Robust Stability of Nonlinear Diffusion Fuzzy Neural Networks with Parameter Uncertainties and Time Delays

In this paper, a class of nonlinear p-Laplace diffusion BAM Cohen-Grossberg neural networks (BAM CGNNs) with time delays is investigated. In the case of p > 1 with p ≠ 2, the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox and deleting the boundedness of the amplification functions. And in the case of p = 2, LMI-based sufficient conditions are also inferred for robust input-to-state stability of reaction-diffusion Markovian jumping BAM CGNNs with the event-triggered control, which is different from those of many previous related literature. In particular, the role of diffusion can be reflected in newly acquired criteria. Finally, numerical examples verify the effectiveness of the proposed methods.


Introduction
In recent decades, reaction-diffusion neural networks have been the subject of research due to the fact that electrons have diffusion behaviors in an inhomogeneous magnetic field, and the role of diffusion items have always been investigated and discussed in many existing results ( [1][2][3][4]).Since the conduction velocity of electrons and components is limited, the phenomenon of time delays inevitably appears in various practical projects.Thereby, time-delay reaction-diffusion systems are relatively common objects of study.For example, in [5], the following time-delay reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with impulse was studied (see [7, (7)]), where ℛ ∘ ∇u is Hadamard product of matrix ℛ and vector gradient ∇u (see [6] for details).In [7], the stability of the following BAM Cohen-Grossberg neural networks (BAM CGNNs) with distributed delays was discussed.The Cohen-Grossberg-type BAM neural network model was initially proposed by Cohen and Grossberg [8] in 1983.
The model not only generalizes the single-layer autoassociative Hebbian correlator to a two-layer pattern matched heteroassociative circuit but also possesses Cohen-Grossberg dynamics, and it has promising application potentials for tasks of classification, parallel computation, associative memory, and nonlinear optimization problems.Since then, a lot of research has been done on BAM CGNNs models ( [7,[9][10][11]).Besides, owing to biological engineering backgrounds and population dynamics, economics, physical engineering, and other reasons, the stability of nonlinear diffusion systems have received widespread attention [11][12][13][14][15][16][17].For example, in [11], the author studied the following nonlinear diffusion fuzzy system, involved to time-delay BAM Cohen-Grossberg neural networks.In recent years, some methods and ideas of related literature ( ) inspire our current work.In this paper, we shall discuss the robust stability of nonlinear p-Laplacian diffusion Takagi-Sugeno (T-S) fuzzy system with discrete delays and distributed delays.Actually, T-S fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems ( [31,32,46]).Besides, there exist parameter errors unavoidable in factual systems due to aging of electronic components, external disturbance, and parameter perturbations.Therefore, the robustness of the system stability should be investigated, too.Our main objectives are as follows: (1) Changing (4) into linear matrix inequalities (LMIs) applicable to computer MATLAB LMI toolbox, which can be adapted to large-scale calculation in practical engineering.
(2) Ensure that the nonlinear diffusion term plays a role in the LMI-based stability criterion while in some existing results ([6, Theorem 3.1], [18, Theorem 3.1], [19, Theorem 3.1],), the role of the nonlinear diffusion term was neglected in their LMI-based criteria.
For these purposes, we need to achieve the following works: (i) Improve [11,Lemma 3.1] and make it adopted to LMI-based criterion, in which the nonlinear diffusion can play roles.
(ii) Construct a novel Lyapunov functional and design comprehensive applications of variational method, Young inequality, and LMI technique so that LMIbased criterion can be derived for the nonlinear diffusion fuzzy system with parameter uncertainties, discrete delays, and distributed delays.
(iii) Relax the restrictions of amplification function a i • so that the boundedness of a i • is not necessary.

Complexity
At the same time, employing LMI technique guarantees structuring LMI-based criterion.
(iv) Explore the input-to-state stability of reactiondiffusion Markovian jumping BAM CGNNs with time delays and the event-triggered control For convenience's sake, we still need to introduce some standard notations: (iii) A = a ij n×n > 0 <0 : a positive (negative) definite matrix.
(viii) Denote C = c ij n×n for any matrix C = c ij n×n ; (ix) u t, x = u 1 t, x , u 2 t, x , … , u n t, x T for any vector u t, x = u 1 t, x , u 2 t, x , … , u n t, x T .
(xii) I: identity matrix with compatible dimension.
(xiv) Denote by λ 1 the lowest positive eigenvalue of the boundary value problem (see [28] for details)

Preliminaries
Consider the following Takagi-Sugeno fuzzy p-Laplace partial differential equations with distributed delay.

