Path differentiability of ODE flows

TL;DR

This paper proves that flows of ODEs driven by path differentiable vector fields are themselves path differentiable, enabling nonsmooth adjoint methods for optimization problems with ODE constraints.

Contribution

It establishes the inheritance of path differentiability in ODE flows and introduces a nonsmooth adjoint method for optimization under ODE constraints.

Findings

Flows inherit path differentiability from vector fields.

A conservative Jacobian for the flow is constructed.

Convergence of first order methods using the nonsmooth adjoint is demonstrated.

Abstract

We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parametrized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.