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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.