Target selection in Lanchester combat: Linear‐law attrition process

Abstract

We develop the solution to a simple problem of target selection in Lanchester combat against two enemy force types each of which undergoes a "linear-law" attrition process.In addition to the Pontryagin maximum principle, the theory of singular extremals is required to solve this problem. Our major contribution is to show how to synthesize the optimal target selection policies from the basic optimality conditions. This solution Synthesis methodology is applicable to more general dynamic (tactical) allocation problems. For constant attrition-rate coefficients we show that whether or not changes can occur in target priorities depends solely on how survivors are valued and is independent of the type of attrition process.

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

“…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.…”

“…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.…”

“…Computing the h t two time derivatives of the switching function we see that on a singular subarc (see [32] for a further discussion) we have [8,211 with the singular control given.by (33) .4s=&l(cl+c2~.…”

Lanchester‐type models of warfare and optimal control

“…Giving consideration to (31), it is seen that for T E [ T~, 7 2 1 one needs only to consider the following equations from the adjoint system (10) 9 = -b k P n + t with P k (7 = 71) = -v k , (32) where (33) Equation (33) follows from the fact that by (8) and (9) at 7'71 we have which may be combined with (27), (28), and (29) to yield the desired result. Equation (34) is readily deduced by observing that according to (19) a "square law" relates the dual variables Pn(7) and P n + l (~) for 0 d 7 G T~ whence follows (34) by use of (27), (29), and (32). It should be noted that all the dual variables may be expressed in terms of P k ( 7 ) (let n = k in (12)) (37) Again, the Equations (32) are routinely solved to yield for T E [ T~, 7 2 1 Subsequent arguments are now similar to those given for the first switch in tactics.…”

“…The analysis details upon which the summary given below is based are given in a companion paper [29] (see also pp. 91-105 of [21]) due to their rather lengthy nature.…”