Fuzzy rule j:
If where  By means of a standard fuzzy inference method, (7) can be inferred as follows, where w t = w 1 t , w 2 t , … , w s * t T , h j w t = w j w t /∑ r k=1 w k w t , and w j w t : R s * → 0, 1 j = 1, 2, … , r 0 is the membership function of the system with respect to the fuzzy rule j. h j can be regarded as the normalized weight of each if-then rule, satisfying h j ω t ≥ 0 and ∑ Particularly in the case of p = 2, the system (8) is the so-called reaction-diffusion impulsive Markovian jumping BAM Cohen-Grossberg neural networks (BAM CGNNs).Inspired by some methods and conclusions of some related literature ( [47][48][49][50][51]), we shall discuss the input-to-state stability reaction-diffusion BAM CGNNs with the event-triggered control in Section 4, for seldom existing literature involved to such complex model with feedback control.Lemma 2.1.a q−1 b ≤ q − 1 /q a q + b q /q , ∀a, b ∈ 0, +∞ , and q > 1.
Note that Lemma 2.1 is the particular case of the famous Young inequality.[52]) Given matrices Q t , S t , and R t with appropriate dimensions, where

Lemma 2.2 (Schur complement
if and only if where Q t , S t , and ℛ t are dependent on t.

Robust Stability on Nonlinear p-Laplacian Diffusion System in the Case of p ≠ 2
Throughout this paper, we assume that , where we denote D j = D j t, x, u and D j = D j t, x, v for short.In addition, we always denote u t, x = u 1 t, x , u 2 t, x , … , u n t, x T and v t, x = v 1 t, x , v 2 t, x , … , v n t, x T .Denote u t, x by u and u i t, x by u i and so do v and v j .
Lemma 3.1.Let p > 1 be a positive real number, and Q = diag q 1 , q 2 , … , q n a positive definite matrix.Let u and v be a solution of (8).Then we have

Complexity
Proof.Since u is a solution of ( 8), it follows by Gauss formula and the Dirichlet zero boundary condition that Another inequality can be similarly proved.And so the proof is completed.
In this section, we suppose (H1) There exist positive definite matrices (H2) There exists positive definite matrices (H3) There are positive definite matrices Remark 4. The condition (H1) implies that the boundedness of amplification functions a i and a i are unnecessary in the case of p > 1 with p ≠ 2, for we may take a i s = a i s p−2 , which is actually unbounded for s ∈ −∞, + ∞ .Below, we denote for convenience where C j t = c ijk t n×n , C j t = c ijk t n×n , M j t = m ijk t n×n , and M j t = m ijk t n×n are diagonal matrices.
Assume, in addition, 5 Complexity and there is a positive definite matrix Q = diag q 1 , q 2 , … , q n such that then there exists the globally asymptotically robust stable unique equilibrium point for (8).
Remark 5. Condition (4) does not complete the matrix form.However, ( 19)-( 20) are complete linear matrices inequalities, which have even gotten better at dealing with the calculation of the large operations involved in the practical engineering by way of computer MATLAB programming.
Proof.There are three steps to the proof.
Step 1.We claim that the null solution is the unique equilibrium point for (8).
In fact, we know from (H2)-(H3) that b i 0 = b i 0 = f i 0 = f i 0 = g i 0 = g i 0 = 0, and hence u = 0 and v = 0 are the equilibrium solution of (8).
Moreover, we prove that the equilibrium point is unique.Indeed, it follows from (H1) that a i s > 0. Let (21) be an equilibrium point for (8) then we get If ( 23) is another equilibrium point of ( 8) we can actually deduce from ( 22) that and then Combining ( 25) and ( 26) implies 6 Complexity and (18) yields Thereby, the null solution is the unique equilibrium point for (8).Remark 6.In ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability.However, ( 8) is a partial differential system, including two different variables: t and x.Since the existence of the equilibrium solution and its global asymptotic stability only determines the equilibrium solution which is unique about variable t, but it may be not unique on variable x.Hence, it is necessary to verify the uniqueness of the equilibrium solution.
Step 2. To derive LMI-based criterion in which the nonlinear diffusion terms can play roles, we need to construct new Lyapunov-Krasovskii functionals as follows: where

