GMFG critical nodes for control affine systems with exponentiated costs

Graphon Mean Field Games (GMFGs) (Caines and Huang, 2021) constitute generalizations of Mean Field Games (MFGs) for which the agents form subpopulations associated with the nodes of large graphs, where infinite cardinality graph node and edge limits are considered with limit graphons g ( α , β ) , ( α , β ) ∈ [ 0 , 1 ] × [ 0 , 1 ] . The work in (Foguen-Tchuendom et al., (2021, 2022) [10],[11]) analyzed the stationarity of Nash equilibrium values with respect to node location for large populations of non-cooperative agents with linear dynamics on large graphs together with their limit graphons. The analysis in (Foguen-Tchuendom et al.,(2021, 2022) [10],[11]) is extended in this investigation to agent systems lying in the class of control affine non-linear systems (see Isidori (1985)). Specifically, control affine GMFG systems in an infinite network are treated where (i) at each node α ∈ [ 0 , 1 ] the drift of each generic agent system is affine in the control function, and (ii) the running costs at each node α are exponentiated negative inverse quadratic functions of the difference between a generic state and the graphon g weighted local mean field Z α , g , which involves μ G ≔ { μ β , β ∈ [ 0 , 1 ] } representing agent state distributions at different nodes. The GMFG equation system is proved to have a unique solution under a contraction condition, and the main result is that Nash equilibrium values V α are stationary with respect to the node location α ∈ [ 0 , 1 ] if the corresponding graphon weighted local mean field Z α , g is stationary with respect to node location; the converse also holds if the model only has cost-coupling.