A Steepest-Ascent Method for Solving Optimum Programming Problem

A Steepest-Ascent Method for SolvingOptimum Programming Programs

A. E. BRYSON AND W. F. DENHAM

AEROSPACE problems influenced considerably the earlydevelopment of optimal control and resulted in the formulationof trajectory optimization problems and methods for solvingthese problems well before the advent of dynamic programmingand the maximum principle. As early as 1919 Goddard posed theproblem of determining the thrust profile of a rocket to maximizealtitude; an analytic solution for this problem, which requires asingular arc, was obtained in 1951 [8] and is described in [4],page 253. Hohmann [5] determined the optimal impulsive ma­noeuvre to transfer a space vehicle from one circular orbit toanother in 1925. These early solutions were of necessity ana­lytic; although the conceptual tools for numerical optimizationwere available, the means were not until the advent of the digitalcomputer in the mid 1950s. The calculus of variations provided,at least implicitly, the concept of a function space gradient, andthe method of steepest descent (or ascent) for optimization datesfrom Cauchy, but it could not be implemented on serious prob­lems prior to the mid 1950s even though it had been well studiedprior to this.

The stimulation provided by aerospace problems such as thedetermination of maximal range rocket trajectories, minimum­time trajectories for space flight from Earth to Mars andminimum-time trajectories for fighter aircraft to reach a highaltitude was intense and resulted in a spate of papers in the late1950s and early 1960s dealing with these problems. Among thefirst were the papers by Bryson and Ross [3] and Breakwell [1]that obtained necessary conditions of optimality for the maximalrange rocket trajectory problem using the calculus of variations,and solved the resultant two point boundary value problem usinga "shooting" algorithm, the initial value of the adjoint servingas the "decision" variable. Bryson and Ross used an IBM 650computer made available by Garrett Birkhoff of the Mathemat­ics Department at Harvard University and later applied the samemethod to the determination of optimal re-entry paths for an or­bital glider. Lawden, Leitmann, Miele, and others also employedshooting algorithms to solve aerospace flight path problems inthe late 1950s and early 1960s.

The extreme sensitivity of the shooting method motivated thedevelopment of gradient methods using the control function u(·)

as the decision variable [6]. The problem studied in Kelley's pa­per and that of Bryson and Denham is the maximization of aterminal performance index (jJ(x(T» subject to constraints im­posed by the differential equation

x = [t», u, t)

of the system, the initial condition rtr-) = xo, a terminal equalityconstraint 1/J(x(T» = 0, and a stopping condition Q(x(T), T) =othat determines the final time T. Kelley [6] replaces the termi­nal constraint by a penalty, making the problem unconstrained(apart from the differential equation); this simplifies the opti­mization procedure but causes numerical difficulties. Ignoringfor simplicity the stopping condition (by regarding T as fixed),Bryson and Denham proceed as follows. Suppose that u(·) isthe current control. The gradients g(jJ(.) and gl/J(.) of the per­formance index and the equality constraint with respect to thecontrol function u(·) are obtained by solving the adjoint differ­ential equation

. T-A = t; (x, u, t)A

with terminal condition A(T) = (jJ~(x(T» (to obtain gq>(.» andterminal condition A(T) = 1/J~(x(T» (to obtain gl/J(.». A controlchange 8u(·) is selected that maximizes (g(jJ(·),8U(·»)2 subjectto (gq>(.), 8U(·»)2 = a and 118u(·)112 ::::: b. Here a is the desiredchange in the value of the terminal constraint and b limits thestep size. The successor control is u'(·) = u(·) + 8u(·). The vari­ables a and b are chosen manually to ensure convergence. Be­cause both performance index and constraint change, the searchdirection is not steepest ascent; indeed the performance indexmay decrease in any given iteration.

What makes these early papers so exciting is the signifi­cant spacecraft and trajectory optimization problems that weresolved. The first aircraft application of the gradient method ap­pears in this paper by Bryson and Denham. One of the problemsconsidered was the determination of the angle of attack profilethat minimized time to achieve a specified altitude of 20 km anda specified speed of Mach 1 at level flight. The solution wassurprising because it was considerably different from and betterthan the best previously achieved (by cut and try). The optimal

219

path consisted of a rapid climb at a speed of just below Mach1 to a height of 30,000 feet (to obtain an increase in potentialenergy) followed by a period in which height is decreased andspeed increased (to gain kinetic energy). A final spurt to thespecified height and speed concludes the manoeuvre. The op­timal path was actually tested in January 1962 at the PatuxentRiver Naval Air Station; the time taken to reach the specifiedaltitude was 338 seconds compared with 332 seconds predictedtheoretically. Contrary to what might have been expected fromsuch a beautiful demonstration of optimal control, the outcomewas not immediately favorable; the two pilots who flew the op­timal path were reprimanded and neither Harvard nor Raytheonreceived an increase in research funding!

However, the long-term effect on applications and researchwas significant. A wide range of spacecraft and aircraft appli­cation studies, well described by Bryson [2], were undertaken.But the work also stimulated theoretical research, particularlyon the mathematical programming problems occurring in opti­mal control. The recent text by Polak [7] reveals the richness ofthe subject and the advances that have been made.

There are several features of the early research by Brysonand Denham and others that deserve comment. Firstly, the re­searchers used the calculus of variations rather than the max­imum principle. This must have been in part owing to timing:the maximum principle was not readily available when the re­search began. There were other reasons: tol this day there areonly one or two algorithms that employ strong variations forthe search "direction" and that generate accumulation points

satisfying the maximum principle rather than weaker optimal­ity conditions. Secondly, the researchers exercised considerableingenuity in obtaining conditions of optimality for control prob­lems involving state and state dependent control constraints,problems that are still being studied. Thirdly, the researchersdeveloped novel algorithms to handle constraints, like that em­ployed by Bryson and Denham. Nowadays the problem wouldprobably use exact penalty functions that provide a single cri­terion in place of the two (performance index and constraintvalue) employed in this paper. Finally, it is clear that this earlyresearch inspired a long and fruitful activity, both theoretical andpractical.

REFERENCES

[1] I.A. BREAKWELL, "The optimization of trajectories," SIAM Journal, 7:215­247,1959.

[2] A.E. BRYSON, "Optimal control-1950 to 1985," IEEE Control Systems,16(3):26-33, 1996.

[3] A.E. BRYSON AND S.E. Ross, "Optimum rocket trajectories with aerody­namics drag," Jet Propulsion, 1958.

[4] A.E. BRYSON AND Y.-C. Ho, Applied Optimal Control-optimization, Es­timation, and Control, Hemisphere Publishing Corp. (Washington), 1969.

[5] W. HOHMANN, Die Erreichbarkeit der Himmelskiirper, Oldenbourg(MUnich), 1925.

[6] H,J. KELLEY, "Gradient theory of optimal flight paths," J. American RocketSoc., 30:948-954,1960.

[7] E. POLAK, Optimization: Algorithms and Consistent Approximations,Springer Verlag (New York), 1997.

