Stability Analysis for Stochastic Markovian Jump Reaction-Diffusion Neural Networks with Partially Known Transition Probabilities and Mixed Time Delays

Abstract

The stability problem is proposed for a new class of stochastic Markovian jump reaction-diffusion neural networks with partial information on transition probability and mixed time delays. The new stability conditions are established in terms of linear matrix inequalities (LMIs). To reduce the conservatism of the stability conditions, an improved Lyapunov-Krasovskii functional and free-connection weighting matrices are introduced. The obtained results are dependent on delays and the measure of the space AND, therefore, have less conservativeness than delay-independent and space-independent ones. An example is given to show the effectiveness of the obtained results.

1. Introduction

During the last decades, neural networks (NNs) with time delays have received considerable attention, because time delays existed in many fields, for instance, finite switching speeds of amplifiers and transmission of signals in a network, which affect the system performance [1–4]. Therefore, the stability analysis of NNs with time delays has attracted more and more attention of the researchers. Various sufficient conditions, either delay-dependent or delay-independent, have been derived to ensure the stability for NNs [1–3]. On the other hand, Markovian jump systems (MJSs) involve both time-evolving and event-driven mechanisms, which can be employed to model the abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. The issues of stability, stabilization, control, and filtering for the systems have been well investigated [5–8]. In such systems, the transition probabilities of the jumping process are crucial, almost all the issues on MJSs have been investigated assuming the complete knowledge of these transition probabilities. However, in most cases the transition probabilities of MJSs are not exactly known. Whether in theory or in practice, it is necessary to further consider more general MJSs with partial information on transition probabilities. Recently, considerable attention has been devoted to the research on MJSs with partial information on transition probability [9–11].

As it is well known, the diffusion phenomena could not be ignored in NNs and electric circuits when electrons are moving in a nonuniform electromagnetic field. Therefore, it is essential to consider the state variables varying with the time and space variables. The NNs with diffusion terms can commonly be expressed by partial differential equations [12–28]. In the real world, a real system is usually affected by unknown disturbances, which may be regarded as stochastic processes [20, 24, 29, 30]. Consequently, it is of significant importance to study stochastic effects for the NNs. In recent years, the dynamic behaviors of stochastic NNs, especially the stability of stochastic NNs, have become a hot study topic. Recently, several kinds of NNs with delays and stochastic effects have been investigated [20, 24, 29]. For example, the authors in [20] have obtained some criteria to guarantee the almost sure exponential stability, and mean square exponential stability of an equilibrium solution for reaction-diffusion neural networks (RDNNs) with continuously distributed delays and stochastic influence. However, in these papers, due to the complicated behavior for stochastic Markovian jump RDNNs with partial information on transition probability and mixed time delays, very few results on such systems have appeared.

Based on the above discussions, we are interested in the asymptotic stability of the equilibrium point in the mean square sense for a new class of stochastic Markovian jump RDNNs with partial information on transition probability and mixed time delays. To the best of our knowledge, till now, this is the first attempt to introduce and investigate stochastic Markovian jump RDNNs with partial information on transition probability and mixed time delays. In this paper, the contribution of our note is two-fold. Firstly, the proposed RDNNs systems will be more general and cover the cases of systems with completely unknown or known transition probabilities. Secondly, based on the Lyapunov stability theory combined with linear matrix inequality (LMI) technique some novel stability conditions in terms of LMIs are derived by introducing some free-connection weighting matrices which can be selected properly to lead much less conservative results. Some asymptotic stability criteria are dependent on delays and the measure of the space, therefore, they have less conservativeness than delay-independent and space-independent ones. Furthermore, the results depend on the reaction-diffusion terms. The criteria of our paper are new and they complement previously known results. Hence, it is shown that the newly obtained results are less conservative and more applicable than the existing corresponding ones. An example is given to illustrate the effectiveness of the proposed method.

Notation. In this paper, the superscript “𝑇” stands for matrix transposition; ℝ^𝑛 denotes the n-dimensional Euclidean space. Vector 𝑋∈ℝ^𝑛, its norm is defined as |𝑋|=√𝑋^𝑇𝑋. For symmetric matrices 𝐴 and 𝐵, the notation 𝐴>𝐵(𝐴≥𝐵) means that the 𝐴−𝐵 is positive definite (semipositive-definite). The symmetric terms in a symmetric matrix are denoted by ∗. Mathematical expectation will be denoted by 𝐸[⋅]. tr(⋅) denotes the trace of the corresponding matrix.

For 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),...,𝑢_𝑛(𝑡,𝑥))^𝑇∈ℝ^𝑛, we define ‖𝑢(𝑡,𝑥)‖₂=^𝑛_𝑖=1𝑢_𝑖(𝑡,𝑥)₂=^𝑛_𝑖=1Ω𝑢_𝑖(𝑡,𝑥)²𝑑𝑥^1/2. (1.1)

Let 𝑃ℂ[(−𝜏,0]×Ω;ℝ^𝑛] denote the family of right continuous function 𝜑 from (−𝜏,0]×Ω to ℝ^𝑛 with the uniform norm ‖𝜑(𝑠,𝑥)‖_𝜏=sup₋𝜏_≤𝑠≤0∑^𝑛_𝑖=1‖𝜑_𝑖(𝑠,𝑥)‖₂. Denote by 𝐿²_𝐹𝑡[(−𝜏,0]×Ω;ℝ^𝑛] the family of all 𝐹_𝑡 measurable, 𝑃ℂ[(−𝜏,0]×Ω;ℝ^𝑛]-valued stochastic variables 𝜉={𝜉(𝜃,𝑥)∶𝜃∈(−𝜏,0]} such that ∫_Ω∫⁰₋𝜏𝐸|𝜉(𝑠,𝑥)|²𝑑𝑠𝑑𝑥<∞.

