A unified optimal control approach for production-depletion trade-off and flight management systems

Abstract

This paper presents a unified control methodology that can be used to formulate, solve, interpret geometrically, and compare the solutions of optimal flight management systems and optimal production planning trade-off problems. The paper focuses on the problem of production of one good at the cost of the depletion of one resource. By production planning it is meant more than just an industrial plant or process. The concept can be extended for example to the management of natural resources when natural forest must be consumed at the expense of agricultural production. The main contribution of this paper is to prove that under strict convexity assumptions such problems have a common and unique feedback solution that depeds on a trade-off parameter called cost index. Furthermore, the solution can be interpreted geometrically using the concept of convex conjugate function and Legendre transformation. The feedback solution yields an analytic expression for the case of zero cost index. When the cost index is positive but small a Taylor series approximation is proposed as a suboptimal analytical solution. It is shown that as the cost index increases the optimal input also increases. This is due to the fact that the tangency point of the supporting line of the rate of depletion that passes through the origin shifts in the positive direction of the input axis. Several examples show the successful application of the theory yielding analytical feedback solutions for flight management systems that can reproduce the maximum range formulas when the cost index is zero.