# Strong stationarity for optimal control problems with non-smooth integral equation constraints: Application to continuous DNNs

arXiv: https://arxiv.org/abs/2302.05318 · HTML: https://mlsanity.com/paper/2302.05318/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2023.

## Built on by (most cited first, 1 of 1)

- [Layerwise goal-oriented adaptivity for neural ODEs: an optimal control perspective](https://mlsanity.com/paper/2601.07397/) (2026)

## Abstract (arXiv)

Motivated by the residual type neural networks (ResNet), this paper studies optimal control problems constrained by a non-smooth integral equation. Such non-smooth equations, for instance, arise in the continuous representation of fractional deep neural networks (DNNs). Here the underlying non-differentiable function is the ReLU or max function. The control enters in a nonlinear and multiplicative manner and we additionally impose control constraints. Because of the presence of the non-differentiable mapping, the application of standard adjoint calculus is excluded. We derive strong stationary conditions by relying on the limited differentiability properties of the non-smooth map. While traditional approaches smoothen the non-differentiable function, no such smoothness is retained in our final strong stationarity system. Thus, this work also closes a gap which currently exists in continuous neural networks with ReLU type activation function.