Finite-Time Robust Stabilization for Stochastic Neural Networks

and Applied Analysis 3


Introduction
Since the first paper of Ott et al. 1 , a large number of monographs and papers studying the stabilization of the nonlinear systems without or with delays have been published 2-5 .These publications have developed many control techniques including continuous feedback and discontinuous feedback.Take 4 for example, the authors studied the pinning stabilization problem of linearly coupled stochastic neural networks, where a minimum number of controllers are used to force the NNs to the desired equilibrium point by fully utilizing the structure of the network.
On the other hand, the well-known Hopfield neural networks, Cohen-Grossberg neural networks and cellular neural networks 6-18 , and so forth have been extensively studied in the past decades and successfully applied in many areas such as signal processing, combinatorial optimization, and pattern recognition.Specially, the stability of Hopfield neural networks has received much research attention since, when applied, the neural network is sometimes assumed to have only one globally stable equilibrium 7-9, 19, 20 .Until now, the stability analysis issues for many kinds of neural networks in the presence of stochastic perturbations and/or parameter uncertainties have attracted a lot of research attention.The reasons include twofold: a in real nervous systems, because of random fluctuations from the release of neurotransmitters, and other probabilistic causes, the synaptic transmission is indeed a noisy process; b the connection weights of the neurons depend on certain resistance and capacitance values that always exist uncertainties.Therefore, the robust stability has been studied for neural networks with parameter uncertainties 21-24 or external stochastic perturbations 7, 19, 25, 26 .However, to the best of the authors' knowledge, most literature regarding the stability of neural networks is based on the convergence time being large enough, even though we eagerly want the argued network states to become stable as quickly as possible in practical applications.In order to achieve faster stabilization speed and hope to complete stabilization in finite time rather than merely asymptotically 27 , an effective method is using finite-time stabilization techniques, which have also demonstrated better robustness and disturbance rejection properties 28 .
In this paper, we will focus on the finite-time robust stabilization for neural networks with both stochastic perturbations and parameter uncertainties.The difference of this paper lies in three aspects.First, based on the finite-time stability theorem of stochastic nonlinear systems 29 , a new continuous finite-time stabilizator is proposed for a stochastic neural network SNN .Moreover, in contrast to 30-33 , we prove finite-time stabilization by constructing a suitable Lyapunov function and obtain some criteria which are easy to be satisfied.Second, the gain parameters in finite-time stabilizator are designed by solving a linear matrix inequality.Finally, a robust finite-time stabilizator for SNNs with parameter uncertainties is designed as well.Moreover, two illustrative examples are provided to show the effectiveness of the proposed designing.
The notations in this paper are quite standard.R n and R n×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices.The superscript "T " denotes the transpose and the notation X ≥ Y resp., X > Y , where X and Y are symmetric matrices, meaning that X − Y is positive semidefinite resp., positive definite .λ max M and λ min M denote the maximal and minimal eigenvalues of real matrix M. Let Ω, F, {F t } t≥0 , P be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions i.e., it is right continuous and contains all P-null sets .E{x} stands for the expectation of the stochastic variable x with respect to the given probability measure P. I and 0 represent the identity matrix and a zero matrix, respectively; diag • • • stands for a block-diagonal matrix.Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Model Formulation and Preliminaries
Some preliminary knowledge is presented in this section for the derivation of our main results.The deterministic NN can be described by the following differential equation: where Because of the existence of environmental noise in real neural networks, the stochastic disturbances should be taken into account in the recurrent NN.For this purpose, we modify the system 2.1 as the following SNN: where ω t ω 1 t , ω 2 t , . . ., ω n t T ∈ R n is an n-dimensional Brownian motion defined on the probability space Ω, F, {F t } t≥0 , P satisfying the usual conditions i.e., the filtration contains all P-null sets and is right continuous .The white noise dω i t is independent of dω j t for i / j.The intensity function h is the noise intensity function matrix satisfying the following condition: where M h is a known constant matrix with compatible dimensions.
In this paper, we want to control the SNN 2.4 to the desired state x * , which is an equilibrium point of NN 2.1 .Based on the discussions in many other papers, the stochastic perturbation will vanish at this equilibrium point x * , that is, h t, x * 0. Without loss of generality, one can shift the equilibrium point x * to the origin by using the translation y t x t − x * , which derives the following stochastic dynamical system: where g y t f x t x * − f x t .
Consider the SNN 2.6 with parameter uncertainties: the parameter matrices A and B are unknown but bounded, which are assumed to satisfy where , where e i ∈ R n is the column vector with the ith element 1 and 0 elsewhere.For i, j 1, 2, . . ., n, let

