Asymptotic behavior of coupled dynamical systems with multiscale aspects

作者：Attouch Hedy; Czarnecki Marc Olivier^*

来源：Journal of Differential Equations, 2010, 248(6): 1315-1344.

DOI：10.1016/j.jde.2009.06.014

• Nonautonomous gradient-like systems
• Monotone inclusions
• Asymptotic behavior
• Time multiscaling
• Convex minimization
• Hierarchical optimization
• Asymptotic control
• Slow control
• Potential games
• Best response
• Splitting methods
• Domain decomposition for PDEs

We study the asymptotic behavior, as time variable t goes to +infinity of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, Phi : H -> R boolean OR {+infinity} and psi : H -> R boolean OR {+infinity} two closed convex functions, and beta a function of t which tends to +infinity as t goes to +infinity, we consider the differential inclusion
(x) over dot(t) + partial derivative Phi(x(t)) + beta(t)partial derivative psi(x(t)) is an element of 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 N(C)
(0)integral(+infinity) beta(t) [psi*(p/beta(t) - sigma(C)(p/beta(t)] dt < +infinity
where psi* is the Fenchel conjugate of psi, sigma(C) is the support function of C = argmin psi and N(C)(x) is the normal cone to C at x. As a by-product, we revisit the system
(x) over dot(t) + epsilon(t)partial derivative Phi(x(t)) + partial derivative psi(x(t)) is an element of 0
where E(t) tends to zero as t goes to +infinity and (0)integral(+infinity) epsilon(t) dt = +infinity, 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.

• 出版日期2010-3-15