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A classic reachability problem for safety of dynamic systems is to computethe set of initial states from which the state trajectory is guaranteed to stayinside a given constraint set over a given time horizon. In this paper, weleverage existing theory of reachability analysis and risk measures to devise arisk-sensitive reachability approach for safety of stochastic dynamic systemsunder non-adversarial disturbances over a finite time horizon. Specifically, wefirst introduce the notion of a risk-sensitive safe set as a set of initialstates from which the risk of large constraint violations can be reduced to arequired level via a control policy, where risk is quantified using theConditional Value-at-Risk (CVaR) measure. Second, we show how the computationof a risk-sensitive safe set can be reduced to the solution to a MarkovDecision Process (MDP), where cost is assessed according to CVaR. Third,leveraging this reduction, we devise a tractable algorithm to approximate arisk-sensitive safe set, and provide theoretical arguments about itscorrectness. Finally, we present a realistic example inspired from stormwatercatchment design to demonstrate the utility of risk-sensitive reachabilityanalysis. In particular, our approach allows a practitioner to tune the levelof risk sensitivity from worst-case (which is typical for Hamilton-Jacobireachability analysis) to risk-neutral (which is the case for stochasticreachability analysis).

A Risk-Sensitive Finite-Time Reachability Approach for Safety of Stochastic Dynamic Systems