[8] H.S. TSIEN AND R.C. EVANS, "Optimum thrust programming for a sound­ing rocket," J. American Rocket Soc., 21(5):99-107, 1951.

D.Q.M.

220

A. E. BRYSONProfessor,

Di~i.ion of engineeringand Applied Physics,

Harvard University,Cambridge, Mass.

ASteepest-Ascent Method for SolvingOptimum Programming Problems

w. F. DENHAMR.search Engineer,

Missiles & Space Division,Raytheon Company,

Bedford, Mass.

A system4lic and ,apid steepest-ascent numerical procedure is described f01 solvingtwo-point boundar,...va1ue problems in the calculus of variations 101' systems gMJtf~tlby a set of nonlinear ordinary differenti4l equations. Numerical e%lJ,mples are /We­sentea for minimum ti~-to-dimb and maximum altuude '/HJths (or a supersonic in­tercePkw and mmmum-range pmhs for an orbittzl glider.

1 Numbers in brackets designate References at end of paper'.

cr := [~I], an m X 1 matrix of e<mlrol voriabk" which we are· free to ehooae, (4)a.

-t == [~J.,a p X J matrix of COMtraint funclionl, each a· known function of x, (6)fI

x := [~JJ an n X 1 matrix of state fJilrlablu, which result from· the choice of II, (5)

%

(I)

(2)

(3)

q, = q,(x),

~ ~ ~(x) = 0,

f( x, a) + '0 = 0

point, finding how badly the specified terminal boundary condi­tions are missed,and then attempting to improve the guess of theunspecified initial conditiOD8. This process must be repeatedover and over until all terminal conditions are satisfied. Thisprocess is not only tedious, expensive, and frustrating, it eome­times does not seem to work at aU {ll.· It is remarkably senai­tive to small chauges in the initial conditioDs; however. it caDbemade to work through great'patience, good guessing, and second­order multiple interpolation [2, 3, 4}.

Recently Kelley [5, 6) and the authors with several coworkers[7, .8] have revived a little-known procedure which olfers a prac­tical, straightforward method for finding numerica1101utioD8 toeven the most complicated optimum programming problems. Itis essentially a steepest-ascent method and it requires the use of ahigh-speed digital computer.

subject to the constraints

where

3 AMaximUM Pr8~1'1I in Ordi.aff CllculusIn order to explain the steepest-aseent method it is helpful to

consider its use in a simpler problem first; namely, the problemof finding the maximum of a nonlinear function of many variablessubject to nonlinear constrainta OD these variables. This is aproblem in the ordinary calculus. A quite general problem of thistype can be stated &8 Iollowa:

Determine Of 80 as to maximize

A lecture presented at the Summer Conference of the Appliedl\.fechanies Division, Chicago, Ill., June 14-16. 1961. of Ta. Aln:RI­CAN SOCIETY OJ' MECHANICA.L ENGINEERS.

Discussion of this paper should be addressed to the Editorial De­partment, ASME. 345 Eut 47th Street, New York, N. Y., and willbe aecepted until July 20, 1962. Discuasion received after the clos­ing date will be returned. Manuacript received by ASME Applied~Iechanies DivisioD. September 6, 1961.

1 SumMaryASYSTJ:at.\TIC and rapid steepest-ascent numerical

procedure IS described for determining optimum progr~ms fornonlinear systems with terminal CODstraints, The procedureuses the concept of local linearization around a nominal (Don:optimum) path. The effect on the terminal conditions of a small~hange in the control variable program is determined by numeri­'Cal integration of the adjoint differential equseione for small per­turbations about the nominal path. Having these adjoint (or in­B"uence) functions, it is then possible to determine the change inthe control variable program that gives maximum increase in tbepay-off function for a given mean-square perturba.tion ofthe control variable program while simultaneoualy changing theterminal quantities by desired amounts. By repeating this proc­e88 in small steps, a control variable program that minimizes onequantity and yields specified values of other terminal quantities'C&I1 be approached as closely sa desired. Three numerical ex­amples t\re presented: (a) The angle-of-attack program for atypical supersonic interceptor to climb to altitude in minimumtime is determined with and without specified terminal velocityand heading. (b) The angle-of-attack program for the sameinterceptor to climb to maximum altitude is determined. (e) TheangJe-of-attack program is determined for a hypersonic orbitalglider to obtain maximum surface range starting from satellitespeed at 300,000 ft altitude.

2 IntroductioriOptimum programming problems arise in connection with proc­

-esses developing in time or space, in which one or more controlvariables must be programmed to achieve certain terminal con­ditions. .The problem is to determine, out of all poasible pro­grams for the control variables, the one program that maximizes(or minimizes) one terminal quantity wbile simultaneously yie1d-.ing specified values of certain other terminal quantities.

The calculus of variations is the classical tool for solvingsuch problems. However, until quite recently, only rather simpleproblems had been solved with this tool owing to computationaldifficulties. Even with a high-speed digital computer theseproblems are quite difficult bec8use,m the. classic&1 formulation,they are two-point boundary-value problems for a set of DonliDearordinary differential equations. Numerical solution requiresguessing the missing boundary conditioDl at the initial point.integrating the differential equations numerically to the terminal

Reprinted with permission from Transactions of the ASME, Journal ofApplied Mechanics, A. E. Bryson and W. F. Denham,"A Steepest-Ascent Method for Solving Optimum Programming Programs," April 1962, pp. 247-257.

221

ip is the pa'll-06 function, a known funetion of x, (7)( -~ ). .;- i.• 'F ; 0, '\1;')

[t} an n X 1 matrix of known functions of x and cr. (8) (b~r ( 1~)(); + l~'F = 0,

t.where

[II] i)Y,1 ~~'!~.()'('J

, .......o«,

fo =. i.. 'an n X 1 matrix of constants (9)~ .. [~ ~_]~'k= I (' 19)()x ():rl ' • • • ().r,. • Ox

These perturbations "..ill cause perturbations dx in the statevariables, where

Taking differentials of equations (3) we obtain to first order inthe perturbatioDs the linear set of equations for dz,

and ( )• iDdieateathat the partial derivatives arc evaluated at thenominal point. Using Lagrange multipliers (Appendix, sec..tion 1), wemay write

dtf> = ~cf"Gdu + ~'dfo (IS}

dq :: ~'Gdcr + '-~'d'o (16)

where the Xmatrices are determined by the linear equatiens

(20)

(22)

(24)

(dP)2 = du'Wda,

Notj~-e if d~ == 0, dlo = 0, equation (23) becomes

del> == =(14). - 1.y.'IH-]I~.)'12dP

and ( )' indicates the transpose of ( ); i.e., rows and columns areinterchanged.

Note that the j., are influence nu"werssince they tell how much4> OT ~ is ehanled by 6m:),U e\\8.nges in the eOD5tra,int levels 10.