2. Problem Formulation and Preliminaries

In this paper, we consider the following stochastic Markovian jumping RDNNs with partial information on transition probability and mixed time delays: 𝑑𝑢(𝑡,𝑥)=^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝐷𝜕𝑢(𝑡,𝑥)𝜕𝑥_𝑙+(𝑟(𝑡𝑢()(𝑟(𝑡𝑓(𝑢((𝑟(𝑡𝑔(𝑢((𝑡)(𝑟(𝑡𝑑𝑡−𝐴))𝑡,𝑥+𝐵))𝑡,𝑥))+𝐶))𝑡−𝑑,𝑥))+𝐸))^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(()((𝑡))𝑠,𝑥))𝑑𝑠+𝐽𝑑𝑡+𝜎𝑡,𝑥,𝑢𝑡,𝑥,𝑢𝑡−𝑑,𝑥,^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠,𝑥))𝑑𝑠,𝑟(𝑡)𝑑𝑤(𝑡),𝑡≥𝑡₀()()∈[)𝑢𝑡≥0,𝑥∈Ω,𝑢𝑡,𝑥=0,𝑡,𝑥0,+∞×𝜕Ω,₀+𝑠,𝑥=𝜑(𝑠,𝑥),(𝑠,𝑥)∈(−𝜏,0]×Ω, (2.1) where {𝑟(𝑡),𝑡≥0} is a right-continuous Markovian chain on the probability space which takes values in the finite space 𝑆={1,2,...,𝑁} with generator Γ=(𝛾_𝑖𝑗)_𝑁×𝑁 given by 𝑃𝑟()(𝑡)=𝛾𝑡+𝛿=𝑗∣𝑟=𝑖^𝑖𝑗𝛿+𝑜(𝛿),if𝑖≠𝑗,1+𝛾_𝑖𝑗𝛿+𝑜(𝛿),if𝑖=𝑗, (2.2) with 𝛿>0 and lim_𝛿→0𝑜(𝛿)/𝛿=0, 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 and 𝛾_𝑖𝑖=−∑_𝑖≠𝑗𝛾_𝑖𝑗. Assume that 𝑥=(𝑥₁,𝑥₂,...,𝑥_𝑚)^𝑇∈Ω, Ω is a compact set with smooth boundary 𝜕Ω and mesΩ>0 in space ℝ^𝑚, Ω=𝜕Ω∪Ω; 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),...,𝑢_𝑛(𝑡,𝑥))^𝑇∈ℝ^𝑛 denotes the state vector associated with the 𝑛 neurons at time 𝑡 and in space 𝑥; the diagonal matrix 𝐴(𝑟(𝑡))=diag(𝑎₁(𝑟(𝑡)),..,𝑎_𝑛(𝑟(𝑡))) has positive entries 𝑎_𝑖(𝑟(𝑡))>0, 𝐵(𝑟(𝑡))=(𝑏_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛,  𝐶(𝑟(𝑡))=(𝑐_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛 and 𝐸(𝑟(𝑡))=(𝑒_𝑖𝑗(𝑟(𝑡)))_𝑛×𝑛 are the interconnection matrices representing the weight coefficients of the neurons; 𝑓(𝑢(𝑡,𝑥))=(𝑓₁(𝑢₁(𝑡,𝑥)),...,𝑓_𝑛(𝑢_𝑛(𝑡,𝑥)))^𝑇,𝑔(𝑢(𝑡,𝑥))=(𝑔₁(𝑢₁(𝑡,𝑥)),...,𝑔_𝑛(𝑢_𝑛(𝑡,𝑥)))^𝑇 and ℎ(𝑢(𝑠,𝑥))=(ℎ₁(𝑢₁(𝑠,𝑥)),...,ℎ_𝑛(𝑢_𝑛(𝑠,𝑥)))^𝑇 are the neuron activation functions, and 𝐽=(𝐽₁,𝐽₂,...,𝐽_𝑛)^𝑇denotes a constant external input vector. 𝑑(𝑡) and 𝜏(𝑡) denote the discrete time-varying delay and the distributed time-varying delay, respectively. Time-delays are assumed to satisfy 0≤𝑑(𝑡)≤𝑑₀,0≤𝜏(𝑡)≤𝜏₀, where 𝑑₀ and 𝜏₀ are positive constants, max{𝜏₀,𝑑₀}=𝜏,  𝜏̇(𝑡)≤𝜇<1,𝑑(̇𝑡)≤𝑑<1, where 𝑑 and 𝜇 are constants; 𝐷=diag(𝐷₁,𝐷₂,...,𝐷_𝑛)𝐷_𝑖=𝐷_𝑖(𝑡,𝑥,𝑢)≥0, 𝑖=1,2,...𝑛 stand for transmission diffusion operator along the ith neurons. The noise perturbation 𝜎(⋅)=𝜎(𝑡,𝑥,𝑢(𝑡,𝑥),𝑢(𝑡−𝑑(𝑡),𝑥),∫^𝑡_{𝑡−𝜏(𝑡)}ℎ(𝑢(𝑠,𝑥))𝑑𝑠,𝑟(𝑡)) is the noise intensity matrix; 𝑤(𝑡)=(𝑤₁(𝑡),...,𝑤_𝑛(𝑡))^𝑇 is an 𝑛-dimensional standard Brownian motion defined on a complete probability space (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) with a filtration {𝐹_𝑡}_𝑡≥0 generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}, where we associate Ω with the canonical space generated by all {𝑤_𝑖(𝑡)} and denote by 𝐹 the associated 𝜎-algebra generated by {𝑤(𝑡)}with the probability measure 𝑃. Moreover, we assume that the Brownian motion {𝑤(𝑡)∶𝑡≥0} is independent from the Markov chain {𝑟(𝑡)∶𝑡≥0}.  𝑛 is the outer normal vector of 𝜕Ω, 𝜑_𝑖(𝑠,𝑥) is given bounded and continuous on (−𝜏,0]×Ω,𝑖,𝑗=1,2,...,𝑛.

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 available, namely, some elements in matrix Θ are unknown in this paper. For example, the transition rate matrix Θ for system (2.1) with 𝑁 operation modes may be expressed as 𝛾Θ=¹¹?𝛾₁₃…𝛾_1𝑁??𝛾₂₃…𝛾_2𝑁⋮⋮⋮⋱⋮?𝛾_𝑁2?…𝛾_𝑁𝑁, (2.3) where “?” represents the unknown transition rate. For notation clarity, for all 𝑖∈𝑆, the set Ξ^𝑖 denotes Ξ^𝑖=Ξ^𝑖_𝑘∪Ξ^𝑖_𝑢𝑘 with Ξ^𝑖_𝑘={𝑗∶𝛾_𝑖𝑗 is known for 𝑗∈𝑆}, Ξ^𝑖_𝑢𝑘={𝑗∶𝛾_𝑖𝑗 is unknown for 𝑗∈𝑆}.

