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154 AIAA JOURNAL V O L . 3, NO. 1Note that En(q, r) [F n(q , r)] is not the Laplace transform of

In these formulas (41-43), ft is used to refer to a particulareigenvalue q = ft n, j, satisfying (39); the temperature (41)involves a sum over all these eigenvalues (j = 1, 2 . . . ,n= 0, 1, . . . ) . In an earlier paper, Warren1 gives a solutionto this problem. The present method has computationaladvantages over Warren's solution, particularly in theneighborhood of the inner boundary r = b and for smalltimes. Indeed Warren's approach converges nonuniformlynear the inner boundary r = b and takes the value zero atthat boundary rather than the nonhomogeneous value of Eq.(32).

References1 Warren, W. E., "A transient axisymmetric thermoelasticproblem for the hollow sphere," AIAA J. 1, 2569 (November1963).2 Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids( O x f or d University Press, 1959), p. 309.3 Watson, G. N., Bessel Functions (Cambridge UniversityPress, 1958),p. 80.

A Steepest-Ascent Solution ofMultiple-Arc Optimization Problems

R O L L Y N G . G R A H A M *Aerospace Corporat ion, El Segundo , Calif.Nomenclature

j f ( ) = n-dimensional vector of known functions of x, u, an d tg = gravitational accelerationh = altitudem = massr = rangeT = thrust magnitudet = independent variable, timeta = corner timesu = r-dimensionalvector ofsteeringfunctionsv = horizontal velocityw = vertical velocityx = n-dimensional vector of state variable histories/3 = mass flow rate(At) = A-dimensional vector of corrections to the corner timesA z = s-dimensional vector of time invariant control variablecorrectionsB = thrust angleX = n-dimensionalvector of Lagrange multipliers< p = performance indexQ = g-dimensional vector of constraint functions, q < n

THE problem of trajectory optimization has received agreat deal of attention in recent years. The problem,as usually stated, involves the determination of one or moresteering functions, such as angle of attack or throttle setting,such that some performance index is optimized, subject tospecified constraints. One of the most successful approachesto this problem has been the so-called steepest-ascent orgradient technique, developed by Bryson and Denham1' 2 andKelley.3 The problems that have been treated with thisapproach, however, are actually special cases in that theyinvolve only one subarc. This note extends the steepest-ascent technique for the simultaneous optimization of timedependent funct ions and time invariant quantities, originallymentioned by Denham,2 to multiple-arc problems with theassociated unknown corner times.

Received April 14, 1964; revision received A ugust 11, 1964.* Member of the Technical Staff, Performance Analysis De-partment. Associate MemberAIAA.

The technique mentioned in R e f . 2 is applicable to the classof problems for which small variations in the constraintfunctions and in the performance index are related to smallchanges in the control variables (either time dependent orinvariant) in the following manner:

A 1 2 = rto, du dt

du dt( 1 )(2 )

In order to apply the technique of Ref. 2, the multiple-arcproblem must be expressed in the form of Eqs.(1) and (2).For general multiple-arc problems, the governing set ofdifferential equations for the state variables can be written

as follows:/(a)=(a)- )=Q

for ta-i

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JANUARY 1965 TECHNICAL NOTES 155

By expanding J(a) in a Taylor series, integrating by parts,.and using the relationship between variations at fixed timepoints and total variations, the following result is obtained:AF =

,0=1- E0=1

A rt- E l ,o = l '"a

5 w dt -

In order to -eliminate the last term, the multiplier functionsA are specified as solutions to the adjoint equations:A ta-1