THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aeronautical and Astronautical Engineering Instrumentation Engineering Program Technical Report A GENERAL THEORY OF MINIMUM-FUEL SPACE TRAJECTORIES Lucien W. Neustadt ORA Project 06181 under contract with: AIR FORCE OFFICE OF SCIENTIFIC RESEARCH OFFICE OF AEROSPACE RESEARCH CONTRACT NO. AF 49(638)-1318 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1964

A General Theory of Minimum-Fuel Space Trajectories by Lucien W. Neustadt t Introduction: This paper is concerned with the trajectories of vehicles moving in free space, i.e., of vehicles that are subject only to gravitational and propulsive forces. The following problem is fundamental in the control of such trajectories: Given the vehicle position, velocity, and mass at a specified initial time, find a propulsion program that brings the vehicle to a prescribed terminal state (in a terminal time which may be free or fixed) with a minimum expenditure of fuelo Such a program will be called optimal. The mathematical treatment of this problem depends very strongly on the model used for the fuel expenditure. In the case of a rocket engine, an excellent approximation is that the rate of fuel consumption is proportional to the magnitude of the thrust vector, and this article will deal exclusively with this representation. We shall assume throughout that no constraints are imposed on the vehicle position and velocity. This assumption is physically reasonable in many (but by no means all) space problems. Further, except for a brief discussion in section 9, we shall always suppose that there is no constraint on the allowed value of the thrust vector. Minimum-fuel thrust programs in the absence of any such constraints generally consist of a finite number of impulses. Although impulsive corrections can never be realized by an actual 1

rocket engine, a knowledge of the optimum impulses will often make it possible to compute the optimum, or near optimum, thrust program in the presence of the thrust amplitude limits which must exist in actual engines. The problem described above is clearly a variational one. In order to permit impulses, and yet have a precise mathematical formulation, it is necessary to place the problem in a somewhat unorthodox framework, and thereby arrive at a non-classical variational problem. This development is carried out in section 2. In section 8 we show that this framework is a reasonable one by proving both an existence theorem for solutions of the resultant variational problem and an approximation theorem which states that solutions of the unorthodox variational problem can be approximated by conventional thrust programs to any desired degree of accuracy. In sections 3-6, necessary conditions that an optimum thrust program and associated trajectory must satisfy are derived. Many of these conditions have been previously obtained by examining the necessary conditions in the presence of a thrust amplitude constraint, and then passing to the limit formally as the maximum allowed amplitude tends to oo (see, eog., Lawden [1]). In section 9 we show that this limiting argument is, in a sense, justified, and also prove an existence theorem for optimum trajectories in the presence of thrust amplitude constraints. Some specific examples of contemporary interest are discussed in section 7. Ewing [2] in 1961 adopted a viewpoint very similar to the one taken in this paper in his investigation of the same problem for the particular case where the gravitational field in which the vehicle moves is uniform. 2

The case where the gravitational field is linear in the space coordinates has been previously treated by the author [3]. While preparing this manuscript, it has come to the author's attention that the problem discussed in this paper has recently been studied, but from a slightly different viewpoint, also by Rishel [4]. 2. Problem formulation: The motion of a vehicle that is subject only to gravitational and propulsive forces can be described by the following differential equations: M(t) where rl, r2, and r3 are the coordinates of the vehicle center of gravity in some inertial, Cartesian coordinate system; G1, G2, and G3 are the components of the vehicle acceleration due to the action of the gravitational force field; T1, T2, and T3 are the components of the vehicle's thrust vector; and M is the vehicle mass. Denoting the vectors (rl,r2,r3), (Ga, G2,G3), and (T1,T2,T3) by r, G, and T, respectively, we may write down the single vector differential equation = G(,t) + (2.1) e- M(t) The rate of change of mass, M, is the negative of the fuel expenditure rate and, for a single rocket engine, is given by (2.2) g Isp 3

where {I |I denoted the Euclidean norm, and g and Isp (the nominal acceleration due to gravity at the earth's surface and the specific impulse of the fuel, respectively) will be assumed to be known positive constants. We denote (g Isp) by A. We shall also suppose that jG, ) is a continuous, bounded function from E3 x E1 to E3 (Em denotes Euclidean m-space) possessing continuous bounded first partial derivatives with respect to all of its arguments. This assumption is consistent with the conventional models of gravitational fields Throughout this paper we shall assume that an initial time to (without loss of generality, and for ease of notation, we shall set to = 0) and initial values for eqs. (2.1) and (2.2) have been given: M(O) = Mo > 0, r( (O) = (0)= o. If (-) is a summable function from [0,oo) to E3 satisfying the inequality lF(t)jIdt < AMo then it follows from standard 0 existence theorems that eqs. (2l1) and (2.2) with T(') = () have a unique solution1 for 0 < t < oo that satisfies the above initial conditions. This solution will be denoted by r(t;F), M(t;F); j(t;F) denotes the time derivative of r(t;F) Finally, we shall suppose that there are given functions hi( ~,,) from E3 X E3 x [0oo) to El, where i = l,...v and v < 6, with the following two properties: 1. The hi are continuous and have continuous first partial derivatives with respect to all of their arguments. 2. If H(t) = = [(x,):xEE3, yeE3, hi(,y,t) = 0 for i = l,..o,v}, then H(t) is a smooth manifold in E6 for each t > 0. For each t > 0, let 3(t) denote the class

of all summable functions F(o) from [O,t] to E3 that satisfy the relations J IF(t)j1dt < AMo and hi(r(t;F), r(tF),t) = 0 for i = l,...,v. Physically o speaking, j (t) consists of all thrust programs that "transfer" the vehicle from the position r and velocity do at t = 0 to a new state (at the time t) that satisfies the boundary conditions hi(rr,) = 0, i = 1,...,v. In the sequel, we shall consider two variational problems. The first, the fixed terminal time problem, consists in finding, for a given tl > 0, an element F( -')(tl) such that M(t;F) > M(t1;F) for every F(- ti). The second, the variable terminal time problem, consists in finding a time ti > 0 and an element F( tey(t)) such that M(tl;F) > M(t;F) for every pair (tV with t > 0 and FEc(t). In concrete terms, the basic problem is to find a thrust program that, for the given initial values, achieves prescribed boundary conditions, and that, in so doing, maximizes the terminal mass. If v = 6 and H(t) consists of a single point for each t, we shall say that the variational problem is a fixed endpoint problem; if v < 6, the problem will be called a variable endpoint problem. Let us now consider variational problems that are derived from, and equivalent to the above problems. Namely, replace eqs. (2.1) and (2.2) by the equations I = G()>t (2.3) p = z +U(t), where z, p, G and u are 3-vectors, G(.,.) is the same function that appears in eq. (2.1),and. ) is assumed to be an absolutely continuous, bounded 5

function from [Oo) to E3. We shall consider solutions z(.), (.) of eqs. (2.3) that satisfy the initial conditions z(0) = vo and p(0) = ro. For a given bounded., absolutely continuous function w(.) from [0,oo) to E3, we shall denote the solution of (2.3) with S-) = w() that satisfies the given initial conditions (it is easily seen that this solution exists and is unique for 0 < t < o) by z(t;w), p(t;W). We shall also say that p(t;w) is the trajectory that corresponds to w. For every t > 0O we denote byk' (t) the class of absolutely continuous functions,.) from [0,t] to E3 for which the relations w(0) = 0 and hi(p(t;w)z(t;w) + w(t),t) = 0 for i = l,...,v are satisfied. Now the derived fixed terminal time problem consists in finding, for a given tl > 0, an element ( o)eV(tl) such that' 11d t - <e dt l t ~o/t for every.uC(tl); the derived variable terminal time problem consists in finding a time ti > 0 and an element u( )~ (tl) such that ^ ii"i a ) t < dIt T I t o dt o dt for every pair (T,U) with T > 0 and UC(T). We shall show that the original and derived variational problems are equivalent. Namely, we shall exhibit a mapping ~ that, for each tl > 0, is one-to-one from'(tl) onto- (tl) (if we identify elements in (tl) 6

that differ only on a set of measure zero), and shall prove that FL(tl) is a solution of the original problem if and only if O(F) is a solution of the derived problem (whether the problem is fixed or variable terminal time). Define the mapping 0 as follows. If F(. /(tl), let u( ) = (F(.)) be the absolutely continuous function from [Otl] to E3 that is given by t F(s) t) = M( d 0 < t < t (2 4) We shall show that 0 is one-to-one fromcY(tl) onto-^d(ti), and that -1 =P,1 where F(-) ='(u( )) for uCO((tl) is defined by F(t) = exp((t;u)) d, < t < ti, (2.5) d.t (t;u) = - - d 1 )11 ds + In yo, 0 < t < ti. (2.6) A / ds0 Note that F(t) is defined by (2.5) for almost all tE[O,tl], since an absolutely continuous function has a derivative almost everywhere. At points where du/dt does not exist, F(t) may be defined arbitrarily. Since du/dt is summable, the integral in (2.6) is finite. Consider eqs. (2.3) with u = O(F), where F/l(ti). It is clear that p(;u) is absolutely continuous in [O,tl], and that, a.e. in [O,tl], p(t;u) = G(p(t;u),t) + (t) (2.7) S inct e M(t;F) Since 7

p(O;) o= O r(O;u) = (0;a) + -0 = v= (2.8) it follows that (replacing u by (F)), for all tE[O,tl], p(t;O(r)) = t;j), (t;~(V+)) + ()(t) = r(tj). (2.9) Hence, D(F~)(tl) by definition of ctl) andatl), or O(Q(tl))CQ(tl). Also (by (2.2), (2.4) and (2.6)), for all t0[O0tl], we have that M(t;F) = exp(ni(t;~D(F))]. (2.10) Let us show thattP((tl))C(ttlt). Thus, let F =(u), where ue tl). It follows from (2.5) and (2.6) that exp(p.( -;u) is absolutely continuous in [O,tl], that exp[i(O;u)) = Mo and that, a.e. in [O,tl], d [exp(k(t;u))] = -A F(t)ll, (2.11) dt tl so that {(F(t)lldt < AMo, and (see (2.2)), for all tC[O,tl], exp(k(t;u)) = M(t;2(u )) (2.12) It is clear from (2.3) thatp(t;u) is absolutely continuous in [O,tl]. Also (see (2.3), (2.5) and (2.12)), p(t;u) satisfies eq. (2.7) a.e. in [O,tl]. By definition of J(tl), u(O) = 0, so that relations (2.8) are satisfied. Hence, for all tE[0,tj], p(t;u) = r(t;u)), t;) + u(t) = (t;()) (2.13) from which it follows that F6E(tl). Now the relation C-1 =-tis a consequence 8

