  We consider the joint design of control and scheduling under stochastic
Denial-of-Service (DoS) attacks in the context of networked control systems. A
sensor takes measurements of the system output and forwards its dynamic state
estimates to a remote controller over a packet-dropping link. The controller
determines the optimal control law for the process using the estimates it
receives. An attacker aims at degrading the control performance by increasing
the packet-dropout rate with a DoS attack towards the sensor-controller
channel. Assume both the controller and the attacker are rational in a
game-theoretic sense. We establish a partially observable stochastic game to
derive the optimal joint design of scheduling and control. Using dynamic
programming we prove that the control and scheduling policies can be designed
separately without sacrificing optimality, making the problem equivalent to a
complete information game. We employ Nash Q-learning to solve the problem and
prove that the solution is guaranteed to constitute an $\epsilon$-Nash
equilibrium. Numerical examples are provided to illustrate the tradeoffs
between control performance and communication cost.
