(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Biodiversity Heritage Library | Children's Library | Advanced Microdevices Manuals | Linear Circuits Manuals | Supertex Manuals | Sundry Manuals | Echelon Manuals | RCA Manuals | National Semiconductor Manuals | Hewlett Packard Manuals | Signetics Manuals | Fluke Manuals | Datel Manuals | Intersil Manuals | Zilog Manuals | Maxim Manuals | Dallas Semiconductor Manuals | Temperature Manuals | SGS Manuals | Quantum Electronics Manuals | STDBus Manuals | Texas Instruments Manuals | IBM Microsoft Manuals | Grammar Analysis | Harris Manuals | Arrow Manuals | Monolithic Memories Manuals | Intel Manuals | Fault Tolerance Manuals | Johns Hopkins University Commencement | PHOIBLE Online | International Rectifier Manuals | Rectifiers scrs Triacs Manuals | Standard Microsystems Manuals | Additional Collections | Control PID Fuzzy Logic Manuals | Densitron Manuals | Philips Manuals | The Andhra Pradesh Legislative Assembly Debates | Linear Technologies Manuals | Cermetek Manuals | Miscellaneous Manuals | Hitachi Manuals | The Video Box | Communication Manuals | Scenix Manuals | Motorola Manuals | Agilent Manuals
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Error bounds for the Lanchester equations with variable coefficients"

LIBRARY 

TECHNICAL REPORT SECTION 
NAVAL POSTGRADUATE SCHOOL 
MONTEREY, CALIFORNIA 93940 



NPS-53Cs77031 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




ERROR BOUNDS FOR THE LANCHESTER EQUATIONS 
WITH VARIABLE COEFFICIENTS 

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



^ved for public release; distribution unlimited, 



FEDDOCS 

D 208.14/2:NPS-53Cs77031 



NAVAL POSTGRADUATE SCHOOL 
Monterey, California 

Rear Admiral Isham Linder Jack R. Borsting 

Superintendent Provost 

Reproduction of all or part of this report is authorized. 



This report was prepared by: 



I CLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Entered) 



REPORT DOCUMENTATION PAGE 


READ INSTRUCTIONS 
BEFORE COMPLETING FORM 


1 REPORT NUMBER 

NPS-53Cs77031 


2. GOVT ACCESSION NO. 


3. RECIPIENT'S CATALOG NUMBER 


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 


6. PERFORMING ORG. REPORT NUMBER 


7. AUTHORf«J 

James G. Taylor 
Craig Corns tock 


8. CONTRACT OR GRANT NUMBERf*) 


9. PERFORMING ORGANIZATION NAME AND AODRESS 

Naval Postgraduate School 
Monterey, CA 93940 


10. PROGRAM ELEMENT. PROJECT, TASK 
AREA & WORK UNIT NUMBERS 


11, CONTROLLING OFFICE NAME AND ADDRESS 


12. REPORT DATE 

March 1977 


13- NUMBER OF PAGES 

13 


14. MONITORING AGENCY NAME 4 ADDRESSf// dltlerent from Controlling Office) 


15. SECURITY CLASS, (ol thia report) 

Unclassified 


15*. DECLASSIFI CATION/ DOWN GRADING 
SCHEDULE 


16. DISTRIBUTION STATEMENT (ol thia Report) 

Approved for public release; distribution unlimited. 


17. DISTRIBUTION STATEMENT (oi the abatract entered in Block 20, It different from Report) 


18. SUPPLEMENTARY NOTES 


19. KEY WORDS i Continue on reverae aide II neceeaary and Identify by block number) 

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°n M 73 1473 EDITION OF 1 NOV 68 IS OBSOLETE 

S/N 0102-014- 6601 

0) 



(CLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whan lata Knterec-; 



UNCLASSIFIED 



fbLUHITY CLASSIFICATION OF THIS P AGEfWhm Datm Entmrtd) 




Form 
1 Jan 73 
S/N 0102-014-6601 

(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) ■ x Q , a _1 (0) ^jf(O) = -y Q 



(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)d s , 




lx. R -l dRdx. x = . 
.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 



d 2 W 



W --£ . 

/R 



1+/R4 (1//R) 



(7) 



W = 



(8) 



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



W ■ c-je" 1 + c 2 e T 



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



t t 

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

x(t)- */*&{*. ° + c 2 e ° ) 



4/b(tj L 

V ^TtT( c i 



(9) 



What is the error in using (9)? A number of error estimates are avail abl 
E 3 J» [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, e T , we take 



w 1 = e T [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 + h 1 (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 




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



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




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 



W 1 = e T £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 + h 2 (x)] 



and then h 2 satisfies 



i 2 - 2h 2 = ij;(l + h 2 ) 



which converts to 



M-0 = i ' \ 



^e 2(x - s) - Ij *(s)[l + h 2 (s)jds 



Now we have 



(16) 
QED 



(17) 



(18) 



1 ft 2(x-s). 