of (2.2), (2.4)-(2.6), and (2.12), and it only remains to show that 0 maps all of the solutions of the original variational problem onto all of the solutions of the derived problem. But it is a consequence of (2.6) and (2.12) that if uiG 1(ti), i = 1,2, then M(t1;(ul)) > M(t2;'(,u2)) if and tzi t2 only if s Ijuj(s)llds < I|u2(s)IIds, and this immediately implies the deo o sired result, Note that eqs. (2.9), (2.12), and (2.13) describe the correspondence between solutions of eqs. (2,1) and (2.2) and solutions of eqs. (2.3) when u and T = F correspond under the mappings 0 and 0 If u(o) is an absolutely continuous function from [Otl] to E3, then dI dt I dt = STV u (2.14) 0 where STV u, the strong total variation of u, is defined (see [5, p.59]) as follows: m STVu sup lu(Ti) -u(Ti9I)l, i=l with the supremum taken over all finite partitions 0 = T0 < Tj <e..< Tm = ti of [O,tl]. For scalar-valued functions (where STV reduces to the total variation), relation (2.14) is well-known. The proof of (2.14) (see, e.g., [6, p.209]) carries over from the scalar-valued to the vector-valued case with only minor modification. Thus, the original fixed terminal t ime problem is equivalent to the problem of finding, for a given ti > 0, an element ~6S(tl) such that 9

STV u = inf STV u; (2.15) u~E$( t1) and the variable terminal time problem is equivalent to that of finding a number tl > 0 and an element "Gx (tl) such that STV[O tl0 = inf STV [O0t], (2.16) ~~~t) t>O where STV[o t] denotes the strong total variation over the interval [O,t]. Unfortunately, there is, in general, no element u (tl) that achieves the infimum in the right-hand side of (2.15) or of (2.16). To circumvent this difficulty, we shall embed the sets B(t) in larger sets 4(t) possessing the following two properties: 1. If u(-) is any element of t), then there exist functions un('t(t), n = 1,2,..., such that un(s) + (s) as n + oo for all sE[O,t], and STV un + STV u as n + oo (see theorem 4 in section 8). 2. There is an element C?4(t) such that STV[ot]' = inf~ (t)STV[o t]u (see theorem 3 in section 8). Consequently, infu j(t)STV[O tt = infuet (t)STV[O,t]. For each t, 0 <t < o, we define p(t) as follows. Let (t) = u( -):u from[0,t] to E3 and continuous from the right in (O,T), u(O) = O, STV[O T] <}. For every wE(t1), eqs. (2.3) with u = w have a unique solution2 in [0,] that satisfies the initial conditions z(0) =, P(O) =ro. We shall also denote this solution by z(t;), p(t;). Then, for each t > O, let ( = w() = 0 for i =,...,v). (2.17) It is obvious that J!(t)C^(t)C(Bt )o 10

We shall denote by 63(oo) the set of all functions from [0,oo) to E3 whose restrictions on [0,t], for every t > O, belong to &3(t). We shall henceforth be concerned with the extended variational problems defined as follows. The extended variable terminal time problem consists in finding a number tl, 0 < tl < oo, and an element`e(t1) such that [STVO t - inf STV[O.t L J4(t) t>O The extended fixed terminal time problem is analogously defined. Sections 3-6 are devoted to the derivation of necessary conditions that solutions of the extended variational problems must satisfy. In section 5, we consider the variable terminal time problem, and in section 6, the fixed terminal time problem. 3. Variational equations: In this section, tl is an arbitrary fixed positive number and,(-) is an arbitrary fixed element of A(tl). Denote p(t;i) and z(t;2), for 0 < t < tl, by $t) and 2(t), respectively. Let./>&) denote the continuous matrix-valued function on [0,ti] whose i,j-th element./\ij is given by j(t) Gi(= t),t); i,j = 1,2,3; 0 < t < ti. brj We shall also use the notation t).)=,(, O t <tl. (3.1) 1r 11

For every function u(.)6E (tl), let bz(;u) and 5x(;u) denote the absolutely continuous functions from [Otl] to E3 that satisfy the equations d__ [t(t;u] =-~(t)Bp(t;u), dt (5.2) [6p(t;] = z (t;u) +u(t) - i(t) dt 14,'it, "I k R -— e almost everywhere in [O,tl], and assume the initial values 6z(O;u) = bp(O;u) =. (3.) We shall refer to eqs. (3.2) as the variational equations associated with (t). Since these equations are linear, their solution is given by the well-known variations of parameters formula, which here takes the form t bp(t;u) = - [Xl(t)Yl(s)+X2(t)Y2(s)][u(s)-/'(s)]ds, 0 5p(t;u) - [X1(t)Yi(s)+X2(t)Y2(s)][u(s)- us)] ds, where the 3 X 3 matrices Xi(t) and Yi(t) (i = 1 or 2) satisfy the differential equations Xi(t) =\(t)Xi(t), Yi(t) = Yi(t ^it), 0 < t < t, i = 1 or 2, and the initial conditions 12

Xl(O) = X2(0) = -Y1(0) = Y2(0) = I, Xi(0) = X2(O) = Y1(o) = Y(0) = 0, (3.6) I being the identity matrix. The matrices Xi, Yi also satisfy the following identity: lXi(t) X2(t)-Yl(t) Yl(t)\ / 0 l I ), o0 t < tl. (3. 7) \Xl(t) X2(tV-Y2(t) Y2(t)/ 0 I/ In conventional physical models of gravitational fields, the function (.,t), for every fixed t, is the gradient of a twice continuously differentiable scalar-valued function on E3. Under this hypothesis,/_(t) is symmetric for every t, 0 < t < ti, in which case (3.5) and (3.6) imply that Y1(t)= T T =-X2(t) and Y2(t) = Xl(t) for 0 < t < ti. This computationally useful result, which is known as Schmidt's theorem, but is apparently originally due to Siegel [7, page 14], was brought to my attention by 0. K. Smith. Integrating eqs. (3.4) by parts, using (3.7) and the fact that u(O) = =^(O) = 0, we obtain t bz(t;u) = [Xl(t)Yl(s)+X2(t)Y2(s)]d[u(s)-(s)] -u(t) +t(t), 0 (3.8) t bp(t;u) = / [X1(t)Y1(s)+X2(t)Y2(s)]d[ u(s)-( s)], 0 the integrals in (3.8) being in the sense of Stieljes. For every u(.)63(tl) and real number a, let ox(u,ca) be the element in E3 X E3 x E1 given by 13

Sx(ua) = (b (tl;u),5z(t,;u)+u(tl)-(tl),STVS [ot) (3.9) and let o= 6x(, 0) = (O,O,STV). (3 10) Now define the set W in E3 x E3 X E1 as follows: W = (5x(ua):ue6(tl), > o. (3.11) Clearly, wo6W. Since, for every u and w in 3(tl) and real number 3, we have STV(u+w) <STV u + STV, STV(u) = IjiSTVu, (3.12) it follows at once that W is convex. The set W is analogous to the cone of attainability described in [8, Chapter 2], and is also patterned closely after the convex set of variations introduced by Warga in [9, section III]. Let r be an arbitrary nonzero row vector in E7. If X = ( 1Q2,rT7), where 1qE3,2,CEs,3 and J17CE1, let p(t;)), for 0 < t < tl, be the row vector defined by 2 p(t;r) = 7 i()Yi(t), i() = rlXi(ti) + r2xi(tl), i = 1 or 2. - ^ LJ -v-W i=l (3.13) It follows at once from (3-7) and (3.13) that p(tit;r) =, P(t1;n) = -. (.14) 14

If we consider p( -;) to be a function from [O,tl] to E3 (for fixed), we conclude, by virtue of (3.5) and (3.13), that p(.;') is twice continuously differentiable and that p(t;) = p(t;n)-t), 0< t <tl. (3-15) Lemma 1. If there is a nonzero vector -T = (i1,...,T7)EE7 such that T-Sx _< f.) for all SxeW, then T7 < 0.,1.W 4O44O 40 Proof. The hypothesis of the lemma, together with the definitions of W, ao, and p(t;-T) (see (3.8)-(3.11) and (3.13)) imply that, for every u6(tl)9 tl tl p(t;Tl)du(t) + r7 STV u < p(t;T)du(t) + T7 STV Uo (3.16) o o We first show that T7 - 0. Suppose the contrary. Then, (3.16) takes the form ti tl / p(t;V)du / p(t;)d. (3 17) o o Since (3s17) must hold for every ugA(t1), it follows that p(t;j) - 0 in [O,tl], Hence, p(t;P) - 0. In particular, p( 0;') = p(O;) = O i.eo (see (3.6) and (3.13)) 01( r) = 2(r) = 0. But this implies that (see (3o7) and (3-13)) ( 1,... -6) = 0, i.e., T = 0, and this contradiction shows that 7r7 # 0, By hypothesis, -5 6x(l1) < T'o, so that (see (3.9)-(3.11))Tl7 < 0. Since T07 # O, r7 < 0. 15

Lemma 2. Suppose that ( *) 7 0. If there is a vector T = (n1,...,) with r7 = -1 such that.bx < fr for allxeW, then max Jlp(t;)ll = 1, (3.18) o_<t_<t and tl p(t; )d((t) = STV[ t. (3 19) Proof. Let Q = maxO<t<tllp(t;n)ll, and let T6[0,tl] be such that IIp(T;-)ll = I. Define w(-)6(ttl) as follows: j0 for 0 < t < T w(t) = T "(t)= K [p(T;)]T for T < t < t (an obvious modification must be made if T = 0), where K > 0 is arbitrary. Then ti P(t;rl)dw(t) = K23o (3020) 0 As in the proof of lemma 1, we can show that (3.16) is satisfied for every u6e(tl) and in particular, for u = w. Since STV ~ = KQ, we conclude that 44, _sn/ /4 tl tl p(t;T)d - STV = KQ(2-l) p(t;j)d - STV - (3.21) 44dw -tV SVw-32 o o But (3.21) must hold for every K > 0O which implies that 2 < 1. 16

