THE LOJASIEWICZ INEQUALITY FOR NONSMOOTH SUBANALYTIC FUNCTIONS WITH APPLICATIONS TO SUBGRADIENT DYNAMICAL SYSTEMS

Given a real-analytic function f : R-n -> R and a critical point a is an element of R-n, the Lojasiewicz inequality asserts that there exists theta is an element of [1/2, 1) such that the function vertical bar f - f(a)vertical bar(theta) parallel to del f parallel to(-1) remains bounded around a. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value +infinity), by establishing an analogous inequality in which the derivative. del f(x) can be replaced by any element x* of the subdifferential. partial derivative f(x) of f. Like its smooth version, this result provides new insights into the convergence aspects of subgradient-type dynamical systems. Provided that the function f is sufficiently regular (for instance, convex or lower-C-2), the bounded trajectories of the corresponding subgradient dynamical system can be shown to be of finite length. Explicit estimates of the rate of convergence are also derived.