Lanchester‐type models of warfare and optimal control

Abstract

The optimization of the dynamics of combat (optimal distribution of fire over enemy target types) is studied through a sequence of idealized models by use of the mathematical theory of optimal controL The models are for combat over a period of time described by Lanchester-type equations with a choice of tactics available to one side and subject to change with time. The structure of optimal fire distribution policies is discussed with reference to the influence of combatant objectives, termination conditions of the conflict, type of attrition process, and variable attrition-rate coefficients. Implications for intelligence, command and control systems, and human decision making are pointed out. The use of such optimal control models for guiding extensions to differential games is discussed.

An examination of the effects of the criterion functional on optimal fire‐support policies

“…The corresponding optimal fire-support policies for Problems 2, 3, and 4 were qualitatively the same and could involve a splitting of supportingJires between the two enemy troop concentrations. This property of the optimal fiedistribution policy is not present in the solution to Problem 1 and was anticipated by our earlier work on optima1 fire distribution against enemy target types which undergo attrition according to a "linear-law" process (see Section 3.1 above) [31,321. The criterion functional for this earlier work was the difference between the overall military worths of friendIy and enemy survivors.…”

“…The only systematic examinations of the irduences of the nature of the criterion function on the structure of optimal time-sequential firedistribution strategies known to the authors are those of Taylor [30][31][32][33]35,37,40,43]. In [31] and [40], however, the influences of the nature of the targettype attrition process on the structure of optimal firedistribution policies were examined.…”

“…In [31] and [40], however, the influences of the nature of the targettype attrition process on the structure of optimal firedistribution policies were examined. I n [30-331 and [40] a linear utility$ was assumed for the military worth of the number of each surviving weapon system type, and the criterion functional (payoff) was taken to be the net military worth of survivors (i.e.…”

“…the difference between the military worths of friendly and enemy forces). Taylor [30][31][32][33]40] has studied how the optimal fire-distribution policy depends on the assignment of these linear utilities. In other words, he examined the sensitivity of the optima& combat policy to parametric variations in the assigned linear utilities for survivors.…”

“…Then by (31) we have where 8+ denotes the "backwards" time derivative & = d~+ / d 7. Considering (31), we may write It follows that S+ (r)>0 and $* (r)=l, V T>O when S+ (0) 2 0 for R>1 (also when S+ (0)>0 for R=l).…”