Asymptotic behavior of coupled dynamical systems with multiscale aspects

We study the asymptotic behavior, as time variable t   goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given HH a general Hilbert space, Φ:H→R∪{+∞} and Ψ:H→R∪{+∞} two closed convex functions, and β a function of t which tends to +∞ as t goes to +∞, we consider the differential inclusionx˙(t)+∂Φ(x(t))+β(t)∂Ψ(x(t))∋0. This system models the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled systems. We show several results ranging from weak ergodic to strong convergence of the trajectories. As a key ingredient we assume that, for every p   belonging to the range of NCNC∫0+∞β(t)[Ψ∗(pβ(t))−σC(pβ(t))]dt<+∞ where Ψ∗Ψ∗ is the Fenchel conjugate of Ψ  , σCσC is the support function of C=argminΨ and NC(x)NC(x) is the normal cone to C at x. As a by-product, we revisit the systemx˙(t)+ϵ(t)∂Φ(x(t))+∂Ψ(x(t))∋0 where ϵ(t)ϵ(t) tends to zero as t   goes to +∞ and ∫0+∞ϵ(t)dt=+∞, whose asymptotic behavior can be derived from the preceding one by time rescaling. Applications are given in game theory, optimal control, variational problems and PDEs.