Path differentiability of ODE flows

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