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Directional differentiability for solution operators of sweeping processes with convex polyhedral admissible sets

Martin Brokate, Constantin Christof

TL;DR

Directionally differentiability of solution operators for sweeping processes with polyhedral admissible sets is rigorously analyzed in the CBV framework. The main result shows that the vector play/stop operators are pointwise Hadamard directionally differentiable if and only if the admissible set $Z$ is non-obtuse; in this case the derivative is uniquely characterized by projection identities and a coupled variational-inequality system, with explicit jump relations $\delta(t)=\pi_{\mathcal{Z}(y(t))}(\delta(t-))$ and $\delta(t+)=\pi_{V^+(t)}(\delta(t))$. If $Z$ is obtuse, no pointwise directional derivative exists even when the input space is restricted to Lipschitz functions, highlighting a sharp geometric dichotomy. The analysis introduces temporal polyhedricity and leverages the Kurzweil-Stieltjes integral to derive a precise derivative characterization, enabling first-order optimality conditions (Bouligand stationarity) for control problems governed by sweeping processes. Collectively, the work clarifies how polyhedral geometry governs sensitivity and opens pathways to derivative-based algorithms for rate-independent variational systems.

Abstract

We study directional differentiability properties of solution operators of rate-independent evolution variational inequalities with full-dimensional convex polyhedral admissible sets. It is shown that, if the space of continuous functions of bounded variation is used as the domain of definition, then the most prototypical examples of such solution operators - the vector play and stop - are Hadamard directionally differentiable in a pointwise manner if and only if the admissible set is non-obtuse. We further prove that, in those cases where they exist, the directional derivatives of the vector play and stop are uniquely characterized by a system of projection identities and variational inequalities and that directional differentiability cannot be expected in the obtuse case even if the solution operator is restricted to the space of Lipschitz continuous functions. Our results can be used, for example, to formulate Bouligand stationarity conditions for optimal control problems involving sweeping processes.

Directional differentiability for solution operators of sweeping processes with convex polyhedral admissible sets

TL;DR

Directionally differentiability of solution operators for sweeping processes with polyhedral admissible sets is rigorously analyzed in the CBV framework. The main result shows that the vector play/stop operators are pointwise Hadamard directionally differentiable if and only if the admissible set is non-obtuse; in this case the derivative is uniquely characterized by projection identities and a coupled variational-inequality system, with explicit jump relations and . If is obtuse, no pointwise directional derivative exists even when the input space is restricted to Lipschitz functions, highlighting a sharp geometric dichotomy. The analysis introduces temporal polyhedricity and leverages the Kurzweil-Stieltjes integral to derive a precise derivative characterization, enabling first-order optimality conditions (Bouligand stationarity) for control problems governed by sweeping processes. Collectively, the work clarifies how polyhedral geometry governs sensitivity and opens pathways to derivative-based algorithms for rate-independent variational systems.

Abstract

We study directional differentiability properties of solution operators of rate-independent evolution variational inequalities with full-dimensional convex polyhedral admissible sets. It is shown that, if the space of continuous functions of bounded variation is used as the domain of definition, then the most prototypical examples of such solution operators - the vector play and stop - are Hadamard directionally differentiable in a pointwise manner if and only if the admissible set is non-obtuse. We further prove that, in those cases where they exist, the directional derivatives of the vector play and stop are uniquely characterized by a system of projection identities and variational inequalities and that directional differentiability cannot be expected in the obtuse case even if the solution operator is restricted to the space of Lipschitz continuous functions. Our results can be used, for example, to formulate Bouligand stationarity conditions for optimal control problems involving sweeping processes.

Paper Structure

This paper contains 20 sections, 61 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Background and relations to prior work
  4. Structure of the remainder of the paper
  5. Preliminaries and preparatory results
  6. Basic notation
  7. Preliminaries on projections
  8. Preliminaries on convex polyhedra
  9. Preliminaries on the Kurzweil-Stieltjes integral
  10. Preliminaries on the vector play and stop
  11. Proof of \ref{['main_sum:item:I']}: directional differentiability in the non-obtuse case
  12. Standing assumptions and basic idea of the proof
  13. Criticality
  14. Identification of the jumps from the left
  15. Identification of the jumps from the right
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $T>0$ be given and let $Z \subset \mathbb{R}^d$, $d \in \mathbb{N}$, be a full-dimensional convex polyhedron. Then the variational inequality eq:EVI_again possesses a unique solution $y \in CBV([0,T]; \mathbb{R}^d)$ for all $(u, y_0) \in CBV([0,T]; \mathbb{R}^d) \times Z$. The associated solutio Further, the following statements hold regarding the directional differentiability properties of $\

Theorems & Definitions (135)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1: projection and distance function
  • Proposition 2.2: properties of $\pi_Z$ and $\operatorname{dist}_Z$
  • proof
  • Definition 2.3: convex polyhedron
  • Remark 2.4: conventions for the case $Z = \mathbb{R}^d$
  • Definition 2.5: set of active/inactive indices, regular description
  • Proposition 2.6: standard descriptions
  • proof
  • ...and 125 more