Further Results on Exponentially Robust Stability of Uncertain Connection Weights of Neutral-Type Recurrent Neural Networks

Further results on the robustness of the global exponential stability of recurrent neural network with piecewise constant arguments and neutral terms (NPRNN) subject to uncertain connection weights are presented in this paper. Estimating the upper bounds of the two categories of interference factors and establishing a measuring mechanism for uncertain dual connection weights are the core tasks and challenges. Hence, on the one hand, the new sufficient criteria for the upper bounds of neutral terms and piecewise arguments to guarantee the global exponential stability of NPRNN are provided. On the other hand, the allowed enclosed region of dual connection weights is characterized by a four-variable transcendental equation based on the preceding stable NPRNN. In this way, two interference factors and dual uncertain connection weights are mutually restricted in the model of parameter-uncertainty NPRNN, which leads to a dynamic evolution relationship. Finally, the numerical simulation comparisons with stable and unstable cases are provided to verify the effectiveness of the deduced results.

EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

Exploration on Robustness of Exponentially Global Stability of Recurrent Neural Networks with Neutral Terms and Generalized Piecewise Constant Arguments

With a view to the interference of piecewise constant arguments (PCAs) and neutral terms (NTs) to the original system and the significant applications in the signal transmission process, we explore the robustness of the exponentially global stability (EGS) of recurrent neural network (RNN) with PCAs and NTs (NPRNN). The following challenges arise: what the range of PCAs and the scope of NTs can NPRNN tolerate to be exponentially stable. So we derive two important indicators: maximum interval length of PCAs and the scope of neutral term (NT) compression coefficient here for NPRNN to be exponentially stable. Additionally, we theoretically proved that if the interval length of PCAs and the bound of NT compression coefficient are all lower than the given results herein, the disturbed NPRNN will still remain global exponential stability. Finally, there are two numerical examples to verify the deduced results’ effectiveness here.

FORCED DIFFUSION EQUATION WITH PIECEWISE CONTINUOUS TIME DELAY

In this article, we study the boundary value problem (BVP) for forced diffusion equation with piecewise constant arguments. We give explicit formula for solving BVP. The problem of finding periodic solutions of some differential equations with piecewise constant arguments (DEPCA) is reduced to solving a system of algebraic equations. Using the method of finding periodic solutions of DEPCA, the solutions of BVP are given in several examples which are periodic in time.

Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments

In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x″(t) + px″(t − 1) = qx([t]) + f(t), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f(t) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.