2.8
Then, through simple manipulations, one has In order to stabilize the SNN 2.4 to the equilibrium point x * , equivalently, one can stabilize the SNN 2.6 to the origin due to the transformation.Hence, in the remainder of this paper, a controller u t will be designed for the stabilization of SNN 2.6 in mean square.The controlled SNN can be described by the following stochastic differential equation SDE : are gain coefficients to be determined, and the real number α satisfies 0 < α < 1.In fact, here the continuous function u t in the SNN 2.10 is the key point for ensuring the finite-time stabilization.
Obviously, when 0 < α < 1, the controller u t is a continuous function with respect to y, which leads to the continuity of controlled system 2.10 with respect to the state y t 30-33 .If α 0, u t turns to be a discontinuous one, which has been considered in 34-36 .If α 1 in the controller 2.11 , then it becomes the typical stabilization issues which only can realize an asymptotical stabilization in infinite time 3-5 .
Similar to the definition of finite-time stability in probability 29 , the finite-time stabilization in probability is given through the following definition.
Definition 2.2.The system 2.6 is said to be finite-time stabilized at the original point by the controller 2.11 in probability, that is, the controlled SNN 2.10 is finite-time stable in probability 37 if, for any initial state x 0 , there exists a finite-time function T 0 such that where T 0 T 0 y 0 , ω inf{T ≥ 0 : y t 0, ∀t ≥ T } is called the stochastic setting time function satisfying E T 0 < ∞.
The following lemmas are needed for the derivation of our main results in this paper.13 with f 0 0 and g 0 0 and assume system 2.13 has a unique global solution.If there exist real numbers η > 0 and 0 < α < 1, such that for the function V x in Lemma 2.3, then the origin of system 2.13 is globally stochastically finite-time stable, and for all V T t V t ≤ I, if and only if there exists a positive constant λ, such that

Main Results
In this section, we first give some theorems in detail to guarantee that the original point of SNN 2.6 is stabilized in finite time, that is, the controlled system 2.10 with 2.

3.4
From condition 2.3 , using the inequality x T y y T x ≤ εx T x ε −1 y T y, where ε > 0 is an arbitrary constant, we have 2y T t PBg t ≤ ε −1 y T t PBB T Py t εg T t g t ≤ ε −1 y T t PBB T Py t εy T t M T My t .

3.8
Thus, based on condition 3.1 , taking the expectations on both sides of 3.3 , we have and E V α 1 /2 y 0 E V y 0 α 1 /2 .

3.9
By Lemma 2.4, V y t stochastically converges to zero in a finite time, that is, the controlled system 3.2 is finite-time stable in probability, and the settle time is upper bounded by