Furthermore, when Ξ^𝑖_𝑘≠𝜙, it is further described as Ξ^𝑖_𝑘=𝑘^𝑖₁,𝑘^𝑖₂…,𝑘^𝑖_𝑛, (2.4) where 1≤𝑛≤𝑁, 𝑛∈𝑁^∗ and 𝑘^𝑖_𝑗∈𝑁^∗,1≤𝑘^𝑖_𝑗≤𝑁, 𝑗=1,2,...,𝑛 represent the jth known element of the set Ξ^𝑖_𝑘 in the ith row of the transition rate matrix Θ.

Remark 2.1. It is noted that the jumping process is commonly assumed to be completely available (Ξ^𝑖_𝑢𝑘=𝜙,Ξ^𝑖_𝑘=Ξ^𝑖) or completely unavailable (Ξ^𝑖_𝑢𝑘=Ξ^𝑖,Ξ^𝑖_𝑘=𝜙). However, in most cases the transition probabilities of MJSs are not exactly known. Whether in theory or in practice, it is necessary to further consider more general jump systems with partial information on transition probabilities. Recently, a considerable amount of attention has been paid to studying the stability and stabilization of general MJSs governed by ordinary differential equations with partly unknown transition probabilities [9–11]. It is well known that the stability analysis of partial differential equations with partial information on transition probabilities is more complicated, very few results on such systems have appeared. In this paper, the new stability criteria for a new class of novel stochastic Markovian jump RDNNs with partial information on transition probability and mixed time delays are investigated.

For the sake of simplicity, we write 𝑟(𝑡)=𝑖∈𝑆, and 𝑑𝑢(𝑡,𝑥)=^𝑛_𝑙=1𝜕𝜕𝑥_𝑙𝐷𝜕𝑢(𝑡,𝑥)𝜕𝑥_𝑙+𝑑𝑡−𝐴_𝑖𝑢(𝑡,𝑥)+𝐵_𝑖𝑓(𝑢(𝑡,𝑥))+𝐶_𝑖𝑔(𝑢(𝑡−𝑑(𝑡),𝑥))+𝐸_𝑖^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠,𝑥))𝑑𝑠+𝐽𝑑𝑡+𝜎_𝑖𝑡,𝑥,𝑢(𝑡,𝑥),𝑢(𝑡−𝑑(𝑡),𝑥),^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠,𝑥))𝑑𝑠𝑑𝑤(𝑡),𝑡≥𝑡₀≥0,𝑥∈Ω, (2.5) Throughout this paper, the following assumptions are made.(A1) There exist positive diagonal matrices 𝐿^𝑓=diag(𝐿^𝑓₁,...,𝐿^𝑓_𝑛),𝐿^𝑔=diag(𝐿^𝑔₁,...,𝐿^𝑔_𝑛),  𝐿^ℎ=diag(𝐿^ℎ₁,...,𝐿^ℎ_𝑛), such that 𝑓0≤^𝑗𝜉₁−𝑓_𝑗𝜉₂𝜉₁−𝜉₂≤𝐿^𝑓_𝑗𝑔,0≤^𝑗𝜉₁−𝑔_𝑗𝜉₂𝜉₁−𝜉₂≤𝐿^𝑔_𝑗ℎ,0≤^𝑗𝜉₁−ℎ_𝑗𝜉₂𝜉₁−𝜉₂≤𝐿^ℎ_𝑗, (2.6) for all 𝜉₁,𝜉₂∈ℝ,𝜉₁≠𝜉₂, 𝑗=1,2,...,𝑛.(A2) There exist diagonal matrices ∑_𝑖1,∑_𝑖2 and ∑_𝑖3(𝑖∈𝑆) such that tr𝜎_𝑖𝑡,𝑥,𝜁₁,𝜁₂,𝜁₃^𝑇𝜎_𝑖𝑡,𝑥,𝜁₁,𝜁₂,𝜁₃≤_𝑖1𝜁₁²+_𝑖2𝜁₂²+_𝑖3𝜁₃², (2.7) for all 𝜁₁,𝜁₂,𝜁₃∈ℝ^𝑛.(A3) There exist constants 𝐷^∗_𝑖 such that 𝐷_𝑖≥𝐷^∗_𝑖≥0,𝑖=1,2,...,𝑛.(A4)𝜎_𝑖(𝑡,𝑥,𝑢^∗,𝑢^∗,∫^𝑡_{𝑡−𝜏(𝑡)}ℎ(𝑢^∗(𝑠,𝑥))𝑑𝑠)=0, where 𝑢^∗ is the equilibrium point of system (2.1).

Remark 2.2. By assumption (A1)–(A4), it is not difficult to prove that there exists a unique equilibrium point 𝑢^∗ for system (2.1) based on Mawhin’s continuation theorem [31].

The main aim of this paper is to investigate the asymptotic stability in the mean square sense of system (2.1). Let 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,...,𝑢^∗_𝑛)^𝑇 be the equilibrium point of system (2.1). Now, we set 𝑧(𝑡)≜𝑢(𝑡,𝑥)−𝑢^∗, which yields the following system: 𝑑𝑧(𝑡)=^𝑛_𝑙=1𝜕𝜕𝑥_𝑙𝐷𝜕𝑧(𝑡)𝜕𝑥_𝑙+𝑑𝑡−𝐴_𝑖𝑧(𝑡)+𝐵_𝑖𝑓(𝑧(𝑡))+𝐶_𝑖𝑔(𝑧(𝑡−𝑑(𝑡)))+𝐸_𝑖^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑡+𝜎_𝑖𝑡,𝑥,𝑧(𝑡),𝑧(𝑡−𝑑(𝑡)),^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑤(𝑡),𝑡≥𝑡₀𝑧(𝑡),𝑥∈Ω,=0,𝑡≥𝑡₀,𝑥∈𝜕Ω,𝑧𝑡₀+𝑠=𝜑(𝑠)−𝑢^∗,(𝑠,𝑥)∈(−𝜏,0]×Ω, (2.8) where 𝑓(𝑧(𝑡))=𝑓(𝑧(𝑡)+𝑢^∗)−𝑓(𝑢^∗),   𝑔(𝑧(𝑡))=𝑔(𝑧(𝑡)+𝑢^∗)−𝑔(𝑢^∗),   ℎ(𝑧(𝑠))=ℎ(𝑧(𝑠)+𝑢^∗)−ℎ(𝑢^∗),   𝑓(𝑧(𝑡))=(𝑓₁(𝑧₁(𝑡)),...,𝑓_𝑛(𝑧_𝑛(𝑡)))^𝑇,  𝑔(𝑧(𝑡))=(𝑔₁(𝑧₁(𝑡)),...,𝑔_𝑛(𝑧_𝑛(𝑡)))^𝑇,    ℎ(𝑧(𝑠))=(ℎ₁(𝑧₁(𝑠)),...,ℎ_𝑛(𝑧_𝑛(𝑠)))^𝑇.

