The Value Function of a Slow Growth Control Problem with State Constraints Dedicated to Aldo Bressan on His Seventieth Birthday

The formal extension of the conventional theory of dynamic programming to control problems with slow growth and a state constraint x 2 encounters two major drawbacks: on one hand the formal Hamiltonian may happen to be discontinuous; on the other hand, just as in the case of bounded controls, the imposition of a state constraint possibly gives rise to a discontinuous value function. On the contrary we provide conditions on the vectogram (at the points of @) which, whenever the L 1 norms of the controls are bounded, guarantee the continuity of the value function. Subsequently , we are able to establish a dynamic programming diierential equation involving a continuous Hamiltonian and enjoying uniqueness properties.