The long time behavior of a class of second-order gradient-like systems with vanishing dissipative term and non-convex analytic potential

Abstract

In this paper, a class of second order dissipative system (1) x ̈ ( t ) + a ( t ) x ̇ ( t ) + ∇ f ( x ( t ) ) = 0 is studied, where f : R N → R is analytic and non-convex, a : R + → R + is continuous and nonincreasing with lim t → ∞ a ( t ) = 0 . We give a sufficient condition for the convergence of global and bounded solutions of (1). The condition shows that the rate of convergence of damping coefficient a ( t ) is related to the Lojasiewicz exponent of the analytic function f .