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Nasa Cr71308 i c Annual Report for 1965 on Research Contract NAS 8-2619 NASA CR71308 APPLICATIONS OF CALCULUS OF VARIATIONS TO TFWECTORY ANALYSIS Submitted to National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama by M. G. Boyce, Principal Investigator J. L. Linnstaedter and G. E. Tyler, Assistants p *ai; ..%-a -k a*. Department of Matnema tics set t Lt Vanderbilt University a Annual Report for 1965 on Research Contract NAS 8-2619 APPLICATIONS OF WCULUS OF VARIATIONS TO TRAJECTORY ANAz;YSIS Submitted to National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama by M. G. Boyce, Principal Investigator J. L. Linnstaedter and G. E. Tyler, Assistants Department of Math ema t ic s Vanderbilt University Nashville, Tennessee January, 1966 INTROD3CTION AND SUMMARY During the year 1965 two reports were made at the two contractor conferences on guidance and space flight theory held at Huntsville in February and August. They were entitled "The Multistage Weierstrass and Clebsch Conditions with Some Applications to Trajectory Optimization," by M. G. Boyce, and "Applications of Multistage Calculus of VEriaticns Theory to Two and Three Stage Rocket Trajectory Problems," by G. E. Tyler. Also a paper was contributed to the Progress Report No. 7 on Studies in the Fields of Space Flight and Guidance Theory, NASA TM X - 53292, pp. 7 - 31, on "Necessary Conditions for a Multistage Bolza - Mayer Problem Involving Control Variables and Having Inequality and Finite Equation Constraints," by M. G. Boyce and J. L. Linnstaedter. A summary of the results in these reports, omitting proofs of theorems, is given in this annual report. Additional work has been done recently on gradient methods and on sufficient conditions, but results are as yet incomplete. The first section of this report contains a modification of the Denbow multistage calculus of variations theory, allowing for discon- tinuities at stage boundaries. This modification is an intermediate step in passing from the classical theory to a form readily applicable to trajectory optimization. The second section summarizes extensions of the multistage theory to problems involving control variables and having inequality and finite equation constraints. The Mayer formulation is used, and differential constraints are taken in normal form since in trajectory problems the equations of motion are in such form. A three stage re-entry problem is treated in Section 111. This example serves to partially illustrate the theory of Section 11. To avoid computational complexity, simple intermediate point constraints are assumed and first order approximation to gravitational attraction is used. 4. 2 Discontinuities will be allowed in the functions appearing in the differential equation constraints and in the dependent variable coordinates defining admissible paths Let t be the independent variable. For fixed p, define a set of variables(to, tl, eO., to be a partition set if and only if t < tlcu. < t Let I - 0 P denote the interval t < t < t and I the subinterval ta,l <- t < ta 0- - p a for a = 1, .*., p - 1 and taWl--< t < ta for a = p. Let z(t> denote the set of functions (z,(t), '.., zN(t)), where each za(t), a = 1, *.., N, is continuous on I except possibly at partition points tl, e.. t At these points right and left limits z (t-),z,(t,), + +'-!'are assumed to exist and we let za(tb) = az,(tb), 1+ b = 1, 0.. z a(t p-1 ..., p - 1. The problem will be to find in a class of admissible arcs t 0-<t<t - p' satisfying differential equations t in I~, f3 = l,*.*,M< N, (1) $aB (t,z,i) = 0, and end and intermediate point conditions one that will minimize Let R a be an open connected set in the 2N+1 dimensional The (t,.,;) space whose projection on the t-axis contains Ia a functions e' B are required to have continuous third partial deriva- I 1 Is assumed of rank M in R tives in Ra and each matrix a 3 I Let S denote an open connected set in the 2Np+p+l dimensional space of points (to, ,t z (to),z ,z *. ,z (t,)) in which the functions P , (ti) it,),+ f , p = 0, 1, e. , K have continuous third partial derivatives and the P matrix is of rank K+1. An admissible -set is a set (t,z,;) in R for some a=l,oo.,po I a An admissible subarc C is a set of functions z(t), t on Ia, with each a (t,z,?) an admissible set and such that z(t> is continuous and k(t) I is piecewise continuous