  In this paper, we analyse Lipschitz continuous dependence of the solution to
Hamilton-Jacobi-Bellman equations on a functional parameter. This sensitivity
analysis not only has the interest on its own, but also is important for the
mean field games methodology, namely for solving a coupled system of
backward-forward equations. We show that the unique solution to a
Hamilton-Jacobi-Bellman equation and its spacial gradient are Lipschitz
continuous uniformly with respect to the functional parameter. In particular,
we provide verifiable criteria for the so-called feedback regularity condition.
Finally as an application, we show how the sensitive results are used to solved
the coupled system of backward-forward equations.
