Commutativity of Control Vector Fields and inf-commutativity

This article analyzes infinitesimal characterizations of commutativity of locally Lipschitz continuous vector fields that depend on a time-varying control parameter. Such a family of vector fields is said to commute if for every choice of control functions the flows of the corresponding time-varying vector fields commute. The article presents and proves a characterization of such commutativity in terms of the vanishing of a set-valued bracket that extends the classical Lie brackets to this nonsmooth case. A second characterization is provided in terms of an invariance condition for lifts of multi-time paths. The results are applied and discussed in terms of commutativity of families of optimal control problems that involve concatenations of different dynamical constraints and different cost functionals