Thus, before discussing the proposed research in detail, it is worthwhile to describe and summarize the main results achieved in the course of the research work under the above contract.The ongoing research has largely been forcused on the development of mathematical models 6f hysteretic nonlinearities with "nonlocal memories". The distinct feature of these nonlinearities is that their current states depend on past histories of input variations. It turns out that memories of hysteretic nonlinearities are quite selective. Indeed, experimentsshow that only some past input extrema (not the entire input variations) leave their marks upon future states of hysteretic nonlinearities. Thus special mathematical tools are need in order to describe nonlocal selective memories of hysteretic nonlinearities. The origin of such tools can be traced back to the landmark paper of Preisach.Our research has been primarily concerned with Preisach-type models of hysteresis.All these models have a common generic feature; they are constructed as superpositions of simplest hysteretic noiilinearities-rectangular loops. During the past four years, our study has been by and large centered around the folloing topics:Further development of scalar and vector Preisacli-type models of hysteresis.e Experimental testing of Preisach-type models of hysteresis.Development of new models for viscosity (aftereffect) in hysteretic systems. Next, I shall briefly described the main scientific results obtained in the areas outlined A §T above.

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This paper introduces a general class of neural networks with arbitrary constant delays in the neuron interconnections, and neuron activations belonging to the set of discontinuous monotone increasing and (possibly) unbounded functions. The discontinuities in the activations are an ideal model of the situation where the gain of the neuron amplifiers is very high and tends to infinity, while the delay accounts for the finite switching speed of the neuron amplifiers, or the finite signal propagation speed. It is known that the delay in combination with high-gain nonlinearities is a particularly harmful source of potential instability. The goal of this paper is to single out a subclass of the considered discontinuous neural networks for which stability is instead insensitive to the presence of a delay. More precisely, conditions are given under which there is a unique equilibrium point of the neural network, which is globally exponentially stable for the states, with a known convergence rate. The conditions are easily testable and independent of the delay. Moreover, global convergence in finite time of the state and output is investigated. In doing so, new interesting dynamical phenomena are highlighted with respect to the case without delay, which make the study of convergence in finite time significantly more difficult. The obtained results extend previous work on global stability of delayed neural networks with Lipschitz continuous neuron activations, and neural networks with discontinuous neuron activations but without delays.

In 1988 Kennedy and Chua introduced the dynamical canonical nonlinear programming circuit (NPC) to solve in real time nonlinear programming problems where the objective function and the constraints are smooth (twice continuously differentiable) functions. In this paper, a generalized circuit is introduced (G-NPC), which is aimed at solving in real time a much wider class of nonsmooth nonlinear programming problems where the objective function and the constraints are assumed to satisfy only the weak condition of being regular functions. G-NPC, which derives from a natural extension of NPC, has a neural-like architecture and also features the presence of constraint neurons modeled by ideal diodes with infinite slope in the conducting region. By using the Clarke's generalized gradient of the involved functions, G-NPC is shown to obey a gradient system of differential inclusions, and its dynamical behavior and optimization capabilities, both for convex and nonconvex problems, are rigorously analyzed in the framework of nonsmooth analysis and the theory of differential inclusions. In the special important case of linear and quadratic programming problems, salient dynamical features of G-NPC, namely the presence of sliding modes, trajectory convergence in finite time, and the ability to compute the exact optimal solution of the problem being modeled, are uncovered and explained in the developed analytical framework.

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