Computation issue of pointwise controls for diffusion Hopfield neural network system

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Abstract

This paper consider the pointwise controls problem of Hopfield neural network equation with diffusion term subject to quadratic criteria in the framework of variational method. Based on established optimal control theory to pointwise case, numerical study is carried out by constructing a semi-discrete algorithm with continuous time. Furthermore, for given point control inputs, the optimal pointwise controls are obtained to achieve the minimization. Experiments demonstration is implemented for verifying proposed algorithm.

Introduction

A great deal contributions have been made concerning with Hopfield neural network (HNN) (cf. [7]). The qualitative analysis of fundamental properties, see for example, stability, convergency and equilibrium for discrete and continuous systems with time delay (cf. [3], [4], [6], [9], [12], [13]). Even as stochastic model is considered in [28], [29]. A question arising here is how to control HNN at finite points? How to formulate the theory and analysis for such a control problem, which most probably happen in various phenomena, for instance, medical science and brain science, computer networking, robotics, telecommunications and information sciences.

This work is aim at investigating the theoretic and numerical issues on pointwise control problem (cf. [14]) for diffusion Hopfield neural network system. The model can be regarded as nonlinear distributed parabolic partial differential equation (cf. [1], [2], [12], [17], [22]).

For mathematics setting, let Ω be an open bounded domain of Rn (assume that n  3) with a piecewise smooth boundary Γ = Ω. Let Q=(0,T)×Ω and Σ=(0,T)×Γ with T > 0. Let yi(x,t) denote the activation potential of the ith neuron for i = 1, 2,  , m. The diffusion Hopfield model is described by simultaneous system of m-numbers neurons with μ-numbers point control inputs:yi(t)t-αiΔyi(t)=-βiyi(t)+j=1mγijFj(yj(t))+j=1μuij(t)δ(x-xj)inQ,yi(t)η=0onΣ,yi(x,0)=y0i(x),i=1,2,,m.Here αi>0 are diffusion constants, βi>0 and γij are connection weight constants. Fj:R=(-,)(-1,1) are nonlinear sigmoidal activation functions, e.g., Fj(s)=tanhs. uij are control inputs on xj, δ is a Dirac function at point xj and uij(t)δ(x-xj) represents the distribution mapping ψ0Tuij(t)ψ(xj,t)dt for ψD(Q), where D(Q) is the basis space on Q.

The content of this paper divided into several sections. In Section 2, mathematics setting for preparation is given for formulating the problems. Section 3 will establish the theoretic results on optimal pointwise control for quadratic cost, that is to prove the existence of optimal pointwise control, and establish the first order necessary conditions of optimality. In Section 4, computational issue for numerical approach is considered by constructing a semi-discrete scheme using finite element approach. Section 5 is to describe the numerical approach for optimal pointwise problems. In Section 6, the experiments demonstration is implemented by Mathematica, simulation results will illustrate the efficiencies and stability of proposed paradigm. Section 7 contains the discussion and conclusions.

Section snippets

Preliminaries

Introduce two Hilbert spaces H = L2(Ω) and V = H1(Ω) according to the Neumann boundary condition of systems (1). They are endowed with the usual inner products (,),((,)) and norms |·|,·, respectively. The dual space of V is denoted by V, and the symbol 〈 , 〉 denotes the dual pairing from V and V (cf. [5], [10]).

Denote ϕ=(ϕ1,ϕ2,,ϕm)T and ψ=(ψ1,ψ2,,ψm)T. Then the Hilbert spaces V=H1(Ω)m and H=L2(Ω)m with the inner products defined by(ϕ,ψ)H=i=1m(ϕi,ψi),ϕ,ψH,(ϕ,ψ)V=i=1m(ϕi,ψi),ϕ,ψV,

Pointwise control problem

Let Uij=L2(0,T) be the Hilbert spaces of control variables uij(t) for i = 1, 2,  , m and j = 1, 2,  , μ, respectively. Let Uj=L2(0,T)m be the Hilbert spaces of uj=(u1j,u2j,,umj) for j = 1, 2,  , μ, respectively. Let U=L2(0,T)mμ be the Hilbert space of u=(u1,u2,,um). Recall diffusion control system (2):y(t)t-AΔy(t)=-By(t)+CF(y(t))+j=1μuj(t)δ(x-xj)inQ,y(t)η=0onΣ,y(x,0)=y0(x)inΩ.

For any uU, by virtue of Theorem 1, there exist a unique weak solution y = y(u) of (4) in W(0,T). Hence, the solution map u  y(u)

Numerical approach based on finite element method

Let 0=x0<x1<<xN<xN+1=l be a partition of interval [0, l] into subinterval Ie=[xe-1,xe] of length he=xe-xe-1 for e=1,2,,N+1. LetVh={bie|bieis quadratic function onIecontinuous on[0,l],i=1,2,3}.Then VhV. The basis functions bieVh defined by quadratic interpolation functionsb1e(x)=1-x-xehe1-2(x-xe)he,b2e(x)=4(x-xe)he1-x-xehe,b3e(x)=-(x-xe)he1-2(x-xe)he.

Here xe is the endpoint of subinterval, and he is the length of the eth subinterval. Using quadratic basis functions to construct the

Numerical approach for optimal pointwise controls

Let uh=(u1h,u2h,,umh) and uih={ui1h(x,t),ui2h(x,t),,uiμh(x,t)}, where uijh(x,t) corresponds to uij(t)δ(x-xj). Refer to (3), the minimization problem for finite element approximation is formulated by: minimize the approximate cost functionsJ(uh)=i=1m0l(yih(T)-zidT)2dx+i=1mj=1μ0T(uijh-uijd)2dtwith yihyi(uh)Vh satisfying0T0ldyihdtϕhdxdt+di0T0lyihϕhdxdt=-ai0T0lyihϕhdxdt+j=1m0T0lFj(yjh)ϕhdxdt+j=1μ0Tuijhϕh(xj)dtfor all ϕhVh.

Theorem 4

By Theorem 2 and [30], there exists at least a

Experiments demonstration

Set Q=(0,1)×[0,T], consider pointwise controls of two simultaneously systems:y1t-d1Δy1=-a1y1+β1(tanhy1+tanhy2)+j=1μu1j(t)δ(x-xj)onQ,y2t-d2Δy2=-a2y2-β2(100tanhy1-2tanhy2)+j=1μu2j(t)δ(x-xj)onQ,y1(0,x)=y10(x),y2(0,x)=y20(x)in[0,1]..

Let N = 20, h = 1/20, t0 = 0.0, T = 1.0, dt = 0.1. d1 = d2 = 0.0001, a1 = a2 = 1.0 and β1 = β2 = 0.3. The initial and desired states are given byy1(x,0)=cos(2πx),z1dT=sin(πx),y2(x,0)=cos(2πx),z2dT=sin(πx).

For their graphics, see Fig. 1, Fig. 2.

Choosing the random points μ = rm for

Discussion and conclusions

This paper presented pointwise optimal control for diffusion Hopfield neural network model. Using the variational method, the existence of optimal pointwise control completely solved, necessary optimality condition is derived. Numerical study for one dimensional case has been demonstrated for two neurons systems. The optimal pointwise controls, the optimal cost functions and necessary optimality condition have been obtained. Finally, the simulation results show employed semi-discrete algorithm

Acknowledgement

The author would like to thank 7th World Congress of Computation Mechanics for presenting the pervious work (cf. [22]).

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    This work is supported by National Natural Science Foundation of China (No. 60474027).

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