It is easily verified that p(t;j)d('-t) < max IIp(t;i)ll STV/ = _'STV, o LO<_t < so that tr p(t;Tb)d - STV < ( —1)STV ~ < 0. (3.22) 0 But (3.16), with u 0, gives rise to the inequality ti p(t;T)d - STV >0. (.23) S 4 ttl ~ ~*yV(3.23)V 0o Combining (3.22) and (3.23), we obtain (3.19)If Q < 1, it follows from (3.19) and (3.22) that STV~u= O, i.e., -O 0. This contradiction shows that Q = 1, i.e., (3.18) holds. This completes the proof of lemma 2. Lemma 3. The interior of W is not empty. Proof. Since W is convex, it is sufficient to show that W does not belong to any flat in E7 of dimension less than seven. Suppose the contrary. Then there is a nonzero vector i = (r1 o...,7) in E7 such that (because TOeW)n.Wo = n-bx for every 5x6W. In particular, = ix(1), so that (see (3.9)-(3.11)) l7 = 0. But this contradicts lemma 1, and thereby proves lemma 3. 17

If ui( ),...,u7( ) are arbitrary fixed functions in Q(tl), we define the function u(, ) from [O,oo) X E7 to E3 as follows: 7 7 (1- 6jj)(t) + 6jj(t) for 0 < t < t, j=1 j=1 u(t,1,...,67) = u(t,6) = (3.24) 7 7 (1 - Xj)t l T) + bjuj(tI) for t < t < oo. j=1 j=1 Note that, for every fixed 6CE7, u(,56 3(oo). For ease of notation, let ^*~" 44- 44p(t,6) = p(t;u(-,6)), z(t,6) = z(t;u(-,5)), 0 < t <o. (3.25) W 44. H'14, 144, 4.- e 44L, *%_ e We shall also consider p(, ) and z(, ) to be functions from [O,oo) E7 to E3. It is easily verified that, for 0 < t < ti, p(t,0) = Pt), z(t,0) = (t). (3.26) It follows from well-known theorems on the dependence of solutions of differential equations on parameters that 6z(t,6)/5bi and ap(t,5)/65i exist and are continuous functions of t and 6 in [O,oo) X E7? In addition, for fixed 56, these derivatives are absolutely continuous functions of t which, for almost all tE[O,ti], satisfy the equations (3.27) d ~ F..(t,5)) Litz( t )q)+ u(t) t) at 6i 6618 18

together with the initial conditions az(0, )/65i = ap(0,)/65i = O. In particular, it follows from (3.26) and (3.1)-(3.3) that, for O < t < tl, z ( t5Pt \('kbL)_ = 0 = 6z(t;ui),) = bp(t;ui), i = 1 7 6 1.8 0 o = o 11 IV" 1 "t (3.28) 4. A fundamental lemma: In this and the next section we shall suppose that q(.) is a solution of the extended variable terminal time problem, and that ti > 0 is the corresponding terminal time. We shall keep the notation introduced in section 3. Let (tl) -= r, (t1) + (te) = v. (4.1) By hypothesis, (r,v)CH(tl), or hi(r,v,ti) = 0 for i = l,...,v, and STV[o, t]< STV[o ta for every ut~( t) and every t > 0. (4.2) For (x,yt) in a neighborhood of (r,,ti), there is a parametric representation of the manifolds H(t) = ((x,y):xEsE3,,yE3 hi(x,y,t) = 0 for i = 1 o.o.v] of the following form: x = (^t) y = Na(ot), where Xi(,) and, 2(-,.) are functions from a neighborhood of (0,ti) in E6-v X E1 (let this neighborhood be of the form N1 x N2, where N-CE6e- and N2CE1) to E3, possessing continuous first partial derivatives with respect 19

to t and. the coordinates of, and r= kl(O,tl), (4.4) V = x2(o,tl). If we have a fixed endpoint problem, so that v = 6, ki and X2 are continuously 44- A'4differentiable functions from a neighborhood to tl to E3 with kl(ti) = r, x2(tl) = v. (4.5) Let Vhi(x,y, t) denote the vector in E6 defined by /h = hi ahi ahi 3hi ahi ahi Vhi =( - -—,, i = 1,.e v \6xi ax2 0x3 ayi ays ayY / By definition of a smooth manifold, the vectors Vhi(x,y,t), i = 1l,.ov, are linearly independent whenever (x,y)EH(t). Let T denote the hyperplane of dimension 6-v that is tangent to H(tl) at (r,v), i.e., T = ((x,y):xeE3,E3,yE,[Vhi(r,v,tl)]e(x-r,y-) = 0, i = 1 oV} (4 6) If v = 6, T consists of the single point (r,v). Now define the set Q in E7 as follows: Q = ((x-r,^, STV +Oa):(x,)T,c< O. (4 7) It is easily seen that Q is convex. If v = 6, Q is the ray L = [bx(),oa):oa 0). Let t =)( (t1)Gtl) ( 1, Gtl) -; tE7; (4.8) 20

and, for every u(-.)C3(tl), oacE and ACE1, let (u,cAa) = 6x(u,a) + A (4.9) and let V = [65(u,OA):ue8(tl), > O0,-oo < A < oo). (4.10) It is clear that RWCV, that V is convex, and that o VTQ (see (3.10), (4.6), (4.7), (4.9), and (4.10)). We now prove a fundamental lemma. Lemma 4. If there is a number \ > 0 such that G.) is continuous in (ti-x,tl), then Q does not meet the interior of V. This lemma is similar to lemma 11 in [8, page 112]. The proof we shall give below is based on the proof given by Warga of an analogous lemma [9, lemma 3.1]. Proof. Let us assume that the lemma is false, so that there is a point qCQ that belongs to the interior of V. Let q = (-rr-vSTV I), (4011) where (x,y)(T and a < 0. If a = 0, we replace a by -e, where E > 0 is sufficiently small, and thereby obtain a point that also belongs to both Q and the interior of V. Thus, without loss of generality, we shall suppose that a< 0 in (4.11). Since q is an interior point of V, there are seven points X1i,^..X79 in V, such that 03, Xi,*. X7 are vertices of a 7-simplex which contains q in 21

its interior. Let X. = 68(wjj,aj), where w.(.)43(tl) and j > 0. For every e, 0 < c < ti, and j = 1,...,7, define j(;E)C3(ti) as follows: wj(t) if 0 < t < ti-e or t = ti u.(t;e) = w (tl) if ti-cE < t < ti. It is easily seen that 6t(uj(;C),CAjA) + X as E - 0 for each j. Hence, for any fixed E > 0 sufficiently small, the points o and 5(uj(;Ec), aj,Aj) for j = 1,...,7 are vertices of a 7-simplex which contains q in its interior. Let Eo be one such e, denote uj(';C) simply by uj(.), and let. = Wu(u.(),cU.A.), j =,..,7. (4.12) 4J1 4 -J' J J Then the vectors (Lj-o), j = 1,...,7, are linearly independent, and there are positive numbers,y1,...,Y77 such that 7 7 q= jt Y7 = 1. (4.13) j=0 j=0 Note that the functions u (H), j = 1,...,7, are constant in [tl-eotl). Let T( ) be the function from E7 to E1 defined by 7 T(1,...,67) = T() = T) = t + jA (4.14) j=1 let ko = min(X,Eo), and let N be a neighborhood of 0 in E7 such that T(_)EN2 and T(5) > tl-0o whenever 5~N. Let u(.,), p(.,), and (.,5) be defined as in section 3 (see (3.24) and (3.25)). Note that u(,.) is continuous in t and 22

b for tC(tl-Xo,oo) and all b5E7. For each EN let (-;b)6(tl) nd i;b)63(T()) be defined as follows: 7 u(t) = ^(t) + bj[uj(t)-(t)], 0 t < t, (4.15) j=l u(t;) = u(t,5) for 0 < t < T(6), U(T(6);5) = u(t1;5). (4.16) It is clear that u(t;0) =q(t) for 0 < t < tl. If T( t) = tl, then u(t;b) = u(t;b) for 0 < t < t1, and obviously STV[O T()Iu(;' ) = STV[ot lu(.;). It easily follows that the same equality holds if T(5) > t, and it is not difficult to show that, if T(5) < ti, then STV[oT()](-;~) < STV[O,tl]u;] ), (4.17) i.e., (4.17) is satisfied for all b5N. Note that (see (4.16) and (3.25)), for 0 < t < T(b), p(t;U(;)) = p(t,), z(t;U(; )) = z(t,5). (4.18) * <1 OV Ifto- ^' * ~ -'* *' Since (QDy) and (r,) both belong to T, the entire line segment joining the two points (assuming that they are distinct) belongs to T. Denote this segment byi; is tangent to H(tl) at (rv). Hence,' is tangent at (rv) to a smooth curve r on H(ti). Let r be represented parametrically as [((l(( (s),tl), 2(c2(s),tl)): -1 < s < 1), where &( ) is a continuously differentiable function from (-1,1) to N1, o(0) = 0, and 23

6-v 6-v X( o t1)td1(01) X (i() 2( 0 ))r-v) (4.19) 6Lj ds a 6j ds j=l j=l If (x.y) = (r,), and this must be true if v = 6, we let F consist of the single point (r,v), or, equivalently, let C(s) 0, in which case (4.19) remains valid. Let 8(., ) be the function from N (-1,1) to E7 defined as follows: (b1,...,67, s) = (,s) = 7 ( fi(5)-gl(5,s),f2(5)+f3(5)-g2(s),' I j [STV uj-STV ]-s, (4.20) j=1 where the:j(:,) are functions from N to E3 defined: as follows: () = p(T); f2() z(T(b)); f3() = (T( 5);5)2 21) and the g.i(..) are functions from N x (-1,1) defined by gk(,s) -= 2k((s),T(.)), k = 1 or 2. (4.22) We shall show that (',' ) is continuously differentiable in N x (-1,1). It follows from (4.15), (4,16) and (4.21) that f3(') is differentiable and that f3(b)= Ui(t1) - q(t). (4.23) It was shown in section 3.hat the partial derivatives p(t,6)/4bi and z(t,6)/30i exist and are continuous functions of t and 5 in [0,oo) E7. 24

