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

2012 13th International Workshop on Cellular Nanoscale Networks and Their Applications

et al. 2012

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The paper considers the full-range (FR) model of cellular neural networks (CNNs) with ideal hard-limiter non-linearities that limit the allowable range of the neuron state variables. It is also supposed that there is a concentrated delay (D) in the neuron interconnections. Due to the presence of multivalued nonlinearities the D-FRCNN model is mathematically described by a retarded differential inclusion. The main result is a rigorous proof that, in the case of nonsymmetric cooperative (nonnegative) interconnections, and delayed interconnections, the semiflow generated by D-FRCNNs is monotone, and that monotonicity implies some basic restrictions on the long-term behavior of the solutions. The result is compared with recent results in the literature on semiflows generated by cooperative standard CNNs, with and without delays

Section: Monotonicity Of Semiflowmentioning

confidence: 99%

Monotonicity of semiflows generated by cooperative delayed full-range CNNs

2012 13th International Workshop on Cellular Nanoscale Networks and Their Applications

et al. 2012

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“…However it is worth remarking that Theorem 7 is to the authors' knowledge the only existing result on convergence for nonsymmetric FR-CNNs with multiple EPs. Indeed, previous papers [22] and [45] have addressed convergence of FRCNNs with multiple EPs in the symmetric case by means of a Lyapunov approach, and an approach based on the principle of trajectories with finite length and the Lojasiewicz inequality, respectively. On the other hand, [46] and [47] have given conditions ensuring global asymptotic stability and global robust exponential stability of the unique EP for some classes of nonsymmetric FRCNNs with and without delays.…”

Section: B Cooperative Frcnnsmentioning

confidence: 99%

Convergent Dynamics of Nonreciprocal Differential Variational Inequalities Modeling Neural Networks

IEEE Trans. Circuits Syst. I

et al. 2013

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The paper addresses convergence of solutions for a class of differential inclusions termed differential variational inequalities (DVIs). Each DVI describes the dynamics of a neural network (NN) evolving in a closed hypercube of $R^n$ and defined by a continuously differentiable, {\em cooperative\/} and (possibly) nonreciprocal vector field $f$. The main result in the paper is that under a new condition on $f$, which is called strong Kamke-Muller condition, the solution semiflow generated by the DVI is strongly order preserving (SOP) and hence it satisfies a {\sc Limit Set Dichotomy} and enjoys generic convergence properties. A characterization of the SKM condition is given in terms of the interconnection properties of the Jacobian matrix of $f$. In the case where $f$ is an affine, or a linear, vector field the considered DVIs include two relevant classes of NNs, namely, the linear systems operating on a closed hypercube, also known as linear systems in saturated mode (LSSMs), and the full-range (FR) model of cellular neural networks (CNNs). By applying the results to LSSMs it is obtained that any cooperative LSSM with a (possibly) nonsymmetric and fully interconnected matrix is generically convergent. Analogous results hold for FRCNNs. All the obtained convergence results hold in the general case where the DVIs, and the LSSMs and FRCNNs, possess multiple equilibrium points

“…On one hand, it has been shown that there are relevant classes of FRCNNs displaying analogous convergence and stability properties as the corresponding SCNNs. These include: (1) FRCNNs with symmetric neuron interconnections, where convergence can be analyzed by means of a generalized Lyapunov approach based on an extended version of LaSalle's invariance principle , or by means of the principle of trajectories with finite length and the Łojasiewicz inequality ; (2) a general class of FRCNNs defined by gradient‐type systems ; (3) FRCNNs with nonsymmetric Lyapunov diagonally stable, or M‐matrices , for which it is possible to prove general results on global asymptotic stability. On the other hand, a counterexample by Corinto and Gilli has shown that there are classes of convergent nonsymmetric SCNNs for which the corresponding FRCNNs display non‐vanishing oscillations .…”

Section: Introductionmentioning

confidence: 99%

A study on semiflows generated by cooperative full‐range CNNs

et al. 2012

Circuit Theory & Apps

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The paper considers the full-range (FR) model of cellular neural networks (CNNs) characterized by ideal hard-limiter nonlinearities with two vertical segments in the current–voltage characteristic. It is shown that when the FRCNNs are cooperative, i.e., there are excitatory interconnections between distinct neurons, the generated solution semiflow is monotone and that monotonicity implies some fundamental restrictions on the geometry of omega-limit sets. The result on monotonicity is a generalization to the class of differential inclusions describing the dynamics of FRCNNs of a classic result due to Kamke for cooperative ordinary differential equations. The paper also points out difficulties to use the standard theory of eventually strongly monotone (ESM) semiflows for addressing convergence of FRCNNs. By means of counterexamples, it is shown that, even assuming the irreducibility of the interconnections, the semiflow generated by a cooperative FRCNN is not ESM; furthermore, also the limit set dichotomy can be violated