On the removal of ill conditioning effects in the computation of optimal controls

Automatica, Vol. 5, pp. 607-614. Pergamon Press, 1969. Printed in Great Britain.

On the Removal of Ill Conditioning Effects in the Computation of Optimal Controls* Sur la suppression des effets d'influence drfavorable dans le calcul des commandes optimales ~Jber die Beseitigung von nachteiligen Effekten bei der Berechnung von optimalen Regelungen 06 ycxpaHenVlVI9qbqbeKToaHe6JlaronpriaTnoro BYIHIIHHflB paccqeTe OnTHMaabHbIX

ynpaa:lenn~

E. POLAKS"

A dual type method for solving discrete optimal control problems with linear plant and convex cost and constraints is presented. This method takes maximum advantage of the dynamic structure of the problem. Summary--First, it is shown how ill conditioning effects may arise when discrete optimal control problems with linear dynamics, and convex cost and constraints are solved by "primal" methods such as linear or quadratic programming, or certain gradient methods. A dual method is then presented, which exploits to the utmost the dynamical structure of the optimal control problem, This method has been found to perform very well and does not suffer from ill-conditioning effects.

exponential nature of solutions of a difference equation. However, we shall not deal with the question as to what modifications may be made to existing non-linear programming algorithms in order to reduce this ill conditioning. Indeed, this is still very much an open question. Second, we shall show that it is possible to make very effective use of non-linear programming results, such as the convergence theory in [3] and the antizigzagging precautions introduced in [4], in conjunction with geometric ideas of optimal control theory, such as those in [5-7], in the construction of very efficient, large step, optimal control algorithms. The algorithms resulting from the approach we are about to present can be used for solving a large variety of problems, such as minimum time, minimum energy and minimum fuel problems for linear dynamical systems with constraints both on the control and on the trajectory. However, in order to keep our exposition simple and to highlight some of the principal ideas, we shall restrict our discussion to a particular class of minimum energy problems. Apart from the mathematically difficult antizigzagging precautions which must be used when the state space constraints have edges and which the interested reader may wish to look up in [8] and [4], the extension of the algorithm as presented in this paper to other problems is essentially trivial and will not be discussed. Finally, it should be pointed out that the algorithm presented in this paper is not the only one to combine both optimal control and non-linear programming ideas. For a comparison, see the algorithm by BARR and GILBERT [9], which also

INTRODUCTION IT HAS become clearer and clearer in recent years, as can be seen from such work as [1, 2], that problems in the calculus of variations, optimal control and non-linear programming can be treated in a unified manner, at least as far as optimality conditions are concerned. It has also become clear that discrete optimal control problems can be recast as standard non-linear programming forms which are solvable by standard algorithms. The purpose of this paper is twofold. First, we shall show that the seeming ease with which nonlinear programming algorithms can be applied to discrete optimal control problems is deceptive and that severe ill conditionning may occur due to the * The research reported herein was supported by the National Aeronautics and Space Administration under Grant NsG-354, Suppl. 5. The original version of this paper was presented at the IFAC Symposium on System Dynamics and Automatic Control in Basic Industries which was held in Sydney, Australia during August 1968. Received 9 December 1968 and in revised form 21 March 1969. It was recommended for publication in revised form by associate editor M. Athans. t Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, California, 94720. 607

608

E. POLAK

uses a geometric approach, but which has entirely different rules for selecting the next point. It is to be hoped that these recent steps in the direction of merging optimal control and nonlinear programming ideas wilt contribute to a new generation of very efficient optimal control algorithms.

which is a standard bounded variable, quadratic programming problem with a unique solution. In Wolfe's method [10], we apply the K u h n - T u c k e r necessary and sufficient conditions [11] to (5), ~o find that h is optimal if and only if for some vectors ~b in R ~ and [11,//2 in R x, with lq >_0, tl2 ~_0

1. A CLASSICAL APPROACH

Statement of the minimum energy problem. We are given a dynamical system described by the difference equation

....... 6 ..... o

o-

(6i

7

kt~2d and

xi+l=Axi-t-bUi+l, i=0,

l .....

