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We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations. Index Terms--Asymptotic stability, differential-difference equations, discontinuous dynamical systems (DDS), exponential stability, hybrid systems, Lyapunov stability, retarded functional differential equations, switched systems, systems with delays.

An optimized method of harvesting vibrational energy with a piezoelectric element using a step-down dc-dc converter is presented. In this configuration, the converter regulates the power flow from the piezoelectric element to the desired electronic load. Analysis of the converter in discontinuous current conduction mode results in an expression for the duty cycle-power relationship. Using parameters of the mechanical system, the piezoelectric element, and the converter; the 'optimal' duty cycle can be determined where the harvested power is maximized for the level of mechanical excitation. It is shown that, as the magnitude of the mechanical excitation increases, the optimal duty cycle becomes essentially constant, greatly simplifying the control of the step-down converter. The expression is validated with experimental data showing that the optimal duty cycle can be accurately determined and maximum energy harvesting attained. A circuit is proposed which implements this relationship, and experimental results show that the converter increases the harvested power by approximately 325%. Index Terms--DC-DC, discontinuous conduction mode, energy harvesting, piezoelectric devices.