Turnpike properties of approximate solutions of nonconcave discrete-time optimal control problems (Q2922469)

Let \((X, \rho)\) be a compact metric space and \(\Omega\) be a nonempty closed subset of \(X\times X.\) A sequence \(\{x_t \}_{t=0}^\infty \subset X\) is called a program if \((x_t, x_{t+1})\in \Omega\) for all \(t \in [0, T-1]\), where \(T\) is a natural number. Consider the problem NEWLINE\[NEWLINE(\mathrm{P3})\quad \max \{ \Sigma_{t=0}^{T-1} v(x_t, x_{t +1}): \{ (x_t, x_{t+1}) \}_{t=0}^{T-1} \subset \Omega \},NEWLINE\]NEWLINE where \(v: \Omega\rightarrow\mathbf{R}\) is a bounded upper semi-continuous objective function. Then, a program \(\{x_t \}_{t=0}^\infty \) is \(v\)-good if the sequence NEWLINE\[NEWLINE\{\Sigma_{t=0}^{T-1} v(x_t, x_{t +1})- Tv(\bar{x}, \bar{x}) \}_{T=1}^\inftyNEWLINE\]NEWLINE is bounded, and it has the asymptotic turnpike property if any \(v\)-good program converges to the turnpike \(\bar{x}\).NEWLINENEWLINEIn this paper, the author shows that for the problem (P3) the turnpike property also follows from the asymptotic turnpike property. Furthermore, it is shown that the asymptotic turnpike property holds for most objective functions in the sense of Baire category.