Error bounds for the Lanchester equations with variable coefficients

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NPS-53Cs77031

NAVAL POSTGRADUATE SCHOOL

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ERROR BOUNDS FOR THE LANCHESTER EQUATIONS
WITH VARIABLE COEFFICIENTS

by
James G. Taylor and Craig Comstock
Final Report for Period
October - December 1976

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4. TITLE (and Subtitle)

Error Bounds for the Lanchester Equations
with Variable Coefficients

5. TYPE OF REPORT & PERIOD COVERED

Final Report

1 Oct - 31 Dec 76

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7. AUTHORf«J

James G. Taylor
Craig Corns tock

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Naval Postgraduate School
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Error Estimates
Lanchester Equation

20. ABSTRACT (Continue on reveree aide II neceeaary and Identity by block number)

Previous error bounds for the classical Liouvil le-Green solutions to second
order ordinary differential equaitons are sharpened. Applications are made
to the Lanchester model for combat between two homogeneous forces.

DD i j°nM73 1473 EDITION OF 1 NOV 68 IS OBSOLETE

S/N 0102-014- 6601

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(2)

We are interested in solving the coupled set of equations

(1)
& - MtU

For the problems we are interested in, these equations do not reduce to any
known special functions. These equations arise in the study of combat models
where x and y are opposing forces and a(t) and b(t) are the attrition
rate coefficients related to the effectiveness of weapons. Lanchester [1]
first looked at these types of models for aircraft battles, using constant
coefficients (in which case the solution is in terms of exponentials) and so
these types of equations are referred to as Lanchester equations. There are
also special cases of the coefficients for which the solutions to (1) are
expressible in terms of the generalized Airy functions [2].

We are interested in problems where the coefficients a(t) and b(t)
are positive and continuous. In that case we expect that we can find solutions
which are linear combinations of monotonic, exponential like functions. If
we have such a set, then the analysis of the forces necessary to guarantee
a win for one side, say x , is yery much simplified. So we are interested
in determining exponential like solutions for (1), and error bounds for the
solutions.

We convert (1) into a single second order differential equations for x(t)

dt (2)

x(0) ■ xQ , a_1(0) ^jf(O) = -yQ

(3)

Equation (2) is still awkward. Let us define two useful quantities, the
intensity of combat I(t) and relative fire effectiveness R(t) ,

I(t) = /a(t)b(t)

™ ^ '

(3)
(4)

A change of independent variables will simplify (2). Let

then (2) becomes

t = / I(s)ds ,
0

lx. R-l dRdx.x = 0 .
.2 dx dt

dt

(5)

(6)

We now reduce (6) to the standard Liouvi lie-Green form by the usual change
of dependent variable

So (6) becomes

d2W

W --£ .

/R

1+/R4 (1//R)

(7)

W = 0

(8)

Then the standard estimates for (8) say that there are two solutions to (8)

W ■ c-je"1 + c2eT

i.e. there are two solutions to (2)

t t

/ I(s)ds -/ I(s)ds)

x(t)- */*&{*. ° +c2e ° )

4/b(tj L

V ^TtT(ci

(9)

What is the error in using (9)? A number of error estimates are avail abl
E3J» [4]. It turns out that they are not quite good enough for our purposes.
The estimates depend upon whether one is interested in the dominant solution
or in the subdominant solution.

(4)

Looking first at the dominant solution, eT , we take

w1 = eT[l + h}(*)l

(10)

Then the error h,(x) satisfies

1 1 i

wnere

h-| + 2h-j = * (1 + h-j)
.2

4, EL /R-Sy (1//R) .

dx^

(11)

(12)

Converting (11) to an integral equation with zero initial conditions we have

^(t) = ^/T[l - e"2(T-s)]Ks)[l + h1(s)]ds . (13)

Our error estimates are based on Willet's generalization of Gronwall's lemma.
Lemma 1 . Let

and let

then

h(x) = / K(s,x>(s)[J(s) + h(s)]ds
0

K(s,t)| < Q(s)P(x)

|h(t)| < P(t) /T|J(s)|Q(s)|Ms)| exp|/Tp(z)Q(z)|^(z)|dz}ds . (14)
0 Is )

Proof: See Willet [5].

An immediate corollary is:

Corollary. Let M(z) = Q(z) max £| J(z)| ,P(z)} then

h(x)| < P(x)

exp}/ M(s) |^(s) |ds> - 1
0

Proof: Integrate (14), using M(z) .
In the case of equation (13)

(15)
QED

- e-2<T " s>) <^(l - e"2T)

_ 1

J(s) = 1

(5)

-2x 1

Thus we can take P(x) = 1 - e and Q(s) = ■*■ . Then lemma 1 gives

Theorem 1 . The dominant solution to (8) satisfies

W1 = eT£l + h^iJJ

< e

T 1

1 + (1 - e"2T)(exp{/ \ |*(s)|ds) - 1)

As for the subdominant solution, we take

2 = e_T[l + h2(x)]

and then h2 satisfies

i2 - 2h2 = ij;(l + h2)

which converts to

M-0 = i ' \

^e2(x-s)- Ij *(s)[l + h2(s)jds

Now we have

(16)
QED

(17)

(18)

1 ft2(x-s).

1 ,Jli

K(sst) -£ e^1 »'- 1 < ^(etl - l)e
Then we have
Theorem 2. The subdominant solution to (8) satisfies

-2s

< e

W2 = e L[l + h2(x)]
1 + (e2T - i) /e"25 |4^-|exp /(l - e"2z) 1^1 dz ds

(19)
QED

0 " s

Observe that we did not use the corollary on (19).