N-l,

(1)

where x~eR" is the state of the system at time i, i = 0 , 1, 2 . . . . . N, Ui+IGR 1 is the system input at time i, i = 0, 1 , . . . , N - I, A is a n x n matrix, and b~R". We are required to find a control sequence u=(fil, fi2 . . . . . UN) which minimizes the cost ½ E u{'

(2)

i=1

subject to the constraints that [ui]~ 1, for i = 1, 2 , . . . , N, and that the corresponding trajectory 2o, 2~ . . . . . xN, determined by (1) must satisfy 20 = ¢o (a given vector in R") and 2Ne92, (a given set in R").

Case 1. The set f~ consists of the point ~NeR" only, i.e. £~= {IN}. This case of the m i n i m u m energy problem can be solved by standard quadratic programming algorithms, such as, say, that of WOLFE [10]. To transcribe the minimum energy problem into a standard quadratic programming problem, we proceed as follows. Solving (1) for xN, we obtain (with x o = ~o) N-1

x v=AN~o + ~ AN-i-lbui+l.

(3)

i=0

Letri=AtC-ibfori= 1 , 2 , . . . , N a n d let d=AN¢o, then (3) becomes N

xu = d + ~ riu I.

(4)

i=1

Letting R be a n x N matrix whose i-th colum is r I and setting ~ = ¢ u - d , the minimum energy problem becomes



=

=0"

(7)

where l = ( l , I . . . . , 1)~R". Wolfe's or, for that matter, many related algorithms, such as LEMKE'S [13], solves (6) by a modification of the Simplex algorithm [12], and which in turn requires the inversion of ( N + n ) x ( N + n ) submatrices of the matrix in (6). Since the top row of R T and the first columns of R, wilt be very close to zero when N is large, it is clear that such submatrices, which to begin with are very large, will be very difficult to invert, necessitating the use of special methods [14] which are costly in time. The severity of this ill conditioning, of course, depends on N, since for N large, not only will the submatrices be large, but also a large number of rows of R r will appear to be zero to a digital computer. Thus, standard quadratic programming algorithms and unmodified computer codes become illconditionned when used for solving certain optimal control problems.

Case 2. The set 92 is a unit ball with center at the origin, i.e. f2= {x I Ilxll--- 1}. Because f2 is a ball, we can solve this case by modifying a gradient method due to J. Plant [15] which he used for continuous time problems. Thus, applying necessary conditions of optimality, which in this case are also sufficient as indicated in [1], directly to the minimum energy problem, we find that the control sequence hi, u2 . . . . . uN, with a corresponding trajectory 40, 21 . . . . . 2N, is optimal if and only if there exist costate vectors Po, Pt, P2, •. •, PN in R" such that pi=Arpi+l for i = 0 , 1. . . . .

N

(8)

* For x=(xl,x2,...,x") any vectors in Rn, we define N

n

minimize ½ ~ t,~ i=1

subject to R u = ( , -1<<.ui<~ +1 for i = 1 . . . . .

N,

(5)

< x,y> = E xlYi. When it will not be possible to tell from i=l the context whether a vector is a row or column vector, we shall use Dirac notation: for a column vector.

On the removal of ill conditioning effects in the computation of optimal controls pu=(//OU if ' ~ if and, for i= 1, 2 . . . . .

=1 for s o m e / 3 < 0 ) <1

(9)

609

and hence, to first order terms, (14) expands as follows: f(/~ + Aft, v + Av) -f(/~, v) = (Xo + Axo) - Xo

N,

=A-NAy-Aft 0~= sat((p~, b)),

(10)

where sat(x)= x for Ixl-< 1, and sat(x)=sgn x for Ix] > 1. Setting Au__Avand making use of (8) and (9), we obtain that pi=(AN-i) T . fly and hence that Oi=sat(/~v, ri) for i=1, 2 . . . . .

N,

A-Nri)(ri, v) A-Nr,)(ri, Av).

(15)

i~I(fl, v)

Now, since v = h(O) and h is obviously differentiable,

Av=C~h(O)Ao, ~0

AX°=I(A-N-- fl 2ieI(IL v)A-Nri)(ri)~

Note. It is rather easy to show that II NII = 1, i.e. that the optimal terminal state is on the boundary of the ball, that ~N must satisfy <~u, ~N--AN~0 > ~<0,*

-fl ~

(11)

where r i = A s- ib, as before.

