带隐式Hessian驱动阻尼的干摩擦惯性动力学：有限时间镇定、跟踪与邻近离散化
Dry-Friction Inertial Dynamics with Implicit Hessian-Driven Damping: Finite-Time Stabilization, Shadowing, and Proximal Discretization
AI总结：
该研究在实希尔伯特空间中分析带粘性阻尼、干摩擦和隐式Hessian驱动阻尼的惯性微分包含，证明其轨迹的适定性、收敛性与有限时间镇定，提出半隐式离散算法并验证其收敛性质。
AI中文摘要：
在实希尔伯特空间$\u210b$中，我们研究如下惯性微分包含：$$\ddot x(t)+γ\dot x(t)+\partialϕ(\dot x(t)) +\nabla f(x(t)+β\dot x(t))\ni0, $$其中$γ>0$为粘性阻尼系数，$ϕ$是在原点处具有尖锐极小值的凸势函数，用于建模干摩擦阻尼（典型形式为$ϕ=r\Vert\cdot\Vert$，其中$r>0$为干摩擦参数）。函数$f$表示待最小化的光滑势函数，而移位梯度求值$\nabla f(x(t)+β\dot x(t))$被称为隐式Hessian驱动模型，其中$β\geq 0$为对应的Hessian驱动参数。已知显式和隐式Hessian驱动动力学均能衰减惯性系统中出现的振荡。我们的贡献涉及在粘性阻尼、干摩擦和隐式Hessian驱动阻尼这三种组合阻尼作用下，对该连续动力学及其时间离散对应形式的定量分析。我们建立了全局适定性，提出了适配隐式Hessian驱动阻尼的精确Lyapunov分析，证明了轨迹具有有限长度，且强收敛到$f$的近似临界点$x_\infty$，满足：$-\nabla f(x_\infty)\in\partial ϕ(0)$。我们在终端力满足严格内部条件的情况下证明了有限时间镇定。我们还比较了显式和隐式Hessian驱动动力学，证明它们的轨迹在有限时间域上的差异为$O(β^2)$。对上述动力学进行时间半隐式离散化，得到了基于单次移位梯度求值的邻近隐式Hessian驱动算法。我们推导了其离散Lyapunov分析、渐近收敛性，以及在严格终端裕度条件下离散迭代的有限收敛性。
英文摘要：
In a real Hilbert space $\mathcal H$, we study the following inertial differential inclusion $$ \ddot x(t)+γ\dot x(t)+\partialϕ(\dot x(t)) +\nabla f(x(t)+β\dot x(t))\ni0, $$ where $γ>0$ is the viscous damping coefficient, and $ϕ$ is a convex potential with a sharp minimum at the origin that models the dry friction damping (typically $ϕ=r\Vert\cdot\Vert$ where $r>0$ is the dry-friction parameter). The function $f$ represents the smooth potential to be minimized, and the shifted-gradient evaluation $\nabla f(x(t)+β\dot x(t))$ is known as the implicit Hessian-driven model. Here $β\geq 0$ represents the corresponding Hessian-driven parameter. Both the explicit and the implicit Hessian-driven dynamics are know to attenuate the oscillations that occurs in inertial systems. Our contribution concerns the quantitative analysis of this continuous dynamic and its temporal discretization counter-part under the action of these three combined dampings: viscous damping, dry friction, and implicit Hessian-driven damping. We establish a global well-posedness, an exact Lyapunov analysis adapted to the implicit Hessian-driven damping, finite length of the trajectory and its strong convergence to an approximate critical point $x_\infty$ of $f$ satisfying: $-\nabla f(x_\infty)\in\partial ϕ(0)$. We show finite-time stabilization under a strict interior condition on the terminal force. We also compare the explicit and the implicit Hessian-driven dynamics and show that their trajectories differ by $O(β^2)$ on finite horizons. A temporal semi-implicit discretization of the dynamic above leads to a proximal implicit Hessian-driven algorithm based on one shifted-gradient evaluation. We derive its discrete Lyapunov analysis, asymptotic convergence and, under a strict terminal margin, finite convergence of the discrete iterates.
