Hamilton--Jacobi equations for controlled gradient flows: cylindrical test functions

This work is the second part of a program initiated in arXiv:2111.13258 aiming at the development of an intrinsic geometric well-posedness theory for Hamilton-Jacobi equations related to controlled gradient flow problems in metric spaces. Our main contribution is that of showing that the comparison principle proven therein implies a comparison principle for viscosity solutions relative to smoother Hamiltonians, acting on test functions that are mere cylindrical functions of the underling squared metric distance and whose rigorous definition is achieved from the Evolutional Variational Inequality formulation of gradient flows (EVI). In particular, the new Hamiltonians no longer require to work with test functions containing Tataru's distance. This substantial simplification paves the way for the development of a comprehensive existence theory.