Path differentiability of ODE flows
This provides a theoretical foundation for applying first-order optimization methods to a broad class of nonsmooth problems with ODE constraints, which is incremental as it extends smooth adjoint methods to nonsmooth cases.
The paper tackles the problem of optimizing nonsmooth functions under ODE constraints by proving that flows of ODEs driven by path differentiable vector fields inherit path differentiability, enabling a nonsmooth adjoint method. This result supports the convergence of small step first-order methods for such optimization problems.
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.