Dynamical systems stability preserving mappings stability of motion finite dimensional systems infinite dimensional systems differential inclusions hybrid dynamical systems stability of hybrid and discontinuous dynamical systems case studies in hybrid and discontinuous dynamical systems.

We first conduct an analysis of the robustness properties of a class of neural networks with applications to associative memories. Specifically, for a network with nominal parameters which stores a set of desired bipolar memories, we establish sufficient conditions under which the same set of bipolar memories is also stored in the network with perturbed parameters. This result enables us to establish a synthesis procedure for neural networks whose stored memories are invariant under perturbations. Our synthesis procedure is capable of generating artificial neural networks with prespecified sparsity constraints (on the interconnecting structure) and with nonsymmetric and symmetric interconnection matrices. To demonstrate the applicability of the present results, we consider several specific examples.

It is well known that Hopfield neural networks without delays exhibit no oscillations and possess global stability (i.e., all trajectories tend to some equilibrium). In the present paper we show that if the bound /spl tau//spl beta//spl par/T/sub 2//spl par/0, interconnection matrix T/sub 2/ associated with delays, and gain of the neurons given by /spl beta/, will exhibit similar qualitative properties as the original Hopfield neural network without delays (/spl par/T/sub 2//spl par/ denotes the matrix norm induced by the Euclidean vector norm).

In this paper we consider retarded and neutral linear, time-delay systems with perturbations. For such systems, we present two types of sufficient conditions for asymptotic stability. One type involves norm conditions while the other type involves corner conditions. Some of our results constitute generalizations of existing results. We demonstrate the applicability of our results by means of three specific examples.

In contrast to the usual types of neural networks which utilize two states for each neuron, a class of synchronous discrete-time neural networks with multilevel threshold neurons is developed. A qualitative analysis and a synthesis procedure for the class of neural networks considered constitute the principal contributions of this paper. The applicability of the present class of neural networks is demonstrated by means of a gray level image processing example, where each neuron can assume one of sixteen values. When compared to the usual neural networks with two state neurons, networks which are endowed with multilevel neurons will, in general, for a given application, require fewer neurons and thus fewer interconnections. This is an important consideration in VLSI implementation.

The authors present a new training algorithm to be used on a four-layer perceptron-type feedforward neural network for the generation of binary-to-binary mappings. This algorithm is called the Boolean-like training algorithm (BLTA) and is derived from original principles of Boolean algebra followed by selected extensions. The algorithm can be implemented on analog hardware, using a four-layer binary feedforward neural network (BFNN). The BLTA does not constitute a traditional circuit building technique. Indeed, the rules which govern the BLTA allow for generalization of data in the face of incompletely specified Boolean functions. When compared with techniques which employ descent methods, training times are greatly reduced in the case of the BLTA. Also, when the BFNN is used in conjunction with A/D converters, the applicability of the present algorithm can be extended to accept real-valued inputs.

A qualitative analysis is presented for a class of synchronous discrete-time neural networks defined on hypercubes in the state space. Analysis results are utilized to establish a design procedure for associative memories to be implemented on the present class of neural networks. To demonstrate the storage ability and flexibility of the synthesis procedure, several specific examples are considered. The design procedure has essentially the same desirable features as the results of J. Li et al. (1988, 1989) for continuous-time neural networks. For a given system dimension, networks designed by the present method may have the ability to store more patterns (as asymptotically stable equilibria) than corresponding discrete-time networks designed by other techniques. The design method guarantees the storage of all the desired patterns as asymptotically stable equilibrium points. The present method provides guidelines for reducing the number of spurious states and for estimating the extent of the patterns' domains of attraction. The present results provide a means of implementing neural networks by serial processors and special digital hardware.

In the present paper we report on the status of stability of complex dynamical systems. We consider those systems which may be viewed as an interconnection of lower order subsystems. We emphasize those types of results which are phrased in terms of the stability properties of the subsystems and in terms of the qualitative properties of the system interconnecting structure. We consider both Lyapunov stability and input-output stability results. In our presentation, we concentrate on selected sample results and we show how some of these results are related. We then use these sample results to identify modifications, extensions, generalizations, and applications. In this way, we are able to provide a reasonably good picture of the entire subject on hand.