3.10
This completes the proof.
Remark 3.2.The two gain parameters k 1 and k 2 in the controller u t play different roles in ensuring the finite-time stability of the controlled system 3.2 .We can see from Theorem 3.1 that, whether or not the controlled system 3.2 could realize the finite-time stability mainly depends on the value of k 1 and satisfies condition 3.1 but nothing on k 2 .However, the size of the settle time depends on the value of k 2 but unrelated to k 1 , the only requirement for the gain k 1 is satisfying condition 3.1 .
Remark 3.3.In 31, 32, 35, 41 , the candidate Lyapunov function V t was chosen as a simple form of V t y T t y t and then the upper bound of settle time turns to be y 0 1−α /k 2 1 − α .In this paper, in order ro reduce some conservation of conditions in Theorem 3.1, a positive definite matrix parameter P is introduced such that condition 3.1 is easier to be satisfied.And the previous conclusions could be included by our results if the matrix P pI is taken, where p is a arbitrary constant, just as shown in the next corollary.
Corollary 3.4.The controlled system 3.2 is finite-time stable in probability, if there exist two constants ε and p such that Moreover, the upper bound of the settle time is Our next goal is to deal with the design problem, that is, giving a practical design procedure for the controller gains: k 1 and k 2 , such that the inequalities in Theorem 3.1 or Corollary 3.4 are satisfied.Obviously, those inequalities are difficult to solve, since they are nonlinear and coupled.A meaningful approach to tackling such a problem is to convert the nonlinearly coupled matrix inequalities into linear matrix inequalities LMIs , while the controller gains are designed simultaneously.
Based on the discussion in Remark 3.2, the parameter gain k 2 is one of the primary factors that affect the size of the settle time, which is unrelated to condition 3.11 .Hence, in the following discussion, we will fix the gain parameter k 2 and mainly focus on the design of control gain k 1 .We claim that the desired controller gain k 1 can be designed if a linear matrix inequality is feasible.Theorem 3.5.For a fixed control gain k 2 , the finite-time stabilization problem is solvable for the SNN 2.6 , if there exist three positive scalars p, K, and ε such that Moreover, the control gain coefficient k 1 p −1 K.
Proof.The result can be proved by pre-and post-multiplying the inequality 3.13 by the block-diagonal matrix diag{I, ε −1/2 I, ε −1/2 I} and then following from the famous Schur complement lemma and Corollary 3.4 and we omit it here.Just as mentioned in Introduction, when modelling a dynamic system, one can hardly obtain an exact model.Specially, in practical implementation of neural networks, the firing rates and the weight coefficients of the neurons depend on certain resistance and capacitance values, which are subject to uncertainties.It is thus necessary to take parameter uncertainties into account in the considered neural network.In the following, we consider the robust finitetime stabilization issue for SNN 2.6 under the parametric uncertainties 2.7 .
Theorem 3.6.The interval SNN 3.2 with uncertain parameters 2.7 is robust finite-time stable in probability, if there exist three constants ε, λ 1 , λ 2 and a positive-definite matrix P ∈ R n×n such that Proof.From Theorems 3.1 and 3.5, we know that the SNN 3.2 is finite-time stable in probability, if there exist a constant ε and a positive-definite matrix P ∈ R n×n such that the following LMI holds: Thus, for the uncertain parameters satisfying 2.7 , we have

10
Abstract and Applied Analysis For the second term in the above equality, it is easy to have ⎠ Ω E T B P 0 0 .

3.17
Then, based on Lemma 2.6, 3.16 and 3.17 , there exist two constants λ 1 and λ 2 such that

3.18
Then the result can be proved by the famous Schur complement lemma and condition 3.14 .
Corollary 3.7.For a fixed control gain k 2 , the finite-time robust stabilization problem is solvable for the SNN 2.6 with 2.7 , if there exist five positive scalars p, K, ε, λ 1 , and λ 2 such that Proof.Let P pI and we can prove the result based on Theorem 3.6.where

