Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

Abstract

Abstract.In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equationswhere ∇φ is the gradient operator of a convex differentiable potential function φ : H → R, A : H → H is a maximal monotone operator which is assumed to be λ-cocoercive, and γ > 0 is a damping parameter. Potential and non-potential effects are associated respectively to ∇φ and A. Under condition λγ 2 > 1, it is proved that each trajectory asymptotically weakly converges to a zero of ∇φ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.Mathematics Subject Classification. 34C35, 34D05, 65C25, 90C25, 90C30.

Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators

“…Inertial dynamics and cocoercive operators. In analogy with (3),Álvarez-Attouch [2] and Attouch-Maingé [7] studied the equation (7)ẍ(t) + γẋ(t) + A(x(t)) = 0, where A is a cocoercive operator. Cocoercivity plays an important role, not only to ensure the existence of solutions, but also in analyzing their long-term behavior.…”

A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

“…In view of the Lipschitz continuity of ∇Ψ, ∇ 2 Ψ and ∇Φ, ∇ 2 Φ, assumed in (H Ψ ) and (H Φ ), respectively, the existence and uniqueness of strong global solutions of (11) is a consequence of the Cauchy-Lipschitz-Picard Theorem (see for example [5,17,25,31]).…”

Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators

“…The similarity between second-order Hamiltonian systems and the corresponding first-order gradient flows is wellknown in mechanical control systems [19], in dynamic optimization [20], [21], [22], and also in transient stability studies [23], [24], [25], but we are not aware of any result as general as Theorem III.1. In [23], [24], [25], statements 1) and 2) are proved under the more stringent assumptions that H λ has a finite number of isolated and hyperbolic equilibria.…”

“…Additionally, if H(x) constitutes an energy function and if a one-parameter transversality condition is satisfied, then the separatrices of system (6) can also be characterized accurately [23], [24], [26]. Also statement 3) can be refined under further structural assumptions on the potential function H(x), and various other minimizing properties can be deduced from the dynamics (6), see [19], [20], [21], [22].…”