The Minkowski-Bellman Equation

This manuscript studies the Minkowski-Bellman equation, which is the Bellman equation arising from finite or infinite horizon optimal control of unconstrained linear discrete time systems with stage and terminal cost functions specified as Minkowski functions of proper C-sets. In regards to the finite horizon optimal control, it is established that, under natural conditions, the Minkowski-Bellman equation and its iteration are well posed. The characterization of the value functions and optimizer maps is derived. In regards to the infinite horizon optimal control, it is demonstrated that, under the same natural conditions, the fixed point of the Minkowski-Bellman equation is unique, in terms of the value function, over the space of Minkowski functions of proper C-sets. The characterization of the fixed point value function and optimizer map is reported.