We study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features.As a benchmark case, given H 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 inclusioṅThis 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 N Cwhere Ψ * is the Fenchel conjugate of Ψ , σ C is the support function of C = argmin Ψ and N C (x) is the normal cone to C at x. As a byproduct, we revisit the systeṁwhere (t) tends to zero as t goes to +∞ and +∞ 0 (t) dt = +∞, whose asymptotic behavior can be derived from the preceding one ✩ With the support of the French ANR under grants ANRby time rescaling. Applications are given in game theory, optimal control, variational problems and PDEs.

Let F: H Q R be a C 1 function on a real Hilbert space H and let c > 0 be a positive (damping) parameter. For any control function e: R + Q R + which tends to zero as t Q +., we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator (HBFC) ẍ (t)+cẋ (t)+NF(x(t))+e(t) x(t)=0.We show that if e(t) does not tend to zero too rapidly as t Q +., then the term e(t) x(t) asymptotically acts as a Tikhonov regularization, which forces the trajectories to converge to a particular equilibrium. Indeed, in the main result of this paper, it is established that, when F is convex and S=argmin F ] ", under the key assumption that e is a ''slow'' control, i.e., > +. 0 e(t) dt=+., then each trajectory of the (HBFC) system strongly converges, as t Q +., to the element of minimal norm of the closed convex set S. As an application, we consider the damped wave equation with Neumann boundary condition

Abstract. We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities 0 ∈ Ax + NC (x) where H is a real Hilbert space, A : H ⇉ H is a maximal monotone operator, and NC is the outward normal cone to a closed convex set C ⊂ H. Let Ψ : H → R be a convex differentiable function whose gradient is Lipschitz continuous, and which acts as a penalization function with respect to the constraint x ∈ C. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn) ∈ ℓ 2 \ ℓ 1 , we consider the diagonal forward-backward algorithm xn+1 = (I + λnA) −1 (xn − λnβn∇Ψ(xn)).Assuming that (βn) satisfies the growth condition lim sup n→∞ λnβn < 2/θ (where θ is the Lipschitz constant of ∇Ψ), we obtain weak ergodic convergence of the sequence (xn) to an equilibrium for a general maximal monotone operator A. We also obtain weak convergence of the whole sequence (xn) when A is the subdifferential of a proper lower-semicontinuous convex function. As a key ingredient of our analysis, we use the cocoerciveness of the operator ∇Ψ. When specializing our results to coupled systems, we bring new light on Passty's Theorem, and obtain convergence results of new parallel splitting algorithms for variational inequalities involving coupling in the constraint. We also establish robustness and stability results that account for numerical approximation errors. An illustration to compressive sensing is given.