  We consider weakly coupled LQ optimal control problems and derive estimates
on the sensitivity of the optimal value function in dependence of the coupling
strength. In order to improve these sensitivity estimates a "coupling adapted"
norm is proposed. Our main result is that if a weak coupling suffices to
destabilize the closed loop system with the optimal feedback of the uncoupled
system then the value function might change drastically with the coupling. As a
consequence, it is not reasonable to expect that a weakly coupled system
possesses a weakly coupled optimal value function. Also, for a known result on
the connection of the separation operator and the stability radius a new and
simpler proof is given.