For steepest ascent, we wish to find the dcr matrix that maxi­mizes dq, in equation (15) for a given value of the positive definitequadratic form

where

and given values of d~ in equations (16). The values of d~ ar»chosen to bring the nomine] solution closer to the specified con­straints,1k &;II O. Choice ofdP is made to insure that the perturba­tiona dB win be small enough for the linearization leading to equa­tions (12) to be reasoeable. W is an arbitrary non-negative d.·fi­nite m X m matrix of weighting numbers, essentially a metri« illthe e byperspace; it is at the disposal of the "optimiser" t.,improve convergence of the procedure.

The proper choiceor d« is derived in section 2 of the Appendixand the result is

[(dP )2 - d~'I",~-Jd6]1/2

du == :f=W-JG'(l~ - j.,.;..I",~ -JI~.) 1 I 'I _I' lit

.4>-"'.~ti' "'.

which is the magnitude of the gradient in the a byperspaee, sincedP is the length of the step in the crhyperspace. .J\S the maximumis approached and the eoaetraints are met (d1k = 0), this gradientmust tend to zero, which results in

d~ =- dtk - l.;'dfo,

I~~ - :A\fI'GW-JG'~#,

I~. == ~~'GW-1G'J...,

I til. == If/)'GW-IG'l.,

( ) -I indicates inverse matrix. and the + sigo is used if '" is to beincreased, the - sign if tIJ is to be decreased. Note that thenumerator under the square root in equation (21) can becomenegative if d~ is chosen too large; thus there is a limit to tbe size:of d~ for a given dP. SincedP is chosen to insure valid lineariza..UOD, the d~ asked for must also be limited. The predicted changein q, for the change in control variables of equation (21) is

dq, - :i=((dP)2 - d~'I~-1d~)(IH - 1.J.'I\it~-JI~o»'/2

+ I",.'I~", -ld~ + l.'dfo• (23)

(11)

(10)

(12)

(13)

(14)

dx ;a x-x·

de ::: ct - cr*

Fdx +Gda = 0

f( lJf. ). (lJ/. )*1F:= ~XI ••••• ~.

l(:~:r·····(~:rJf( b/l ). (~)·l~1 , •••• baM

. .

. .

l(~lr,····(~:rJ.G

where

4 Steepest-Ascent Method in Ordilary CalculusThe maximum problem stated in the preceding section caD be

solved systematically and rapidly on & high...speed digital com­puter using the steepest-ascent method. This method startswith a nominal control variable matrix ex*, and then imp~ves

this estimate by determining the direction of steepest ascent, inthe Cl hyperspace; this determination is made by a linearizationabout the nominal point in the u hyperspace .. The method pro­ceedsas follows:

(a) Guess some reasonable control variables 0·, and use themin equations (3) to calculate Dumerically the state variables x·that c.orrespond to this choice. In general, this nominal "pointU

willnotsatisfy the constraint conditions~ - 0, or yield the maxi..mum value of q,.

(b) Consider small perturbations da, about the nominal controlvariable point where

222

(25) •This relation shows how much the maximum pay-otJ functionchanges for ~mall changes in the constraint levels.

(c) A new control variable point is obtained 88

UN EW == Q'OLD + d«

where d« is obtained from equation (21). fiNn' is used inthe original nonlinear equatioDs (3) and the whole process isrepeated several times until the t == 0 constraints are met andthe gradient is nearly zero in equation (24). The maximumvalue of. has then been obtained.

This proeess can be likened to climbinga mountain in a densefog. We cannot see the top but we ought to be able to get thereby always climbing in the direction of steepest ascent. If we dothis in steps, climbing in one direction until we have traveled acertain horizontal distance, then reassessing the direction ofsteepest ascent, climbing in that direction, and so 00, this is theexact analog of the procedure suggested here in a space of m-di­menaiona where ~ is altitude and QJ, Qz are co-ordinatea in thehorizontal plane, Fig. 1. There is, of course, & risk here in thatwe may climb a secondary peak and, in the fog, never becomeaware of our mistake.

5 O,limu.. Prolramming, a Problem in Ih. Calculus ofVarlallons

An optimum programming problem of considerable generalityCADbestated &I follows, Fig. 2:

Determine Cl(t) in the interval t. 5 t :S T, so as to maximize

Fig. 1 Finc!I"I .......IIum of a functfon of two v.ria..... a,y ,....d­a,centmeth...

TERMINAL POINTt-T

n.fCX..."ctt

FII. 2 SytIIMIlc sketch Df optlmulil progr.m",'n. probl ....

[~(t)] , an n X 1 matrix of stale va.riable programs, (32)

x(t) =. which result from a choice of a(t) and• giveD values of .(1.0),z,,(t)

cr(t) =- [~I(t)] r an m X 1 matrix of control rJari4b~ pro- (31)• gramll, which we are free to choose,a".(t)

[til]. , a l' X 1 matrix of terminal comtraint June-~ -. titmI, each of which is a known function of (33)

• aCT) &DdT,

tft If all inteeraJ ~ to be mazimiaed, simply introduce an additional

state variable Sf and aD additional differential equation ., - q(x, a,e) where 9 is the iDtqrand of the intepal. :t,(r) is then mazimiJedwith ~(Ie) - O.

I In lOme problema not all of the state variables are specifiedinitially: in this cue the. unspecified state variables may be deter­mined a10DI with a(t) to maximize tP.

q, is the pay-offJunction and is known function of x(T) and T,(34)

f =a [~]J au n X 1 matrix of known functions of x(I), (~i5). u(l), and t,

I"n = 0 is the stopping condition tAattktmnim, final time T,

and is a known function of x(T) end T (36)

The formulation of the neceseary conditions for an extremalsolution to this problem has been given by Breakwell [2J withthe added complexity of inequality cODstraints OD the controlvariables.. The present paper is concerned with the efficient andrapid solution of sucb problems using a 8teepes~ascent procedure.

& Slee"st·Ascenf Melbld in Calculus of VariationsThe optimum programmiuK problem stated in the preceding

section can be solved systematically and rapidly OD a high-speeddigital computer using the steepeet-ssceat technique. Thisteehoique starts with a Dominal control variable program w·(t),and then improves this program in steps, using informatioD ob­tained by a mathematical diagn06is of the program for the pre­vious step. Conceptually. it is a process of local linearizationaround the path of the previous step. The method proceeds asfollows:

(a) Guess some reasonable control variable programs a*(t),and use·them with the initial conditions (29) and t~e ditfereutialequatioD8 (28) to calculate, numerically, the sta~ variable pro­grams x·(t) until n =- o. In general, this nominal "path" winnot satisfy the terminal eonditicns 'f! = 0, or yield the maximumpossible value of q,.