Further, az(,')/6t exists and is continuous in [0,oo) E7, and inasmuch as u(,') is continuous in (tl-koo) X E7, ap( -,)/6t exists and is continuous in (t-\o,o) X E7 (see (3.25), (2.3) and footnote 2). It now follows from (4.21), (4.14), and (2.3) that fl(-) and f2(') have continuous derivatives in N, and that, for tN, ai (( + [ (T ), )+u( T( ),)]Ai a5 i asi^ t = T) (4.24) abz (e) + G2( T(5 ),b ),} T() )Ai. ~5i = \ ~'5i't T Because of the differentiability properties that we have assumed for the functions ac(*), Ki(. ), and k2(-,*), it follows from (4.22) that x1(', ) and. 2(*,,) have continuous first derivatives in N x (-1,1), and ow 6-v 8 Sk(^s) V k(c(s),T(6)) dfj(s) as LJd as j=l (4.25) (,S)~- = k((S),T(~~)) Ai, k = 1 or 2. ai at It follows from (4.20)-(4.25) that 8(',') has continuous first partial derivatives in N X (-1,1). In addition, by virtue of (4.14), (3.28), (3.26). (3.24), (4.1), (4.8), (3.9), (3.10), (4.9) and (4.12), ( _) 0 = Xi - Xo (4.26) s = 0 25

Also, it follows from (4.20), (4.25), (4.14), (4.19), (4.11), (3.10), and (4.13) that 7 ( s ) = r- Zj(j-Wo'. (4.27) s = O0 0 o j=l If v = 6, the Xi are functions only of t, and obvious notation changes must be made in (4.8), (4.22), and (4.25), after which eqs. (4.26) and (4.27) follow in the same way. Now consider the vector equation e(s ) = 0 (4.28) for the unknown 6 as a function of s. For s = O, it is easily seen (see (4.20)-(4.22), (4.14)-(4.16), (3.26), (4.1) and (4.4)) that (4.28) has the solution 5 = O. Because of (4.26), the Jacobian of eq. (4.28) at 6 = 0, s = 0 is the determinant of the matrix whose columns are the vectors (wci-o). This Jacobian does not vanish inasmuch as these vectors are, by hypothesis, linearly independent. Since E(,.) has continuous first partial derivatives with respect to all of its arguments in a neighborhood of (0,O)6E7 X Ei, we can appeal to the implicit function theorem and solve eq. (4.28) for 5 as a continuously differentiable function of s in a neighborhood of s = O, Say b =J(s) = ((l(s),...,~ r(s)). ThenO(O) = 0 and da/ds can be obtained by differentiating (4.28) implicitly: 26

X ^(s),s) ~(s) + a( s),s) o a5sj (is as j=l In particular, for s = 0, we obtain, by virtue of (4.26) and (41.27), 7 7 Z(Uj-W) d (0)-o (4.29) j=l j =l Since the vectors (Wj-o), j = 1,.,7, are linearly independent, (4.29) implies that dj(0)/ds = 7y for each j. Recalling that 7j > 0 and Xj(O) = O for each j, we conclude that /j(s) > 0 if s is positive and sufficiently small. Also, (s)- 0 as s 0. Thus, let s, 0 < s < 1, be sufficiently small that ~(s)eN, ij(s) > 0 for each j, and (s) < 1. Denote ((s) and. j(s) by and 5j, respectively, j=1 so that (65,s) = 0. Let T = T(). It follows from (4.20)-(4.22) that p(T,5) = xi(a(s),T), (4.30) z(T,) + (T;) = x2((s),T), (. 31) 7 sj[STV uj-STV'-~j] = sa. (4.32) j=l But (4.30), (4.31), (4.18), and the representation (4.3) of H(t) imply that hi(p(;U(.;~)),z(T;U(.;)) + U(T7;),T) = 0, i = l,..., v, (43) 27

i.e. (see (2.17)), ~u(.;)C(T). But it follows from (4.17), (4.15), (3.12), (4.32), the non-negatively of the aj and. j, and the relations s > 0, c < 0, that STV[oT]U(;) < STV[O, t1l ) + a < STV[O, t 11( ) contradicting (4.2), and thereby proving lemma 4. 5. The necessary conditions: We now prove the following theorem which provides necessary conditions for the extended variable terminal time problem. Theorem 1. Let.(' ) be a nontrivial (i.e., q Z O) solution of the extended variable terminal time problem, let tl be the corresponding terminal time, and let $t) be the corresponding trajectory. Suppose that the points of discontinuity of ( d) do not cluster at ti. Then there exists a twice differentiable (column) vector-valued function (') from [O,tl] to E3 such that T (- ( t) ( t) (t) (5.1) for all t[Otl] r, for all t6[0,ti], max 1V(t)11 = 1, (5.2) O<t<t 1 and Ptl T [v(t)] d(t) = STV[o (53) 28

et (4.3) and (4.4) be a parametric representation of H(t) in a neighborhood of ($(tl), z(tl;-)i tl),tl) = (r,v,tl), and let T be the hyperplane tangent to H(tl) at (r,v). Then (.) satisfies the following transversality conditions: (ti).(x-r) = (r(ti) (y-v) for all (x,y)ET, (5.4) or, equivalently, v ('v(tl), -r(ti)) = liVhi(=rvtl). for some real constants l,).. v,) i=l and a(i) 0, ~.. // ~.. ti)~)). -_r(tl) [G(r tl) - aO t1)1+ (tl)F -FO t) ] = (tl) [/(t)(t~)]>o + tcL. at t If 11((t)11 < 1 in some interval [t',t"t][0,tl], then /( ) is constant for t' < t < t". If (T) /QT-) for some TE(Otl], then 11U(T)U = 1, and there is a.number K such that U T) - T- ) = K (T), K >. (5.6) If 0 = (0) 0+), then I1(0)11 = I, and(O+) = o j(0), where _o > 0. In particular, if D = (t: 11(t)11 = 1, O < t < ti) is a finite set, then( i) s a step function whose points of discontinuity all belong to D, and whose jumps are given by (5.6) or the modification thereof if T = 0. 29

Note that eq. (5.4) is satisfied trivially if v = 6, since, in this case, T consists of the single point (r,v). Also, if v = 6, aki(0,tl)/t ( = 1 and 2) in (5.5) should be replaced by d\i(tl)/dt. Proof. By lemma 4, Q does not meet the interior of V. Since <oCVTQ., is a boundary point of V. Now both V and Q are convex, and VDW, so that, by virtue of lemma 3, the interior of V is not empty. Hence, there is a supporting hyperplane to V at o that separates V from Q. Let T = (1L,)277) i O, where 1 3E3, saE3,and T7EE1, be a normal to this hyperplane directed so that fl ~ < n-'o < r9- for all 5EV, Q (5.7) 44 4 ~~' 4 1 i By virtue of lemma 1, relations (5.7) allow us to conclude that -T7 < O. WithT out loss of generality we shall assume that T7 = -1. If we let'jt) = [p(t;j)] y/4V eqs (5.2) and (5.3) follow from lemma 2 (see (3.18) and (3.19)), and eq. (5.1) follows from (3.15) and (3.1). Let (x,y) be an arbitrary point of T. Then (see (4.6) and (4.7)), (x-r,y-v,STV/\)EQ, and (-x+r,-y+v,STV )EQ. Consequently, by virtue of (5.7) /4i 44 4,.41*. AV *t 4g /4 * Al and (3-10), rll X-r + (2(y-v) = 0a A-I AV AX Taking (3.14) into account, we obtain (954). The equivalence of (5.4) and (5.4T) follows at once from the definition of T (see (4.6)). Consider the points 6(uy,0,~1) = St+ of V (see (4.9), (4.10), and (3.10)). It follows from (5.7) that T.~(~t) < O0 i.e.~, T = 0, and the equality in (5.5) follows from (4.8), (4.1), and (3o14). 30

The final conclusions of the theorem are direct consequences of (5.2) and (5.3) (see [3, theorem 3]). It only remains to prove the inequality in (5.5). If tli) =(tlI), the inequality is obvious. If tl1) # (tj), then Ilr(tl)| = 1, and, since 11r(t)1l < 1 for 0 < t < ti, =to 0<(.It)t t_ 24(t1) (t1). (5.8) d tt =_ -ti Relations (5.8) and (5.6) imply the inequality in (5.5), completing the proof of theorem 1. The vector-valued function (4(*),-"( )) is analogous to the adjoint variable in the formulation of the Pontryagin maximum principle, or to the Lagrange multipliers of the classical calculus of variations. Relation (5.3) corresponds to the maximum principle itself, or to the Weierstrass E-condition. If the set D is finite-say D = (TIl,...,T)-~then (o ) is determined by 6+i scalar parameters: six initial values for eq. (5.1), and the I constants KT in (5.6). Indeed, given the values of these parameters and the i initial values z(0), p(0), it is possible to "simultaneously solve" eqs. (2.3) with u = i and eq. (5.1), and determine /i through (5,6). /t /w1 -VI. 6. The fixed terminal time problem: In this section we shall derive the necessary conditions for the extended fixed terminal time problem, Thus, let ti > O be fixed, and let (') be a solution of the corresponding extended fixed terminal time problem. We shall keep the notation introduced in sec31

tion 3. Let (r,v) be defined by (4.1), so that, by hypothesis, hi(r,v,ti) = 0 for i = 1,...,v, and STV[O,tl ] STV[Otl]u for every uC(ti). (6.1) We define T and Q as in section 4 (see (4.6) and (4.7)). Corresponding to lemma 4, we have the following proposition. Lemma 5. The set Q does not meet the interior of W. Note that here it is not necessary to assume that (.) is continuous in (tl-k,tl) for some X > 0. Proof. The derivation is almost identical to that of lemma 4, with certain simplifications, and we shall only outline the necessary arguments. Assuming the contrary, we show that there are points wj = x(uj,aj), j = 1,...,7,such that each oCGW and wo0,l,-...aY7 are the vertices of a 7simplex that contains a point q of the form (4.11), with a < 0Y in its interior. We let T(5) = ti for all E7, and consider solutions of eq. (4.28) near s = 0, where E(, ) is again given by (4.20)-(4.22) and (4.16), and the function (.) has all of the properties described in section 4. Relations (4.26) and (4.27) can now be derived as in section 4, except that (4.26) can be obtained without having to show that ap(t,6)/at exists in a neighborhood of tl (since T( ) - tl). The continuity of u in (tl-X,tl) was used only in showing the existence of this derivate, and consequently the extra continuity hypothesis can here be dispensed with. It then follows as before that there is a vector 5 possessing the same properties as in section 4 such that eqs. (4.32) and (4.33)9 with T = ti, are satisfied. 32

