Lagrange problems with variable endpoint as optimal control problem

Sagan, H.

English

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LAGRANGE PROBLEMS 
WITH A VARIABLE ENDPOINT 
AS OPTIMAL CONTROL PROBLEMS 
by Hans Sagm 
Prepared by 
NORTH CAROLINA STATE UNIVERSITY 
Raleigh, N. C. 
f Or 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION . WASHINGTON, D:.C.’ . -‘dlY 1967 
https://ntrs.nasa.gov/search.jsp?R=19670022003 2020-03-12T11:00:46+00:00Z
TECH LIBRARY KAFB, NM 
NASA CR-837 
LAGRANGE PROBLEMS WITH A VARIABLE ENDPOINT 
AS OPTIMAL CONTROL PROBLEMS 
By Hans Sagan 
Distribution of this report is provided in the interest of 
information exchange. Responsibility for the contents 
resides in the author or organization that prepared it. 
Prepared under Contract No. NGR-34-002-032 by 
NORTH CAROLINA STATE UNIVERSITY 
Raleigh, N. C. 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - CFSTI price $3.00 
LAGRANGE PROBLEMS WITH A VARIABLE 
ENDPOINT AS OPTIMAL CONTROL PROBLEMS 
Hans Sagan 
SUMMARY 
L. S. Pontryagin et al. have shown ([l], pp. 248-256) that the 
maximum principle leads, as is to be expected, to the multiplier rule and 
the Weierstrass inequality for the problem of Lagrange. 
In this paper, we will demonstrate how the transversality conditions 
for the Lagrange problem with a variable endpoint may be obtained from 
Pontryagin's maximum principle and transversality conditions for an 
optimal control problem with a variable endpoint ([l], pp. 45-50, 62-63). 
We consider the Lagrange problem of finding a trajectory 
(y,(t) ,...,y,(t)) E C'[a,b] (optimal trajectory) so that 
b 
f(t ,Yl , l **,Yn,Y;,...,y;)dt + minimum 
a 
under the constraints 
. 
. . 
Y; = 4p,Y, .....Yn.Y~+l’....Y~). j.l<n 
which emanates from the given point 
yi(aj = y:, i = 1,2,...,n 
and terminates for some b on the (smooth) (n + 1 - k)-dimensional manifold T 
(1) . . . 
X,tt SY, ,...,yn) = 0, k < n. 
We will assume that f, I$~, af/at, aflay j, a4i/at, aftp3Yj are 
continuous in an open set of the (t,yl,...,yn)-space, that contains the 
optimal trajectory, and that f, $J~, are continuous for all yi,...,yi. We 
will further assume that axi/%, axi/ayj are continuous for all t,yl ,***, 'n 
and that for every fixed t, grad xk are linearly independent for all 
(Y,¶Y,,... ,Y,)* 
2 
The transversality conditions as stated by G. 9. Bliss ([2], p. 311) 
and as adopted to our specific problem and notation are as follows: 
There have to exist (k + 1) constants (pO,pl,...,uk) # (O,O,...,O) 
such that 
(2) 
(3) 
where 
(4) 
Jl jib = (q)bs J = 1,2,.. . ,n 
f:i (2) b = - ‘o(f)b - ,%,(%), Y;(b) 
h=- Xof + E 
i=l 
x&y' i 
and where (Xo,A1,...,Xp) # (O,O,..., 0) are the Lagrange multipliers with 
A0 = lJ,* 
In order to show how these conditions may be obtained from Pontryagin's 
maximum principle and transversality conditions, we first formulate the 
Lagrange problem as an optimal control problem, introducing 
t 
y,(t) = as, Y,(S) ,. . . ,yJd, y;(s) ,. l . ,y;(s))ds, 
a 
1 1 
u1 = yU+l,...,~m = y,, m = n - ur 
Then the problem will read as follows: To be found is a trajectory 
(ye(t), y,(t),...,y,(t)) and a control (u,(t),...,u,(t)) so that 
3 
(5) 
y:, = 4,h Y1,-*sYn, U1,“‘,Um) . . . 
I 
yv = 4,(t, Y1¶” l ,y,, U1,’ l * ,u,, 
t 
yp+1= u1 
