Average cost-per-stage

An alternative to discounted cost is to use the average cost-per-stage model, which keeps the cumulative cost finite by dividing out the total number of stages:

$$L({\tilde{x}},{\tilde{u}},{\tilde{\theta}}) = \lim_{K \to \infty} \left( {1 \over K} \sum_{k=0}^{K-1} l(x_k,u_k,\theta_k) \right).$$ (10.68)

Using the maximum per-stage cost bound $c$, it is clear that (10.69) grows no larger than $c$, even as $K \to \infty$. This model is sometimes preferable because the cost does not depend on an arbitrary parameter, $\alpha$.