e1(-PiAi-Ai⊺Pi+∑j=1N‍μijPj)e1⊺+2e1PiBie3⊺+2e1PiCie4+e1Q1ie1⊺-(1-di-∑j=1N‍μijτj)e2Q1ie2⊺+e3Q2ie3⊺-(1-di)e4Q2ie4⊺+∑j=1N‍μijτje4Q2ie4⊺+τe1W1e1⊺+τe3W2e3⊺+e1ΓsMiΓse1⊺-e3Mie3⊺+e2ΓsNiΓse2⊺-e4Nie4⊺.
By the fact that∑j∈𝒩‍μij=0, we can rewrite Ξi as
(27)Ξi=e1(-PiAi-Ai⊺Pi+∑j=1N‍μijPj)e1⊺+2e1PiBie3⊺+2e1PiCie4+e1Q1ie1⊺-(1-di-∑j=1N‍μijτj)e2Q1ie2⊺+e3Q2ie3⊺-(1-di)e4Q2ie4⊺+∑j=1N‍μijτje4Q2ie4⊺+τe1W1e1⊺+τe3W2e3⊺+e1ΓsMiΓse1⊺-e3Mie3⊺+e2ΓsNiΓse2⊺-e4Nie4⊺-e1∑j=1N‍μij(PiAi+Ai⊺Pi+Sυ)e1⊺.
Thus, from (6), we have
(28)Ξi=e1(1+∑j∈𝒩𝒦i‍μij)(-PiAi-Ai⊺Pi)e1⊺+e1∑j∈𝒩𝒦i‍μijPje1⊺+2e1PiBie3⊺+2e1PiCie4+e1Q1ie1⊺-(1-di-∑j∈𝒩𝒦i‍μijτj)e2Q1ie2⊺+e3Q2ie3⊺-(1-di)e4Q2ie4⊺+∑j=1N‍μijτje4Q2ie4⊺+τe1W1e1⊺+τe3W2e3⊺+e1ΓsMiΓse1⊺-e3Mie3⊺+e2ΓsNiΓse2⊺-e4Nie4⊺-e1∑j∈𝒩𝒦i‍μijSυe1⊺+∑j∈𝒩𝒰𝒦i‍μij[e1(-PiAi-Ai⊺Pi+Pj-Sυ)e1⊺+e2τjQ2ie2⊺].
Then, forj∈𝒩𝒰𝒦i and if i∈𝒩𝒦i, Ξi<0 can be guaranteed. On the other hand, for j∈𝒩𝒰𝒦i and if i∉𝒩𝒦i, Ξi can be further expressed as
(29)Ξi=e1(1+∑j∈𝒩𝒦i‍μij)(-PiAi-Ai⊺Pi)e1⊺+e1∑j∈𝒩𝒦i‍μijPje1⊺+2e1PiBie3⊺+2e1PiCie4+e1Q1ie1⊺-(1-di-∑j∈𝒩𝒦i‍μijτj)e2Q1ie2⊺+e3Q2ie3⊺-(1-di)e4Q2ie4⊺+∑j=1N‍μijτje4Q2ie4⊺+τe1W1e1⊺+τe3W2e3⊺+e1ΓsMiΓse1⊺-e3Mie3⊺+e2ΓsNiΓse2⊺-e4Nie4⊺-e1∑j∈𝒩𝒦i‍μijSυe1⊺+∑j∈𝒩𝒰𝒦i,j≠i‍μij[e1(-PiAi-Ai⊺Pi+Pj-Sυ)e1⊺+e2τjQ2ie2⊺]+μii×[e1(-PiAi-Ai⊺Pi+Pj-Sυ)e1⊺+e2τjQ2ie2⊺].
Similarly, (18) and (19) can be rewritten, respectively, as
(30){∑j∈𝒩𝒦i‍μijQ1j-(1+∑j∈𝒩𝒦i‍μij)W1+∑j∈𝒩𝒦i‍μijQ1i}+∑j∈𝒩𝒰𝒦i,j≠i‍μij[Q1j-W1+Q1i]+μii[Q1j-W1+Q1i]<0,{∑j∈𝒩𝒦i‍μijQ2j-(1+∑j∈𝒩𝒦i‍μij)W2+∑j∈𝒩𝒦i‍μijQ2i}+∑j∈𝒩𝒰𝒦i,j≠i‍μij[Q2j-W2+Q2i]+μii[Q2j-W2+Q2i]<0.
It is well known thatμii=-∑j=1,j≠iN‍μij<0; according to (6), one can also obtain
(31)£V(zt,i)<0.
On the other hand, from (32) and the needed constant η>0, it yields that
(32)𝔼{£V(zt,rt)}<η𝔼{V(zt,rt)},
from which we can easily get that
(33)e-ηt𝔼{V(zt,rt)}<𝔼{V(z0,r0)}.
Note that 0≤t≤T; we can obtain the following inequality:
(34)𝔼{V(zt,rt)}<eηt𝔼{V(x0,r0)}=eηt[z⊺(0)Prtz(0)+∫-τrt(t)‍z⊺(s)Q1rtz(s)ds+∫-τrt(t)‍g⊺(z(s))Q2rtg(z(s))ds+∫-τ0‍∫θ0‍z⊺(s)W1z(s)ds+∫-τ0‍∫θ0‍g⊺(z(s))W1g(z(s))ds]<eαt[maxi∈𝒩λmax(Pi)+τmaxi∈𝒩λmax(Q1i)+τγ¯s2maxi∈𝒩λmax(Q2i)+τ2λmax(W1)+τ2γ¯s2λmax(W2)maxi∈𝒩]×sup-τ≤s≤0{x⊺(s)x(s)}≤c1eηT(λ2+τλ3+τγ¯s2λ4+τ2λ5+τ2γ¯s2λ6).
On the other hand, from (16), we can get
(35)𝔼{z⊺(t)Piz(t)}≥maxi∈𝒩λmin(Pi)𝔼∥z(t)∥2.
Then, we can obtain
(36)𝔼∥z(t)∥2<c1eηT(λ2+τλ3+τγ¯s2λ4+τ2λ5+τ2γ¯s2λ6)λ1.
By condition (14), we can obtain
(37)𝔼∥z(t)∥2<c2.
By Definition4, we conclude that Markovian jump system (1) is stochastically finite-time bounded with respect to (c1,c2,T).Remark 7.
In this paper, it is in contrast with existing results for delay-dependent Markovian jump systems with partly unknown transition probabilities, and another different method is presented to tackle the unknown elements in the transition matrix. Compared with [33], some slack matrix variables Sυ are introduced in this paper based on the probability identity ∑j=1N‍μij=0, which leads to less conservativeness than [33].Remark 8.
Theorem6 develops a finite-time bounded criterion of Markovian jumping neural networks with time-varying delays and partially known transition rates. Theorem 6 makes full use of the information of the subsystems’ upper bounds of the time-varying delays, which also brings us the less conservativeness.Remark 9.
In our paper,τi(t) and τ˙i(t) may indicate the different upper bounds during various time-delay intervals which satisfies condition (2), respectively. However, in existing work, for example, [17], τi(t) and τ˙i(t) are always extended to τi(t)≤τ=max{τi,i∈𝒩} and 0≤τ˙i(t)≤d=max{di,i∈𝒩}, respectively, which may inevitably lead to the conservativeness. Therefore, in order to reduce the conservatism, the cases above are taken into account by employing the stochastic Lyapunov-Krasovskii functional (16).