We can see that our estimates depend wery strongly on $ , given by (12).

For the Lanchester equations (1), the coefficients a(t) and b(t) , for
many applications, can be expressed by

a(t) = k ft + c)p
a

b(t) = kb(t + c + a)

(20)

(6)

where c is the "starting" parameter, and A is an "off-set" parameter
(so y can have a different firing range than x ). These are referred to
as the "power rate" coefficients. For future use, let

B = AX
a b

6 = t5 + u + 2

We first consider the case of no offset, i.e. A = 0

t

t = B / (t + c) * dt
0

.,[i

t + c)5 c6

and

u-jj (u + 3u + 4)

T5~

(5t + c6)Z

We see that for -1 < u < u that i> is negative, and

exp /

Us)

ds

- 1

■ "I ♦ exp fe# * 3" + 4V exp

325B

A'"

Then (see (5))

u - uYu + 3y + 4

+ 6t

326B An5

Let

~= ^A (« + 3y + 4)

325B

an

d D1 = exp (Y/A6)

Then (16) becomes

2t

-2t

(21)

(22)

(23)

w] < eT(D] exp (-Y/(A6 + 6x)) + e~dT - D]e i exp (-y/(A6 + 5x))

as an error term for the dominant solution.

For the subdominant solution the corollary gives too crude an estimate.
Even with using the lemma, the error estimate is not good. Integrating by
parts we get

(7)

w2 = e" L [1 + h2( t)J

< e"T[l + (e - 1)11 - exp

JO -e"2z)|||dz

/T|||^/T(l -e-2z)|^f|dz]ds

Working this out we get

w2 < e

TpT+0(expW))

(25)

This is not a very exciting bound, but we are unable to do any better, using
the Willet result.

We can, however, alter the Willet result. The subdominant solution error
also satisfies the equation

h2 -£/"[l - e"2(s " t)>(s)[1 + h2(s)]ds

T

We now prove

Lemma 2. If h( x) satisfies

h(x) < / K(s,t)*(s)[J(s) + h(s)]ds

and

|K(s,t)| < P(t)Q(s)
where K, ^, and J are all > 0 , then

r s

h(x) < P(t) / ^s)Q(s)J(s)exp[~/ P(a)Q(cH(a)da]ds
x It J

Proof. Eq. (27) can be written, using (28)

h(x)

pUJ * / Q(s)u,(s)J(s)ds + / Q(s)^(s)h(s)ds

Differentiating

(26)

(27)

(28)

(29)

(8)

(£) < -Q(x)*(x)J(x) - Q(x)*(x)h(x) U-

?UT

Therefore

(£) + Q*P(£) * -Q^J •

Integrating we get

£[i}< /"Q(s)*(s)J(s)exp

/ P(a)q(oMo)da

ds

+ K exp

But h(«) = 0 , so K = 0 .
For equation (26) we have

/ P(a)Q(a)*(a)da

LT

K(s,x) = M - e

[,.

,-2(s - x)

Thus

h2</°^exPr/Si(a)da
X L x

= exp j/'iMdaj - 1 .

,1 .*

ds

QED

(30)

This is the same bound found by Olver [3].

For the case of the power attrition coefficients with no offset we have

w9 < e"Texp(

\A + 6t/
which is a fairly reasonable bound.

For the case where y = u = 1 , the linear case, we can also get some
explicit results with offset. That is

(31)

Then

a(t) = k (t + c)
a

b(t) = kfa(t t c + A) .

x = BAijn^T7_£n(n + ^r7T)

(32)

(9)

where

Then

= -i + 2U + c)

j I ^(a)da

■o

U(t + c)3 - 6A(t + c)2 - 12A2(t + c) - 7A3

16BA£

[(t + c)(t + c + A)J

3 2 2"?

4c - 6Ac - 12A^c - 7AJ

£C(C + A)]3/2

3/2

while

j / if/(a)da

L.jd 4(t + cY - 6A(t + cp - 12A*(t + c) - 7A

16BA" ( [(t + c)(t + c

Both integrals are monotone increasing.
For the general case

2-/T*(s)ds

To

TTrfiz

(33)

(34)

1 n j (y + fu2)(n2 + An + A2/4) - (u + u2/4)(ri2 - An + A2/4

2virf L I (n - A/2)2 + y/2 (n + A/2)2 + u/2

'a,xb "0

1
2 uu

(n - A/2)1 + y/2 (n + A/2)1 + u/2

dn

(35)

where n = t + c + A/2 .

(10)

REFERENCES

[1] Lanchester, F. W. , "Aircraft in Warfare: The Dawn of the Fourth Arm - No. V. ,
The Principle of Concentration," Engineering 98, (1914) 422-423.

[2] Taylor, J. G. and Comstock, C. , "Force Annihilation Conditions for Variable
Coefficient Lanchester-type Equations of Modern Warfare" to appear Naval Res.
Logistics Q. 1977.

[3] Olver, F. W. J., Asymptotics and Special Functions, Academic Press, New York
and London, 1974.

[4] Willner, B. and Rubenfeld, L. A., "Uniform Asymptotic Solutions ...," Comm.
Pure and Appl . Math 29 (1976) 343-367.

[5] Willet, D. , "A Linear Generalization of Grenwall ' s Inequality", Proc. Ams.
16 (1965) 774-778.

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