  The standard approach to risk-averse control is to use the Exponential
Utility (EU) functional, which has been studied for several decades. Like other
risk-averse utility functionals, EU encodes risk aversion through an increasing
convex mapping $\varphi$ of objective costs to subjective costs. An objective
cost is a realization $y$ of a random variable $Y$. In contrast, a subjective
cost is a realization $\varphi(y)$ of a random variable $\varphi(Y)$ that has
been transformed to measure preferences about the outcomes. For EU, the
transformation is $\varphi(y) = \exp(\frac{-\theta}{2}y)$, and under certain
conditions, the quantity $\varphi^{-1}(E(\varphi(Y)))$ can be approximated by a
linear combination of the mean and variance of $Y$. More recently, there has
been growing interest in risk-averse control using the Conditional
Value-at-Risk (CVaR) functional. In contrast to the EU functional, the CVaR of
a random variable $Y$ concerns a fraction of its possible realizations. If $Y$
is a continuous random variable with finite $E(|Y|)$, then the CVaR of $Y$ at
level $\alpha$ is the expectation of $Y$ in the $\alpha \cdot 100 \%$ worst
cases. Here, we study the applications of risk-averse functionals to controller
synthesis and safety analysis through the development of numerical examples,
with emphasis on EU and CVaR. Our contribution is to examine the
decision-theoretic, mathematical, and computational trade-offs that arise when
using EU and CVaR for optimal control and safety analysis. We are hopeful that
this work will advance the interpretability and elucidate the potential
benefits of risk-averse control technology.
