1736 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 54, NO. 8, AUGUST 2007 Dynamical Analysis of Full-Range Cellular Neural Networks by Exploiting Differential Variational Inequalities Guido De Sandre, Mauro Forti, Paolo Nistri, and Amedeo Premoli Abstract—The paper considers the full-range (FR) model of cellular neural networks (CNNs) in the case where the neuron nonlinearities are ideal hard-comparator functions with two vertical straight segments. The dynamics of FR-CNNs, which is described by a differential inclusion, is rigorously analyzed by means of theoretical tools from set-valued analysis and differential inclusions. The fundamental property proved in the paper is that FR-CNNs are equivalent to a special class of differential inclusions termed differential variational inequalities. A sound foundation to the dynamics of FR-CNNs is then given by establishing the existence and uniqueness of the solution starting at a given point, and the existence of equilibrium points. Moreover, a fundamental result on trajectory convergence towards equilibrium points (complete stability) for reciprocal standard CNNs is extended to reciprocal FR-CNNs by using a generalized Lyapunov approach. As a consequence, it is shown that the study of the ideal case with vertical straight segments in the neuron nonlinearities is able to give a clear picture and analytic characterization of the salient features of motion, such as the sliding modes along the boundary of the hypercube defined by the hard-comparator nonlinearities. Finally, it is proved that the solutions of the ideal FR model are the uniform limit as the slope tends to infinity of the solutions of a model where the vertical segments in the nonlinearities are approximated by segments with finite slope. Index Terms—Cellular neural networks (CNNs), differential in- clusions, full-range (FR) model, sliding modes, trajectory conver- gence, variational inequalities. NOTATION Real -space. , Square matrix. Transpose of . Inverse of . , Column vector. Manuscript received September 25, 2004; revised November 29, 2006 and January 14, 2007. This paper was recommended by Associate Editor L. Tra- jkovic. G. De Sandre is with STMicroelectronics, 20041 Agrate Brianza, Italy (e-mail: guido.de-sandre@st.com). M. Forti and P. Nistri are with the Dipartimento di Ingegneria dell’Infor- mazione, Università di Siena, 53100 Siena, Italy (e-mail: forti@dii.unisi.it; pnistri@dii.unisi.it). A. Premoli is with the Dipartimento di Elettronica e Informazione, Politec- nico di Milano, 20133 Milan, Italy. Digital Object Identifier 10.1109/TCSI.2007.902607 , Scalar product of . , Euclidean norm of . , -dimensional open ball with center 0 and radius . Closure of set . Interior of . Boundary of . , Distance of from . Tangent cone to at . Normal cone to at . Projector of best approximation of on . Difference of sets . Element of with the smallest norm. Linear subspace of spanned by vectors . Direct sum of subspaces and . Empty set. (Conventional) single-valued function. (Conventional) gradient of . Extended-valued function. Graph of . Epigraph of . Generalized gradient of . I. INTRODUCTION T HE standard (S) cellular neural networks (S-CNNs) intro- duced by Chua and Yang [1] have been one of the most investigated neural paradigms for real time signal processing. 1549-8328/$25.00 © 2007 IEEE