  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 via
Conditional Value-at-Risk (CVaR). The objective function represents the maximum
extent of constraint violation of the state trajectory, averaged over a given
percentage of worst cases. This problem is well-motivated but difficult to
solve tractably because the temporal decomposition for CVaR is
history-dependent. Our primary theoretical contribution is to derive
computationally tractable under-approximations to risk-sensitive safe sets. 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 via numerical
examples.
