Model Predictive Control: Why the horizon size, $N$, must be equal or larger than 2?

Question

If you read "Nonlinear Model Predictive Control" by L. Grune and J. Pannek (and anywhere else), everyone says that the prediction horizon size $N$ must be larger or equal to $2$,$ N\geq2$.

Why?

Answer 1

The $N\geq 2$ condition in the referenced book is due to a slightly non-standard notation in the definition of the problem, using the objective $\sum_{k=0}^{N-1} \ell(x_k,u_k)$, which still makes sense for $N=1$ but typically would lead to the trivial solution $u = 0$ as only the current input and no future state is penalized in the objective. It would still be a well-defined problem, but perhaps a bit silly.

A more common form (in papers with a theoretical bias) is $\sum_{k=0}^{N-1} \ell(x_k,u_k) + \Psi(x_{N})$ for some terminal penalty $\Psi$, and then $N=1$ would make sense as the one-step ahead predictive control setup. Another form which would make $N=1$ reasonable is $\sum_{k=0}^{N-1} \ell(x_{k+1},u_k)$, as it typically makes no sense to add the current state to the objective.