(b) Consider small perturbatioDs BaCt) about the nominel eOD­

trol variable programs, where

(28)

(27)

(30)'

(26)1

(29)1

t/J ::II q,(x( T), T),

te and x(lo)given,

tt = «x(T), T) := 0,

dxdi =- f(x(t), Q(t), 0,

T determined by {} 31!1 fl(x(T), T) la 0

The nomenclature of the problem is as follows:

subject to the constraints,

223

au := u(t) - u*(t) (37) and

(4~)

(·tH)

(51)

(53)

(()!i <>U)"n= ---·+-1 •<>t ~x ,::: T

(o~ ()~).t!r = --~ -to -- f

.. ()t ()x t _ T 1

I""" = f,.T lll-ll'GW-IG'l~l,

('i'Iu = J to l~o'GW-1G').~ndtt

l~ =- f T )..n'GW-1G/~.udt,s,

(dP)1 = f,.T lio:'(t)W(llOo:(t)dt,

• ( (xjJ i)q,) ..4>= -+-t J

()t ()x c- T

where

( )' indicates the transpose of ( ); i.e., rows and columns are ill­terchanged.

Note that the '-are injf:u.enr.elunctions since they tell how mucha certain terminal condition is changed by a small change in 80Dle

initial state variable. Note also that the adjoint equations (4;)must be integrated backward since the boundary conditions aregiven .a.t the terminal point, t = T.

For steepest ascent we wish to find the Oct(t) programs tha.tmaximize dq, in equation (42) for a J!:iven value of the integral

Notice that if d1k :Ill' O,oxCio) = 0, equation (52) becomes

d~ = ±(If/). _ 1~.'I~.y -II~~)I/tdP

given values ofdq in equations (43) and dn = 0 in equation(44). The values or d~ arc chosen to bring the nominal solutioncloser to the desired terminal constraints, tt! = o. Choice of tiPis made to insure that the perturbations on(i) will be 8n1:111 enoughfor the linearization leading to equations (46) to be reasonable.W(t) is an arbitrary symmetric Til X 171 matrix of weighting fune­tiona chosen to improve convergence of the steepest..ascent pro­cedure; in some problems it is desirable to subdue Ocr in certainhighly sensitive regions in favor of larger 00: in the less sensitiveregions..

The proper choice of ocr(l) is derived in section .. of the Ap ..pendix and the result is

[<dPP - dll'I¥r~ -'d~]'/t

Oct(t) = ±W-1G/(~a - ~~nl~4--1r",~) -------;--:--.-I.tIJ - I",. 1-;.,;,· 'I",.

+ W-1G/l~nlJ.l/--ld~, (.50)

( )-1 indicates inverse matrix, and the + sign is used if 4> is to beincreased, the - sign is used if q, is to be decreased. Note thattbe numerator under the square root in equation (50) can becomenegative if d~ is chosen too large; thus there is a limit to the sizeof dlJ for a given dP. Since dP is chosen to insure valid linearisa...tlon, the dO asked for must also be limited. The predicted changein q, for the change in control variable program (50) is

d<p :I: ± [«dP)t - d~'I~.y -ld~)(I~. - 1¥-~'ltJ.p -JI~~) j'h

+ J~.'I~", -ld~ + J-~n'(to)Ox(to) (52)

(39)

(40)

(41)

(46)

(45)

F(t) =s

ero

where

These perturbatioDs will cause perturbations in the state variableprograms clx(t), where

elx ::a x(t) - x*(I) (38)

Substituting these relations into the differential equations (28).we obtain, to first order in the perturbations, the linear differen­tial equations for elx,

ddt (8x) = F(t)8x + G(t)8cr,

~/(T) .. (?Jq,) * ,~'(T) .. ()~).. .'ax ,- 7' ()x , - T

j.g'(T) = (()!1).()x 1- T

d4- == -F'(t)~dt

~ = [:~. · · · ::. J. (47)

where the elements of the 1 matrices are determined by numericalintegration of the differential equations adjoint to equations(39); namely,

aDd ( ). indicates that the partial derivatives are evaluated alongthe nominal path. From the theory of adjoint equations, section3 of the Appendix, we may write

dlfJ ... J,:r ~'(')G(t)8a(,)dt + ~'(ta)8x(ta) + .jW.7', (42}

d1\! = J,.7' l",'(')G(')8a(,)dt + ll(Ie)8x(lo) + +l7', (43)

dO - J,.7' l'o(')G{')8a(,)dt + 19'(te)Ox(le) + Ud7', (44)

where

with boundary conditions

224

which is a "gradient" in function space, since dP is the "length"of the step in the control variable programs. As the optimumprogram is approaehed and the terminal cODStraints are met (dtk== 0), this gradient must tend to zero, which results in

(k) Obtain & new nominal path by using aNSW' == crOLD + ocrand repeat processes (a) through (k) until the terminal con­straints tt :II: 0 are satisfied and the squareof the gradient, [HI ­I'",J~~-II",., tends to zero.

This relation shoWl how much the maximum pay-off functionchanges for small changes in the terminal constraints and forsmall changes in the initial conditions.

(c ) New control variable programs are obtained as

where c5a is given in equation (50).. «NEW(I) is used in theoriginal nonlinear differential equations (28) and the whole processis repeated several times until the terminal constraints are metand the gradient is nearly zero in equation (53). The optimumprogram has then been obtained.

Note that miniminng final tiIM T fits into this pattern con­veniently. For this case we let q, == -t which implies'" = -1,,.,. == 0, and hence

1J.,..n =t -;- 10

n

(57)

(58)

(59)

(60)

(61)

ZERO-UFTAX'SYELOCITY.V

h

h = V sin 'Y

:t = V cos 'Y

WEIGHT

x~

DRAG

• £(1&, AI, a) + F(A, M) • 9 cos 'Y'Y == ---SID at - --

mV mV V

L ". CL(a, M) P~'S is lift

Cd,a., M}, CL(a, l{) are given as tabular functions, Fig.. 4(b)

p p(h.) is air density, given as a tabular function

g acceleration due to gravity (taken as CODSta.Dt here)

VJf - is Mach number

a

ALTITUDE Lin

8 Example 1A".......Attaclc ,........ su Iftferc""" .. cu•• Fr....

Sea Level to • 0IYeft AIIHUII. III n e. A typical 8upersonic in-terceptor is considered with lilt, drag, and thrust characteristicsas shown in Fig. 4. The vehicle is eonaidered u a m&88 point(short period pitching motions are thus neglected) and the Domen­clature used is shown in Fig. 3. The problem is to find the angle­of-attack program a(t), using maximum thrust, that takes theinterceptor toa given final altitude in the least time, atartiDg justafter take-ofJ at M - 0..22, 'Y - 0, at sea level (11 == 0).. Fint theproblem is solved with DO terminal coDStr&iDta, then with M ­0.9 specified at the termiDal point, then with M == 0.9 and '1 .lD 0at the terminal point. The differential equatioDs for the inter­ceptor path are

V F(h,M) D(h, M, tX) •c: -m-- cos a - m - 9 81n ..,

m = m(h, M)

where ( · ) means ~ ( ), and

F = F(h, M) is thrust, given &8 a tabular function, Fig. 4(4)

pV28

D = C»(at, ill) -- is drag2

(55)

(56)

ClNEW(t) Jar «OL!>Ct) + ,suet)

i T 1 1de; = -dT = -;- ~ll'Gd.dt + -;- 4a'(lo)&x(ft)

to n 0

1 Compaling ProceduresThe computing procedures evolved over a period of time in

solving many types of problems are summarized here. The com­puter used for all these problems was an IBM 704.

