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 behaviour. 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.
Highlights
In recent years, dissipativity as introduced into systems theory by Willems [20,21] has turned out to be a highly useful concept in order to understand the qualitative behaviour of optimally controlled systems
In mathematical economy, where discounted optimal control problems are an important modelling tool, this is a well known fact at least since the pioneering work of Skiba [18] and Dechert and Nishimura [2], and since it was observed in many other papers, see, e.g., [13] and the references therein. It is the goal of this paper to show that a local version of the strict dissipativity property for discounted optimal control problems can be used for obtaining local convergence results to optimal equilibria
The number β ∈ (0, 1) is called the discount factor. For such problems it was shown in [5] that if the optimal control problem is strictly dissipative at an optimal equilibrium xβ, for sufficiently large β ∈ (0, 1) all optimal trajectories converge to a neighbourhood of xβ
Summary
Dissipativity as introduced into systems theory by Willems [20,21] has turned out to be a highly useful concept in order to understand the qualitative behaviour of optimally controlled systems. In mathematical economy, where discounted optimal control problems are an important modelling tool, this is a well known fact at least since the pioneering work of Skiba [18] and Dechert and Nishimura [2], and since it was observed in many other papers, see, e.g., [13] and the references therein It is the goal of this paper to show that a local version of the strict dissipativity property for discounted optimal control problems can be used for obtaining local convergence results to optimal equilibria.
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