Dynamical approach of renewal equation for optimal control model

This article establishes a process for the solution of an optimal control model. The model has been prepared over the situation of orders at different phases and proceeds to a penalty cost due to surplus demand. Independent of time interval and the rebate over cost has been discussed. The model has been studied to solve some real life situations. The situation has been observed by using the lemmas to verify the stability of the solutions. This article has strategy to determine the optimal condition, reduce the predicted total cost. The decision of a single and bounded solution over the negative variables is used in various renewal equations. The result for the renewal equation of the model has been derived by using fixed point theorem.