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State-dependent Delay Differential Equations on $H^1$

Johanna Frohberg, Marcus Waurick

TL;DR

This paper develops a global, contraction-mapping-based solution theory for state-dependent delay equations on the Sobolev space $H^1$, addressing limitations of traditional spaces of continuous or $C^1$ functions that rely on solution manifolds. By employing exponentially weighted spaces and a pre-history mapping, the authors extend Lipschitz right-hand-sides and delay functionals to $H^1$, with a key Lipschitz estimate that enables a fixed-point argument. The framework yields existence and uniqueness of solutions for equations of the form $x'(t)=g(t,x(t),x(t+r(x_t)))$ under Lipschitz conditions, and extends to general functional differential equations with $G$ almost uniformly Lipschitz; projections in Hilbert spaces allow extensions from subsets to the whole space via Kirszbraun-type results. The approach is illustrated with applications from mathematical biology and a geometric positioning problem, demonstrating robustness to non-autonomous and multiple-delay settings. Overall, the work provides a versatile, manifold-free, Hilbert-space-based methodology that broadens the class of state-dependent delay equations amenable to rigorous analysis and practical modeling.

Abstract

Classically, solution theories for state-dependent delay equations are developed in spaces of continuous or continuously differentiable functions. The former can be technically challenging to apply in as much as suitably Lipschitz continuous extensions of mappings onto the space of continuous functions are required; whereas the latter approach leads to restrictions on the class of initial pre-histories. Here, we establish a solution theory for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on the Sobolev space $H^1$. The provided solution theory is independent of previous ones and is based on the contraction mapping principle on exponentially weighted spaces. In particular, initial pre-histories are not required to belong to solution manifolds and the generality of the approach permits the consideration of a large class of functional differential equations even for which the continuity of the right-hand side has constraints on the derivative.

State-dependent Delay Differential Equations on $H^1$

TL;DR

This paper develops a global, contraction-mapping-based solution theory for state-dependent delay equations on the Sobolev space , addressing limitations of traditional spaces of continuous or functions that rely on solution manifolds. By employing exponentially weighted spaces and a pre-history mapping, the authors extend Lipschitz right-hand-sides and delay functionals to , with a key Lipschitz estimate that enables a fixed-point argument. The framework yields existence and uniqueness of solutions for equations of the form under Lipschitz conditions, and extends to general functional differential equations with almost uniformly Lipschitz; projections in Hilbert spaces allow extensions from subsets to the whole space via Kirszbraun-type results. The approach is illustrated with applications from mathematical biology and a geometric positioning problem, demonstrating robustness to non-autonomous and multiple-delay settings. Overall, the work provides a versatile, manifold-free, Hilbert-space-based methodology that broadens the class of state-dependent delay equations amenable to rigorous analysis and practical modeling.

Abstract

Classically, solution theories for state-dependent delay equations are developed in spaces of continuous or continuously differentiable functions. The former can be technically challenging to apply in as much as suitably Lipschitz continuous extensions of mappings onto the space of continuous functions are required; whereas the latter approach leads to restrictions on the class of initial pre-histories. Here, we establish a solution theory for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on the Sobolev space . The provided solution theory is independent of previous ones and is based on the contraction mapping principle on exponentially weighted spaces. In particular, initial pre-histories are not required to belong to solution manifolds and the generality of the approach permits the consideration of a large class of functional differential equations even for which the continuity of the right-hand side has constraints on the derivative.
Paper Structure (10 sections, 21 theorems, 129 equations)

This paper contains 10 sections, 21 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Functional analytic prerequisites and the pre-history map
  3. A key estimate for the solution theory
  4. Solution theory for state-dependent delay equations
  5. Solution theory for functional differential equations
  6. Applications
  7. On a problem from mathematical biology
  8. A positioning problem
  9. Further comments on other approaches concerning functional differential equations
  10. Conclusion

Key Result

Theorem 1.1

Let $h,T>0$, $G\colon [0,T]\times H^1(-h,0;\mathbb{R}^n)\to\mathbb{R}^n$ continuous. Assume that for all $\beta>0$ there exists $L\geq 0$ such that for all $\phi,\psi \in H^1(-h,0;\mathbb{R}^n)$ with $\|\phi'\|_\infty,\|\psi'\|_\infty\leq \beta$ and $t\in [0,T]$ Then for all Lipschitz continuous $\phi\colon (-h,0)\to \mathbb{R}^n$ there exists $0<T_1\leq T$ and a unique solution $x\in H^1(-h,T_1;\

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 2.1: Sobolev-embedding theorem
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 3.1
  • ...and 37 more