Control of Recurrent Neural Networks Using Differential Minimax Game: the Stochastic Case

As a continuation of our study, this paper extends our research results of optimality-oriented stabilization from deterministic recurrent neural networks to stochastic recurrent neural networks, and presents a new approach to achieve optimally stochastic input-to-state stabilization in probability for stochastic recurrent neural networks driven by noise of unknown covariance. This approach is developed by using stochastic differential minimax game, Hamilton-Jacobi-Isaacs (HJI) equation, inverse optimality, and Lyapunov technique. A numerical example is given to demonstrate the effectiveness of the proposed approach. INTRODUCTION The past two decades have witnessed enormous advances in engineering and in computer science to build artificial computational systems [1], among which recurrent neural networks have been applied to many scientific and engineering fields, such as system identification and control, pattern recognition, image processing, and modeling biological sensor-motor systems. Therefore, theoretical studies on both stability and controllability of recurrent neural networks have been heavily investigated in the last few years [2] [10]. However, these studies primarily focused on deterministic recurrent neural networks. In the mathematical models of these aforementioned networks, they do not consider the noise process that is fraught with signal transmission particularly in biological systems. On the other hand, Werbos [1] pointed out that in order to develop mathematical neural network specifications which have dual uses as models of intelligence in the brain, and as highly effective artificial intelligent systems when implemented in computers and chips, we must consider the stochastic environment. Unfortunately, with regard to the analysis of stochastic recurrent neural networks, there has been little work in the literature until the very recent years [11]. Hence, it is important to analytically explore the characteristics of stabilization and controllability for recurrent neural networks under the influence of stochastic perturbation. As a continuation of our study in [12], we present in this paper a theoretical analysis for stochastic recurrent neural networks to achieve stochastic input-to-state stabilization in probability under an optimal control strategy, and to attenuate incremental covariance of stochastic perturbation to a predefined level within stability margins. By applying the theory of differential minimax game to the stochastic networks, the approach is developed by considering the vector of external inputs as a player and the vector of stochastic disturbance as the opposing player. Therefore, a minimax equilibrium can be achieved by properly controlling stochastic recurrent neural networks. It should be pointed out that this paper develops a stochastic counterpart of the disturbance attenuation results of those in [12]. PROBLEM FORMULATION Based on the standard formulation of stochastic recurrent neural networks [13], we consider the following stochastic recurrent neural network, which is derived from the model of deterministic recurrent neural networks defined in [12] plus an additive white noise. Mathematically, it can be described by the following Ito-type compact form Ψ + + + − = d dt u W x S W Ax dx ) ) ( ( 2 1 (1) 1 Copyright © 2010 by ASME Proceedings of the ASME 2010 Dynamic Systems and Control Conference DSCC2010 September 12-15, 2010, Cambridge, Massachusetts, USA