  Recent results in the literature have provided connections between the
so-called turnpike property, near optimality of closed-loop solutions, and
strict dissipativity. Motivated by applications in economics, optimal control
problems with discounted stage cost are of great interest. In contrast to
non-discounted optimal control problems, it is more likely that several
asymptotically stable optimal equilibria coexist. Due to the discounting and
transition cost from a local to the global equilibrium, it may be more
favourable staying in a local equilibrium than moving to the global - cheaper -
equilibrium. In the literature, strict dissipativity was shown to provide
criteria for global asymptotic stability of optimal equilibria and turnpike
behavior. In this paper, we propose a local notion of discounted strict
dissipativity and a local turnpike property, both depending on the discount
factor. Using these concepts, we investigate the local behaviour of
(near-)optimal trajectories and develop conditions on the discount factor to
ensure convergence to a local asymptotically stable optimal equilibrium.
