Lyapunov Method and Convergence of the Full-Range Model of CNNs

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

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“…System (F) is a special case of the class of differential variational inequalities (DVIs) [28,17,18]…”

Section: Semiflows Generated By Frcnnsmentioning

confidence: 99%

A study on semiflows generated by cooperative full‐range CNNs

Marco

Forti

Grazzini

et al. 2012

Circuit Theory & Apps

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“…Accounting for Property 1, (2) is equivalent to the differential variational inequality (DVI) [19], [22], [26] …”

Section: D-frcnn Modelmentioning

confidence: 99%

“…We refer the reader to [27], and references therein, for an account of research results along this line. We refer the reader also to [22], [28], for other related results on convergence of FRCNNs with symmetric interconnections, and to [26] for an extended Lyapunov approach to study convergence of generalized gradienttype FRCNN models.…”

Section: Monotonicity Of Semiflowmentioning

confidence: 99%

Monotonicity of semiflows generated by cooperative delayed full-range CNNs

Marco

Forti

Grazzini

et al. 2012

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

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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

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“…During recent years, techniques from nonsmooth analysis have been widely applied to study neural networks with high-gain nonlinearities (ideal diodes, hard-comparators, or hard-limiters), such as linear and nonlinear programming networks [24]- [28], networks for global optimization, synchronization and identification [29]- [39], and full-range cellular neural networks [40]. Some features and advantages of nonsmooth analysis can be briefly pointed out as follows.…”

Section: Introductionmentioning

confidence: 99%

“…Nonsmooth analysis is based on the concept of Filippov solutions, which are good approximations of solutions of actual dynamic systems with high-gain nonlinearities [41]. The needed mathematical machinery, which relies on subgradient calculus and nonsmooth Lyapunov method, is rigorous [40], [41]. Moreover, nonsmooth analysis is able to highlight salient features of the dynamics as the presence of sliding modes along discontinuity surfaces or the phenomenon of convergence in finite time [29].…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Neural Networks for Finite-Time Solution of Time-Dependent Linear Equations

Marco

Forti

Nistri

et al. 2016

IEEE Trans. Cybern.

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This paper considers a class of nonsmooth neural networks with discontinuous hard-limiter (signum) neuron activations for solving time-dependent (TD) systems of algebraic linear equations (ALEs). The networks are defined by the subdifferential with respect to the state variables of an energy function given by the L norm of the error between the state and the TD-ALE solution. It is shown that when the penalty parameter exceeds a quantitatively estimated threshold the networks are able to reach in finite time, and exactly track thereafter, the target solution of the TD-ALE. Furthermore, this paper discusses the tightness of the estimated threshold and also points out key differences in the role played by this threshold with respect to networks for solving time-invariant ALEs. It is also shown that these convergence results are robust with respect to small perturbations of the neuron interconnection matrices. The dynamics of the proposed networks are rigorously studied by using tools from nonsmooth analysis, the concept of subdifferential of convex functions, and that of solutions in the sense of Filippov of dynamical systems with discontinuous nonlinearities.

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Lyapunov Method and Convergence of the Full-Range Model of CNNs

Cited by 30 publications

References 34 publications

A study on semiflows generated by cooperative full‐range CNNs

A study on semiflows generated by cooperative full‐range CNNs

Monotonicity of semiflows generated by cooperative delayed full-range CNNs

Discontinuous Neural Networks for Finite-Time Solution of Time-Dependent Linear Equations

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