Prediction of Zero Points of Solutions to Lanchester-Type Differential Combat Equations for Modern Warfare

Abstract

This paper considers an important mathematical problem of modern operations research, determination of conditions under which one homogeneous military force can be annihilated by another in combat modeled by so-called Lanchester-type equations for modern warfare, i.e. a deterministic system of two first-order linear differential equations. Equivalently, it investigates the classical-analysis problem of determining the real zeros of nonoscillatory (in the strict sense) solutions to the general second-order differential equation, but results are given in a form appropriate for analysis of the Lanchester-type differential combat model. It is shown that the occurrence of the single real zero of such a solution, which corresponds to the annihilation of one force by the other, may be predicted by a deterministic inequality involving the initial conditions and a so-called parity-condition parameter, which is a functional depending on only the model’s time-dependent coefficients. This parity-condition parameter is shown to be equal to the limiting value of the ratio of a certain pair of canonical hyperbolic-like general Lanchester functions (GLF), which may be used to represent the combatants’ force levels as functions of time, denoted as $x( t )$ and $( y )t$. The asymptotic behavior of solutions as $t \to + \infty $when $( x )t$ and $( y )t > 0$ for all finite $t \geqq 0$ is investigated, and its dependence on the integrability properties of the coefficients is determined.