In this study, a control theoretic description of resilience is provided to quantify the characteristics of a resilient system. The aim is to establish a paradigm for resilient control design based on tangible control objectives that yield desirable attributes for safety-critical systems. In that regard, durability and recoverability properties are identified as key components of the proposed resilience framework and, to offer a methodology to enforce these attributes, the notion of finite-time robust control barrier function (FR-CBF) is introduced. Furthermore, to offer a comprehensive treatment of the problem, resilient control design is investigated for both continuous and sampled-data systems. To that end, FR-CBF-based design conditions for both continuous and piece-wise constant zero-order hold (ZOH) control inputs are included. Moreover, to provide a concrete example of how the proposed framework could be adopted for safety-critical control applications, in this study we also investigate the voltage regulation problem for inverter-interfaced radial power distribution networks subject to adversarial injections. In that regard, sufficient conditions for both the continuous and sampled-data ZOH control are derived to guarantee finite-time recovery and safe operation of the distribution grid in accordance with the proposed resilience framework. Finally, the efficacy of the proposed results is advocated using a simulation study showing resilient grid performance in the presence of the 'worst-case' power injection attack, as reported in (Lindstr & ouml;m et al. 2021).