But these equations are inconsistent with relation (6.1), and we have a contradiction. This completes the proof of lemma 5. We now have the following theorem. Theorem 2. Let( i.) be a nontrivial (i.e., I 0) solution of the extended fixed terminal time problem, ti being the terminal time, and let / ) be the corresponding trajectory. Then there exists a function r(.) from [O tl] to E3 such that,( -) anda. ~) satisfy all of the conditions stated in theorem 1 with the possible exception of (5.5). If the points of discontinuity of ( ~) cluster at tl, then, in addition, (^Jfl) = 0, lll(t)ll = 1. (6.2) d t = ti Proof. Just as the existence of a vector r = (i1,wt2,Ti7) which satisfies ^t^ ^ tAl -_ (5T7) followed from lemma 4, it is here a consequence of lemma 5 that there is a vector M = (l,Ma2,,T7) such that Tbx < _0Wo < 7o for all bxEW and GCQ (6~3) With the exception of (5.5),the conclusions of theorem 1 now follow from (6.3) as in section 5. Since 11r(t)11 = 1 at all points of discontinuity of, and 11r(t)l) is differentiable (and certainly continuous) at t = tj, the last sentence of theorem 2 follows at once. Note that if ~( ) is a solution of the extended variable terminal time problem, it is a fortiori a solution of a fixed terminal time problem~ Therefore, theorem 2 also provides necessary conditions for the case excluded in 33

theorem 1; i.e., if the points of discontinuity of.() do cluster at ti, the conclusions of theorem 1 remain valid, with the exception that (5.5) must be replaced by (6.2). 7. Examples: Let us apply theorems 1 and 2 to some problems that are of contemporary interest. First consider the variable terminal time, fixed endpoint problem (sometimes referred to as the "rendezvous" problem) in which il(t) and X2(t) (the functions that describe H(t)) represent the position and velocity, respectively, of an actual or fictitious target at the time t. The equations of motion of such a target can be written in the form.t = 2(t) t G(%l(t),t) + (t), (71) dt, t dt where at) is the non-gravitational acceleration experienced by the target. Since (7.1) must hold, in particular, when t = t1, relations (5.5) in this case take the form (see (4.5)) *(tl) a(tl) = (tl) [Stl)- tl)] > O. If a(t) - 0, i.e., if the target is in a "free-fall" trajectory, we obtain Ar(t )-[tti)-ttl)] = 0, (7.2) which implies that either Stl) =1(tT), or that II(ti)ll = 1 and (see (5.6)) V(ti)4((ti) = 0, i.e., relations (6.2) are satisfied. Also consider the following three variable endpoint problems. 34

The first, the so-called "intercept" problem, is the problem in which the vehicle terminal position is specified (but may depend on the terminal time), and the terminal velocity is arbitrary. In this case, eqs. (4 c3) can be put in the foir x = k1(t), y = ad and H(ti) = T = (xy): x = 1(t)). The transversality condition (54' ) in this case implies that _(tl) = 0. Consequently, Ilr(t)ll < 1 for t' < t < tj and some t' < tl, so that, by theorem 1, U(t) is constant for t' < t < ti. This conclusion is valid whether the problem is with fixed or variable terminal time. For the variable terminal time problem, relations (5.5) take the form (since (tl) = (tl) and 4(tl) = 0) dt f(tl)oL- ii) ( ^ )[t )J As a second example, consider the case where the terminal velocity is specified (but may depend on the time) and the terminal position is arbitrary. Then, eqs. (4.3) can be put in the form x =, y = k2(t), and the transversality condition (5.4') implies that t(tl) = 0 whether the problem is for fixed or variable terminal time. For the variable terminal time problem,(5.5) takes the form *(tl) (r,t) _ a (t0) O ^ 1^^ dt J In the third example, the "transfer to a specified orbit" problem, we shall assume that G(rt) is independent of to Here, the vehicle terminal 35

position and velocity are to be the same as the position and velocity at any point on a specified solution curve (i.e., orbit) of eq. (2.1) with T - 0. Thus, for every t > 0, s G(.x(s) c )) H(t) = (x(s),y(s)):-o<s<oo d d (x(s)), x() a (0) I~ ~l.41L ds' ds ]t - % where a and b are given vectors in E3 that specify the orbit. Then, in the notation of section 4, T = ((r+dV, v+cG(r)): -oo < < o), and the transversality condition (5.4) takes the form..(t l) = G-()) tr(tl). (7.3) Since H is independent of t, the functions Xi and X2 can be chosen to be independent of t, so that, for the variable terminal time problem, we have (by virtue of (5.5) and (7-3)) that (7.2) holds, i.e. either (tl) =/(tl) or eqs. (6*2) hold. 8. Existence and approximation theorems: In this section we shall prove that the sets %/(t) possess the two properties described in section 2. We first prove an almost self-evid.ent, but nevertheless interesting,lemma. We shall say that a function h(*) from [0,1] to E3 is regular if h(o) is continuous and piecewise smooths and if dh/dt is of strong bounded variation in [0,1]. Lemma 6. Let h(-) be a regular function from [0,1] to E3, t an arbitrary positive number, and Zo and zl arbitrary vectors in E3. Then there exists a ~~' ~ ^~- ~ ~ ~~ ^^ ~< ^ ~~ ^ ~~ " ~~ ~ "

function u( )EG(t) such that the solution (.), p() of eqs. (2.3), with u = u and initial values p(0) = h(0) and z(0) = z satisfies the equation p(t) = h(t/t) for all tE[O,T] as well as the boundary condition z(t) + (F) = Z1i Proof. For each tE[O,t], let t:(t) = G(h(s/t),s)ds + z, (81) 0 and let the function wt ()3(t) be defined as follows: (d 1s) + for 0 < t < t as + s = (t/t) w(t) = 0 for t = 0 tzl for t = t Since the function G(',) is bounded, it follows that the function:( ) defined by (8.1) is of strong bounded variation in [0,t]. Let u( ) = (l/T)(o)-5(.). Then e3(t). If we set z(t) =:(t) and p(t) = h(t/t) for t[0O,t], it can be verified directly that the functions z( ~) and p( ~) are a solution of eqs. (2.3) with u = u that satisfy the initial and boundary conditions prescribed in the statement of the lemma. Let K(.) be a function whose domain is [Ooo) and whose range is the class of subsets of E3. Preserving the notation of section 2, we shall in addition denote, for every t > 0, by l(t;K( )) the following subset of W(t): g( t;K(.)) = [w(~.): w(.))( ), p(t;w)K(t) for every tE[0,t] ) 37

We now prove the following existence theorem. Theorem 3. Let ti be a given positive number and let K( ) be a function from [0,oo) into the class of all closed subsets of E3 with the property that there is at least one regular function h( ) from [0,1] to E3 such that h(t/tl)CK(t) for every tE[Otlj, h(O) =, and (h(l),y)EH(t) for some yEE3 ThenN(tl;K( )) is not empty, and there is an element.(.)tl;K( )) such that STVtl]' ) = inf STV[otl]u) (8 2) ~_~tl;K(')) Proof. The fact that /(ti;K( )) is not empty follows immediately from lemma 6. If (tl;K(-)) is a finite set, the theorem is trivial. Thus, let u('), n = 1,2,..., be a sequence of elements of (t,;K( )) such that lim STV[o,tl]n = inf STV[o,tl. (8.3) n-oo /u((tl;K(. )) Denote the functions z(;un) and P(;n) by Zn() and n( ), respectivelyo Because the function G(,.) is bounded, the derivatives zn(o) as well as the functions zn( ) themselves are uniformly bounded, and the zn(~) are equicontinuous on [O,tt]. Since un(O) = 0 for each n, and the numbers STV[to jt].n n = 1,2,.,., are bounded, it follows that the functions jn( ) are uniformly bounded on [Otl]. Consequently, the derivatives n (.), which exist almost everywhere in [O,tl], are uniformly bounded and the functions P(') are themselves uniformly bounded and equicontinuous on [0,t1]. Appealing to Arzela's theorem [6, p. 122] and the Helly selection theorem [11, p. 222], we conclude

that there is a subsequence of the Un( ) (which we shall continue to denote by un, without loss of generality), and functions u )( )),( ), and p(.) from [Oti] to E3, where u, is of strong bounded variation and z, and po are continuous, such that, for every t6[O,ti], lim un(t) = U ) lim (t) = (t), limn z(t)= z(t).(8.4) n->oo n-oo n-oo Also, it is easily seen that limn STVn u STV[limn n], i.e. (see (8.3) and (8.4)), STV u < inf STV u. (8.5) - uIG ti; K(.)) Now define the function *(') from [O,ti] to E3 as follows: u (t ) for 0 < t < tl Ugt) = (8.6) LO(t) for t = 0 or t = ti Since u( ) is of strong bounded variation, uo(t+) exists for each t(0O,t1), and since there are at most a denumerable number of points at which a function of bounded variation is discontinuous, (t) = (t) for almost all t in [Otl]. It is easily seen that STV < STV u, so that, by virtue of (8.5), STV < inf STV u. (8.7) u(t1; K(*)) It follows from (8.4) and (86) that O), so that )C(t). In addition, because of (8.4), (8.6), and the continuity of the functions a n hi we have that p,(O) = 0, (0) =o, and hi(p,(tl),z(tl)tl),tl) = O 39

for i = l,...,v. Since the sets K(t) are closed by hypothesis, it follows from (8.4) that po(t)K(t) for all t[O,tl]. Consequently, if we can show that poO() = p(,~') and that z () = z(-;), we can conclude that /~ (tl;K(.)), and (8.2) is then an immediate consequence of (8.7). By virtue of (2.3), t Zn(t) v +f G(n(s),s)ds, 0 t n(t) O r +o [zn pn(t) = ro + [Ln(s)+un(s)]ds, 0 < tj 0 Since the functions G, Zn, and un are uniformly bounded, we can appeal to the Lebesgue dominated convergence theorem, and conclude that t zo(t) = Zo + G( oo(s),s)ds, 0 t p(t) =rO jo [z (s)-)(s)]ds, < t ti, where we have also used the fact that S(t) = (~t) for almost all tC[O,tl]. It now follows immediately (see footnote 2) that z () = z(';i) and po( ) = = p(.;/), completing the proof of theorem 3. If K(t) = E3 for every t > 0, then i(tl;K( )) =/(tl), and it is evident that there exists a regular function h(*) with the required properties. The existence theorem promised in section 2 then follows at once from theorem 3. We now prove the following theorem. aa hn O)+1 n -n1 nT7 + n nc P —r In -:~n7

