**MSC:**- 90C39 Dynamic programming, See also {49L20}
- 49M37 Nonlinear programming, See also {90C30}

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 we derive from this

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.

submitted