  This article considers the problem of risk-optimal allocation of security
measures when the actuators of an uncertain control system are under attack. We
consider an adversary injecting false data into the actuator channels. The
attack impact is characterized by the maximum performance loss caused by a
stealthy adversary with bounded energy. Since the impact is a random variable,
due to system uncertainty, we use Conditional Value-at-Risk (CVaR) to
characterize the risk associated with the attack. We then consider the problem
of allocating the security measures which minimize the risk. We assume that
there are only a limited number of security measures available. Under this
constraint, we observe that the allocation problem is a mixed-integer
optimization problem. Thus we use relaxation techniques to approximate the
security allocation problem into a Semi-Definite Program (SDP). We also compare
our allocation method $(i)$ across different risk measures: the worst-case
measure, the average (nominal) measure, and $(ii)$ across different search
algorithms: the exhaustive and the greedy search algorithms. We depict the
efficacy of our approach through numerical examples.