1 ,Jli 



K(s s t) -£ e^ 1 »'- 1 < ^(e tl - l)e 
Then we have 
Theorem 2. The subdominant solution to (8) satisfies 



-2s 



< e 



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



(19) 
QED 



" 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) = k b (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 = 

t 



t = B / (t + c) * dt 




.,[i 



t + c) 5 c 6 



and 



u-jj (u + 3u + 4) 



T5~ 



(5t + c 6 ) Z 



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



exp / 



Us) 



ds 



- 1 



■ "I ♦ exp fe # * 3 " + 4 V exp 



325B 



A'" 



Then (see (5)) 



u - uY u + 3y + 4 



+ 6t 



326B A n 5 



Let 



~= ^A (« + 3y + 4) 



325B 



an 



d D 1 = exp ( Y /A 6 ) 



Then (16) becomes 



2t 



-2t 



(21) 



(22) 



(23) 



w ] < e T (D ] exp (- Y /(A 6 + 6x)) + e~ dT - D ] e i exp (-y/(A 6 + 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) 



w 2 = e" L [1 + h 2 ( 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 



w 2 < e 



T p T+0 ( exp W)) 



(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 



h 2 -£/"[l - e" 2(s " t) >(s)[1 + h 2 (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 > , 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(«) = , so K = . 
For equation (26) we have 



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

LT 



K(s,x) = M - e 



[,. 



,-2(s - x) 



Thus 



h 2 </°^ex P r/ S i(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 



w 9 < e" T exp( 



\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) = k fa (t t c + A) . 



x = BAij n ^T7_ £n ( n + ^r7T) 



(32) 



(9) 



where 



Then 



= -i + 2U + c) 



j I ^(a)da 



■o 



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



16BA £ 



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

3 2 2"? 

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

£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 

T o 



TTrfiz 



(33) 



(34) 



1 n j (y + fu 2 )(n 2 + An + A 2 / 4 ) - (u + u 2 / 4 )(ri 2 - An + A 2 / 4 

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



'a ,x b "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. 



(11) 



DISTRIBUTION LIST 

No. of Copies 

Advanced Research Projects Agency 1 

Department of Defense 
Technical Library 
Washington, DC 20301 

Chairman, Joint Chiefs of Staff 1 

Studies Analysis and Gaming Agency 
Washington, DC 20350 

Defense Documentation Center 2 

Cameron Station 
Alexandria, VA 22314 

Mr. Alvin F. Andrus, Code 230 1 

Dr. Toke Jayachandran, Code 431 1 

Mr. James F. Smith, Code 431 1 

Office of Naval Research 
Department of the Navy 
Arlington, VA 22217 

Naval War College 1 

Technical Library 
Newport, RI 02840 

Naval Ordnance Laboratory 1 

Technical Library 
Silver Spring, MD 20910 

Naval Research Laboratory, Code 2029 1 

Washington, DC 20390 

Director, Army Research 1 

Office of the Chief for Research and Development 
Department of the Army 
Washington, DC 20310 

Center for Naval Analyses 1 

Technical Library 
1400 Wilson Boulevard 
Arlington, VA 22202 

Institute for Defense Analyses 1 

400 Army-Navy Drive 
Arlington, VA 22202 



(12) 



Professor Joseph Engel 1 

Systems Engineering Department 

Box 4348 

College of Engineering 

University of Illinois at Chicago Circle 

Chicago, IL 60680 

Professor Robert M. Thrall 1 

Department of Mathematical Sciences 
Rice University 
Houston, TX 77701 

Mr. H. K. Weiss 1 

P.O. Box 2668 

Palos Verdes Peninsula 

Pal os Verdes, CA 90274 

Dudley Knox Library, Code 0142 2 

Naval Postgraduate School 
Monterey, CA 93940 

2 
Dean of Research, Code 012 

Naval Postgraduate School 

Monterey, CA 93940 

Professor James G. Taylor, Code 55Tw 10 

Professor Craig Comstock, Code 53Cs 10 

Professor Samuel H. Parry, Code 55Py 1 

Professor Michael G. Sovereign, Code 55Zo 1 

Professor Gerald G. Brown, Code 55Bw 1 

Department of Operations Research 
Naval Postgraduate School 
Monterey, CA 93940 



(13) 



DUDLEY KNOX LIBRARY ■ RESEARCH REPORTS 



5 6853 01067326 2