Clearly, 𝑓_𝑗(𝑧_𝑗(𝑡)),𝑔_𝑗(𝑧_𝑗(𝑡)), and ℎ_𝑗(𝑧_𝑗(𝑠)) satisfy (A1). Thus, the stability problem of system (2.1) is equivalent to the stability problem of system (2.8).

Definition 2.3. The equilibrium point 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,...,𝑢^∗_𝑛)^𝑇 of system (2.1) is said to be asymptotic stability in the mean square sense if there exists a constant 𝛿>0 and for any 𝜀>0 satisfying that 𝐸‖𝑢(𝑡,𝑥)−𝑢^∗‖²₂<𝜀 and lim_𝑡→∞𝐸‖𝑢(𝑡,𝑥)−𝑢^∗‖²₂=0 when 𝑡≥𝑡₀≥0 and ‖𝜑(𝑠,𝑥)−𝑢^∗‖_𝜏<𝛿, where ‖𝜑−𝑢^∗‖_𝜏=sup₋𝜏_≤𝑠≤0‖𝜑(𝑠,𝑥)−𝑢^∗‖₂.

Lemma 2.4 (see [2] (Jensen inequality)). Consider the following. (i) For any constant matrix Ξ∈ℝ^𝑛×𝑛,Ξ>0, scalar function 𝑑(𝑡)∶0<𝑑(𝑡)<𝑑, and vector function 𝜔∶[0,𝑑]→ℝ^𝑛 such that the integrations concerned are well defined, then ₀^𝑑(𝑡)𝜔(𝑠)𝑑𝑠^𝑇Ξ₀^𝑑(𝑡)𝜔(𝑠)𝑑𝑠≤𝑑(𝑡)₀^𝑑(𝑡)𝜔(𝑠)^𝑇Ξ𝜔(𝑠)𝑑𝑠. (2.9)
(ii) For any constant matrix Ξ∈𝑅^𝑛×𝑛,Ξ>0,Ω⊂ℝ^𝑛,mesΩ>0, if 𝜔∶Ω→ℝ^𝑛 is vector function such that the integration is well defined, then _Ω𝜔(𝑠)𝑑𝑠^𝑇Ξ_Ω𝜔(𝑠)𝑑𝑠≤|Ω|_Ω𝜔(𝑠)^𝑇Ξ𝜔(𝑠)𝑑𝑠. (2.10)

Lemma 2.5 (see [24]). Let Ω be a bounded open domain in ℝ^𝑚 with smooth boundary 𝜕Ω. If ℎ=ℎ(𝑥) define on Ω is a smooth function with ℎ|_𝜕Ω=0, then the following inequality holds: _Ωℎ²(𝑥)d𝑥≤𝛼_Ω𝜕ℎ(𝑥)𝜕𝑥²d𝑥, (2.11) where |Ω| denotes the volume of Ω, 𝛼=(|Ω|/𝜔_𝑚)^1/𝑚>0, 𝜔_𝑚 denotes the surface area of unit ball in ℝ^𝑚.

3. Main Results

Theorem 3.1. Under assumptions (A1)–(A4), if there exist positive definite diagonal matrices 𝑃_𝑖,  Λ_𝑖=Λ^𝑇_𝑖, positive definite symmetry matrices 𝑄₁,𝑄₂,𝐺,W,𝐾₁,𝐾₂, positive definite diagonal matrices 𝑀₁,𝑀₂ with appropriate dimensions, and scalar 𝜌_𝑖>0, such that the following Linear matrix inequalities (LMIs) hold: Ξ₁=𝛼₁₁00𝛼₁₄𝛼₁₅𝛼₁₆00∗𝛼₂₂000000∗∗𝛼₃₃00000∗∗∗𝛼₄₄0000∗∗∗∗𝛼₅₅000∗∗∗∗∗𝛼₆₆00∗∗∗∗∗∗𝛼₇₇0∗∗∗∗∗∗∗𝛼₈₈𝑃<0,_𝑖≤𝜌_𝑖𝑃𝐼,_𝑗−Λ_𝑖≤0,𝑗∈Ξ^𝑖_𝑢𝑘𝑃,𝑗≠𝑖,_𝑖−Λ_𝑖≥0,𝑗∈Ξ^𝑖_𝑢𝑘,𝑗=𝑖, (3.1) where 𝛼₁₁=−𝑃_𝑖𝐴_𝑖−𝐴_𝑖𝑃_𝑖−𝑃_𝑖𝑅−𝑅𝑃_𝑖+𝑄₁+𝑄₂+𝐿^𝑔𝐺𝐿^𝑔+_{𝑗∈Ξ𝑖𝑘}𝛾_𝑖𝑗𝑃_𝑗−Λ_𝑖+𝜌_𝑖^𝑇_𝑖1𝑖1+𝜏₀𝐿^ℎ𝑊𝐿^ℎ+𝜏₀|Ω|𝐿^ℎ𝐾₁𝐿^ℎ+𝑑₀|Ω|𝐿^𝑔𝐾₂𝐿^𝑔,𝛼₁₄=𝑃_𝑖𝐵_𝑖+𝐿^𝑓𝑀^𝑇₁,𝛼₁₅=𝑃_𝑖𝐶_𝑖,𝛼₁₆=𝑃_𝑖𝐸_𝑖,𝛼₂₂=−1−𝜇𝑄₁,𝛼₃₃=𝐿^𝑔𝑀₂𝐿^𝑔+𝜌_𝑖^𝑇_𝑖2𝑖2−(1−𝑑)𝑄₂,𝛼₄₄=−2𝑀₁,𝛼₅₅=−(1−𝑑)𝐺−𝑀₂,𝛼₆₆=𝜌_𝑖^𝑇_𝑖3𝑖3−𝜏₀𝛼𝑊,₇₇=−𝜏₀⁻¹𝐾₁,𝛼₈₈=−𝑑₀⁻¹𝐾₂, (3.2)𝑅=diag{−𝐷^∗₁/𝛼,...,−𝐷^∗_𝑛/𝛼}, then equilibrium point 𝑢^∗ of system (2.1) with a partly known transition rate matrix (2.3) is an asymptotic stability in the mean square sense.

