# On the relation between discounted and average optimal control problems

L. Grüne:
**On the relation between discounted and average optimal control problems**

*Journal of Differential Equations*
**148**,
65 - 99,
1998

DOI:
10.1006/jdeq.1998.3451

Keywords: deterministic nonlinear optimal control; discounted optimal control; average optimal control; Morse decomposition; viable sets; chain control
sets; control sets; viability

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## Abstract:

We investigate the relation between discounted and average 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-Tauberian type theorem, controllability properties of the system and the Morse decomposition of the corresponding control flow.