Force-Annihilation Conditions for Variable-Coefficient Lanchester-Type Equations of Modern Warfare, I. Mathematical Theory

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 attrition equations for combat between two homogeneous forces with temporal variations in fire effectiveness as expressed by the Lanchester attrition-rate coefficients. It introduces a cannonical 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.