# On the rate of convergence of infinite horizon discounted optimal value functions

L. Grüne, F . Wirth:
**On the rate of convergence of infinite horizon discounted optimal value functions**

*Nonlinear Analysis: Real World Applications*
**1**,
499 - 515,
2000

DOI:
10.1016/S0362-546X(99)00288-6

Keywords: rate of convergence; optimal value function; infinite horizon
Discounted optimal control problem

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## Abstract:

In this paper we investigate the rate of convergence of the optimal value function of an infinite horizon discounted optimal control problem as the discount rate tends to zero. Using the Integration Theorem for Laplace transformations we provide conditions on averaged functionals along suitable trajectories yielding at most quadratic pointwise convergence. Under appropriate controllability assumptions from this we derive criteria for at most linear uniform convergence on control sets. Applications of these results are given and an example is discussed in which both linear and slower rates of convergence occur.