The Łojasiewicz 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 R^n, the Łojasiewicz inequality asserts that there exists 12, 1) such that the function |f-f (a) |^\, f^-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 +), by establishing an analogous inequality in which the derivative f (x) can be replaced by any element x^ of the subdifferential 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.