Optimal control problems over an infinite Horizon

Optimal control problems over an infinite number of decision stages are considered with emphasis on the deterministic scenario. Both the open-loop and the closed-loop formulations are given and conditions for the existence of a stationary optimal control law are provided. Unless strong assumptions are made on the dynamic system and on the random variables (if present), the design of the optimal infinite-horizon controllers is an almost impossible task. Then, the well-known "receding-horizon" (RH) approximation is considered and the optimal control problem is restated accordingly. In the second part of the chapter, we consider the fundamental issue of closed-loop stability that arises owing to the infinite number of decision stages. More specifically, we address the stability properties of the closed-loop deterministic system under the action of approximate RH control laws obtained by the "extended Ritz method" and implemented through fixed-structure parametrized functions containing vectors of "free" parameters. Conditions are established on the maximum allowable approximation errors so as to ensure the boundedness of the state trajectories.