A Doubly Nonlinear Evolution System with Threshold Effects Associated with Dry Friction

In this paper, we investigate the asymptotic behavior of inertial dynamics with dry friction within the context of a Hilbert framework for convex differentiable optimization. Our study focuses on a doubly nonlinear first-order evolution inclusion that encompasses two potentials. In our analysis, we specifically focus on two main components: the differentiable function f that needs to be minimized, which influences the system’s state through its gradient, and the nonsmooth dry friction potential denoted as \(\varphi = r\Vert \cdot \Vert \). It’s important to note that the dry friction term acts on a linear combination of the velocity vector and the gradient of f. Consequently, any stationary point in our system corresponds to a critical point of f, unlike the case where only the velocity vector is involved in the dry friction term, resulting in an approximate critical point of f. To emphasize the crucial role of \(\nabla f(x)\), we also explore the dual formulation of this dynamic, which possesses a Riemannian gradient structure. To address these dynamics, we employ the recently developed generic acceleration approach by Attouch, Bot, and Nguyen. This approach involves the time scaling of a continuous first-order differential equation, followed by the application of the method of averaging. By applying this methodology, we derive fast convergence results for second-order time-evolution systems with dry friction, asymptotically vanishing viscous damping, and implicit Hessian-driven damping.