Continuous-time fully distributed generalized Nash equilibrium seeking for multi-integrator agents

We consider a group of (multi)-integrator agents playing games on a network, in a partial-decision information scenario. We design fully distributed continuous-time controllers, based on consensus and primal-dual gradient dynamics, to drive the agents to a generalized Nash equilibrium. Our first solution adopts fixed gains, whose choice requires the knowledge of some global parameters of the game. Therefore, to adapt the procedure to setups where the agents do not have any global information, we introduce a controller that can be tuned in a completely decentralized fashion, thanks to the use of integral adaptive weights. We further introduce algorithms, both with constant and dynamic gains, specifically devised for generalized aggregative games. For all the proposed control schemes, we show convergence to a variational equilibrium, under Lipschitz continuity and strong monotonicity of the game mapping, by leveraging monotonicity properties and stability theory for projected dynamical systems.