Theorem 4. Le-t'(.)UW(tl) for tl > 0. Then there exist functions un(')* (tl), n = 1,2,..., such that: 1. the derivatives dun/dt are essentially bounded, 2. un(t) -/(t) for each tc[0,tl] as n oo, and 3. lim STV u STV /U n->oo -wn -IP Proof. We first prove two lemmas. Lemma 7. Let wl( ), w2( ),... denote a sequence of uniformly bounded functions in 3(tl) such that wn(t) +/(t) as noo for each it[0,tl], where'i.)-&(t). Then t p(t; and z(t; a.+t ) z(t-Ga) as n-oo uniformly in [o,tl] Proof. Let Gn( ) be the scalar-valued function on [O,tl] defined by an(t) = IIP(t;wn)-p(t;/| + z(t;wn)-z(t;)|ll. Then it follows from (2.3) and the boundedness of the partial derivatives aG./ar. that 0 (t) < RGn(t) + lt)-(t)l- for some positive constant R and all t{[0,tl]o Now Gn(0) = 0, so that, for 0 < t < tl, t ttl Gn(t) < R Gn(T)dT + IT)-n(T) dT. 0 0 tti It follows from the Lebesgue dominated convergence theorem that o 1 -T)-wn(T)lldT + 0 and n -+ o, and the lemma is now an immediate consequence of Gronwall' s inequality. Lemma 8. For every E > 0 there exists a 6 > 0, depending only on E, such that the following proposition holds: If w( ) is any absolutely continuous function in 3(tl) whose derivative is essentially bounded, and A and /\ are any vectors in E3 that satisfy the inequality I11 + 1 Kl < 6, then 41

there exists an absolutely continuous function u( )66(tl) with essentially bounded derivative such that STV(u-w) < E and p(tl;u) = p(tl;w) + $, z(tl;u) + u(tl) = z(tl;w) + w(t) +. (8.8) Proof. Fix E > O. Let G = sup ltG(r,t)11, E = min (e/(7+24), 1,tl), and 6 = (). Let' ) be an arbitrary, fixed, absolutely continuous function in 3(tl), and let s and e be arbitrary fixed vectors in E3 such that II|l + + 1l1z < 6. Define the element () of ((ti) as follows: 0 for 0 < t < tl-e (t) = V (t-tl+E)J, for tl-e < t < ti-E/2 (t-tl)m2+x for tl-E/2 < t < t, 2 2 where ml (4a/ )-2/E and m2 = (x/E)- 4/E It can be immediately verified that (.) is absolutely continuous, that its derivative is essentially bounded, and that tl wti) =, / t t)dt = ^. (8.9) 0 Also, STV ~ *) = I [ ll2] < + 2') < + 26 = 4E + 2E < 7 [1I111 * 111_2 11 <j tN, 2 ^ 2^ - c E (8.10) Define the absolutely continuous functions p(~), z( ), ~(')} and u( ) from [ 0, t] to E3 as follows: 42

p(t) = rO + [Z(T;Z)+(T)T)]dT, 0 t z(t) = vO +/ 2G (T),T)dT, 0 (811) 5(t) = z(t;w) - z(t), u(t) = ~t) + w(t) + _(t) Since the function G(-,~) is bounded, it follows that the derivatives of z(.;w) and of z( ), and consequently of ~(.) and of u( ), are essentially bounded. Now (8.11) and (2.3) imply that,for 0 < t < tl, p(t) = p(t;, z(t) = z( t;u), and eqs. (8.8) therefore follow from (8.9), (2.3) and (8.11). Further, u()-w(),) =(.*)+'(-), so that (see (8.10)) STV(u-w) < STV w + STV 5 < 7~ + STV 8. (8.12) Since ^(t) = 0 for 0 < t < tl-E, it is a consequence of (8.11) and (2.3) that, in this interval, p(t;w) = T(t) and z(t;) =(t); i.e., S(t) = 0 for 0 < t < tl-E. Because i(*) is absolutely continuous, rtlr STV j = -(<)ll?,tl I<(T)ll~d o O th-E But II5 < 2G (see (8.11) and (2.3)), so that STV 5 < 2Ge; i.e. (see (8.12)),

STV(u-w) < (7+2G)E < E. This completes the proof of lemma 8. 4/. We now turn to the proof of theorem 4. Since( )6E4(tl), ^ is of bounded variation and continuous from the right in (O,ti). Consequently, ^(.) has the representation *) = ()+r2() were ) a ae in 3(tl), il(') is continuous, and/'2(.) is the jump function of ~.). Let T1,T2,... denote the points of discontinuity of () (or, equivalently, of ~2( )). Without loss of generality we shall assume that they are infinite in number. Let vi = 2(Ti)-2(Tj) if Ti 7 0; Vi = 2(0+)-u2(0) if Ti = 0. Then 00 STV u = STV^Il + STV, STV~ = 1ill < ~ i=l Let s(;T) denote the unit step function at t = T: O for t < T I 1 if T f 0 s(t;T) = < S(T;T) = 1 for t > Tif T 0 for t > T}'[ 0 if T = 0 For each positive integer n, define the absolutely continuous functions (-), ^ W^ (.)e6(t1) as follows: Wn(t) = /1(itl/n) + [(nt/tz)-i]L[l((i+l)tl/n) - ~l(itl/n)] itl ti for ~ < t < (i+l) -, i = 0 n-l n n n 2"n() = 1 is';Ti) ) i=l 44

Let n( ) =n(')+n() It is clear that -n(t) +-(t) uniformly in [Otl] as n oo, that each (')((t1), and that n STV n STSTV w+ l[i < STV 1 + STV STSTV. (8-13) *fn -l -~. -~.-' i=l For every E > 0 and positive integer j, define open intervals IjE as J E follows: I = (0,E) if T = 0; I = (T.-eTj) if T. > 0, JE J j,, J J J and let r(; Tj-r ) denote the absolutely continuous, real-valued functions defined as follows: 0 for 0 < t < Tj-E r(t;T.j,) = - - (t-Tj.+) for T. - E < t < T if T. > 0 J - - J1 for Tj < t < ti t for for 0 < t < E r(t;Tj,e) = if Tj = 0 1 for E < t < ti Let e1,eC2... be a strictly decreasing sequence of positive numbers, with En < l/n for each n, such that, for every n, the intervals IlEno...,I nen are mutually disjoint and all contained in [0,ti]. Let the absolutely continuous function w )n(')(tl) be defined as follows:

n Wn() = wn() + r(.;Tin)i i=l It is easily seen that the derivative of w (.) is essentially bounded, that ~~On ~~n n(t) =~n(t) for toJIi~ I (t)(t)IlI < Ii for tI STV n <STVw + I STV (8.15) j~~~i=l and that (see (8.13)) STV w < STV w + L[~i- < STV /.^ (8.15> Now, ill + 0 as i -> o; i.e., for any fixed E > O, there is an i' > 0 such that Ivil < for all i > i'. Consequently, it follows from (8.14) that i' lln(t)- (t)ll < for all tUIi n. Since, for each i, Ii DIi E n+ and i=l 00 0n Ii =, we conclude that(wn(t)-n(t))+ 0 as n + oo for every tE[Otl]. n ~<^^ <n n=l Recalling that /n(t) - t), we obtain that lim w (t) = it) for all t[0.tl]. (8.16) n-oo Further, since wn(0) = 0 for each n, it follows from (8.15) that the wn(~) are uniformly bounded. Appealing to lemmas 7 and 8^ we can assert that there exist absolutely continuous functions un( )E13(tl) with essentially bounded derivatives such that 46

P(ti;un) = P(tl;/) z(tl;un) +Un(ti) = z(t1;U) + tl), and STV (un-n) - 0 as n o. Thus, eachunX(lti), and (-(t)-un(t)) O0 as n - oo for all tE[O,tl], which, by virtue of (8.16), implies that lim n(t) = /t) for all tE[O,ti]. (8.17) n-oo Now STV u, < STV +STV- (see(8.15)),<STV'u1. Now STV un < STV wn+V(un-)-. Therefore (see (8.15)), lim supnSTVn < STV/. But it follows from (8.17) that lim inf STV u > STV~. Consequently, limnSTV u exists and is equal to STVY. This completes the proof of n- oo ^n,wn theorem 4. According to theorems 1 and 2, there is good reason to expect that a solution (') of the extended variational problem is a step function. The result of theorem 4, as well as the method for constructing the approximating functions in the proof thereof, together with eq. (2.5) indicate that, loosely speaking, a minimum-fuel thrust program generally consists of a finite number of impulses. 9. Bounded thrust problem and limit theorem: In this section we shall consider the original fixed terminal time variational problem described in section 2 in the presence of the additional constraint that H|F(t)!I < i1 for all t, where p < co is a given positive constant. This constraint is a mathematical representation of the physical fact that the magnitude of the thrust vector of a rocket engine is always limited. 47

The problem may be precisely formulated as follows. Preserving the notation of section 2, for given positive numbers tl and A, let (tl,[) = = F(.): FC((tl), I F(tt)ll < i for all t-[O,tl]). Then we shall consider the problem of finding (for given ti and p) an element F(')E (tl,t) such that (tl;F) > M(tl;F) for every FE9(til,). We shall refer to this problem as the p-bounded problem. (It is to be understood in this section that all problems are fixed terminal time.) We shall show that the p-bounded problem always has a solution if [i is sufficiently large; that, in a certain sense, the solutions of the A-bounded problem tend to a solution of the extended. fixed terminal time problem when.+ -; o and that the necessary conditions for the latter problem can, in a sense, be obtained by passing to the limit in the necessary conditions for the p-bounded problem. We shall first prove the existence theorem. Theorem 5. If p is sufficiently large, then S(ti,[) is not empty and there exists an element P o)6IF(tl,) such that M(t; > M(tF) for every FEJ(tl,.); i.e., the >-bounded problem admits a solution. Proof. Let us show that (tl,") is not empty for p. sufficiently large. According to theorems 3 and 4, there exists a function ( ~)(.(tl) with essentially bounded derivative. Let STV u = a and let ildu(t)/dtll < 3 < o for almost all tE[O,tl]. If F(.) is now defined by means of relations (2.5) and (2.6), it follows that F(.)(7(tl) and (F(t)11 < (MB) for almost all tC[O,tl]. Changing the values of F(t) for t in a set of measure zero if necessary, we conclude that F(.)C(tl,) whenever p. > Mot 48

