The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems

Given a real-analytic function $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ and a critical point $a \in \mathbb{R}^{n}$, the Łojasiewicz inequality asserts that there exists $\theta\in\lbrack\frac{1}{2},1)$ such that the function $|f-f(a)|^{\theta}\,\Vert\nabla f\Vert^{-1}$ remains bounded around $a$. In this paper, we extend the above result to a wide class of nonsmooth functions (that possibly admit the value $+\infty$), by establishing an analogous inequality in which the derivative $\nabla f(x)$ can be replaced by any element $x^{\ast}$ of the subdifferential $\partial f(x)$ 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.