Computation issue of pointwise controls for diffusion Hopfield neural network system☆
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 (assume that n ⩽ 3) with a piecewise smooth boundary Γ = ∂Ω. Let and with T > 0. Let 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:Here are diffusion constants, and are connection weight constants. are nonlinear sigmoidal activation functions, e.g., . are control inputs on , δ is a Dirac function at point and represents the distribution mapping for , where 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 , and the symbol 〈 , 〉 denotes the dual pairing from V and (cf. [5], [10]).
Denote and . Then the Hilbert spaces and with the inner products defined by
Pointwise control problem
Let be the Hilbert spaces of control variables for i = 1, 2, … , m and j = 1, 2, … , μ, respectively. Let be the Hilbert spaces of for j = 1, 2, … , μ, respectively. Let be the Hilbert space of . Recall diffusion control system (2):
For any , 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 be a partition of interval [0, l] into subinterval of length for . LetThen . The basis functions defined by quadratic interpolation functions
Here is the endpoint of subinterval, and is the length of the eth subinterval. Using quadratic basis functions to construct the
Numerical approach for optimal pointwise controls
Let and , where corresponds to . Refer to (3), the minimization problem for finite element approximation is formulated by: minimize the approximate cost functionswith satisfyingfor all . Theorem 4 By Theorem 2 and [30], there exists at least a
Experiments demonstration
Set , consider pointwise controls of two simultaneously systems:
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 by
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).