Summary: Starting with Hotelling (1931), the stock of non-renewable resources have been treated as fixed. Along the line of Pindyck (1978) and Greiner and Semmler (in press) we treat the stock of oil resources as time varying, depending on new discoveries. The resource is finite and only a part of the resource is known while the rest has not yet been discovered. The discovery leads to a rise of known oil resource which can then be optimally exploited. The optimal control model has two state variables, the known stock of the resource and the cumulated past extraction. The control variable is the optimal extraction rate. The optimal control model assumes a monopolistic resource owner who maximizes intertemporal profits from exploiting the resource where the price of the resource depends on the extraction rate, the known stock of the resource, and the cumulated past extraction. The model is solved for a finite time horizon using NUDOCCCS, a numerical solution method to solve finite horizon optimal control problems. Various parameter constellations are explored. For certain parameter constellations the price path becomes U-Shaped as some empirical research, see Greiner and Semmler (in press), have found to hold for actual price data. This holds if the stock of the initially known resource is small.