(a) Compute the nominal path by integrating the nonlinearphysical differential equations with a nominal control variableprogram and appropriate (here assumed fixed) initial conditionsand store the solution on tape.

(b) Compute the ~., ~~ 1, J.~ J, ••• ~ II' J.o functions all at tMS01M time by integrating the adjoint differential equations back­wa.rd, evaluating the partial derivatives on the nominal path byreference to the tape in (a).

(e) Simultaneously with (b), calculate the quantities '-.n, l~rn,... l~pg and store ).c;o'G and l~Q'G on tape.

Cd) Also simultaneouely with (b) and (c) perform the inte­grations (backward) leading to the numbers 1.., I~.J I~~.

(e) Print out the values of <P, 1/Ih "'" ..• y,p achieved by thenominal path.

(I> Select desired terminal condition changes dt/l., d.p2, ....d"';pto bring the next solution closer to the specified values t = 0than were achieved by the nominal path.

(g) Select a reasonable value of (dP)'/(T - to), which is amean-square deviation of the control variable programs fromthe nominal to the next step.

(h) Use the values of dq and dP to calculate (dP)t. ­d~'IH -ld,p; if this quantity is negative, automatically scale downdl/l to make this quantity vanish. If the quantity is positive,leave it as is.

(i) Using the values of dP and dtk (modified by (h) if necessary)calculate &a(') from equation (50), (&x(lo) :IS 0).

(j) If final timet T, is not specified or being extremalized, com­pute the predicted change, dT, for the next step:

dT = _ ~ r T In'GOadl{} J~

If IdTI is greater than a preselected maximum allowable value,scale down &a(t) to achieve this maximum value.

)wr."""'''J''''P''f''''''''''''''''''''''''''''''''''''''~''''''''''''P7''I''......,.~,.....HORIZONTALDISTANCE

Fig_ 3 Nomenclature used in analysis of sup......nlc interceptor

225

2.5LO 1.5.,"CH IiIUMII!l'

.5

II_ ALT1TUO£

~ H_ IlAXI_ ALTIlU)( FOR

IT£AIlT HOltlZOIlTAL /

V/

oo

.2

.1

1.0

••41:

~..S .44

o

6 Co.c:a.+" lIICL.2.2

lIIo ·

5-'Ci:"~ • I1II.

f",..- I---

"• lIICL;

If" <,1_•.. r--

II r\. .-" <, r---

2 4

CO.X.OZ ~I ["

0 0

RtI.4(a) Sv""'C In"c~1ftand d.... coefflcl.nt yen... Mach Fig. 4(c) 5II.....olllc Int..c.pto.-Itltudo y.nvI Mach nVlllbor fo,nv.......nd a"I" ......ck Itoady I...... flight

where H is the final altitude desired, taken in this call as themaximum altiwde for steady horizontal flight (the service ceil­ing). Initial conditions were

\ ( D F cos (1) (L F sin (1)1\", +),1' - - --- + >..,. - - - -- ~ 0

m' Ttl i mlV mlV(66)

(67)

(68)

nh-Hh-Hh-H"Y,

gm "'"

M = 0.22

"Y = 0

h - 0x = 0

Wo

•.1/- O.ll,.If - 0.9,

-t,-t,-t,

In the nomenclature of Sections 5 and 6, three caBes were cal­culated:

Cue123

where W. is the initial weight.The resultll are shown in Fig. 5. The trajectories all show the

"zoom" characteristic found by other investigators [91; the air­plane first climbs, then dives accelerating through sonic speed,then pulll up, trading kinetic for potentia! energy in the earth'sgravitational field. Thill characteristic appears to be caused bythe sharp transonic drag rille, the rapid throat attenuation withaltitude, and the thrust increase with Mach number. Thesesurprising trajectories show the danger in using claaBical per­{ormance methode on high-per(ormance airplanes. The quasi­steady analYl'is to detennine local maximum rate-o{-dimb showsan almost COD8tant slightly subsonic Machnumber to be the best{or climb. If the airplane is flown this way. it takes nearly tWlllllaa long to reach the altitude H &8 it does with the tmieetoriesshown in Fig. S. This quaei-steady ftight path was used &8 thenominal path in the steepest-aacent method and the optimumpath W&8 reachedin six or seven "steps."

Note that the addition of the M = 0.9 terminal constraint in­creased the minimum time to climb by 8 per cent. Further addi­tion of the 'Y - 0 terminal con8traint increased the minimumtime to climb less than 1/ , per cent. The angle-of-attaek pro­gram8 are not at all unusual and should not be difficult to ap­proximate for actual flight situations.

QA aa I.Z .. 2DIIIACH l1li• .,.

Q*----=~-+~~-I--l

a - a(A) ill speed of Bound.given &8 a tabular functionJh(h. M) iI fuel COllIumption,given &8 a tabular function'" it maasof vehicle

The adjoint differential equatioDllare:

x, + x,(~ ~ _~ _ !!... 1.. (JCD)ma oM mV ma CD e>M

+ x (g COl"Y +~ +~ ..!... llCL _ F sin aT V' mV' mVa Cz. ?:>M mV'

(62)sin a OF) 1 Cnh

+ mYa oM + >'. lin "Y + ),~ COlI "Y + X.. ~ llM = 0

X., + >'1' (-g COl"Y) + x, (g~ "Y) + >'_V COB"y

- >'sV sin "Y = 0 (63)

x, + >'1' (_ !! ~ r!!! + ~ ..!.. C>CD M ~ +~ lJF)'" p d1I m CD c>M II d1I m c>h

+ x (..!:..! ~ _..!:....!. ()Cz. ~ ~ + sin a C>F) (64)T mV p d1I mV Cz. llM a d1I mV lJh

Fig. 4(.) hponoftlc InIwc............hNIt venVI Mach nVIII"" and altl.tuft

(65)

226

43

ANGLE·OF-ATTACK• IDEGREES)

5 ;=:::::':"T£R=:::M:::lNAL:;";t"~MACH==NO:".-;;:CCHS=;;TItAI=-:;IIED;:;;-:TO=O....:.•:-.=TEIt=":::_=L--'FUGH1' MTNANGt.£ CONSTMlNEDTO ZERO

---T!MIlNALltACHNO.CIlNSTIWIIfD TOo"--NO T£AlIIlNAL CCHSTRAlNTS

3t+----f--~+~.L--+-_T~_f

432

tOI-l-...d~--4-----+-----+-----~

o

"it- TEItMlNAL MACH NO.CONSTlWNm TO0", TUlIIlNAL

,LIGHT MTM MeLJ: CONSTlWNED TOZEIlO"J------1.--TEIlMItIAI. IIMCMIIIlUXlNITllAlNm TO0...-NO TDIMINAL: CONITItAIIn'S

"oALTITUlIE I IHolUIlIttUII ALTITIIIlE 'Oft

STOOY HOItlZONTAI. FUOHTXoHOflIZON1M. DISTANCE

fit. 5(.) Sup.,. IftIWC......-.u.ht ....h for ..laI tint. to dlmlt.. ',""Ie celli rr. lev.'-ItIIuII. v_ .

