The aim of this paper is to formulate and solve an optimal control problem under finite time horizon. Here, the produced items are deteriorating and meet the time-dependent demand for a finite time period having no stack at both ends. The rate of deterioration of goods is looked upon as an exogenous variable, which is not subject to control. In the real system, firms reduce the rate of deterioration of items by means of effective capital investment through preservation technology. The unit production cost is a function of production rate and also depends on raw material cost. The objective function which consists of revenue from the sale of deteriorated items at low price, production cost, holding cost and preservation technology cost is formulated as a Fixed-Final Time and Fixed State System optimal control problem with finite time horizon. Here, production rate is unknown and considered as a control variables and stock level is taken as a state variable. The problem is formulated to optimize the production rate so that total cost is minimum. The models are formulated as optimal control problems and solved by using conventional variational principle. Proposed problems are numerically illustrated and some managerial decisions are derived. It is observed that initially, the stock level is zero and then increases with time, but while demand meets the production rate, after that the stock level decreases and becomes zero at the end of the cycle. This investigation shows that a higher or lower preservation technology investment does not lead to a minimum level of total cost; there exists an optimal value for the extreme benefit of the model.