Global straightening of solution manifolds for state-dependent delay equations

Abstract

Let $U$ be an arbitrary open subset of $C^1([-h,0],\mathbb R^n)$, and let $f:U\to\mathbb R^n$ be continuously differentiable. Suppose that each derivative $Df(ϕ)$, $ϕ\in U$, admits a continuous linear extension $D_ef(ϕ):C([-h,0],\mathbb R^n)\to\mathbb R^n$, such that $D_ef(ϕ)χ$ depends jointly continuously on $ϕ\in U$ and $χ\in C([-h,0],\mathbb R^n)$. We construct a $C^1$ diffeomorphism of $U$ onto an open subset of $C^1([-h,0],\mathbb R^n)$ 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 $τ\in(0,h)$, 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.