## Optimal joint control and triggering strategies against denial of service attacks: a zero-sum game

This study is concerned with the optimal control and scheduling problem for linear networked control systems with communication constraints and security issues. The communication channel, which connects remote sensors with a controller, has limited transmission capability and may suffer denial of service attacks from an external attacker. Unlike most existing cyber-attack-related literature which presumed a prior well-designed control or estimating strategy on which triggering and jamming strategies were established, in the present work, optimal strategies of the controller, the trigger as well as the external attacker are considered simultaneously under the renowned linear quadratic criteria. A designer, who can jointly devise the controller and the trigger, aims to improve the system's performance while the attacker's goal is just the opposite. With the assumption that the trigger and the attacker have limited opportunities to send or attack, a zero-sum static game of complete information is first utilised to investigate the optimal strategies for both players, i.e. the designer and the attacker. The existence of Nash equilibrium (NE) point, i.e. the combination of optimal strategy for each player, is proven afterwards. Finally, the property of none pure-strategy NE in a special case is provided with rigorous proofs. A numerical example is employed to demonstrate how to derive the optimal strategies for the designer and the attacker.