On global exponential stability of standard and full‐range CNNs

Abstract

SUMMARYThis paper compares the dynamical behaviour of the standard (S) cellular neural networks (CNNs) and the full-range (FR) CNNs, when the two CNN models are characterized by the same set of parameters (interconnections and inputs). The FR-CNNs are assumed to be characterized by ideal hard-limiter nonlinearities with two vertical segments in the i-v characteristic. The main result is that some basic conditions ensuring global exponential stability (GES) of the unique equilibrium point of S-CNNs, with or without delay, continue to ensure the same property for FR-CNNs for the same set of parameters. The significance of this result is discussed with respect to the results in a paper by Corinto and Gilli addressing the similarity of the qualitative behaviour of S-CNNs and FR-CNNs. FR-CNNs are analysed in this paper from a rigorous mathematical viewpoint by means of theoretical tools from set-valued analysis and differential inclusions. In particular, GES is investigated via an extended Lyapunov approach that is applicable to the differential inclusion describing the dynamics of FR-CNNs.

Synchronization of identical neural networks and other systems with an adaptive coupling strength

“…Among them, identical synchronization of coupled complex networks has attracted more attention, since 632 X. LIU AND T. CHEN synchronization can not only explain many natural phenomena [7], but also have many applications, such as neural networks, image processing, secure communication, etc., see [8][9][10][11][12][13][14].…”

“…Recently, an increasing interest has been devoted to the study of complex networks, see [1][2][3][4][5][6], which can be regarded as a composition and interaction of several dynamical nodes. Among them, identical synchronization of coupled complex networks has attracted more attention, since 632 X. LIU AND T. CHEN synchronization can not only explain many natural phenomena [7], but also have many applications, such as neural networks, image processing, secure communication, etc., see [8][9][10][11][12][13][14].…”

Monotonicity of semiflows generated by cooperative delayed full-range CNNs

“…Moreover, we allow for the presence of a concentrated delay (D) in the neuron interconnections. The D-FRCNNs, introduced in [9], are expected to have advantages over delayed SCNNs analogous to those obtainable with FRCNNs. We also stress that the presence of delays is important in view of applications typical of the CNN paradigm to the solution of motion related problems in real time.…”

Łojasiewicz inequality and exponential convergence of the full‐range model of CNNs

“…Results obtained so far along this line fall in two groups. On one hand it has been shown that there are classes of neuron interconnection matrices, such as symmetrizable matrices, matrices with non-negative off-diagonal entries [7] and Lyapunov Diagonally Stable matrices [8] for which the qualitative behavior of the solutions of FRCNNs is indeed analogous to that of the SCNNs for the same set of parameters. On the other hand, there are also cases where the global dynamical behavior of FRCNNs is basically different from that of SCNNs with the same set of parameters.…”