on An admissible arc E is a parti- ‘a * I tion set (to, t ) together with a set of admissible subarcs ..., P + -. ‘a’ a = 1, ..., p, such that the set (to,..., t ,z(t ),z(t;),z(tl) ,...,z(tp)) I PO is in S e Multiplier Rule. An admissible arc E’ that satisfies equations (l), (2), (3) is said to satisfy the multiplier rule if there exist con- stants e not all zero and a function P F(t,z,i,h) = h BB$a(t,z,i), t in Ia, with multipliers h,(t) continuous except possibly at corners or dis- continuities of E’, where left and right limits exist, such that the fol- lowing equations hold: + Every minimizing --arc must satisfy -the multiplier -rule. An extremal is defined to be an admissible arc and set of muitipiiers satisfying equations (1) and (5) and such that the functions ia(t), (t) have continuous first derivatives except possibly at par- B tition points, where finite left and right limits exist. An extremal is -non-singular in case the determinant is different from zero along it. An admissible arc with a set of multi- pliers satisfying the multiplier rule is called normal if e = 1. With 0 this value of eo the Set Of multipliers is unique. Weierstrass Condition. An admissible arc E’ with a set of multi- liers (t) is said to satisfy the Weierstrass condition if -B 1 holds at every element (t,z,i,h) -of E’ for all admissible sets (t,z,i) satisfying the equations fl; = 0. Every normal minimizing arc must satisfy the Weierstrass condition. ,?‘Lebsch Condition. An admissible arc E’ with a set of multipliers is said to satisfy the Clebsch condition if B (t) F. (t,z,i,h) flu% 2 0 za% *. 5 holds at every element (t,z,i,X) -of E' for all sets (nl,, a e ,%) satisfying the equations a .- r\ ,z ) 31a '- J. @Pia (t ,i Every normal minimizing -arc -must sati s f y -the Clebsch condition. 6 By the introduction of new variables and by notational transformations the theory of Section I can be utilized to es- tablish necessary conditions for the more general formulation of this section. As before, let t be the independent variable and define a set of variables (t ..., t ) contained in the range of 0' P t to be a partition set if and only if t < tl < < t e Let I 0 P denote the interval to -< t -< t p' and let Ia denote the sub-interval t < t < ta for a = 1, p 1 and ta,l < t < ta for a = pa a.-1 - ..., - - - Let x(t) denote the set of functions (xl (t) , ..*, x n (t)). For each i, i = 1, n, assume x (t) to be continuous on I ..., i except possibly at partition points tb, b = 1, p - 1, where finite left and right limits exist; denote these limits by xi(tb) + and xi(tb), respectively. The amount of discontinuity of each member of x(t) at each partition point will be assumed known, and we write + with each dib a known constant. Also let xi(tb) = xi(tb)* Thus x.(t) is continuous at t, if and only if dib = 0. 1 Let y(t) denote the set (yl(t), ym(t)), where y.(t) is J piecewise continuous on I, .j = 1, .*., m, finite discontinuities being allowed between, as well as at, partition points. In the formulation of the problem the y.(tj will occur only as undifferentiated variables J and will not occur in the function to be minimized nor in the end and intermediate point constraints. Such variables are called control variables, while the x. (t) are called state variables. 1 The problem is to find in a class of admissible arcs which satisfy differential equations a 2 = Li(t,x,y,), t in I~, a = 1, p, i = 1, n, i *.., 7 finite equations Mi(t,x,y) = 0, g = 1, 0.0, q, inequalities N&X,Y) ,> 0, h = 1, ..*, r, q+r5m, and end and intermediate point conditions Jk(to, t x(t ), x(t;), ~(~1,+ x(tp)) = 0, ..., P' 0 ..., k = 1, ..., s C (n + 1) (p + l), + xi($)- - dib = 0, b = 1, ..., p - 1, one that will minimize In order to state precisely the properties of the functions in- be an open connected set in the m + n + 1 volved in the problem, let Ra dimensional. (t,x,y) space whose projection on the t-axis contains the interval Ia, and let S be an open connected set in the 2np + p + 1 dimensional space of points The functions L:, Mi, Nt are assumed continuous with continuous partial derivatives through those of third order in R a.nd a'. Jok J are to have such continuity properties in S. For each a, the matrix is assumed of rank q + r in R a,' where Da1 is an r by r diagonal a matrix with Nl, , Na as diagonal elements e The matrix .
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