Preserving Global Exponential Stability of Hybrid BAM Neural Networks with Reaction Diffusion Terms in the Presence of Stochastic Noise and Connection Weight Matrices Uncertainty

We study the impact of stochastic noise and connection weight matrices uncertainty on global exponential stability of hybrid BAM neural networks with reaction diffusion terms. Given globally exponentially stable hybrid BAM neural networks with reaction diffusion terms, the question to be addressed here is how much stochastic noise and connection weights matrices uncertainty the neural networks can tolerate while maintaining global exponential stability.The upper threshold of stochastic noise and connection weights matrices uncertainty is defined by using the transcendental equations. We find that the perturbed hybrid BAM neural networks with reaction diffusion terms preserve global exponential stability if the intensity of both stochastic noise and connection weights matrices uncertainty is smaller than the defined upper threshold. A numerical example is also provided to illustrate the theoretical conclusion.


Introduction
The bidirectional associative memory (BAM) neural networks were first introduced by Kosko in which the neurons in one layer are fully interconnected to the neurons in the other layer, while there are no interconnection among the neurons in the same layers [1][2][3].The BAM neural networks widely have applications in pattern recognition, robot, signal processing, associative memory, solving optimization problems, and automatic control engineering.For most successful applications of BAM neural networks, the stability analysis on BAM neural networks is usually a prerequisite.The exponential stability and periodic oscillatory solution of BAM neural networks with delays were studied by Cao et al. [4,5].Moreover, in BAM neural networks, diffusion phenomena can hardly be avoided when electrons are moving in asymmetric electromagnetic fields.The BAM neural networks with reaction diffusion terms described by partial differential equations were investigated by many authors [6][7][8][9][10][11]. Sometimes, it is necessary to assess the parameters of the neural network that may experience abrupt changes caused by certain phenomena such as component failure or repair, change of subsystem interconnection, and environmental disturbance.The continuous-time Markov chains have been used to model these parameter jumps [12][13][14].These neural networks with Markov chains are usually called hybrid neural networks.The almost surely exponential stability, moment exponential stability, and stabilization of hybrid neural networks were also researched; see, for example, [15][16][17].By making use of impulsive control, Zhu and Cao [18] considered the stability of hybrid neural networks with mixed delay.
For neural networks with stochastic noise, the system is usually described by stochastic differential equations.The stability of stochastic neural networks with delay or reaction diffusion terms was extensively analyzed by using the Itô formula and the linear matrix inequality (LMI) methods [18][19][20][21][22].As is well known, stochastic noise is often the sources of instability and may destabilize the stable neural networks [23].For stable hybrid BAM neural networks with reaction diffusion terms, it is interesting to determine how much noise the stochastic neural networks can tolerate while maintaining global exponential stability.
Moreover, the connection weights of neurons depend on certain resistance and capacitance values which include uncertainty.The robust stability about parameter matrices uncertainty in neural networks was investigated by many authors [24,25].If the uncertainty in connection weights matrices is too large, the neural networks may be unstable.Therefore, for stable hybrid BAM neural networks with reaction diffusion terms, it is also interesting to determine how much connection weights matrices uncertainty the neural networks can also tolerate while maintaining global exponential stability.
In this paper, we will study the impact of stochastic noise and connection weight matrices uncertainty of hybrid BAM neural networks with reaction diffusion terms.We give the upper threshold of stochastic noise and connection weights matrices uncertainty defined by using the transcendental equations.We find that the perturbed hybrid BAM neural networks with reaction diffusion terms preserve global exponential stability if the intensity of both stochastic noise and connection weights matrices uncertainty is smaller than the defined upper threshold.
The remainder of this paper is organized as follows.Some preliminaries are given in Section 2. Section 3 discusses the impact of the stochastic noise on global exponential stability of these neural networks.Section 4 discusses the impact of the connection weight matrices uncertainty and stochastic noise on global exponential stability of these neural networks.Finally, an example with numerical simulation is given to illustrate the effectiveness of the obtained results in Section 5.

Preliminaries
Throughout this paper, unless otherwise specified, let (Ω, F, {F  } ≥0 , P) be complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all Pnull sets).Let () be a scalar Brownian motion (Wiener process) defined on the probability space.Let   denote the transpose of .If  is a matrix, its operator norm is denoted by ‖‖ = sup{|| : || = 1}, where | ⋅ | is the Euclidean norm.Let (),  ≥ 0, be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1, 2, . . ., } with the generator Γ = (  ) × given by where Δ > 0. Here,   > 0 is the transition rate from  to  if  ̸ =  while We assume that the Markov chain (⋅) is independent of the Brownian motion (⋅).It is well known that almost every sample path of (⋅) is a right-continuous step function with finite number of simple jumps in any finite subinterval of R + := [0, +∞).
In addition, we assume that the neural networks (3) have an equilibrium point The initial conditions and boundary conditions are given by Hence, the origin is an equilibrium point of (6).The stability of the equilibrium point of ( 3) is equivalent to the stability of the origin of the state space of (6).From (5), we give the assumption about activations functions  and .
Assumption (H1).The neuron activation functions  and  are global Lipschitz continuous; that is, there exist constants  > 0 and  > 0, such that We consider the following function vector space: ) is continuous on  and twice continuous differentiable on .
For every pair of (V, ) in  and every given  ∈ R + , define inner product for V and  with Obviously, it satisfies inner product axiom, and the norm can be deduced by Definition 1.The neural networks ( 6) are said to be global exponentially stable if for any , , there exist  > 0 and  > 0, such that For the purpose of simplicity, we rewrite (6) as follows:

Mathematical Problems in Engineering
The initial conditions and boundary conditions are given by where Here, ∘ denotes Hadamard product of matrix  and ∇ and  * and ∇V.

Noise Impact on Stability
In this section, we consider the noise-induced neural networks (6) described by the stochastic partial differential equations The initial conditions and boundary conditions are given by where  is the noise intensity.We rewrite (16) as follows: For the globally exponentially stable neural networks (6), we will characterize how much stochastic noise the neural networks (16) where ( ū (, ;  0 , ), V (, ;  0 , )) is the state of neural networks (16).
From the above definitions, it is clear that the almost sure global exponential stability of the neural networks (16) implies the mean square global exponential stability of the neural networks (16) (see [26,27]) but not vice versa.Theorem 4.Under Assumption (H1), the mean square global exponential stability of neural networks (16) implies the almost sure global exponential stability of the neural networks (16).
Therefore, the neural networks ( 16) are mean square globally exponentially stable, and by Theorem 4, the neural networks ( 16) are also almost surely globally exponentially stable.
We rewrite (59) as follows: For the global exponential stability of neural networks (6), we will characterize how much the intensity of both the self-feedback matrix (, )  uncertainty and stochastic noise the stochastic neural networks (59) can tolerate while maintaining global exponential stability.Theorem 6.Let Assumption (H1) hold and let the neural networks (6) be globally exponentially stable.Then, the neural networks (59) are mean square globally exponential stability and also almost sure globally exponential stability, if there exists   > 0, ( ∈ S), and (, ) is in the inner of the closed curve described by the following transcendental equation: where ], ‖ Â‖ = max ∈S ‖()‖, and so forth and μ = max ∈S   and μ = min ∈S   .