.... ICe) SupenonIe 11IMrafIIe for """'...... HtII. to eIImltto nrvIn celli....... _ 1eveI_ ' d1 v.,.., rant•

ANGLE-O'-ATTACK.IDEGREESI

5 _ TERMINAL MACH NO.CONSTRAINED TO0.',TERMINAL'LIGHT Pll'TN ANGLE CONSTAAlH£DlO ZERO

, --TERMINALMACHNO.CONSTltAINEDTOO.9'_-+-_--!, --NO TDlMI!"ALCOHSTRAIHTS,

• "TIME I+---=~-~c....j,o· MAXIMUM RATE-OF-CUM'

ATSEALEVELI

I' 2~btMENSlONLESS TIMEI

..... '(11) ........e Intwc.,........ht path for ........._ H•• t.dltlllo

...,""Ie celli..hili _ levcJ-ql........Ic~•• M....

No IIAXU'''M ALTITUOt:'Oft ITEADY LEVEL'L•...,.

" 0 ALTlTUllC

1,0

,.••

"" •

.2

00 .5 ID lIS

MACH NU...,.

fl. sa) S.,.,..nie In'.e ,.....ht path f.r ..1_uIIIII_ .. cllmltto nntce cell.... frllII leve'-lIItucie v.,... Mleh _Iter

hH .. 0.81

9 Elillpl12An~ek ProtIra.. for • Su..-.le In!ere.,..., .. CIInIIt ..

MaIM... AIIIIvtI.. The same airplane considered in 8ec:tlOIl 8 iauaed here with the pay-oft' function, ~, being altitude, A. TheItopping condition ia () - 'Y - 0; Le., flight.-path angle aero.The initial conditiOD wa.s taken as the maximum eI1e1'1)' condi­tion for eteady horizontal flight, where tlIllll'lY per unit mall ill(V'/2) + gh; i,e., kinetic plus potential enel'l)'. This maximumenergy condition occurred at

oo

II 0 ALnTUD€ I

H 0 IIAX.IIUIIALTITUDI!: fOR STEADYHORIZONTAL fLIGHT ,-

X 0 HORIZONTAL DIsTANCE

/V-

'Y =s 0

M = 2.1

2.0LO 1.5MACH HUMlI(R

5

.... t'--...<,

r-,'\~

II 0 ALTITUMH • MAXIMUM ALTITUDE FOR STEADY

HOAlfNTAL f'l,IGHT Ioo

.5

2.0

1.5

In thia caae DO terminal constraints were used. The physical andadjoint equatiOlll are the same as those in Section 8.

The reaultiDg flight path is BhOWD in Fig. 6. It again contaiDaa pntliminary cUmb and dive, followed by a zoom to the mui·mum altitude which is 64 per cent higher than the aervice c:eiliug(DIAllimum altitude for steady horisontal flight). Ifall the energy

74:5X

H

2

0.5

""1.0

2.0

1.5

2rt

I IH· MAXIMUM ALTITUDE FOR- STEADY HORIZONTAL- FLIGHT

- X • HORIZONTAL DISTANCE /'_./

./V

»:....

12

:: 10...II:

'"~ ..~ &:!

~ 4,~ 2

'"ZC 0

o 2X

H

·6 7

LIFT

___GLIDEII ZERO-LIn AXIS

llRA6!.-"+~;""'~::·7::VELOCITY.V;- -

Fl.. 'Cc) Supersonic Intwceptor-fllllht path for ...axl ...um altilud_.""I..,.....clc venu. ra"".

Fi.. 1 NOlllondaturo us'" In anolysl. of hyporaonlc mllal gilder

oo

- ·ANGLE OF ATTACK

A .& IDDIWl COEFFICIENTICoI

.2

::.~ ·1S'r--t---:Pt""-+--i--+--~t ?l:'--~-~~

~ .4r-~'---r--+--t--+---+---Cl~-~..:i . 2'r..,..-t--+--+--i--+---+--+-\-~

CL.CLoS1H. COS.ISIH_I

Co'Coo +CoL,SINJ.)

.8~-~--..--.._-,-_-,-_--.·

z

t. TIME /'i... MAXIMUM RATI:'OF-CLIMeATSEA LEVEL /H • MAX/YUill ALTITUDE FORSTEADY HOlIIZONTAL /FLIGHT

/ ir ~

/-- /

• 7......'" 6IIIe e•~ 4;'!!C 3

~ Z...i 1c

•

'1II,r6(e11 Supor.onic Int..c.pl.r-fl1Ilht path for m.xlmum altitude­.ngl........aclc "'..v. II....

Fig.. HypI ..oltic orbital glldor-llft and drall coefficl.nt. a. fvndionsof angl_Httack

(69)

(71)

(72)

(73)

(70)

(74)

g(h) cos -yVI

g(h) sin -yV

Ilt 1ds ". V

d" = sin 'Yds

~+ U.h,M,a)R + h mY'

UD ( R : h )', acceleration due to gravity

radius oC earth (344{) nautical miles)

1CIi,a) '2P(h)Y'S, drag Coree

CL(a) ~ p(h)VtS, lift force

d'YdB

U

L

R

D

where

convenient to use distance along the flight path, " as the indepen­dent variable in solving the differential equatioDll. The physicaldifferential equatioDll used are;

dV D{h, lIf, a)---;t; = - mV

wherem mass of vehicleUD '"' acceleration of gravity at earth's surface8 - wing plan-form ares

The initial conditions used were

V 25,920 ft sec-I'Y = 0.18 degh -= 300.000 ft

Owing to the "ide range of velocities encountered in thisproblem (landing speeds were around 200 ft sec-I) it was found

";D = 27.31b ft-2

10 Elample 3Anille-of-Attadr ............ for. Hyponoltic Orbital Gild.. 10 Achl.....

Maxlmum Ra"g.. The problem here is to find the angle-of-attackprOl1'am a(t) for a hypersonic glider to achieve maximum rangeon the eurface of the earth, starting from the point where it hasbeen injected into a low-altitude satellite orbit by a rocket booster;The nomenclature for the analysis is shown in Fig, 7, and the lift­drag characteristics are shown in Fig. 8. The wing loading usedwas

could be converted into potential energy, the maximum altitudewould be 97 per cent higher than the service ceiling. Note thatthe thrust essentially vanishes near h/H = 4/3: actually theturbojet engines will "blowout" somewhere near this latteraltitude.