~ i~l(fl, v)

(12)

and that the optimal control sequence ill, fi2. . . . . fiN, and, consequently, also the corresponding optimal trajectory, 40, Xl, xz . . . . . xN, are unique. To apply the gradient method, we first express the initial state Xo in terms of a terminal state v~S, where S = {vsR"] ]l ll = 1}, by means of(11) and (3), (4). Thus, N

xo=A-Uv - ~ A-urisat(flv, ri>Af(fl, v) (13) i=1

If/3 < 0 and yeS are chosen properly, then x o = Go, and (11) gives the desired solution. Now, the functionf(/~, v) can be shown to be one-to-one for (/3, v)in the set {(/~,v)[ < v, v - AN~o> ~<0, [lvl[= 1 and []~< 1, for at least one iE{1, 2 . . . . , N}}. Hence it can be inverted to find the/0, ~ such that ~o=f(/3, ~) and hence, by (11) the optimal control sequence. We shall now discuss the solution of the equation ~o=f(/3, v). Since v is a point on a unit sphere, v = h(O), where 0 are its (n - 1) spherical co-ordinates. Let 1(/3, v)c {1, 2 . . . . . N} be an index set such that ieI(fl ,v)if and only if{[ [~< 1}. Suppose that we have guessed a /3<0 and a v satisfying I[vI[=1, <<,0,such that/(/3, v)is not empty, let xoAf(fl, v). Then

-

Z A-",'i)(,'. i~I(ll, v)

v)

F]

'

(16

Since, f(fl, v) is one-to-one, the matrix in (16) will be nonsingular almost always. Hence, if we choose Axo =2(~o-Xo) with 2 > 0 sufficiently small for (16) to remain valid, we can compute (A0, Aft) by inverting the matrix in (16) and move from Xo to Xo + Axo, which is closer to 4o than x o was. Again, by inspection of (16) and the way I(fl, v) was defined, we see that matrix inversion will incapacitate this method when N is large, since the matrix in (16) may have an extremely small determinant. Incidentally, apart from this ill-conditioning, the above approach also suffers from the fact that it cannot be demonstrated to converge. Thus, once again we find that the straightforward, textbook type approach to a simple problem like our minimum energy problem, may not get one very far on the way to finding a solution. 2. A P A R A M E T R I C

APPROACH

(14)

We shall now use the two problems discussed in the previous section as a vehicle to describe a new and very general algorithm which suffers from none of the defects we have encountered in the previous section. The price one pays for this in Case 1 is that (5) can no longer be solved in a finite number of steps. However, there seems to be no finite algorithm for Case 2, so we only have gains here. The trick, of course, is to avoid matrix inversions by means of parametrization. Thus, we shall assume that ~ = {x] !lx-~]] ~
where i is the complement of I. Now, for small perturbations in/~ and v, i(/~, v) does not change,

Definition. For ~e[0, ~/N], let cg(~) be the set of states which the system (1) can reach from the given initial state 4o at time N, with energy consumption

xo=f(fl, v)=A-Nv -

~

A-Uri)sgn(flv, ri>

~

/~A-Nr~)(r,, v),

iJ(ILv) id(fl, v)

N

* This will become clear in section II.

} Z u?=½ i=1

i.e.

610

E. POLAK N

(g(e)={xlx=d+ E u,ri, lull<~1, i=1 N

Z u~=cd, ee[0,

~/N]}.

(17)

i=1

Note that c#(~) is a closed and bounded convex set and that c#(e~)= c#(e2) whenever e~ > c%.

subject to [u~]~< 1 and the taking of the system (1) from 4o to a given hyperplane, is essentially trivial to solve, as we shall see in section 3, procedure P2. We therefore propose to solve the optimal control problem which takes system (1) from ~'o to by solving a parametric family of optimal control problems which take the system (I) from ~o to a suitably chosen hyperplane.

