An optimal control problem in determining the optimal reinforcement schedules for the Lanchester equations

Abstract

An optimal control problem for the Lanchester equations by utilizing the iterative regularization method, i.e. a nonparametric conjugate gradient method (NP-CGM), with adjoint equations is illustrated to determine the optimal reinforcement schedules based on the desired (or required) force strengths at some specified time. The numerical simulations are performed to test the validity of the present algorithm by using three different types of force strength requirements. Results show that the method is powerful enough to yield desired strengths at all points during the battle. Moreover, the advantages of applying the NP-CGM in the optimal control calculations lie in that (i) the initial guesses of the reinforcement schedules can be chosen arbitrarily and (ii) the optimal time-dependent reinforcement schedules can be determined within a very short computer time. Scope and purpose The Lanchester square law has been used as a warfare model for a long time. Many research papers in determining the attrition coefficients can be found in the literature. For instance, Robert L. Helmbold has discussed the inverse problem for the Lanchester equation in estimating the constant attrition coefficients. More recently, Hsi-Mei Chen used a nonparametric conjugate gradient method (NP-CGM) in predicting the time-dependent attrition coefficients and obtained good estimation. However, the discussion regarding the determination of optimal reinforcement schedules for the Lanchester square law has never been found, to the author's best knowledge, in the literature. For this reason, an optimal control algorithm for the Lanchester square law based on the NP-CGM is examined in the present study. The optimal reinforcement schedules are to be defined such that the desired force strengths at any specified time in the battle can be satisfied. Moreover, constraint for the reinforcement schedules is also required. For this reason, the objective function should be defined as the combination of the square of difference between desired and computed forces and the square of reinforcement schedules. Finally, three numerical experiments will be illustrated to show the validity of the present optimal control algorithm in estimating the optimal reinforcement schedules for the Lanchester square law.