Two Numerical Examples
h t, x t diag tanh x 1 t , tanh x 2 t , tanh x 3 t , and the activation function is taken as f s tanh s .Then, it is obvious that M M h I 3 , where I 3 is a 3 × 3 identity matrix.The SNN 4.1 with the above-given parameters is depicted in Figure 1 with initial values x 0 1, −1, 3 T .
The stabilization controller is designed as where the parameter α is chosen as 0.5 and the initial value x 0 1, −1, 3 T .Then, ||x 0 || 3.3166.
According to Theorem 3.5 and using Matlab LMI toolbox, we solve the LMI 3.13 , and obtain p 2.8118, K 10.8900, and ε 10.1114.Then by Theorem 3.5, the desired controller parameter can be designed as k 1 3.8730.
By choosing an arbitrary fixed gain k 2 , SNN 4.1 can be stabilized in finite time in probability.Taking k 2 1, for example, we can obtain the upper bound of the settle time T x 0 1−α /k 2 1 − α 3.6423.Simulation result is depicted in Figure 2, which shows the states x 1 t , x 2 t , and x 3 t of the controlled SNN 4.1 .The simulation result has confirmed the effectiveness of our main results.By choosing an arbitrary fixed gain k 2 , SNN 4.1 can be robustly stabilized in finite time in probability.Taking k 2 1.5, for example, we can obtain the upper bound of the settle time T x 0 1−α /k 2 1 − α 1.5856.Simulation result is depicted in Figure 3, which shows the states x 1 t and x 2 t of the second-order controlled SNN 4.1 .The simulation result has confirmed the effectiveness of our main results.

Conclusions
In this paper, we have investigated the issue of finite-time stabilization for SNNs with noise perturbations by constructing a continuous nonlinear stabilizator.Meanwhile, Based on the Lyapunov-Krasovskii functional method combining with the LMI techniques, a sufficient criterion is derived for the states of the augmented system to be global finite-time stable in probability.Subsequently, for SNNs with parameter uncertainties, the robust finite-time stabilizator could be designed well.Finally, two illustrative examples have been used to demonstrate the usefulness of the main results.It is expected that the theory established in  this paper can be widely applied in delayed systems, particularly in those discontinuous cases.It will be an interesting topic in our future research.

Figure 1 :
Figure 1: Trajectories of SNN 4.1 without any controller in Example 4.1.

Figure 2 :
Figure 2: Trajectories of SNN 4.1 under the controller 4.3 with k 2 1 in Example 4.1.

Figure 3 :
Figure 3: Trajectories of SNN 4.1 under the controller 4.3 with k 2 1.5 in Example 4.2.
a 2 , . . ., a n , E B b 11 e 1 , . . ., b 1n e 1 , . . ., b n1 e n , . . ., b nn e n n×n 2 , and F B b 11 e 1 , . . ., b 1n e n , . . ., b n1 e 1 , Lemma 2.3 see 38 .It ô's formula .Let x t ba an n-dimensional Itô's process on t ≥ 0 with the stochastic differential Let V x t , t ∈ C 2,1 R n × R ; R .Then, V x t ,t is a real-valued Itô's process with its stochastic differential given by dV x t , t LV x t , t dt V x x t , t g t dω t , LV x t , t V t x t , t V x x t , t f t 1 2 trace g T t V xx x t , t g t , 2.14 where C 2,1 R n × R denotes the family of all real-valued functions V x t , t such that they are continuously twice differentiable in x and t.Lemma 2.4 see 29 .Consider the stochastic differential equation 2.
5 see 39 .If a 1 , a 2 , . . ., a n are positive number and 0 < r < p, then Boyd et al. 40 .If U, V t , and W are real matrices of appropriate dimension with N satisfying N N T , then 11 is finitetime stable in probability.Then, for SNN 2.6 with parameter uncertainties, we provide a sufficient condition under which the controlled system 2.10 is robust finite-time stable in probability.Finally, the control gains k 1 and k 2 are designed by solving some linear matrix inequalities.
Theorem 3.1.The controlled system 2.10 with 2.11 is finite-time stable in probability, if there exist a constant ε and a positive-definite matrix P ∈ R n×n such that−2PA − 2k 1 P ε −1 PBB T P εM T M λ max P M T h M h < 0. 3.1Moreover, the upper bound of the stochastic settling time for stabilization can be in terms of the initial errors asλ max P /λ min P • y 0 1−α /k 2 1 − α .Proof.Consider the controlled system 2.10 with the controller 2.11 , we have dy t − A k 1 I y t Bg y t − k 2 sign y t y t α dt h t, y t dω t .T t P sign y t y t α .