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

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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 DD i j°n M 73 1473 EDITION OF 1 NOV 68 IS OBSOLETE S/N 0102-014- 6601 0) (CLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Whan lata Knterec-; UNCLASSIFIED fbLUHITY CLASSIFICATION OF THIS P AGEfWhm Datm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. 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