Denoting ri by xi+3 and ri by xi (where i. = 1,2, or 3), and (-M) by x7, eqs. (2.1) and (2.2) may be rewritten as follows: xi(t) = xi_3(t), i = 4,5,6 xi(t) = Gi(X4,X5,X6,t) (t) i = 1,2,3 (9.1) T - x7(t)' (9.1) 7(t) - A, T =(T1,T2,T3) In order to show the existence of an element F in. (tli) that max44L imizes M(tl;F), we shall first consider a system which, in a sense, is more general than (9.1), and which, following Warga [12], we shall refer to as the relaxed system. Namely, we consider the following system of equations for the scalar variables yi,... y7' Yi(t) = yi3(t), i = 4,56, Yi(t) = Gi(Y4,y5,yt) - t i = 1,2,3, (9.2) 1 y7()(t) Y7(t) )l+(t) T = (T,T2,T3) A ^ If F( ) is a measurable function from [0,oo) to E3 and F( ) is a summable 00 function from [0,oo) to E1 such that the inequality [lF(t)!l+F(t)]dt < AMo 0 holds, we shall denote by (yl(t;F,F),...,y7(t;F,F)) =y(t;F,F) the solution for t > 0 of eqs. (9.2),with T(t) = F(t) and T(t) = F(t), that assumes the initial values (yl(0;F,F),,..,y3(0;F,F)) = v, (y4(0;FF),...,Y6(0;FF)) = r y7(O;F,F) = -Mo. Let ~R(tl, ) denote the class of all pairs (F( ), F(.)) 49

such that lo F( ) is a measurable function from [O,tl] to E3, 2. F(' ) tis a summable function from [O,tl] into [0,oo), 3. [lF(t)+ll(t)t]dt < AMo, 4. llF(t)ll+F(t) < k for all tO[O,tl], and 5. hi(y4(tl;FF),...,y6(tl;F,F), Yl(tl;F,F),...~y3(tl;F,F),tl) = 0 for i = l,...,v. Then we shall consider the relaxed kl-bounded problem which consists in finding an element (F(~), F( )) P3R(t,i) such that y7(tl;FF) < y7(tl;FF) for every (F(.), F('))6~eR(tlj,). According to a theorem of Warga [12, Theorem 3.3], such a minimizing element always exists so long as YR(tliP) is not empty, and we shall show below that this set is not empty whenever p. is sufficiently large. Indeed, it follows at once from (9.1) and (9.2) that, if F ( ) denotes the function which vanishes identically, then, for every measurable function`ti F( ) from [O0tl] to E3 with / F(t)lldt < AM, 0 To (Yl(t;FFo)..Y3(t;FFo)) = r(t;F) (y4(tt;,Fo)... y 6(t;F,Fo0)) = r(t;F) y7(t;F,FO) = - M(t;F) for every tE[0,tl] It then follows at once that F( )' (tl,) if and only if (F( ), Fo('))E5R(tlp). Inasmuch as we have shown that ~(tl,i) is not empty for p sufficiently large, we conclude that 5R(tl,.) is not empty for i large enough. We shall prove below that if ('), F(*)) is a solution of the relaxed p-bounded problem (and we have shown that such a solution exists if \i is sufficiently large), then F(t) = Fo(t) = 0 for almost all te[O,tl]. By what was said above, this will imply that F )ES (t,p.) and that M(tl;F) 50

= -y7(tl;F ) > -y7(tl;FFO) = M(tl;F) for every F(').(tl,i); i.e., (.) is a solution of the k-bounded problem. It easily follows from the Pontryagin maximum principle [8, Chapter 1] that if (F "), F(.)) is a solution of the relaxed k-bounded problem, then;here exists a twice differentiable function r(') from [O,tl] to E3, and an absolutely continuous function i(~) from [0,tl] to El, with llr(')ll+( ) not vanishing identically, such that d2t) - G(y4(, F).. ly6(t; tF)')' dt2 K (r v)) (t) 0 <t t ti; (9,3) de(t) - [y7(t;,F)] 2[r(t)' -t)] for almost all t, 0 < t < tl; dt (9.4) (tl) < 0; (9)5) v(t) Ft)-(t) +(t t) t) m+v- (t).1 - max * lt_ A y7(t;F) \EE3 A y7(t;F lvll v+v_< for almost all t, O < t < ti (9.6) It follows from (9.6) that, for almost all t6[0,ti], either F(t) = 0 or 11r(t)11 = 0. Hence, if the zeros of IU( ")l are isolated, then F(t) = 0 for almost all t. If the zeros of II( -)II in [O,tl] are not isolated, then r(t) O. For, if t2 is an accumulation point of zeros of 11('), then r(t2) = -(t2) = 0, which, because of the uniqueness of solutions of eqs. (9.3), implies that Z5t) - O. If 4(t) -=, it follows from (9.4) and (9.5) 51

and the fact that 4 and. cannot both vanish identically, that t(t) = const.<O and (9.6) then implies that 11Ft)jj = F(t) = 0 almost everywhere in [O,tl]. Q.E.D. According to the Pontryagin maximum principle as applied to eqs. (9ol), if ( ) is any solution of the p-bounded problem, there exists a twice differentiable function 4(') from [O,tl] to E3 and an absolutely continuous function 4( ) from [O,tl] to El, with llr( $)l+( ) not vanishing identically, such that T *(t) 5=;;.o1)) (t) 0< <t t; (9~7) )(t) = - [M(t; )] -2[(t (t)t)] for almost all t, 0 < t < ti; (9.8) 4(ti) < 0; (9.9) tl _t)_ _t) _ m: a)x..... + (t) t)l + t) max (t)l + (t), for almost all t, O<t<t; A Mt;t) V L A M(t;)J - - ) llvll_< t(tl)'[x-r(tz;)] = t(tl)[y-r(tl;~)] for all (x,y)CT, (9-11) /W -w4~'* 4- k, 4.04- 1*, where T is the hyperplane tangent to H(tl) at (r(t;if), r(tl;9)). We shall say that such a pair (r(.), 4( )) is an adjoint function which corresponds to ). It follows from (9.10) that, for almost all tE[O,tl], 52

0 if Ihl(t)ll < (t) i(t) = Cr(t), where 0 < a < I/ilJ(t)ll, if i )(t)ll = v(t) i 0 4"' - -( 9. 1 2 ) k(t)/lIZ(t)ll if o i~(t)ll > (t), Il(t)!l = if 0 = Il(tt) 1 > (t) where v(t) = - M(t;1')(t)/A. (9.13) It is easily verified that r(.) is differentiable on [Ot,] and that (see (9.8) and (2.2)) [O if ll<(t)ll-r(t) < 0 der(t) _ ^d(t) - - (9.14) dt ~~$ [ll~(t)ll -(t)] if lk(t)ll-(t) > 0 AM(t;F) - - Hence, d. (t)/dt > 0 for all t, O < t < ti. Thus, let') be a solution of the pi-bounded problem, let (4('),(' )) be a corresponding adjoint function, and let (.) be defined by (9.13). It follows as before that if r(t) ~ O on [O,ti], then the zeros of lr( )|l are isolated. Further, if V(t) 0- on [O,ti], then (see (9.8), (9.9), (9.13) and (9.12)) 4(t) = const. < 0, r(t) > 0 for all t6[Otl], and(t) = 0 almost everywhere in [O,tl]. Note that (0o( ),cV( )) is an adjoint function cor4PI responding tot( ) for every a > 0. Thus, if (t: F(t) # 0 0 < t < tjj is of positive measure, then r(t) / 0 for some ti[O,ti], and, multiplying (9('),('*)) by a suitable positive constant if necessary, we may assume that 53

max 4lf(t)ll = 1. (9.15) O<t<t Adjoint functions ((.),^(.)) that satisfy (9.15) will be said to be nornmalized. Let Fo( ) denote the function from [O,tl] to E3 defined by Fo(t) = 0 for all t, 0 < t < t. We now prove the following limit theorem. Theorem 6. ~et 1, 2,... be a sequence of positive numbers such that pi + ~ as i ~ Co. Then, if Fo( ) ()~(tl), there exists a subsequence 1.i112 of the pi' solutions F.( ) of the ji-bounded problems, normalized adjoint functions (i.( ), i.()) corresponding to the F., and functions i( *) and v( a) from [Otl] to E3 such that ( () is twice differentiable and 1. ( * ) is a solution of the extended fixed terminal time problem, -'"? 2. %(t) -jt), (9.16) i-ioo 3. -M(t;Fi)Ti(t)/A + 1, (9 17) 4. ^ M(s;) (9)18) 4t. F(s i))ds + t )( M(s;Fj) 5. r(t;Fi) p(t;^), (9 19) i->oo 6. r(t;Fi) + z(t;$)-(t), (9.20) i-J oa 7. M(t;Fi) + Mo exp[-A'STV ], (9.21) i-soo 54

where 2, 3, and 5 hold for every tC[0,tl] and the convergence is uniform with respect to t[0O,t1]; and 4 and 6 hold for almost all tO[0,tl] including 0, tl and the points of continuity of ~('). Also, ( *) and 4r( ) satisfy eqs. (5.l)-(5.4), where (t) = p(t;^), and r, v, and T are defined as in the statement of theorem 1. Proof. According to theorem 5, solutions of the u-bounded problem exist when 1 ~> i*, where p* < oo is a sufficiently large positive constant. For every p > k*, let F( )Cy(tl) be a solution of the k-bounded problem. Since Fo( -)y(tl), t: t (t) # 0, 0 < t A tl} is of positive measure for every > 1 * and, by virtue of the immediately preceding discussion, there exists a normalized adjoint function, which we shall denote by ( (), $ (')) corresponding to each F with p > *. Let u( e) = ( *)); i.e. U(t) = ds 0 < t < t. (9.22) It follows from the discussion in section 2 that, for each p > k*, uk(')u(tl) and (see (2.6), (2.10), and (2.14)) STV u ) A in (923) Since 2F.*( )&9(tj, ) when k > k*, M(ti;f) > M(tl;l*.) for k > 1* (9.24) By virtue of (9.23) and (9.24), we can conclude that STV _ < STV u*. for all 55

