  A classic reachability problem for safety of dynamic systems is to compute
the set of initial states from which the state trajectory is guaranteed to stay
inside a given constraint set over a given time horizon. In this paper, we
leverage existing theory of reachability analysis and risk measures to devise a
risk-sensitive reachability approach for safety of stochastic dynamic systems
under non-adversarial disturbances over a finite time horizon. Specifically, we
first introduce the notion of a risk-sensitive safe set as a set of initial
states from which the risk of large constraint violations can be reduced to a
required level via a control policy, where risk is quantified using the
Conditional Value-at-Risk (CVaR) measure. Second, we show how the computation
of a risk-sensitive safe set can be reduced to the solution to a Markov
Decision Process (MDP), where cost is assessed according to CVaR. Third,
leveraging this reduction, we devise a tractable algorithm to approximate a
risk-sensitive safe set, and provide theoretical arguments about its
correctness. Finally, we present a realistic example inspired from stormwater
catchment design to demonstrate the utility of risk-sensitive reachability
analysis. In particular, our approach allows a practitioner to tune the level
of risk sensitivity from worst-case (which is typical for Hamilton-Jacobi
reachability analysis) to risk-neutral (which is the case for stochastic
reachability analysis).