228

d>.,,+>. (2g sin'Y) + >. (29COS'Y)+>. (-~)=o

cia" V' ., V' lV'

(75)

p • P(h), density of atmosphere, a tabular function (ARDestandard atmosphere used)

OD :2 Olla +ODLlsin'al, Olla "" 0.042, ODt co 1.46, Newtoniandrag coefficient

Or, = Or. sin a Isin al cos a, OLt '" 1.82, Newtonian lift co­efficient

The adjoint differential equations for the influence functionsare:

4

."

2" - )"'AHCll.f~':TTIlCICr IL/~~2

( ~- .J...~I .ANClLE·OF·ATTACX

/ INlTlALCOMOCTIOIlv,,·zs,tZO"1SII:

I,J....:SOO.OOO" r-'0 ·OJlIlED

II!!/'.27.3LllSin

Fig. 9(&) Hypenonlc orbital gllde~ngl...f-a"ock versus range

Fig. 9(0) HYflersonlc orbital glld_hllvde venus range

ti...~ 400 I..13,OO-~-+----+-'!~~~"I:::?o~c-~...! 200-r--r--;;t;;::.;!;;;=~~r- ~-:.---+-~:d

~!: 100+---+-4~cI...

o 5 10 15 20 25 50 35 40X'SURFAa; RANGE IN THClUSUlDS OF NAUTICAL MILES

o4--+--L........-....,...--..~---1I--+-~+--Go

(77)

(76)

(787dh. = 0cis '

d>.., x .(_ gcos'Y)+>. (Dsin'Ych+" VI ., V'

sin 'Y ) ( sin 'Y )- --- - >. --- + A. cos 'Y ee 0,R+h • l+~-

R

d>.. x ( 2g sin 'Y _..!!.......!. dP)ch + .. V'(R + h) mV' P dh

(2gcos'Y cos'Y L 1 dP)

+ h., V'eR + h) - (R + h)' + mV' -; dh

References

v .lD "" log V; (V. some reference velocity)

The resulting flight path is shown in Fig . 9(a) . For comparisonthe flight path for a '" 20.5 deg is as shown ; this is the angle-of­attack for maximum lift-to-drag ratio, which is in this case(LID)... "" 2.0. It can be seen that the optimum a(t) programdill'ers from the a "" 20.5 deg path most significantly in the first10 min of the flight; this is truly the critical part of the 8ight.The optimum path achieves 15 per cent more range than themaximum LID path. Again the optimum control variable pro­gram, Fig. 9(b) should not be difficult to approximate in practice.

1 C. R. Faulders, "Low Thrus~ Rocket S~eering Program forMinimum Time Transfer Between Planetary Orbits." Sociel.y ofAutomotive Engineen, Paper 88A. October. 1958.

2 J. V. Breakwell, " T he Optimization of Trajectories," Journal0/ the Socidl/ 0/ IMu.mal4M Applied Ma1hemDtic" vol. 7, 1959, pp.215-247.

3 A. E. Bryson and 8. E. ROII5, "Optimum Trsicctorics WithAcrodynamic Drag," Jet Pr01/Ul8ion (now Journal 0/ the AmericanRDcket SocieM . vol. 28, 1958. pp. 46l1-!169.

4 ~. J. Kulakowski and R. T. Stancil, " Rockct Bo09~ Trajec­tories for Maximum Burnout Velocity," JournoJ, of tile AmericanRockd &cidu, vol. 30,1960, pp. 612-619.

!5 H. J . Kelley, "Gradient Theory of Optimal Flislht Paths,"Journal 0/ tTte Ammicon Rocket &cUll/' vol. 30, 1960, pp. 947-953,

6 H. J. Kelley, "Method or Gradients," chapter 6, "Optimiza­tion Techniques," edited by G. Leitmann, Academic Press (to bepublished).

(81)

(82)

<I> = tf>{x) +~'(f(x. a) + f.)

where ;),Ijo' is a row matrix of Lagrange multipliers to be deter­mined for our convenience. Note that the term multiplying l.'is zero by equation (3) so that <I> so t/>. Take the differential ofthis quantity:

(bt/> Of) , bfd<l> - -+ ;),,,'_.. dx + ;),. - da + ;),/ df.,bx bx ~a

APPENDIX

and evaluate the partial derivutives at the nominal point a. ­x·:

I U.e of LaOrange Multipliers for Small Perturbation. Abaul a Give"COn.....1Variable Palnl.

For the maximum problem stated in Section 3 of the paper, wewiah to determine the changes in t/> and 1/1 for a small perturbation<ia in the control variables from 110 nominal point a·. To do thisconsider first thc quantity

7 A. B. Bryson, W. h'. IJenh ..m, 10'. J. Carroll, and K. Mikami," Lift or Drag Programs That Minimize Re-entry Heating," JOIl,nolof the Aero,pacll Scillncu, vol. 29, April. 11162, pp, 420-430.

8 A. E. Bryson, "A Gradien~ Method (or Optimising Multi­Stage Allocation Proeeeses," Proceedings, Harvard Univcrsity Sym­posium on Digital Computers and Their Applications, April, 1961 (tobe published).

9 H. J . Kclley, "An Investigation of Optimal Zoom Climb Tech­niques," Journal 0/ Aero,pace Science" vol. 26, 1959. pp. 794-803.

to G. A. Bliss, "Mathematics for Exterior Ballistics, " John Wiley&: Sons, Inc. , New York , N. Y., 1944.

11 H. S. Tsien, "Engineering Cybernetics," chapter 13, "ControlDesign by Perturbation Theory," McGraw·HiII Book Company,Inc., New York, N . Y., 1954.

(80)

(79)

where

229

d+ - # - ((~). + ~'F ) d. + ~'Gd« + ~.'dfo, (83) (98)

or

Thus the maximum of ~ occurs when the coefficient of iliavanishes in equation (93). This will be the case if (103)

ddI. (elx) == F(t)6x + G(t)8cr

where

1.. == ~et»'GW-1G'~ (99)

Substituting (97) and (98) iDUJ (94) and (89) we obtain the resultsgiven in SectiODS 4 of the text in equations (21) and (23), respec­tively.

These results have a simple geometric interpretation in the cr­hyperspace. Equations (89) and (90) with clio = 0 may bewritten

dtj) - V</Klu == ('VcPW_ 1/ t )W1h dcr (1 0)

dq -= V~da =a (Vv.W-I/')W'ltdo. (101)

i.e., ~'G ezVq, is the gradient of tP in the a-hyperspace, and~.p'G !:2 V1k is the gradient of 'l!. For the moment we will con­sider q to be a lingle scalar quantity rather than a columnmatrix. If W is the metric in the a-hyperspace then dP is theinfinitesimal distance from the present nominal point to a neigh­boring point a + d«. We wish to choose d« to maximize dq"keeping d1/J = 0, for a given value of

dP =: IW-'hdal.

