A Mathematical Framework for Resilience: Dynamics, Uncertainties, Strategies, and Recovery Regimes

Abstract

Resilience is a rehashed concept in natural hazard management—resilience of cities to earthquakes, to floods, to fire, etc. In a word, a system is said to be resilient if there exists a strategy that can drive the system state back to “normal” after any perturbation. What formal flesh can we put on such a malleable notion? We propose to frame the concept of resilience in the mathematical garbs of control theory under uncertainty. Our setting covers dynamical systems both in discrete or continuous time, deterministic or subject to uncertainties. We will say that a system state is resilient if there exists an adaptive strategy such that the generated state and control paths, contingent on uncertainties, lay within an acceptable domain of random processes, called recovery regimes. We point out how such recovery regimes can be delineated thanks to so-called risk measures, making the connection with resilience indicators. Our definition of resilience extends others, be they “à la Holling” or rooted in viability theory. Indeed, our definition of resilience is a form of controllability for whole random processes (regimes), whereas others require that the state values must belong to an acceptable subset of the state set.