Abstract

We investigate the relation between discounted and average deterministic optimal control problems for nonlinear control systems. In particular we are interested in the corresponding optimal value functions. Using the concepts of Viability, Chain Controllability, and Controllability, a global convergence result for vanishing discount rate is obtained. Basic ingredients for the analysis are an Abelian type theorem, controllability properties of the system, and the Morse decomposition of the corresponding control flow.

Highlights

  • In this paper we investigate the relation between average and discounted deterministic nonlinear optimal control problems for discount rate tending to zero

  • In the deterministic setup, which we will consider in this paper, Colonius 8] in 1989 published a convergence result for vanishing discount rate on invariant control sets, a similar result for arbitrary control sets has been obtained in 1993 by Wirth 27]

  • (i) A uniform lower bound for v0t related to chain control sets by Theorem 6.4 (ii) A uniform upper bound for v0t related to control sets by Theorem 7.4 (iii) A convergence result for trajectories staying inside control sets by the Corollaries 7.5 and 7.6

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Summary

Introduction

In this paper we investigate the relation between average and discounted deterministic nonlinear optimal control problems for discount rate tending to zero. In the deterministic setup, which we will consider in this paper, Colonius 8] in 1989 published a convergence result for vanishing discount rate on invariant control sets, a similar result for arbitrary control sets has been obtained in 1993 by Wirth 27] These results have in common that assumptions on the optimal trajectories are made which are di cult to check and in general not satis ed even for simple one-dimensional systems. This kind of approach was inspired by the analysis of the Lyapunov spectrum of bilinear control systems as carried out in 13]. In particular for the analysis of the complete asymptotic behaviour of a system a global convergence result is needed; the result of the present paper closes the gap in the convergence analysis in 18]

Problem statement
Discounted and averaged functionals
The total time of a chain is given by
Now x k0 2 N such
Md 1 obvious implying p 62 the assertion
Viable sets
Chain control sets
The total time of a chain is given
Control sets
The complete picture
Restriction of the state space
Full Text
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