  This paper deals with the regularity of solutions of the Hamilton-Jacobi
Inequality which arises in H-infinity control. It shows by explicit
counterexamples that there are gaps between existence of continuous and locally
Lipschitz (positive definite and proper) solutions, and between Lipschitz and
continuously differentiable ones. On the other hand, it is shown that it is
always possible to smooth-out solutions, provided that an infinitesimal
increase in gain is allowed.