## 4. Illustrative Example

Example 1.
Consider a class of delayed Markovian jumping neural networks (9) with two operation modes in [33]:
(38)A1=[2001],A2=[3002],B1=[0.51-0.20.5],B2=[1.11-0.20.1],C1=[0.90.1-0.10.1],C2=[0.3-0.80.10.2],Γs=I2.
The mode switching is governed by a Markov chain that has the following transition rate matrix:(39)Ω=[-0.50.50.3-0.3].
In this paper, let the initial values forc1=0.25, T=2, η=1, and time-varying delay be τ1(t)=τ2(t)=0.2×|cost|, which means that τ=0.2 and d=0.2. Through Theorem 6 and optimization over value c2, it yields that delayed Markovian jumping neural networks (9) are finite-time bounded with respect to (c1,c2,T) with minimal c2=5.0312 while minimal c2 in [33] is 5.4296, which shows the less conservative result in this paper.Example 2.
Consider a class of delayed Markovian jumping neural networks (9) with partially known transition rates and operation modes described as follows:
(40)A1=[2002],A2=[2.2001.5],A3=[2.3002.5],B1=[11-1-1],B2=[10.60.10.3],B3=[0.30.20.40.1],C1=[0.88111],C2=[1-0.10.10.2],C3=[0.50.70.70.4],Γs=I2.
The three cases of the transition rates matrices are considered as(41)CaseI:Ω=[-0.80.30.50.1-0.80.70.70.4-1.1],CaseII:Ω=[-0.8??0.1-0.80.70.70.4-1.1],CaseIII:Ω=[-0.8???-0.8?0.70.4-1.1].
With the same mode switching rates, initial values and time-varying delays, through Theorem6 and optimization over value c2, it yields that in Case I, c2=4.8124; in Case II, c2=4.6121; in Case III, c2=4.5372. Therefore, the delayed Markovian jumping neural networks (9) are finite-time bounded with respect to (c1,c2,T).Remark 10.
The accessibility of the jumping process{rt,t≥0} in the existing literature is commonly assumed to be completely accessible or completely unaccessible. Note that the transition probabilities are still viewed as accessible in this paper. Therefore, the transition probabilities matrix considered in this paper is more general assumption than Markovian jump systems.


## 5. Conclusions

Unlike most existing research results focusing on Lyapunov stability property of Markovian jump system, our paper investigated finite-time stability which concerns the boundedness of state during the delayed Markovian jump interval. In this paper, we have examined the problems of finite-time boundedness for a class of delayed Markovian jumping neural networks with partly unknown transition probabilities. Based on the analysis result, the static state feedback finite-time boundedness is given. Although the derived result is not in LMIs form, we can turn it into LMIs feasibility problem by fixing some parameters. At last, numerical examples are also given to demonstrate the effectiveness of the proposed approach.



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*Source: 597298-2014-01-06.xml*