  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.

Optimization and Control;Systems and Control