  In this paper, we consider a multi-objective control problem for stochastic
systems that seeks to minimize a cost of interest while ensuring safety. We
introduce a novel measure of safety risk using the conditional value-at-risk
and a set distance to formulate a safety risk-constrained optimal control
problem. Our reformulation method using an extremal representation of the
safety risk measure provides a computationally tractable dynamic programming
solution. A useful byproduct of the proposed solution is the notion of a
risk-constrained safe set, which is a new stochastic safety verification tool.
We also establish useful connections between the risk-constrained safe sets and
the popular probabilistic safe sets. The tradeoff between the risk tolerance
and the mean performance of our controller is examined through an inventory
control problem.
