Local turnpike analysis using local dissipativity for discrete time discounted optimal control

TL;DR

This paper introduces a local notion of discounted strict dissipativity and turnpike property for discrete-time optimal control, analyzing how trajectories converge to local stable equilibria depending on the discount factor.

Contribution

It develops a local analysis framework for discounted optimal control, linking dissipativity, turnpike behavior, and stability in the presence of multiple equilibria.

Findings

Conditions on the discount factor ensure convergence to local equilibria.

Local dissipativity criteria guarantee stability of optimal trajectories.

Analysis applies to economic models with discounted costs.

Abstract

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.