Force‐annihilation conditions for variable‐coefficient lanchester‐type equations of modern warfare

This paper develops a mathematical theory for predicting force annihilation from initial conditions without explicitly computing force‐level trajectories for deterministic Lanchester‐type “square‐law” áttrition equations for combat between two homogeneous forces with temporal variations in fire effectivenesses (as expressed by the Lanchester attrition‐rate coefficients). It introduces a canonical auxiliary parity‐condition problem for the determination of a single parity‐condition parameter (“the enemy force equivalent of a friendly force of unit strength”) and new exponential‐like general Lanchester functions. Prediction of force annihilation within a fixed finite time would involve the use of tabulations of the quotient of two Lanchester functions. These force‐annihilation results provide further information on the mathematical properties of hyperbolic‐like general Lanchester functions: in particular, the parity‐condition parameter is related to the range of the quotient of two such hyperbolic‐like general Lanchester functions. Different parity‐condition parameter results and different new exponential‐like general Lanchester functions arise from different mathematical forms for the attrition‐rate coefficients. This theory is applied to general power attrition‐rate coefficients: exact force‐annihilation results are obtained when the so‐called offset parameter is equal to zero; while upper and lower bounds for the parity‐condition parameter are obtained when the offset parameter is positive.