A Cauchy-Lipschitz setting for control problems in complete CAT(0) spaces

We provide a framework for optimal control problems in CAT(0) spaces, in which one can recover the existence, uniqueness and estimates of the classical theory. Our proposition allows to extend the Aumann-Filippov theorem with a suitable extension of convex-valued dynamics. This formulation is shown to be compatible with the recent theory of viscosity solutions of Hamilton-Jacobi equations in CAT(0) spaces.