Optimality-Oriented Stabilization for Recurrent Neural Networks

2011 ◽  
pp. 93-115
Author(s):  
Ziqian Liu
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Jacobi Equation ◽  
Control Design ◽  
Optimal Stabilization ◽  
Numerical Examples ◽  
Differential Game Theory ◽  
Inverse Optimality ◽  
Hamilton Jacobi Equation ◽  
Lyapunov Technique

This chapter presents an approach of how optimality-oriented stabilization is achieved for recurrent neural networks, which includes both the input-to-state stabilization for deterministic recurrent neural networks and the noise-to-state stabilization for stochastic recurrent neural networks. Owing to the difficulty in solving the Hamilton-Jacobi equation for nonlinear systems, optimal regulation seems to be an unachievable goal in control design for recurrent neural networks. However, a methodology proposed in this chapter solves the problem and obtains optimal stabilization by using the knowledge of Lyapunov technique, inverse optimality, and differential game theory. Numerical examples demonstrate the effectiveness of the proposed design.

Design of Globally Robust Control for Biologically-Inspired Noisy Recurrent Neural Networks

2011 ◽  
pp. 116-135
Author(s):  
Ziqian Liu
Keyword(s):  
Neural Networks ◽  
Robust Control ◽  
Recurrent Neural Networks ◽  
Disturbance Attenuation ◽  
Biologically Inspired ◽  
Inverse Optimality ◽  
Modeling Errors ◽  
Minimax Game ◽  
Biological Motor ◽  
Lyapunov Technique

This chapter presents a theoretical design of how a global robust control is achieved in a class of noisy recurrent neural networks which is a promising method for modeling the behavior of biological motor-sensor systems. The approach is developed by using differential minimax game, inverse optimality, Lyapunov technique, and the Hamilton-Jacobi-Isaacs equation. In order to implement the theory of differential games into neural networks, we consider the vector of external inputs as a player and the vector of internal noises (or disturbances or modeling errors) as an opposing player. The proposed design achieves global inverse optimality with respect to some meaningful cost functional, global disturbance attenuation, as well as global asymptotic stability provided no disturbance. Finally, numerical examples are used to demonstrate the effectiveness of the proposed design.

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

2010 ◽  
Author(s):  
Ziqian Liu ◽  
Nirwan Ansari
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
New Approach ◽  
Numerical Example ◽  
Inverse Optimality ◽  
Stochastic Case ◽  
Stochastic Input ◽  
Minimax Game ◽  
Unknown Covariance ◽  
Lyapunov Technique

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.

scholarly journals On a New Faster Implicit Fixed Point Iterative Scheme in Convex Metric Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Renu Chugh ◽  
Preety Malik ◽  
Vivek Kumar
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Metric Spaces ◽  
Iterative Scheme ◽  
Mann Iteration ◽  
Numerical Examples ◽  
Convex Metric Spaces ◽  
Networks Analysis ◽  
Implicit Iteration ◽  
Convergence Stability

The purpose of this paper is to consider a new implicit iteration and study its strong convergence, stability, and data dependence. It is proved through numerical examples that newly introduced iteration has better convergence rate than well known implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster as compared to corresponding explicit iterations. Applications of implicit iterations to RNN (Recurrent Neural Networks) analysis are also presented.

scholarly journals Further Stability Criterion on Delayed Recurrent Neural Networks Based on Reciprocal Convex Technique

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Guobao Zhang ◽  
Tao Li ◽  
Shumin Fei
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Upper And Lower Bounds ◽  
Sufficient Condition ◽  
Time Varying ◽  
Functional Theory ◽  
State Delay ◽  
Numerical Examples ◽  
Delay Partitioning ◽  
Partitioning Technique

Together with Lyapunov-Krasovskii functional theory and reciprocal convex technique, a new sufficient condition is derived to guarantee the global stability for recurrent neural networks with both time-varying and continuously distributed delays, in which one improved delay-partitioning technique is employed. The LMI-based criterion heavily depends on both the upper and lower bounds on state delay and its derivative, which is different from the existent ones and has more application areas as the lower bound of delay derivative is available. Finally, some numerical examples can illustrate the reduced conservatism of the derived results by thinning the delay interval.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

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