Proof. Consider the Lyapunov-Krasovskii functional: 𝑉(𝑡,𝑧(𝑡),𝑖)=_Ω𝑧(𝑡)^𝑇𝑃_𝑖𝑧(𝑡)𝑑𝑥+_Ω^𝑡_{(𝑡)𝑡−𝜏}𝑧(𝑠)^𝑇𝑄₁𝑧(𝑠)𝑑𝑠𝑑𝑥+_Ω^𝑡_{(𝑡)𝑡−𝑑}𝑧(𝑠)^𝑇𝑄₂𝑧(𝑠)𝑑𝑠𝑑𝑥+_Ω^𝑡_{(𝑡)𝑡−𝑑}𝑔(𝑧(𝑠))^𝑇𝐺𝑔(𝑧(𝑠+))𝑑𝑠𝑑𝑥_{Ω0−𝜏0𝑡𝑡+𝜃}ℎ(𝑧(𝑠))^𝑇𝑊ℎ(𝑧(𝑠))𝑑𝑠𝑑𝜃𝑑𝑥+⁰_−𝑑0^𝑡_𝑡+𝜃Ω𝑔(𝑧(𝑠))𝑑𝑥^𝑇𝐾_2Ω𝑔(𝑧(𝑠+))𝑑𝑥𝑑𝑠𝑑𝜃_{0−𝜏0𝑡𝑡+𝜃Ω}ℎ(𝑧(𝑠))𝑑𝑥^𝑇𝐾_1Ωℎ(𝑧(𝑠))𝑑𝑥𝑑𝑠𝑑𝜃. (3.3) Define an operator 𝐿 associated with (2.8) acting on 𝑉(𝑡,𝑧(𝑡),𝑖) by 𝐿𝑉(𝑡,𝑧(𝑡),𝑖)=𝑉_𝑡(𝑡,𝑧(𝑡),𝑖)+𝑉_𝑧((𝑡))×𝑡,𝑧,𝑖_𝑛𝑙=1𝜕𝜕𝑥_𝑙𝐷𝜕𝑧(𝑡)𝜕𝑥_𝑙𝑑𝑡−𝐴_𝑖𝑧(𝑡)+𝐵_𝑖𝑓(𝑧(𝑡))+𝐶_𝑖𝑔(𝑧(𝑡−𝑑(𝑡)))+𝐸_𝑖^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠+1))𝑑𝑠2trace𝜎_𝑖^𝑇(⋅)𝑉_𝑧𝑧(𝑡,𝑧,𝑖)𝜎_𝑖(⋅)+^𝑁_𝑗=1𝛾_𝑖𝑗𝑉𝑡,𝑧(𝑡),𝑗, (3.4) where 𝑉_𝑡(𝑡,𝑧(𝑡),𝑖)=𝜕𝑉(𝑡,𝑧(𝑡),𝑖)/𝜕𝑡,𝑉_𝑧(𝑡,𝑧(𝑡),𝑖) = (𝜕𝑉(𝑡,𝑧(𝑡),𝑖))/𝜕𝑧₁,…,(𝜕𝑉(𝑡,𝑧(𝑡),𝑖))/𝜕𝑧_𝑛, 𝑉_𝑧𝑧(𝑡,𝑧(𝑡),𝑖) = (𝜕²𝑉(𝑡,𝑧(𝑡),𝑖)/𝜕𝑧_𝑖𝜕𝑧_𝑗)_𝑛×𝑛.
The weak infinitesimal generator 𝐿𝑉(𝑡,𝑧,𝑖) along with (2.8), leads to 𝐿𝑉(𝑡,𝑧,𝑖)=_Ω2𝑧(𝑡)^𝑇𝑃_𝑖𝜕𝑧(𝑡)+𝜕𝑡^𝑁_𝑗=1𝛾_𝑖𝑗𝑧(𝑡)^𝑇𝑃_𝑗𝑧(𝑡)+trace𝜎_𝑖^𝑇(⋅)𝑃_𝑖𝜎_𝑖(⋅)𝑑𝑥+_Ω𝑧(𝑡)^𝑇𝑄₁𝑧(𝑡)−(1−𝜏̇(𝑡))𝑧(𝑡−𝜏(𝑡))^𝑇𝑄₁𝑧((𝑡+𝑡−𝜏))𝑑𝑥_Ω𝑧(𝑡)^𝑇𝑄₂𝑧(𝑡)−1−𝑑̇(𝑡)𝑧(𝑡−𝑑(𝑡))^𝑇𝑄₂𝑧((𝑡+𝑡−𝑑))𝑑𝑥_Ω𝑔(𝑧(𝑡))^𝑇𝐺𝑔(𝑧(𝑡))−1−𝑑̇(𝑡)𝑔(𝑧(𝑡−𝑑(𝑡)))^𝑇𝐺𝑔(𝑧((𝑡+𝑡−𝑑)))𝑑𝑥_Ω𝜏₀ℎ(𝑧(𝑡))^𝑇𝑊ℎ(𝑧(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))^𝑇𝑊ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑥+⁰_−𝜏0Ωℎ(𝑧(𝑡))𝑑𝑥^𝑇𝐾_1Ωℎ(𝑧(𝑡−))𝑑𝑥𝑑𝜃_0−𝜏0Ωℎ(𝑧(𝑡+𝜃))𝑑𝑥^𝑇𝐾_1Ωℎ(𝑧(+𝑡+𝜃))𝑑𝑥𝑑𝜃_0−𝑑0Ω𝑔(𝑧(𝑡))𝑑𝑥^𝑇𝐾_2Ω𝑔(𝑧(𝑡−))𝑑𝑥𝑑𝜃_0−𝑑0Ω𝑔(𝑧(𝑡+𝜃))𝑑𝑥^𝑇𝐾_2Ω𝑔(𝑧(𝑡+𝜃))𝑑𝑥𝑑𝜃. (3.5)
Considering the situation that the information of transition probabilities is not accessible completely, the following equalities satisfy for arbitrary matrices Λ_𝑖=Λ^T_i, due to ∑^𝑁_𝑗=1𝛾_𝑖𝑗=0−𝑧(𝑡)^𝑇𝑁_𝑗=1𝛾_𝑖𝑗Λ_𝑖𝑧(𝑡)=0, (3.6) From (A1), we have _Ω𝑓(𝑧(𝑡))^𝑇𝑀₁𝐿^𝑓𝑧(𝑡)−𝑓(𝑧(𝑡))^𝑇𝑀₁𝑓(𝑧(𝑡))𝑑𝑥≥0,_Ω𝑧(𝑡−𝑑(𝑡))^𝑇𝐿^𝑔𝑀₂𝐿^𝑔𝑧(𝑡−𝑑(𝑡))−𝑔(𝑧(𝑡−𝑑(𝑡)))^𝑇𝑀₂𝑔(𝑧(𝑡−𝑑(𝑡)))𝑑𝑥≥0, (3.7) where 𝑀₁ and 𝑀₂ are positive definite diagonal matrices.
According to Lemma 2.4, we derive ⁰_−𝜏0Ωℎ(𝑧(𝑡))𝑑𝑥^𝑇𝐾_1Ωℎ(𝑧(𝑡−))𝑑𝑥𝑑𝜃_0−𝜏0Ωℎ(𝑧(𝑡+𝜃))𝑑𝑥^𝑇𝐾_1Ωℎ(𝑧(𝑡+𝜃))𝑑𝑥𝑑𝜃≤𝜏₀|Ω|_Ωℎ(𝑧(𝑡))^𝑇𝐾₁ℎ(𝑧(𝑡))𝑑𝑥−𝜏₀^−1𝑡_{(𝑡)𝑡−𝜏Ω}ℎ(𝑧(𝑠))𝑑𝑥𝑑𝑠^𝑇𝐾₁^𝑡_{(𝑡)𝑡−𝜏Ω}ℎ(𝑧(𝑠))𝑑𝑥𝑑𝑠_0−𝑑0Ω,𝑔(𝑧(𝑡))𝑑𝑥^𝑇𝐾_2Ω𝑔(𝑧(𝑡−))𝑑𝑥𝑑𝜃_0−𝑑0Ω𝑔(𝑧(𝑡+𝜃))𝑑𝑥^𝑇𝐾_2Ω𝑔(𝑧(𝑡+𝜃))𝑑𝑥𝑑𝜃≤𝑑₀|Ω|_Ω𝑔(𝑧(𝑡))^𝑇𝐾₂𝑔(𝑧(𝑡))𝑑𝑥−𝑑₀^−1𝑡_{(𝑡)𝑡−𝑑Ω}𝑔(𝑧(𝑠))𝑑𝑥𝑑𝑠^𝑇𝐾₂^𝑡_{(𝑡)𝑡−𝑑Ω}𝑔(𝑧(𝑠,−))𝑑𝑥𝑑𝑠_{Ω𝑡(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))^𝑇𝑊ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑥≤−𝜏₀⁻¹_Ω^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠^𝑇𝑊^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑥. (3.8)(3.9)(3.10) According to the boundary condition, 𝐷_𝑖≥0 and Lemma 2.5, applying Green’s formula, we get _Ω^𝑚_𝑙=1𝑧_𝑖(𝑡)𝜕𝜕𝑥_𝑙𝐷_𝑖𝜕𝑧_𝑖(𝑡)𝜕𝑥_𝑙𝑑𝑥=^𝑚_𝑙=1Ω𝜕𝜕𝑥_𝑙𝐷_𝑖𝜕𝑧_𝑖(𝑡)𝜕𝑥_𝑙𝑧_𝑖(𝑡)𝑑𝑥−^𝑚_𝑙=1Ω𝐷_𝑖𝜕𝑧_𝑖(𝑡)𝜕𝑥_𝑙²=𝑑𝑥_𝜕Ω𝐷_𝑖𝑧_𝑖(𝑡)𝜕𝑧_𝑖(𝑡)𝜕𝜈𝑑𝑠−^𝑚_𝑙=1Ω𝐷_𝑖𝜕𝑧_𝑖(𝑡)𝜕𝑥_𝑙²𝐷𝑑𝑥≤−^∗𝑖𝛼_Ω𝑧_𝑖(𝑡)²𝑑𝑥, (3.11) where 𝛼=(|Ω|/𝜔_𝑚)^1/𝑚>0.
From (3.6)–(3.11), we have 𝐿𝑉(𝑡,𝑧(𝑡),𝑖)≤_Ω𝑧(𝑡)^𝑇−𝑃_𝑖𝐴_𝑖−𝐴_𝑖𝑃_𝑖−𝑃_𝑖𝑅−𝑅𝑃_𝑖+𝑄₁+𝑄₂+𝐿^𝑔𝐺𝐿^𝑔+_{𝑗∈Ξ𝑖𝑘}𝛾_𝑖𝑗𝑃_𝑗−Λ_𝑖+𝜌_𝑖^𝑇_𝑖1𝑖1+𝜏₀𝐿^ℎ𝑊𝐿^ℎ+𝜏₀|Ω|𝐿^ℎ𝐾₁𝐿^ℎ+𝑑₀|Ω|𝐿^𝑔𝐾₂𝐿^𝑔𝑧(𝑡)+𝑑𝑥_Ω𝑧(𝑡)^𝑇𝑃_𝑖𝐵_𝑖+𝐿^𝑓𝑀^𝑇₁𝑓(𝑧(𝑡+))𝑑𝑥_Ω𝑧(𝑡)^𝑇_{𝑗∈Ξ𝑖𝑢𝑘}𝛾_𝑖𝑗𝑃_𝑗−Λ_𝑖𝑧(𝑡)𝑑𝑥+_Ω𝑓(𝑧(𝑡))^𝑇𝐵^𝑇_𝑖𝑃_𝑖+𝑀₁𝐿^𝑓𝑧(𝑡)+𝑑𝑥_Ω𝑧(𝑡)^𝑇𝑃_𝑖𝐶_𝑖𝑔(𝑧(𝑡−𝑑(𝑡)))𝑑𝑥+_Ω𝑔(𝑧(𝑡−𝑑(𝑡)))^𝑇𝐶^𝑇_𝑖𝑃_𝑖𝑧(𝑡)+𝑑𝑥_Ω𝑧(𝑡)^𝑇𝑃_𝑖𝐸_𝑖^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑥+_Ω^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠^𝑇𝐸^𝑇_𝑖𝑃_𝑖𝑧(𝑡)𝑑𝑥+_Ω𝑧(𝑡−𝜏(𝑡))^𝑇−1−𝜇𝑄₁𝑧((𝑡−𝑡−𝜏))𝑑𝑥_Ω𝑔(𝑧(𝑡−𝑑(𝑡)))^𝑇(1−𝑑)𝐺+𝑀₂𝑔(𝑧(𝑡−𝑑(𝑡)))𝑑𝑥−2_Ω𝑓(𝑧(𝑡))^𝑇𝑀₁𝑓(𝑧(𝑡))𝑑𝑥+_Ω𝑧(𝑡−𝑑(𝑡))^𝑇×𝐿^𝑔𝑀^𝑇₂𝐿^𝑔+𝜌_𝑖^𝑇_𝑖2𝑖2−(1−𝑑)𝑄₂𝑧(𝑡−𝑑(𝑡))𝑑𝑥+_Ω^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠^𝑇𝜌_𝑖^𝑇_𝑖3𝑖3−𝜏₀⁻¹𝑊^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠𝑑𝑥−𝜏₀^−1𝑡_{(𝑡)𝑡−𝜏Ω}ℎ(𝑧(𝑠))𝑑𝑥𝑑𝑠^𝑇𝐾₁^𝑡_{(𝑡)𝑡−𝜏Ω}ℎ(𝑧(𝑠))𝑑𝑥𝑑𝑠−𝑑₀^−1𝑡_{(𝑡)𝑡−𝑑Ω}𝑔(𝑧(𝑠))𝑑𝑥𝑑𝑠^𝑇𝐾₂^𝑡_{(𝑡)𝑡−𝑑Ω}𝑔(𝑧(𝑠=))𝑑𝑥𝑑𝑠_Ω𝜍^𝑇₁Ξ₁𝜍₁𝑑𝑥, (3.12) where 𝜍₁=𝑧(𝑡)^𝑇,𝑧(𝑡−𝜏(𝑡))^𝑇𝑧(𝑡−𝑑(𝑡))^𝑇,𝑓(𝑧(𝑡))^𝑇𝑔(𝑧(𝑡−𝑑(𝑡)))^𝑇_{𝑡(𝑡)𝑡−𝜏}ℎ(𝑧(𝑠))𝑑𝑠^𝑇𝑡_{(𝑡)𝑡−𝜏Ω}ℎ(𝑧(𝑠))d𝑥𝑑𝑠^𝑇𝑡_{(𝑡)𝑡−𝑑Ω}𝑔(𝑧(𝑠))d𝑥𝑑𝑠^𝑇𝑇. (3.13) From the conditions of Theorem 3.1, noting that 𝛾_𝑖𝑖=−∑^𝑁_{𝑗=1,𝑗≠𝑖}𝛾_𝑖𝑗 and 𝛾_𝑖𝑗≥0 for all 𝑗≠𝑖, that is 𝛾_𝑖𝑖<0 for all 𝑖∈𝑆. Hence, when 𝑖∈Ξ^𝑖_𝑘, from (3.1) and 𝜍₁≠0, we can derive 𝐿𝑉(𝑡,𝑧(𝑡),𝑖)<0. (3.14) On the other hand, if 𝑖∈Ξ^𝑖_𝑢𝑘, according to (3.1) and 𝜍₁≠0, we have (3.14) holds.
Therefore, if ‖𝑢(𝑡₀,𝑥)−𝑢^∗‖₂<𝛿 and 𝑘→∞, lim_𝑡→∞𝐸‖𝑢(𝑡,𝑥)−𝑢^∗‖₂=0, which implies that system (2.1) is asymptotic stability in the mean square sense. This completes the proof.

