Sequential gradient dynamics in real analytic Morse systems

TL;DR

This paper studies a sequential gradient process in real analytic Morse systems, proving convergence to a unique minimum under certain conditions, with implications for robotics navigation and trajectory length bounds.

Contribution

It introduces a sequential hybrid gradient process in Morse systems and proves its convergence to the system's unique minimum under specified conditions.

Findings

The process converges to a critical point if each coordinate is chosen infinitely often.

The critical point reached is the unique minimum for a dense subset of navigation functions.

An upper bound for the total length of trajectories near critical points is provided.

Abstract

Let $\Omega$ in $R^M$ be a compact connected $M$-dimensional real analytic domain with boundary and $\phi$ be a primal navigation function; i.e. a real analytic Morse function on $\Omega$ with a unique minimum and with minus gradient vector field $G$ of $\phi$ on the boundary of $\Omega$ pointed inwards along each coordinate. Related to a robotics problem, we define a sequential hybrid process on $\Omega$ for $G$ starting from any initial point $q_0$ in the interior of $\Omega$ as follows: at each step we restrict ourselves to an affine subspace where a collection of coordinates are fixed and allow the other coordinates change along an integral curve of the projection of $G$ onto the subspace. We prove that provided each coordinate appears infinitely many times in the coordinate choices during the process, the process converges to a critical point of $\phi$. That critical point is the unique minimum for a dense subset in primal navigation functions. We also present an upper bound for the total length of the trajectories close to a critical point.