k> kt*. The functions 4(.) are uniformly bounded by definition, and satisfy eq. (9.7) with F replaced by. Since G(, ) has bounded first partial derivatives the functions' ( ) are also uniformly bounded. But' if the 4' and the 4 are uniformly bounded, then the functions 4 (.) must also possess this property. Consequently, the functions At and r for p > i* are equicontinuous as well as uniformly bounded. Just as was done in the proof of theorem 3, we can now show with the aid of relation (2.9) that there exist functions'), (') and a subsequence iP 2,... of the pi such that ~u~6(tl), and, denoting,. by and F. by Fi, such that: (a) (9.16) and (9.19) are satisfied uniformly in [Otl], (b) (9.18) and (9.20) hold for almost all t([O,tl] including 0, tl and the points of continuity ofu' ), (c) (5.1), (5.2), and (5.4) are satisfied, and (d) AIV, lim. STV u exists and STV u < lim STV u. (9.25) - i ->0oo i We now shall verify (9.21). Let ~(.)X/(tl) be a solution of the extended 44-' fixed terminal time problem. Then (see theorem 4) there exist functions ui(*)c*(tl), i = 1,2,...,and positive constants M1,M2,... such that limn STV u = STV u, and Idu.(t)/dtll < Mi < o for almost all ti[Oj and n->c."n *Z -~~.~i -L every i = 1,2,.... Setting Fn =iun), we conclude on the basis of (2.5) and (2.6) that l1Fn(t)11 < MoMn almost everywhere in [O,til], i.e., (modifying the 56

values of F (t) for t in a set of measure zero, if necessary) F C(tlp) for every [ > MoMn. Consequently, for each n = 1,2,..., there is an integer I, depending on n, such that M(tl;Fi) > M(tl;F ) for every i > I(n), or (see (9.23)), STV ui < STVun for every i > I(n). Therefore (see (9.25)), STV' < lim STVu < lim STVu = STV u i->oo i Q0^ But ~u'Ctg(tl) and u is a solution of the extended fixed terminal time problem, so that STV u < STV. Therefore, STV`V = STV, and " is a solution of this problem also. Consequently, (9.25) is actually an equality, and (9.21) now follows at once from (923). We now verify (9.17). Denote i(') by ni( d), and let Ui(t) = - M(t;Fi) 4i(t)/A, 0 < t < ti, i = 1,2,... (9.26) We shall show that.i(0) -* 1 as i - oo, and that ri(t) < 1 for every i = 1,2,... and tE[O,tl]. Since each ri is differentiable, and d\i(t)/dt > 0 for each i and t, this will imply that ri(t) - 1 uniformly in [O.tl] as i - oo, i.e., that (9-17) holds uniformly in [O,ti]. Let Ei = (t: lli(t > i( t), 0 < t < tl), and let IEij denote the Lebesgue measure of Ei. According to (9.12), 11Fi(t)11 = eli when tEEi, so that (see (2.2)) 0 < M(ti;Fi) < Mo - PiiEiJ/A. Consequently, ilEil < AMo i 1,2,,., (9.27) and, since limipli = oo, IEi) + 0 as i o. 57

Now suppose that ri(T) > 1 for some tE[O,tl] and some i = 1,2,... Since 4i( ) satisfies eq. (9.14) with F replaced by Fi, p by ii, and by ti, andli (t) l < 1 for every te[O,tl], it follows that Vi(t) = 4i(t) > 1 for for every tE[O,tl], and, by virtue of (9.12), that Fi(t) = 0 for almost all tE[O,tl]. This implies that FoE.`(tl), which is a contradiction, so that *i(t) < 1 for every i and t. Because M(t;Fi) > M(tl;F ) > M(ti;F ) = M(see (9.24)), and l(i(t)II< 1 for each i and tC[O,tl], it follows from (9.14) that, for every i and t, i(t- <AM i [1-i(t)]cE (t), dt AM* i where cEi(t) is the characteristic function of Ei. Thus, d~i(t) _Li dit AM [1- iti (t)] (t) i(t), 0 < t < tl, (9028) dt AM* - t i i - where Ui(t) > 0 for all tE[O,t1]. But the solution of (9.28) is given by 4i(t) = 1- xp i (s)ds. ()+ (s)exp i cE (T)dTj ds A CEiAM* i i UAM* Ei 0 so that *i(t) < 1 - [l-ii(0)]exp[ 1 - i EI - [l-i(O)]A, 0 t ti (9.29) where A* = exp [-Mo/M*] > 0 (see (9.27)). We shall show that (9.29) implies that Ii(0O) - 1 as i + oo. 58

For each i, let Ti be any value in[O,tl] such that K{7i(Ti)(i = 1 (Such values exist by definition of the V = ri). If ti(Ti) = 1, it follows from (9.29) and the fact that 4i(0) < 1 that *i(0) = 1. Now suppose that *i(Ti) < 1. Then TiCEi, and let ( TiT i) be the largest open interval contained in E. which contains Ti in its closure. We shall suppose that i is sufficiently large that (T T' ) 7 (0, t). Then, Vi.(T') =t i (T) and/or i.() = (T Without loss of generality, we shall assume that II (T')II = ti(T[). Since 0 < Ti-Ti < JEi., and there is a constant K > 0 such that Id7Vi(t)/dtJl < K for each i = 1,2,... and tE[O,ti], we have that Ti 0 < 1 - 11^i(Ti)11 = llti(Ti)IHI^tT(T')ll =, -r ll3i(t)llct < l [1i(t)ll<itKlE But JEi +- 0 as i - o, so that 1 - 11i(T)1 = l-4i(Tj) will be non-negative and arbitrarily small if i is sufficiently large. Consequently, we can conclude, by virtue of (9.29), that limio[l-1i(0)]A* = 0, i.e.i that limi i(0) = 1. This completes the verification of (9.17). It only remains to prove that (5.3) holds. Let Gi =t: Ii(t)lI >'i(t), 0 < t < tl}. It follows from (9.22) and (9.12) that du (s)/dt = 0 when soGi, 0 < s < 1, and that, for almost all sEGi, either dug (s)/dt is a non-negative scalar multiple of (s s) or (s)= O. Hence, [(t)] dui(t) dt = F1V~i(t)11 ui d (9.t0) ~dt ~dt 59

If tEGi, 1 > llTi(t)ll ~> i(t) ~> i(O), so that, by virtue of (9.30), Vi(~O) l; u i(t) dT (t) (t) ^(0) <['i (t)] d u ci dt < (9.,l) dt - -i ddt - 3dt Since (9.25) has been shown to be an equality, we have, by virtue of (2.14), that du i(t) dti dKu (t) dit td~ tt = /id STV u STV STV (9.32). 01 Also, ri(O) -+ 1 as i - oo, so that (9-31) and (9.32) yield that tj - T, (t) t [i(t)] T d STV)'/4, dt + STV (9-33) dt i9o3 Finally, it follows from (9.16), (9.18), (9.22) and the Helly-Bray theorem [6, p. 288] that )t T dui /t) /t i i Tti T t[i(t)] dd(t)t = [?i(t)] dui(t)i. [(t)] d~ t), o dt -o oL (9 34) and (5.3) is now an immediate consequence of (9.33) and (9.34). Corollary 1. Let be a solution of the extended fixed terminal time problem, and suppose that u F. Then, if is any solution of the pbounded problem (for every u sufficiently large), we have that 1. M(tl;F ) +o M exp[-A 1STV ]. 60

Further, if *) is the unique solution of the extended fixed terminal time problem, and u is given by (9.22), then 2. u(t) t) 3' r(t;/) F $~ z(t;/u) +SU(t), 4. r(t;/) P(t;/^)^ where 4 holds uniformly in [O,tl], and 2 and 3 hold for almost all t[[O,tl] including O,tl, and the points of continuity of'). If, in addition, the function t(.) that satisfies relations (5.1)-(5.4) is unique, and (i,8p) is any normalized adjoint function corresponding to F, then the following limits exist uniformly in [O,tl]: 5. E ~o(t) poo*Nt, 6. -M(t;F /A+1. The proof of the corollary is straightforward and is therefore omitted. 61

REFERENCES 1. D. F. Lawden, Optimal Trajectories for Space Navigation, Butterworths, London, 1963. 2. G, M. Ewing, A fundamental problem of navigation in free space, Quart. Appl. Math., 18 (1961), pp. 355-362. 3. L. W. Neustadt, Optimization, a moment problem, and nonlinear programming, J. SIAM, Ser, A.: Control, 2 (1964), pp. 33-53. 4. R. W. Rishel, An extended POntryagin principle for control systems whose control laws contain measures, Mathematical Analysis Note No. 48 (1964), Aero-Space Division, Boeing Co., Seattle, Washington. 5. E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc., Providence, R.I., 1957. 6. L. M. Graves, The Theory of Functions of Real Variables, 2nd edition, McGraw-Hill, New York, N.Y., 1956o 7. C. L. Siegel, Vorlesungen uber Himmelsmechanik, Springer-verlag, Berlin, 1956. 8. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and Eo Fo Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Division of John Wiley, New York, N.Y., 1962. 9. J. Warga, Necessary conditions for minimum in relaxed variational problems, J. Math, Anal, Applo, 4 (1962), pp. 129-145o 10. A. N, Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Vol. 1, Graylock, Rochester, N.Y., 1957. 62

REFERENCES (Concluded) 11. I. P. Natanson, Theory of Functions of a Real Variable, Vol. 1, Ungar, New York, N.Y., 1955. 12. J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), pp. 111-128.

FOOTNOTES This research was begun while the author was with the Aerospace Corporation, El Segundo, California, but was supported primarily by the United States Air Force through the Air Force Office of Scientific Research under Contract No. AF 49(638)-1318. t Instrumentation Engineering Program, The University of Michigan, Ann Arbor, Michigan. 1. By a solution of eq. (2.2) we here mean an absolutely continuous function M(.) that satisfies (2.2) for almost all t > O. The inequality 00 J lIldt < AM implies that M(t) > M for some positive constant M. and all o t > 0. Physically, the first inequality signifies that the rocket can not provide thrust once the fuel has been consumed. By a solution of eq. (2.1) we mean a function r(-), whose time derivative 9( ) exists (for all t > 0) with r(-) absolutely continuous, that satisfies eq. (2.1) for almost all t > 0. 2. If w()6y3(t), w has at most a denumerable number of points of discontinuity, and the discontinuities of H are of the first kind. By a solution of eqs. (2.3), with u(t) = w(t), we here mean a continuously differentiable function z(-) that satisfies the first equation everywhere, and an absolutely continuous function p(^) that satisfies the second equation at all points of continuity of w( ). 6h