  This paper studies the control of safety-critical dynamical systems in the
presence of adversarial disturbances. We seek to synthesize state-feedback
controllers to minimize a cost incurred due to the disturbance, while
respecting a safety constraint. The safety constraint is given by a bound on an
H-inf norm, while the cost is specified as an upper bound on the H-2 norm of
the system. We consider an online setting where costs at each time are revealed
only after the controller at that time is chosen. We propose an iterative
approach to the synthesis of the controller by solving a modified discrete-time
Riccati equation. Solutions of this equation enforce the safety constraint. We
compare the cost of this controller with that of the optimal controller when
one has complete knowledge of disturbances and costs in hindsight. We show that
the regret function, which is defined as the difference between these costs,
varies logarithmically with the time horizon. We validate our approach on a
process control setup that is subject to two kinds of adversarial attacks.
