  In practical applications, decision-makers with heterogeneous dynamics may be
engaged in the same decision-making process. This motivates us to study
distributed Nash equilibrium seeking for games in which players are mixed-order
(first- and second-order) integrators influenced by unknown dynamics and
external disturbances in this paper.
  To solve this problem, we employ an adaptive neural network to manage unknown
dynamics and disturbances, based on which a distributed Nash equilibrium
seeking algorithm is developed by further adapting concepts from gradient-based
optimization and multi-agent consensus. By constructing appropriate Lyapunov
functions, we analytically prove convergence of the reported method.
Theoretical investigations suggest that players' actions would be steered to an
arbitrarily small neighborhood of the Nash equilibrium, which is also testified
by simulations.
