Hadamard-Lévy theorems for maps taking values in a finite dimensional space
Yacine Chitour, Zhengping Ji, Emmanuel Trélat
TL;DR
The paper develops Hadamard–Lévy type global surjectivity theorems for maps $F:X\to\mathbb{R}^n$ from a Hilbert space $X$ to a finite-dimensional space, using second-order conditions encoded by the Gramian $G(u)=dF|_u dF|_u^*$ and a growth bound controlled by a nondecreasing function $\xi$. It introduces a homotopy continuation framework with a path-lifting equation $\partial u/\partial s = (dF|_u)^* G(u)^{-1} \dot{\gamma}(s)$, analyzes the spectrum $\lambda_i(u)$ and the least eigenvalue $\lambda_1(u)$ to derive global existence, and proves two main results: (i) a global surjectivity theorem under a second-order lower bound and integral divergence condition $\int_R^{\infty} ds/\xi(s)=\infty$, and (ii) a path-lifting result that holds even when approaching the singular set $\tilde S$. The results are then applied to nonlinear motion planning by examining the endpoint map and establishing well-posed continuation methods at critical values, with implications for robust trajectory design. The work offers a framework to handle singularities in infinite-dimensional domains with finite-dimensional targets, linking second-order analysis to global coverage guarantees in the context of control and optimization.
Abstract
We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional Euclidean space, when its second-order differential has uniform upper and lower bounds, it has a global path-lifting property in the presence of singularities. This is then applied to the nonlinear motion planning problem, establishing in some cases the well-posedness of the continuation method despite critical values of the endpoint maps.