35
one can find it impossible that the sufficient conditions of stability criterion can be derived.In addition, Lyapunov functions (33) and (34) help us to derive the complete linear matrix inequality condition for the stability criterion of nonlinear diffusion system (8).
Step 3. We claim that the null solution is globally asymptotically robust stable.
Evaluating the time derivation of V1 t along the trajectory of the (8), we can derive from Lemma 3.1

41
where C j t = c ijk t n×n .

42
where M j t = m ijk t n×n .So we have 8 Complexity Besides, we get by ( 32)

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One can deduce from (33) that Similarly, we can deduce from V 2 , V 4 , and V 6 that Therefore, ( 17), (19), and (20) yield It follows by the standard Lyapunov functional theory that the null solution of ( 8) is globally asymptotically robust stable.And the proof is completed.Remark 8.There have been some other approaches removing boundedness of amplification functions.For example, in [53], an appropriate Lyapunov-Krasovskii functional is set up to derive the LMI-based μ-stability for discrete timedelay system.This is really a good result.However, in this paper, our system model ( 8) is the continuous system different from the discrete system ([54, (1)]).Of course, the main results of this paper are inspired by some methods and ideas of these documents.
Remark 9.The Lyapunov functionals (33) and ( 34) are similar to the quadric form different from those of [11,13,14,20].Actually, the quadric form and matrix form help us to derive the LMI-based criterion.
If the diffusion phenomena are ignored, (8) degenerates into the following BAM CGNNs with discrete and distributed time-varying delays: Since in ordinary differential systems, the uniqueness of the equilibrium solution can be determined by the existence of the equilibrium solution and its global asymptotic stability, and the diffusion items disappear, we can directly deduce the following corollary from Theorem 3.2: and there is a positive definite matrix Q = diag q 1 , q 2 , … , q n such that then there exists the unique globally asymptotically robust stable equilibrium point for (49).

Input-to-State Stability of Markovian Jumping Reaction-Diffusion BAM CGNNs with Event-Triggered Control in the Case of p = 2
In this section, we consider the following Markovian jumping reaction-diffusion BAM CGNNs with event-triggered control under Dirichlet zero-boundary value.
Let Ω , ϒ, ℙ be the given probability space where Ω is the sample space, ϒ is σ, the algebra of subset of the sample space, and ℙ is the probability measure defined on ϒ.Let S = 1, 2, … , n 0 and the random form process r t : 0, +∞ → S be a homogeneous, finite-state Markovian process with right continuous trajectories with generator Π = γ ij n 0 ×n 0 and transition probability from mode i at time t to mode j at time t + Δt, i, and j ∈ S.
where γ ij ≥ 0 is transition probability rate from i to j j ≠ i and Let e t, x = e 1 t, x , … , e n t, x T and e t, x = e 1 t, x , … , e n t, x T be the error signal defined by
In this section, we assume that the conditions (H1)-(H3) hold still in the case of p = 2.
Besides, suppose that For any mode r ∈ S,

61
which do not have to be diagonal matrices or other special matrices.
11 Complexity In addition, we assume that which can guarantee that u = 0, and v = 0 is a trivial solution of (53).Besides, there are nonnegative matrices C r * and C r * such that Before giving the man result of this section, we need the following lemma: and ε > 0. Then we have 53) is called robust stochastic inputto-state in mean square stable for all admissible uncertainties satisfying (63), if for t > 0, there exist function β ∈ Kℒ and γ ∈ K such that where where I represents the identity matrix with suitable dimension under different cases for convenience.
then ( 53) is a robust stochastic input-to-state stable mean square.
Proof.Construct the Lyapunov-Krasovskii functionals as follows:

