  This paper develops a safety analysis method for stochastic systems that is
sensitive to the possibility and severity of rare harmful outcomes. We define
risk-sensitive safe sets as sub-level sets of the solution to a non-standard
optimal control problem, where a random maximum cost is assessed using the
Conditional Value-at-Risk (CVaR) functional. The solution to the control
problem represents the maximum extent of constraint violation of the state
trajectory, averaged over the $\alpha\cdot 100$% worst cases, where $\alpha \in
(0,1]$. This problem is well-motivated but difficult to solve in a tractable
fashion because temporal decompositions for risk functionals generally depend
on the history of the system's behavior. Our primary theoretical contribution
is to derive under-approximations to risk-sensitive safe sets, which are
computationally tractable. Our method provides a novel, theoretically
guaranteed, parameter-dependent upper bound to the CVaR of a maximum cost
without the need to augment the state space. For a fixed parameter value, the
solution to only one Markov decision process problem is required to obtain the
under-approximations for any family of risk-sensitivity levels. In addition, we
propose a second definition for risk-sensitive safe sets and provide a
tractable method for their estimation without using a parameter-dependent upper
bound. The second definition is expressed in terms of a new coherent risk
functional, which is inspired by CVaR. We demonstrate our primary theoretical
contribution using numerical examples of a thermostatically controlled load
system and a stormwater system.