Remark 3.2. In proof of above Theorem 3.1, the new Lyapunov functional to construct is more general. It is easy to see that our results expanded the model in [23]. The obtained results in [23] are independent on the measure of the space and diffusion effects. The results of this paper are dependent on the measure of the space and diffusion effects. Besides delays and stochastic effects, it is necessary to further consider more general MJSs with partial information on transition probabilities. Searching for stability analysis issue for stochastic Markovian jump RDNNs with partial information on transition probability and mixed time delays is obviously more preferable and challenging. However, no such results have been available up to now. Moreover, in this paper, the idea of free-connection weighting matrix is introduced, this method may lead to derive an improved feasible region for delay-dependent and space-dependent stability criteria. Therefore, it is shown that the newly obtained results are less conservative and more applicable than the existing corresponding ones.

Remark 3.3. In [8, 26], free weighting matrix was introduced via Newton Leibniz formula and system transformation, respectively. Theorems  3.2 and 4.1 in [26] and Corollary  3.2 in [8] have no restriction on the derivative of time-varying delays. The same way can be directly applied into our paper to cancel the restricted on the derivative of the time-varying delays. In order to simplify and avoid repeating the technique in [26], we do not do so. While new free-connection weighting matrices are introduced in (3.6) by using the relationship of the transition rates among various subsystems, that is, ∑^𝑁_𝑗=1𝛾_𝑖𝑗=0, for all 𝑖∈𝑆, in this paper, which may develop less-conservative results. This is quite different from free weighting matrices in [8, 26], we introduced free-connection weighting matrices lead to the obtained criteria are dependent on delays, the measure of the space and the reaction-diffusion terms, therefore, have less conservativeness than delay-independent, space-independent, and diffusion terms independent ones [15, 22, 23, 25], respectively.

When the smooth operator 𝐷_𝑖=0,𝑖=1,2,...,𝑛, model (2.2) is reduced to the following model: 𝑑𝑢(𝑡)=−𝐴_𝑖𝑢(𝑡)+𝐵_𝑖𝑓(𝑢(𝑡))+𝐶_𝑖𝑔(𝑢(𝑡−𝑑(𝑡)))+𝐸_𝑖^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠))𝑑𝑠+𝐽𝑑𝑡+𝜎_𝑖𝑡,𝑢(𝑡),𝑢(𝑡−𝑑(𝑡)),^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠))𝑑𝑠𝑑𝑤(𝑡),𝑡≥𝑡₀𝑢𝑡≥0,₀+𝑠=𝜑(𝑠),𝑠∈(−𝜏,0]. (3.15)

Remark 3.4. The system (3.15) is a special case of system (2.1). In [32], the authors consider the following model: 𝑢̇=−𝐴𝑢+𝐵𝑓(𝑢(𝑡))+𝐶𝑔(𝑢(𝑡−𝑑(𝑡)))+𝐸^𝑡_{(𝑡)𝑡−𝜏}ℎ(𝑢(𝑠))𝑑𝑠+𝐽. (3.16) However, To the best of our knowledge, the LMI technique to guarantee the asymptotic stability of stochastic Markovian jump NNs with partial information on transition probability and discrete and distributed time-varying delays is not studied which is very important in both theories and applications and is also a very challenging problem.

