  We prove comparison principle for viscosity solutions of a
Hamilton-Jacobi-Bellman equation in a strong coupling regime considering a
stationary and a time-dependent version of the equation. We consider a
Hamiltonian that has a representation as the supremum of a difference of two
functions: an internal Hamiltonian depending on a control variable and a
function interpreted as a cost of applying the controls. Our major innovation
lies in the use of a cost function that can be discontinuous, unbounded and
depending on momenta, enabling us to address previously unexplored scenarios
such as cases arising from the theory of large deviations and homogenisation.
For completeness, we also state the existence of viscosity solutions and we
verify the assumptions for an example arising from biochemistry.
