Prescribed-time distributed Nash equilibrium seeking for hybrid networked game with disturbances

This paper investigates the prescribed-time distributed Nash equilibrium (NE) seeking problem of networked games for continuous-discrete-time hybrid systems with unknown disturbances. Multi-player optimisation and decision-making are widely applied in practical engineering scenarios such as smart grid dispatching, sensor networks, and distributed cooperative control. In contrast to existing works, the continuous-discrete action dynamics, hybrid unknown disturbances, as well as prescribed-time convergence are addressed simultaneously for networked games. First, a prescribed-time hybrid extended state observer is constructed for each agent to simultaneously estimate the unknown disturbances and actions. Then, based on the hybrid observer, a prescribed-time hybrid NE seeking algorithm is developed, combining gradient-play dynamics with consensus-based estimation methods. To ensure the prescribed-time convergence of the hybrid strategies, we design a class of sampling time sequences that establish an explicit mapping between the sampling intervals and the prescribed settling time. These intervals are then incorporated as time-varying gains into the hybrid NE seeking algorithm. Finally, illustrative examples are presented to verify the proposed algorithms' effectiveness.