Assumption. In order to facilitate exposition, we shall assume that the set f~ has points in the interior of the maximal reachable set, c#(x/N ). However, this assumption is not necessary for the algorithm to be applicable. With ~ either a point or a closed ball, the optimal control sequence fi~, fi2. . . . . fin is unique and so is the optimal terminal state 2u. Let

,~ic¸ ~;.

J

i ,

J N

Z

FtG. 1. The geometry of the problem.

i=1

then it is easy to see that the following must be true: a = min{ee[0,

,/N]I~(~)~n~ ¢},

(18)

where q5 denotes the empty set, and

{2

(19)

i.e. 2 N is the only point in the intersection of cg(a) and f~. Thus 2N is on the boundary o f ~ and, since both ~f(~) and f~ are convex, there must exist a support hyperplane to ~ at xs which separates f~ from cg(a). Let P be such a support hyperplane, then, since by assumption ~ has points in the interior of cg(x/N--), cg(a)c@ also contains only the one point 2N. To see this, observe that since f~ has point in the interior of ¢g(~/N) and ~(~1)~cg(~2) whenever ~ >c%, we must have that &< x/N. Now suppose that P has more than one point in common with cg(a). Then it follows, from the geometry of the sets cg(~), that P must be a boundary hyperplane for c~(~/N), which contradicts the fact that f~ has points in the interior of c~(\,/N). We therefore conclude that 2~ is also the optimal terminal state for the problem of minimizing N i=l

subject to ]u,I ~< 1 and the requirement of taking the system (1) from the given initial state 4o to P. Obviously, ill, fi2. . . . . ~ , the optimal control sequence for taking the system (1) to ~ , is also optimal for taking the system (1) to P. Now, it so happens that the problem, of minimizing

The gist of the algorithm we are about to describe is as follows:* If def~ then ~ = 0 for i = 1, 2 , . . . , N is the optimal control sequence. Hence, let us assume that d¢~, then (i) Insert a hyperplane P between the point cg(0) = {d} and the set ~, so that P is also a support hyperplane to f~ at a point v; (ii) Increase e until cg(e) touches P at a point w; (iii) If v= w then we are done, since we must have found the smallest e satisfying (#(c0c~ ~ qS. If v ¢ w, then we can rotate the hyperplane P in such a way that it stays in contact with f~ but breaks away from c#(e). We can then increase and check again if v--- w. Thus, we should stop if either v = w, in which case v is the optimal terminal state 2N, or else, if we have established that v = 2 N by independent means. Note that our algorithm hanlzlles the situation where does not have points in the interior of ~f(,,/N) by first checking the optimality of the initial guess by independent means and then proceeding as if v = w were the case for the optimal solution. Obviously, the rotation of the plane P cannot be done in an arbitrary way if one wishes to insure convergence. Hence we shall need the following machinery to make it work.

Definition. Let P(v, s) denote the hyperplane in R" which passes through the point veR", with unit normal s, i.e. P(v, s)=(xl(x-v,

(20)

Definition. Let S = {s[ Ilsll = 1} be a unit sphere in R" and let v: S~0f~ (the boundary of ~ ) be the contact function defined by the relation (x-v(s), s)<<.Ofor

N

i=1

s)=0}.

* See Fig. 1.

all xefL

(21)

On the removal of ill conditioning effects in the computation of optimal controls is a closed ball of radius Since!GI={xl/x-<116P} p and center 5, D(S)= t + ps. Thus, if p = 0, Z)(S)= 5 for all s. Definition. Let VcaQ be the set of all points in aR through which can pass a hyperplane separating Q from d, i.e. for each UEVthere is a ES, such that v(s) = II and

(d-u(s),

s)>O.

Definition. For a~[0, JN], let c : T+[O, ,/N] be the function defined by

c(s)= min(alP[u(s), s]n%‘(cl)# 4, CE[O,,/N]}

(23)

i.e. c(s) is the smallest c( for whjch the intersection of %?(a)with P[v(s), s], the tangent hyperplane to Q at u(s), is not empty. Observing now that P[u(s), s]n%?[c(s)] is the set of terminal states for the minimum energy problem with Q replaced by P[u(s), s], and that the solution to this new problem is also unique, we conclude that p[r(~)~ sln@c(s)l

i.e. it is the point s’ on a(s) which maximizes c(s’). Simple geometric considerations lead one to believe that the function c[a( - )] is continuous. That this is indeed so is proven in [8]. Theorem. Let s,,, sl, s2, . . . , be a sequence in T generated according to the law Si+l

(22)

Clearly, the optimal terminal state Z.N must belong to V. Furthermore, let T be the set of all points s in S such that D(s)EV. Note: T is a closed set.

