The Reduction to Finite Rewards

This chapter proposes a unifying approach to be used when one is dealing with infinite horizon optimization problems with unbounded rewards. This approach which is called the reduction to finite rewards encompasses the turnpike property that was the main tool used in Chapter 4. It is also more general as it permits the consideration of problems with slowly decreasing discount rates, problem of tracking periodic trajectories and also problems which are non convex w.r.t. the state variable x. An important aspect of the method developed here is that it exploits a connection which exists between continuous and discrete-time control systems. This link is particularly attractive when one considers that many early results on the turnpike property have been obtained for discrete time economic growth models (see e.g. Gale [81]). Also the method establishes a link with the Dynamic Programming approach.