  The popularity of Conditional Value-at-Risk (CVaR), a risk functional from
finance, has been growing in the control systems community due to its intuitive
interpretation and axiomatic foundation. We consider a nonstandard optimal
control problem in which the goal is to minimize the CVaR of a maximum random
cost subject to a Borel-space Markov decision process. The objective represents
the maximum departure from a desired operating region averaged over a given
fraction of the worst cases. This problem provides a safety criterion for a
stochastic system that is informed by both the probability and severity of the
potential consequences of the system's behavior. In contrast, existing safety
analysis frameworks apply stage-wise risk constraints or assess the probability
of constraint violation without quantifying the potential severity of the
violation. To the best of our knowledge, the problem of interest has not been
solved. To solve the problem, we propose and study a family of stochastic
dynamic programs on an augmented state space. We prove that the optimal CVaR of
a maximum random cost enjoys an equivalent representation in terms of the
solutions to these dynamic programs under appropriate assumptions. For each
dynamic program, we show the existence of an optimal policy that depends on the
dynamics of an augmented state under the assumptions. In a numerical example,
we illustrate how our safety analysis framework is useful for assessing the
severity of combined sewer overflows under precipitation uncertainty.