From (3.15), we have the following corollary.

Corollary 3.5. Suppose that, (A1)-(A2) and (A4) hold. If there exist positive definite diagonal matrices 𝑃_𝑖,  Λ_𝑖=Λ^𝑇_𝑖, positive definite symmetry matrices 𝑄₁,𝑄₂,𝐺,W,𝐾₁,𝐾₂, positive definite diagonal matrices 𝑀₁,𝑀₂ with appropriate dimensions, and scalar 𝜌_𝑖>0, such that the following LMIs hold: Ξ₂=𝛼₁₁00𝛼₁₄𝛼₁₅𝛼₁₆0∗𝛼₂₂00000∗∗𝛼₃₃0000∗∗∗𝛼₄₄000∗∗∗∗𝛼₅₅00∗∗∗∗∗𝛼₆₆0∗∗∗∗∗∗𝛼₇₇𝑃<0,_𝑖≤𝜌_𝑖𝑃𝐼,_𝑗−Λ_𝑖≤0,𝑗∈Ξ^𝑖_𝑢𝑘𝑃,𝑗≠𝑖,_𝑖−Λ_𝑖≥0,𝑗∈Ξ^𝑖_𝑢𝑘,𝑗=𝑖, (3.17) where 𝛼₁₁ = −𝑃_𝑖𝐴_𝑖−𝐴_𝑖𝑃_𝑖−𝑃_𝑖𝑅−𝑅𝑃_𝑖+𝑄₁+𝑄₂ + 𝐿^𝑔𝐺𝐿^𝑔+∑_{𝑗∈𝑈𝑖𝑘}𝛾_𝑖𝑗(𝑃_𝑗−Λ_𝑖)+𝜌_𝑖∑^𝑇_𝑖1∑_𝑖1 + 𝜏₀𝐿^ℎ𝐾₁𝐿^ℎ+𝑑₀𝐿^𝑔𝐾₂𝐿^𝑔,  𝛼₆₆=𝜌_𝑖∑^𝑇_𝑖3∑_𝑖3−𝜏₀⁻¹𝐾₁,  𝛼₇₇=−𝑑₀⁻¹𝐾₂, the other notations are the same as those in Theorem 3.1, then system (3.15) with a partly unknown transition rate matrix (2.3) is asymptotic stability in the mean square sense.

Proof. By constructing the following Lyapunov functional: 𝑉(𝑡,𝑧(𝑡),𝑖)=𝑧(𝑡)^𝑇𝑃_𝑖𝑧(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑧(𝑠)^𝑇𝑄₁𝑧(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝑑}𝑧(𝑠)^𝑇𝑄₂𝑧(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝑑}𝑔(𝑧(𝑠))^𝑇𝐺𝑔(𝑧(𝑠+))𝑑𝑠_{0−𝑑0𝑡𝑡+𝜃}𝑔(𝑧(𝑠))^𝑇𝐾₂𝑔(𝑧(𝑠+))𝑑𝑠𝑑𝜃_{0−𝜏0𝑡𝑡+𝜃}ℎ(𝑧(𝑠))^𝑇𝐾₁ℎ(𝑧(𝑠))𝑑𝑠𝑑𝜃, (3.18) Similarly, calculate the derivative along the system (3.15). By a minor modification of the proof of Theorem 3.1, we can easily obtain that system (3.15) is asymptotic stability in the mean square sense.

4. An Illustrative Example

In this section, we provide the effectiveness of the proposed stability criteria through solving a numerical example. Here, we consider the system (2.1) with three modes described as 𝐴₁=60.10.613,𝐴₂=150.60.710,𝐴₃=,𝐵100.50.611₁=0.10.20.20.1,𝐵₂=0.50.30.50.1,𝐵₃=,𝐶0.30.40.20.1₁=𝐶₂=𝐶₃=,𝐸0.5000.5₁=𝐸₂=𝐸₃=,Σ0.1000.1₁₁=Σ₁₂=Σ₁₃=1.2𝐼,Σ₂₁=Σ₂₂=Σ₂₃=0.8𝐼,Σ₃₁=Σ₃₂=Σ₃₃𝜏=1.1𝐼,₀=𝑑₀=0.6,𝜇=𝑑=0.2,|Ω|=𝜔_𝑚𝐿=𝜋,_𝑓=𝐿^𝑔=𝐿^ℎ=𝐼,𝐷₁=𝐷₂=𝐷₃=𝐷^∗₁=𝐷^∗₂=𝐷^∗₃=1,𝑅=−𝐼. (4.1)

The transition probability matrices of form (2.5) are given by Θ=−0.3???0.1???0.15. (4.2) In (3.1), by applying the MATLAB LMI Control Toolbox, we obtain the feasible solution as follows: 𝑃₁=1.9466001.9466,𝑃₂=1.9072001.9072,𝑃₃=,𝑄1.7064001.7064₁=1.31240.11230.11233.3850,𝑄₂=,Λ5.19030.10190.10197.1036,𝐺=0.81890.09840.09842.6072₁=Λ3.65420.30050.300522.2842²=0.9530−0.0010−0.00100.9875,Λ₃=,0.84930.00140.00140.8382𝑊=5.26830.14240.14247.9714,𝐾₁=0.69090.05990.05991.8031,𝐾₂=,𝐴0.70280.05970.05971.8035₁=3.9211003.9211,𝐴₂=.1.2151001.2151 (4.3) Therefore, it follows from Theorem 3.1 that system (2.1) is asymptotic stability in the mean square sense.

5. Conclusions

In this paper, we have dealt with an interesting and important problem of asymptotic stability in the mean square sense for a new class of stochastic Markovian jump RDNNs with unknown transition rates and mixed time delays based on LMI and free-connection weighting matrix techniques. By applying the Lyapunov stability analysis approach, the novel stability criteria for that system are proposed. The obtained criteria are dependent on delays, the measure of the space, and the reaction-diffusion terms, therefore, they have less conservativeness than delay-independent, space-independent, and diffusion terms independent ones, respectively. So, it is shown that the newly obtained results are less conservative and more applicable than the existing corresponding ones. A numerical example has been given to show the effectiveness of the obtained results.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China under Grant no. 60974139 and the Fundamental Research Funds for the Central Universities no. 72103676.