= {w(s))

(24)

DeJinition.

We define the function

i=O,1, 2, . . .

=Ll(SJ,

Proof. First, we observe that for every SETsuch that u(s)# w(s), we must have c[a(s)]> c(s). Since c(si+ 1)> c(si), i=O, 1, 2, . . . , is a monotonic increasing sequency which is bounded from above by JN, we must have,

c(si)+c*< JN, k{O, 1, 2, . . .}.

(25)

Now, it is not difficult to see that small changes in s produce correspondingly small changes in u(s), w(s) and c(s), i.e. that all these functions are continuous.

(29)

Since T is closed and bounded, the sequence {si} must contain a convergent subsequence {Si}, kK, with K contained in (0, 1, 2, . . . }, whose limit point s*eT. Hence, since c( *) and c[a( *)I are continuous, c[a(s*)] = c(s*) = c*.

(30)

But this implies that u(s*) = w(P).

w : T-+R"

(28)

Then every converging subsequence of {Si} converges to a point S*ETsuch that u(s*) = w(s*).

i.e. that it must consist of one point only.

by (24).

611

(31)

Note. When p > 0, u( a) is continuous and one-toone on T. Since the intersection %[c(s*)]nQ consists of exactly one point w(s*)= u(s*) it then then follows that the limit point s*, above, is the only point to which a subsequence of (si) can converge, and hence (si} itself converges to this point.

Dejinition. For any s E T, let G(S) be the arc in Corollary.

T defined by

defined by

s+A[w(s)-u(s)] J/S +Xw(s)- U(S)1 11’

a(s) = s’ETIs’ = ,

Again we see that small variations in s cause only small variations in a(s), i.e. that it is a continuous map from T into the set of all subsets of T with respect to the Hausdorff metric. We now have all the parts we need to define our algorithm. be the function defined

by c[a(s)] = max c[s’J,

15 ‘E(I(S)

1) and let aL. : T-+T be (32)

s + A[a(s) - s]

(26)

Definition. Let a : T+T

Let k(0,

(27)

a”(S)=ils+l[a(s)-s]~i

(33)

where a( *) is defined by (27). Then any sequence (Si) in T generated according to the law si + i =

ak(%>

(33)

converges to point S*ETsuch that u(s*) = I+*). Proof. We simply note that c[a ( .)] is continuous and that c[a (s)]>c(s) for all SET such that u(s)# w(s). Hence the proof is exactly as for theorem (28).

612

E. POLAK

Remark. We conclude from the above corollary (since 2 > 0 may be taken to be quite small without affecting convergence) that even a very approximate evaluation of a(s) should be compatible with convergence. (34) We shall now give the algorithm for carrying out the computation of the optimal control sequence h=(fil, u2 . . . . . uN) for the minimum energy problem. 3. THE ALGORITHM P1. Initial guess procedure Set So=(d-~)/l[d-~II, where ¢ is the center of f~={x[llx-~[[<~p} and d=A-N~o, as before. Clearly, So is in T.

Step 3. Set sI =y.

Note that it is usually best to start with M = 2 and to increase M only if (40) cannot be satisfied. P4.

Verification of feasibility

Suppose that for some so~T, we find that the set cg(\,/N) does not intersect the plane P[v(so), So], then it is clear that there is no admissible control sequence which will take the system from Xo = ~o to ~ in N steps, i.e. the problem has no solution.

Step 1. Compute a f l * < 0 such that

Computation of V(So), W(So),C(So),a(So) Step 1. By (21),

I(fi So, "i>i ~>l for i = l

P2.

V(So)= ~ + pso.

(35)

(41)

~

N.