Hence we must subtract the component of V¢W- 1/ , that is

parallel to V.pW-I/s from Vq,W- l /a. Using the Pythagoreantheorem we have simply then

(c14J)Z == (V~W_l/J)(W-'''V4J') _ [(VepW-l/t)(W_I/tV~')]'dP 'P (Vl/IW-1/t)(W_1/tVt/!')

= VcPW-1VlIIt. - (VtPW-IVI/I')f (102)." VJ/lW-1V~'

:::: I.et» - I~'I#-J'i4

which gives a geometric interpretation of equation (24). Clearlythis quantity is positive unless Vt/J is parallel to V"'. in which caseit iSlero.

3 Ad;oi"' D ntia' Equations for Small Perturbation, AboutGiven Control V.rIo "..,.",••

For the optimum programming problem stated in Section 6 ofthe text, we wish to determine the changeJ!l in ., ~ and n forsmall perturbatioDS, 8cr(t), in the control variable programs aboutgiven Dominal eontroJ variable programs ••(t), where n -= 0 isthe stopping condition. To do this we consider the linear equa­tions (39) describing small perturbations about the nominalpath:

(84)

(85)

(87)

(89)

(90)

(91)

(86)

(88)

dq, -= ~/GdQ + ~'dfo

d~ == ~'Gda. + 'J..~/df.

(dP)' cz da.'Wda.

where the nomenclature is explained in Section 4 of the peper,Now let us ehOOle ~' so that the eoefficientof dx vanishes; Le.,

(~). + ~'F" 0,

This reduces equation (83) to the expression

dq, == :>-.. 'Gdcr + '-.,/ til.

An exactly similar procedure yields

d'k == ~'Gdcr + ~'dfo,

where

2 St t-A.c..t M....... in Orcllftary Calculu. Us_... a WeJ.hteclSctu.. ' 1... of the Control Varia"'.. to D......ln. Step Slz••

The problem, 88 stated in Section 4 of the text, is to choose du80 88 to maximize dtP for given values of d~1 diG (usually zero), andtlP, where

where v' is a 1 X fJrow matrix of eoDstants, and p is a constant, allto bedetermined for our convenience. Note that the quantitiesmultiplying .' aad #j are both zero, by equations (90) and (91).Take the second differential of this quautity:

d't/J := (~'G - .''J..",'G - 2lUlcr/W)d t a (93)

We use the process of Seetion 1 of the Appendix again; Le., we.coDlider a linear combination of the three foregoing equa.tions:

~ - ~'Gdcr + ~'dfQ + Y'(d~ - l.,,'dfo - '-p'Gda.)

+ P«dP)1 - da'Wd.) (92)

(94) To determine the changes in tj), ~I and 11 we introduce the lineardifferential equations adjoint to (103) defined 8S

SubltitutinR '-97)aud (94.1 into (jill and 101YiD2 for u.we obtain

( 104)

(105)d- ().'6x) = ).'GSude

d:>--:- = - F'(t)lat

where ~. is an n X 1 matrix, 4J, is an n X p matrix, and A.o is anft X 1 matrix. If (103) is premultiplied by).' and (104) is pre­multiplied by (8x)' and the transpose of the eeeond product isadded to the first product, we have

~I d(8x) + rD.' ~x == l'F8x - ~'Fax + ~'G6uell tit

or

(96)

(95)

(97)

wbere

1da. a& 2~ W-J(G/~ - G~.)

Substituting (94) into (90) we have

1d~ ~ - (IH - I"'I/f~)

2p.

d~ :::I dq - ~/dI.

I~ =- J.p/GW-iG/~

IH == ~/GW-l(;').#

Solving equatioD (95) for Y we obtain

" -= -2.uleJ",-ld~ + IH-ll~.

230

(115)

(116)

(118)

(119)

(122)

dll - d1i - ~D/(,.)&X(,.)

.## - /..'1'~'GW-JG~

" - -2pIH- ldO+ I~~-JI~. (120)

4) :!: [1•• - IH'I~.,,-JI~ ] (121)...p. =- (dP)1 - d~/IH-Id~

dt/I - f..T ~'G81f1lt + ).,.o'e4)8.(4)

dtl' - j',.' ~'G8" + ~0'(to)8.l")

Subetitutinl this into equations (110) and (111), we have the re­latioDl

Now we take the variatioD of this relation (117):

6(d4J) ... f..1' (~'G - v'~'G - 2~e'W)6lcrd~

where 1.(Ie), d~, and dP are coDlidered CODltaDta. The maxi­mum ~ will occur if the coef&eieDt of 'III in the integrand of(118) vauiaheaj i.e.,

where

18ex &II 21£ W-I G'(~ -~,,)

Subltitutinl (119) back into (116) and then (113) we CaD solvefor the CODltaDta " and 1':

II> ... f..1'~'GW-JG'~1

I .. - f..T ~o'GW-JG'~ndt

SubstitutinC (120) and (121) into (119) and (115) we obtain therelU1ta pven in Section 6 of the text in equatioDl (SO) and (51),respectively.

(106)

(109)

(110)

(111)

(112)

(113)

~:: 8.+~T

d~ - 8q + tkdTdn=80+DdT

(T .dt; = J" ~/G8crdl+ ~'(I0)&.(Ie) + <lJdT

1: '1' •dt - ~'G8crdt + ~/(1o)8x(lo) + +ITt.

o ... dO ... /,.7'19'G6..u + '-0'(10)6.(10) + OtlT

(tIP)1 ... f,.T 8«'WBad,

II we let

11 we integrate equatioDl (105) 'rom It to T, the result is

().'h)1-2' - f..1').'GBIfIlt + ().'Ix),_••

~'(T) - (~). ; ~'(T) ... (!t). ,()X ,- T ()X '-T where the nomenclature is pveD iD equatioDS (51). Next we con.

'-o'(T) .. (=r (107) eider & linear combiDatioD of (116) and (113) with (1115); i.e.,

d4J - (,.,'1'(~'G - v'~o'G - p&e'W)lcrdtwhere the nomenclature is given in equatioDB (47), it is clear that J,

-+ (~/(Io) - y'l~'(Ie»8x(,.) + Y'd~ + lA(dP)1 (117),. - (~'lx),-2'; 8t- (~'aX)"'T; an .. (~'8.)...2' (108)

For unall perturbatioDS the value of T will be changed only asmall amount d'l't 10 that

where the nomenclature is given in equatioDS (48).SuhetitutiDl (107) and (109) into (106) we obtain equatioDl

(42), (43), and (44) of the text.

4 II-A.cent M ln Calcvlu... y n. U W••hWM 01 Centrol V•.., D SI•••

The problem 88 stated iD Section 6 of the text is to choose 8u(t)10 as to maximise d4J for liven values of d.J '-<Ie) (usually zero).and dP, with dO -= 0 where

We use the analog of the process of Section 2 of the Appendixwith only small differences arising from the additional term intiT in the foregoiDl equatioDl. The first step is to elimiDate dTfrom equation (112):

liT 1dT a - -. 19'G8adt - - 10'(10)6.(10) (114)D " n

231