Global straightening of solution manifolds for state-dependent delay equations
Abstract
Let U be an arbitrary open subset of C1([−h,0],ℝn), and let f:U→ℝn be continuously differentiable. Suppose that each derivative Df(ϕ), ϕ∈ U, admits a continuous linear extension Def(ϕ):C([−h,0],ℝn)→ℝn, such that Def(ϕ)χ depends jointly continuously on ϕ∈ U and χ∈ C([−h,0],ℝn). We construct a C1 diffeomorphism of U onto an open subset of C1([−h,0],ℝn) which maps the solution manifold of x'(t)=f(xt) onto an open subset of the closed subspace defined by ϕ'(0)=0. This resolves an open problem posed by Walther concerning regularization of the entire solution manifold. The diffeomorphism preserves the value at zero and agrees with the original history on [−h,−τ] for any prescribed τ∈(0,h), and is uniformly arbitrarily close to the identity in the supremum norm. It is connected to the identity by a jointly C1 family of diffeomorphisms. The same conclusions hold under the weaker assumption that the finite signed measures representing the derivatives of f have no atom at zero. We also establish a corresponding theorem for separable real Hilbert-valued histories.
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BibTeX
@misc{dipaulo2026global,
title = {{Global straightening of solution manifolds for state-dependent delay equations}},
author = {Lily DiPaulo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01482},
eprint = {2611.01482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01482},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global straightening of solution manifolds for state-dependent delay equations
AU - Lily DiPaulo
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01482
UR - https://arxiv.org/abs/2611.01482
AB - Let U be an arbitrary open subset of C^1([−h,0],ℝ^n), and let f:U→ℝ^n be continuously differentiable. Suppose that each derivative Df(ϕ), ϕ∈ U, admits a continuous linear extension D_ef(ϕ):C([−h,0],ℝ^n)→ℝ^n, such that D_ef(ϕ)χ depends jointly continuously on ϕ∈ U and χ∈ C([−h,0],ℝ^n). We construct a C^1 diffeomorphism of U onto an open subset of C^1([−h,0],ℝ^n) which maps the solution manifold of x'(t)=f(x_t) onto an open subset of the closed subspace defined by ϕ'(0)=0. This resolves an open problem posed by Walther concerning regularization of the entire solution manifold. The diffeomorphism preserves the value at zero and agrees with the original history on [−h,−τ] for any prescribed τ∈(0,h), and is uniformly arbitrarily close to the identity in the supremum norm. It is connected to the identity by a jointly C^1 family of diffeomorphisms. The same conclusions hold under the weaker assumption that the finite signed measures representing the derivatives of f have no atom at zero. We also establish a corresponding theorem for separable real Hilbert-valued histories.
ER -