Suppose you are the owner of a mine in Copper Harbor, MI. The known reserves of the mine are 60 (It would probably be measured in MTons, but let's leave off units here). Suppose the price of copper ore is constant over the time period we are studying. Your quarterly profits are directly proportional to the amount of ore you mine (minus the costs of mining it). Your control variable is , (probably measued in tons/day, but again, let's leave off the units for this problem). In other words, your managers have the ability to change how much ore is mined each day by increasing overtime, hiring new workers, upgrading equipment, etc. However, each of these incremental output increases will increase your costs quadratically: 1/2 u^2 (where the 1/2 is for notational convenience). You, as the owner, would like to maximize the profits that you reap from the mine. Another way of saying this is that you would like to minimize the cost of mining the ore while still digging it out (we could have just left the mine alone if we only prefer to minimize our costs). The total cost over the time interval where we mine the ore (which we will normalize between 0 and 1 here): =1/2 u^2 dt Assume that you empty the mine by the end of the time period. Use Pontryagin's Maximum Principle to find the optimal contol You'll see this optimal control referred to as * in the literature. Check your work using fmincon in MATLAB (or use your favorite convex optimization software)