(42)

If N

(d-d,-PSo+ ~sgn(fi*So, ri)ri, So}>O

Step 2. Note that W(So)is the point on P[v(so), So], which is the terminal state for the minimum energy problem when f~ is set equal to P[v(so), So]. Hence, from necessary and sufficient conditions, we obtain [as in (10), (I 1)] that

then there is no solution to the minimum energy problem. P5.

N

W(So)=d+ ~ sat(floSo, Fi)ri

(36)

i=1

for some fl0<0. To compute rio, substitute (36) into the expression for P[v(So), So], and set flo to satisfy [see (20)],

Verification of optimality of So

Step 1. Minimize with respect to fi the function N

g(/~,

ri, S0)=0.

so)Aild-~-pSo- Z sat
,',>r, lrz.

(44)

,=i

If min

N

(43)

i=1

g(fi, So)=0 then So is optimal.

(37)

i=l

P6.

This computation is quite easy since (37) is a piecewise linear expression.

Computation of the optimal control sequence

Suppose s o satisfies either V(So)=W(So) or rain g(fl, So)=0. Then So is optimal and

Step 3. By inspection of (36) and (23), ~i=sat
N

(45)

where flo is determined either by (36) or by

g(3o, So) = O.

Step 4. By definition of a(so) (26), a(So)= {s'eTIs'= s+ 2[w(s°)-~-PS°]

ils+

-pso]ii'

0~<2~< (39)

Computation Q/'s~ Step 1. Compute M + 1 points Yo, )q, • - •, YM, in a(So), by setting 2 = 0 , l/M, 1/2M. . . . . 1, in P3.

(39), [assuming of course, that for 2 = 1, the point is in a(So)].

Step 2. Compute c(y~) for i = 0 , 1, 2 . . . . . M by so=yl in P2. Find a j ~ { l , 2 . . . . . M} such

setting that

c(yj)>~c(y~) for all i{0, 1, 2 . . . . .

M}.

(40)

The manner in which these six procedures are combined into an algorithm is best illustrated by the flow chart in Fig. 2. Note that when d~f~, fi=(0, 0, . . . , 0) and hence the problem becomes trivial. Also note that the use of P5 can be omitted since the chances of guessing right the very first time are very slight indeed. However, if f~ had no points in the interior of cg(\/N) and So were the optimal solution, the algorithm would not recognize it if P5 were omitted. It would yield an s~ CSo and would then proceed to construct a sequence s2, s3 . . . . which would converge to So. The use of a truncation error ~.> 0 is introduced to stop computations after a finite number of steps when IlV(So)- W(So)ll~ e.

On the removal of ill conditioning effects in the computation of optimal controls

613

FLOW CHART Read In [)ata print Data 7"ect~ Data

L

1

/

T

E)qTS /-i") U~dr:,en respense inside target set ~ .

L

¢ompute the mop r( )and an iaten or point

I

NO-~S Chose e > 0 smlOl~

OPPn ol caqlroIs :0 '~

s **,,'s~)-[s} S~ fwfS~'l ....So}_ u ' u u ~ S,,, t w(SO, ~'(53 I

Target set not reachable.

(~) Problem s0;ved

C)

r

-

compote v (So), so

Prin| oplirral cc"trc!s as gi~ert by P 6

_

j

~-Use P 2 , Io compute

I

I

W(So),C(So }

use P2 !a | ,_ompate v (sl . _ ~

'Ti: ;

J

' . T ,~

~,D

__1

Set__

U=So÷~(s'-s d i

( L' 4) ) ~ - , ~ < ~ def Fit i no "~-~ ~ solahonsj

0

!c ,TampJ'£ .. (S']:

i]se P 2

"-.u

j

ltJ Set

.....

I s,:so+~(s' Sa: 1

..e:

i [

. . . . . . . . . . .

Z ,A,b, :,/i

,'is:,> cts'l

:

I

t~ ] soma,ire . s')

Use F2 ~

I' !

[

"11

FIG. 2. Flow chart.

