  Cyber-physical systems are conducting increasingly complex tasks, which are
often modeled using formal languages such as temporal logic. The system's
ability to perform the required tasks can be curtailed by malicious adversaries
that mount intelligent attacks. At present, however, synthesis in the presence
of such attacks has received limited research attention. In particular, the
problem of synthesizing a controller when the required specifications cannot be
satisfied completely due to adversarial attacks has not been studied. In this
paper, we focus on the minimum violation control synthesis problem under linear
temporal logic constraints of a stochastic finite state discrete-time system
with the presence of an adversary. A minimum violation control strategy is one
that satisfies the most important tasks defined by the user while violating the
less important ones. We model the interaction between the controller and
adversary using a concurrent Stackelberg game and present a nonlinear
programming problem to formulate and solve for the optimal control policy. To
reduce the computation effort, we develop a heuristic algorithm that solves the
problem efficiently and demonstrate our proposed approach using a numerical
case study.
