Convergence rates of damped inerial dynamics from multi-degree-of-freedom system

Abstract

In this article, we investigate the convergence rate of the following dynamic system in ${\mathbb {R}}^{n}$

$$\begin{aligned} \ddot{x}(t)+\frac{A}{t^\theta }{\dot{x}}(t)+\nabla F(x(t))=0,\quad t>0, \end{aligned}$$

where A denotes the constant positive definite matrix and the potential function $F:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is continuous differentiable. This system is of vital importance, especially in optimization and engineering. This article presents new convergence rates of the above dynamics when F(x) satisfies some local geometrical properties by constructing a proper Lyapunov function. Finally, some numerical experiments were performed to explain the convergence results.

Appendix: Existence and uniqueness of the solution of (1)

Theorem 4

For any initial value $(x_0$, $v_0)$, there is a unique solution of system (1) defined on $[0,+\infty )$ satisfying $x(0)=x_0$, ${\dot{x}}(0)=v_0$, if F(x) is bounded from below.

Proof

Let $X(t)=(x(t),{\dot{x}}(t))^T$, $X(0)=(x_0, v_0)^T$, $G(u,v)=(v,-\frac{Av}{t}-\nabla F(u))^T$. So the system is equivalent to the following system:

$$\begin{aligned} {\dot{X}}(t)=G(X(t)). \end{aligned}$$

(31)

Because of the Lipschitz continuity of $\nabla F$ on a bounded subset of ${\mathbb {R}}^n$, it is easy to obtain the uniqueness and existence of a local solution associated with the system (31), it is also ture for system (1).

Assume that the maximum existence interval of the solution of (1) is $[t_0,T_{\max } )$, where $0<T_{\max }\le +\infty$. Let $W:[t_{0},+\infty )\rightarrow {\mathbb {R}}$ be defined by

$$\begin{aligned} W(t)=\frac{1}{2}\Vert {\dot{x}}(t)\Vert ^2+F(x(t)). \end{aligned}$$

For each $t\ge t_{0}$,

$$\begin{aligned} {\dot{W}}(t)=\langle -\frac{A}{t}{\dot{x}}(t),{\dot{x}}(t) \rangle \le 0. \end{aligned}$$

So W(t) is nonincreasing, we can get:

$$\begin{aligned} \frac{1}{2}\Vert {\dot{x}}(t)\Vert ^2+F(x(t))\le \frac{1}{2}\Vert {\dot{x}}(t_0)\Vert ^2+F(x(t_0)). \end{aligned}$$

Since the function F is bounded from below,

$$\begin{aligned} \frac{1}{2}\Vert {\dot{x}}(t)\Vert ^2\le \frac{1}{2}\Vert {\dot{x}}(t_0)\Vert ^2+F(x(t_0))-\inf F(x(t)). \end{aligned}$$

So ${\dot{x}}(t)\in L^{\infty }([t_0,T_{\max }); {\mathbb {R}}^n)$, assume $T_{\max }<+\infty$,

$$\begin{aligned} \Vert x(t_1)-x(t_2)\Vert \le \int _{t_1}^{t_2}\Vert {\dot{x}}(t)\Vert dt \le \vert t_1-t_2\vert \cdot \max \limits _{t\in [t_1,t_2]} \Vert {\dot{x}}(t)\Vert . \end{aligned}$$

Note that $T_{\max }<+\infty$, hence $x(t)\in L^{\infty }([0,T_{\max });{\mathbb {R}}^n)$ and $x_\infty :=\lim \limits _{t\rightarrow T_{\max }}x(t)$ exists in ${\mathbb {R}}^n$. Similarly, we define ${\dot{x}}_\infty :=\lim \limits _{t\rightarrow T_{\max }}{\dot{x}}(t)$, now we take $(x_\infty ,{\dot{x}}_\infty )$ as the initial value for the system, so the solution can be extended to a larger intervals, this is impossible. Henece, $T_{\max }=+\infty$. $\square$