4. CONCLUSION

The algorithm presented in section 3 is the least obvious and least direct one of the three methods presented. However, its computational behaviour is considerably better than that of the other two. As a reference point of comparison, the author would like to indicate that on a number of specific runs on a problem with N = 50 and n = 10, a tenth order system with eigenvalues of A ranging from 0"9 to 0-5, the parametric method took 3-5 sec

of IBM 7094 time to obtain a solution, while the gradient method of section 1 required 40-60 sec to compute. The reason for the good behavior of the parametric method is obvious: it requires no matrix inversions, and the substitute step, the solution of (37), which it introduces is easy to carry out. With non-linear programming becoming more and more relevant to control problems, it is hoped that the present work will facilitate the task of modifying and adapting standard algorithms to the specific structure of optimal control problems.

614

E. POLAK

REFERENCES [1] M. CANNON, C. COLLUM and E. POLAK; Constrained minimization problems in finite dimensional spaces. J. S I A M Control 4, 528-547 (1966). [2] H. HALKIN and L. W. NEVSTADT: General necessary conditions for optimization problems. Proc. Nat. dcad. Sci. 56, 1066-1071 (1966). [3] E. POLAK: On the convergence of optimization algorithms. Revue Francaised Informatique et de Recherche Operationelle, Serie Rouge 16 (1969). [4] G. ZOrdTENDIJK: Methods of Feasible Directions: a study in linear and nonlinear programming. Elsevier, Amsterdam (1960). [5] N. N. KRASOVSKU: On the theory of optimal control. PrikL Mat. Meh. 23, 625-639 (t959). English translation in J. Appl. Math. Mech. 23, 899-919. [6] L. W. NEOSTADT: Synthesizing time optimal control systems. J. math. Analysis Applic. 1, 484-493 (1960). [7] J. H. EATON: A n iterative solution to time-optimal control. J. math. Analys& dpplic. 5, 329-344 (1962). [8] E. POLAK and M. DEPARIS: An algorithm for Minimum Energy Control. University of California, Electronics Research Laboratory, Berkeley, California, ERL Memorandum M225, November 1, 1967. [9] R. O. BARR and E. G. GILBERT: Some Iterative Procedures for Computing Optimal Controls. Third Congress of the International Federation of Automatic Control, London 20-25, Paper No. 24.D, June, 1966. [I0] P. WOLFE: The simplex method for quadratic programming. Econometrica 27, 392-398 (1959). [11] H. W. KUHN and A. W. TUCKER: Nonlinear Programming, Proc. of the Second Berkeley Symposium on Mathematic Statistics and Probability. University of California Press, Berkeley, California, pp. 481-492, 1951. [12] G. HADLEY: Linear Programming. Addison-Wasley, 1963. [13] C. E. LE~aKE: A method for solution of quadratic programs. Management Seience 8, 442-453 (1962). [14] G. FORSYTHE and C. B. MOLER: Computer Solution of Linear Algebraic Systems. Prentice Hall, Section II (1967). [15] J. B. PLANT: Some lterative Optimal Control Solutions. Monograph, MIT Press (1966).

R6sum6--11 est montr6, d'abord, comment des effets d'influence d6favorable peuvent prendre naissance lorsque des probl6mes de commande optimale discrete avec une dynamique lin6aire et des cofits et contraintes conv6xes sont r6solus fi l'aide de m6thodes "primaires", telles que la programmation lin6aire ou quadratique ou certaines m6thodes de gradient. 11 est pr6sent6, ensuite, une m6thode duale qui exploite au maximum la structure dynamique du probl~me de commande optimale. 11 a 6t6 trouv6 que cette m6thode fonctionne tr6s bien et n'est pas soumise ~ des effets d'influence d6vavourable.

Zusammenfassung--Zuerst wird gezeigt, wie nachteilige Effekte auftreten k6nnen, wenn diskrete OptimalwertRegelungsprobleme mit linearer Dynamik und Beschr/inkung durch "primal" Methoden, wie linearer oder quadratischer Programmierung oder gewissen Gradient-Methoden geliSst werden. Eine duale Methode wird angeftihrt, die die dynamische Struktur des optimalen Regelungsproblems bis zum ~iuBersten auswertet. Dieses Verfahren l[iBt sich gut durchffihren trod leidet nicht dutch nachteilige Effekte.

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