69
where each P r r ∈ S is positive definition diagonal matrix.-Due to v t, x = η v t, x + e t, x , 70 + 2α 2 e t, x T η 2 L2 e t, x dx,

72
where u = u t, x and v = v t, x .Similarly, we get

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Let L be the weak infinitesimal operator, then we get where

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and Φr = max λ max α 3 LP r P r L , λ max α 1 LP r P r L 80 In addition, for any t ∈ t k , t k+1 , the definition of t k+1 derives Furthermore, Dynkin's formula yields where Applications of the Comparison principle to (84) reaches

86
which derives

Numerical Examples
Example 5.1.Consider (8) with the following parameters: x ∈ Ω = 0, π , and then the first eigenvalue      Let θ = 0 01, ε = 0 001, and W = 0 5 and then we can compute and verify that (68) is satisfied.Now using computer MATLAB LMI-toolbox to solve LMI (66) gives the feasibility data as follows:  53) is robust stochastic input-to-state stable in mean square.
Remark 16.This paper is inspired by the methods and conclusions of the previous literature [55][56][57][58][59].But the sufficient conditions of Theorem 4.3 is easier to be verified than those of existing results.

Conclusions
In this paper, we mainly provided two novel conclusions for p-Laplace diffusion BAM CGNNs.In the case of p > 1 with p ≠ 2, the authors construct novel Lyapunov functional to overcome the mathematical difficulties of nonlinear p-Laplace diffusion time-delay model with parameter uncertainties, deriving the LMI-based robust stability criterion applicable to computer MATLAB LMI toolbox, deleting the boundedness of the amplification functions.On the other hand, when p = 2, LMI-based sufficient conditions are also inferred for robust input-to-state stability of reactiondiffusion Markovian jumping BAM CGNNs with the eventtriggered control, which is different from those of many previous related literature.As far as we are concerned, seldom literature involved a reaction-diffusion stochastic system with time delays and the event-triggered control.It is the first time to explore the method for the stability analysis of this system.Finally, numerical examples illustrate the effectiveness and feasibility via computer MATLAB LMI toolbox.

4
and other conditions, a stability result ([11, Theorem 3.2]) was given, whereD = min jk inf t,x,u D jk t, x, u , D = min ji inf t,x,v D ji t, x, v 5

Remark 1 .
Lemma 3.1 improves [11, Lemma 3.1] and [18, Lemma 2.3] for the first time, which makes a contribution to the final LMI criterion.

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According to Theorem 4.3, ( s * is the premise variable and µ jk j = 1, 2, … , r ; k = 1, 2, … , s * is the fuzzy set that is characterized by membership function.r is the number of the if-then rules, and s is the number of the premise variables.
x is the neuron activation function of the ith unit of time t − τ i t in space variable x and so is g i u i t − τ i t , x .Both τ i t and τ i Complexity appropriately behavior functions.C j , C j , M j and M j are connection weight strength coefficient matrices, and ΔC j t , Δ C j t , ΔM j t and ΔM j t are real-valued matrix functions which represent time-varying parameter uncertainties.
T, andB u t, x = b 1 v 1 t, x , … , b i v i t, x , … , b n v n t, xT, in which both b i u i t, x and b i v i t, x are 3 The uncertainty of parameters brings a difficulty to design the Lyapunov functions.If imitating the previous Lyapunov functions in existing literature, for example, let

Table 1 :
Comparisons of amplification function a j u j t, x in related results.
j ≤ a j r ≤ a j , ∀r ∈ R 10 Complexity x T 2r 0 α 2 η 2 L2 e t, x dx x T 2r 0 α 4 ξ 2 L2 e t, x dx the best of our knowledge, it is the first time to investigate input-to-state stability of reaction-diffusion time-delay system with event-triggered control.Especially, the diffusion items play roles in the criterion.
Remark 14.From Table1, we know, our new results (Theorem 3.2 and Corollary 3.3) is novel because the boundedness of amplification functions becomes unnecessary.Remark 15.From Table2, we know, our Theorem 3.2 is novel, different from those of existing results.

Table 2 :
Comparisons of our results with other results related to reaction-diffusion.