Y,: 
=u m 
where 
y,(a) = 0, yi(a) = yy, i = 1,2,,..,n, 
xjb, ylb),...,ynb)) = 0, j = 1,2,.-o&. 
for some b, and 
yob) + minimum. 
In the spirit of the problem, as originally posed, the entire 
(u,, . . ..um)-space is to be taken as the control region. Then the following 
relations between the Lagrange multipliers ho,Al,...,hv and the solutions 
IcIo,JIy.*, $n of the conjugate system to (5) follow from the maximum principle 
([ll, p* 59251,252) 
(6) Jlp = Ai(t) - 
(7) 
. (8) 
$,+jw = - $, -z- 
ay ’ lJ+j 
afJ, 
ay; O’ 
i = O,l,...,v 
. f 3L Xi, j. = 1,2,...,m. 
i=l ay’ lJ+j By (4) 
1 
- A K+ A., 
ah -= O ay; 1 
i = 1,2,...,n 
ay; -A arm j A, Z.5 , i = u+l, p+2,...,n. 
O ay; j=i j ay; 
4 
Hence, in view of (6) and (7) 
(9) ah - = Jli 
a$ 
1 
along the optimal trajectory. 
By Pontryagin's transversality condition, ([l], p. 631, it is necessary 
that at the termination point on T 
$l(b)pl + l .- + Jln(b)pn = 0 
for any vector (p, ,....,pn) that lies in the tangent plane to the (n-k)- 
dimensional (smooth) manifold T* 
xj(b,yl ,...,y,) = 0, j = 1,2,...,k 
at the termination point. There are exactly k linearly independent vectors 
that are orthogonal to Tt at the termination point, namely 
(grad xllb,. . . ,(grad x.k)b' 
Hence, by necessity 
(10) qjb) = i Pi( 3 )‘, j = 1,2,...,n 
i=l J b 
for some constants P~,)A~,...,LI~. Equations (9) and (10) yield the 
transversality conditions (2). 
The remaining condition (3) is obtained as follows: Let YN$,Y,t) 
denote the maximum in ul,...,um of 
for fixed t,y,JI. If i = (l,q1,*4J is a tangent vector to T at 
b,ylb),...,yn(b)), we have to have ([l], p. 62) 
(11) mtJ,b) ,y(b) rb) = i qi9i(b)* 
i=l 
Since i is a tangent vector to T, we have 
We consider first the case where (P~,P~,...,P~) # (O,O,...,O), &.e., 
(4+), +2(b),...,$nb)) # (W,...,O). Then we obtain after multiplication 
by P j, summation over j and observation of (10) 
(12) j~l~j( ~ ) = - f QiJli(b)* 
b i=l 
In view of (7), which is a consequence of aj&'aui = 0, i = 1,2,...,m, 
we obtain along the optimal trajectory 
mldJ,Y,t) = Jl,f + J1191 + -*- + $9, +Ybo? +pf 
i=l LIZ1 aYp+i 
xJ)yi+i, 
p+i 
which because of (8) yields after cumbersome manipulations 
‘r/C(J,,y,t) = $,f + ; ah y.I'. 
i=l ay; 1 
This together with (6), (11) and (12) leads directly to (3) with 
(P~,P~,...,P~) # (O,O,...,O) and hence (P~,v~,~~~,~~) # (O,O,...,O). 
The case where (p1,p2,...,pk) = (O,O,...,O) is easily taken care of. 
If (p1,u2,...,pn) = (O,O,...,O), then ($l(b),--.,$n(b)) = (O,O,...,O) but 
JI, < 0 Ull, pa 18,uL 
Then we have from (2) that (ah/ayi), = 0, j = l,...,n and hence, 
-r/C($b) a(b) $1 = $o(f)b- By (11) we obtain instead of (3) 
- 4J,(f), = 0 
and since J1, < 0 and 9, = X0 = u,, we have again (u,,P~,...,P~) # (O,O,...,O). 
6 
Bibliography 
[l] L. S. Pontryagin, et al. The Mathematical Theory of Optimal Processes, 
Interscience Publishers, New York, 1963. 
[2] G. A. Bliss. "The problem of Mayer with Variable endpoints',', Trans. 
Am. Math. Sot., vol. lg(lg18), pp. 305-314. 
North Carolina State University 
Raleigh, North Carolina 
August 9, 1966 
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Lagrange problems with a variable endpoint as optimal control problems

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