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Chapter II The Formulation of Optimization Problems With Vector Functionals
CHAPTER I1 THE FORMULATION OF OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
1.
The General Mathematical Formulation of the Problem
We formulate a new statement of the problem of optimizing a vector functional. The idea of such a formulation was first presented in the author's works.(142,143). [ Z ] , Let each element of some space Z define a system S characterized by some vector performance index I
=
which is
-
I(I l,. . ,Ik).
In such a situation each element of the space Z, defining the system S [ Z ] ,
will have associated with it a definite point with
.,Ik (Z) in the space of performance
Coordinates I (Z) I (Z),.. 1
indices I(Z).
2
So, 'Eor example, for the element z E Z, the system
will have the performance indices I1(z),. .. ,I (2) dek fining a point I(z) E I ( Z ) .
Sz
E
S [ Z ] ,
We let z a
E
Z be an element of the space 2 for which
with a = 1, ...,k. This means that for the element za
E
Z, only the performance
index I ( z ) assumes its optimal value. a We consider the euclidean space I(2) with coordinates I (Z),..., 1 Ik(Z). Let the values I ( z ) (a=l,...,k) be given. It is always a a possible to say that there exists an element z* E 2, forming a system
Sz*
E
S [ Z ]
which, in the space I (Z), the values of the
functionals I (z*) 1
.,Ik ( z * )
will be the best (in some agreed
29
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
30
upon sense) approximation to the true optimal values of these functionals I ( z ),I ( z ),...,I ( z ) . As a criterion for such an 1 1 2 2 k k approximation, it is possible to take any positive-definite function
of the variables I (Z),I (Z),... ,Ik(Z). We let R(Z) denote such a 1 2 function which forms one or another metric in the space I(Z). Definition. A n element z*
E
Z optimizes the system S in the
R(Z) -sense if R(z*) = inf R(2). ZEZ
Consequently, the problem of optimizing the system S with a vector criterion may be formulated as: find z*
E
given S [ Z ] ,
I(Z), R(Z),
Z.
As noted in the last chapter, the vector optimization problem was considered in much the same spirit in ( 8 9 ) ; however, it is important to emphasize the essential difference between the problem statement given above and that in (89).
Recently, these ideas have
been developed in the works (91-99).
2.
A Statement of Optimal Trajectory Programming Problems with a Vector-Valued Criterion
We present a statement of the problem in the same form as it was given in (142). Let the system dynamics be given by the differential equations
which are defined in some region N(x(t),U(t))
x(xll...,xn and control space u(ul ,...,um 1 , t
0 of state space E
T, T =
Po
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
31
The right side of the expression (2.2) is a vector with coordinates
...,fn (x,u,t),
fl(xru,t),f2(x,urt),
which are assumed to be con-
tinuously differentiable functions in the variables x, u and t. Let some admissible class of controls U be given where u assumes values in the region N
2 0.
E
U
Also, let there be given a
vector functional
with components
It is assumed that the functions @[email protected] have continuous n first partial derivatives with respect to all arguments in the region N
20
for t
E
T.
Let the boundary values for the vector x(t) be x(t 0) = xor where T is free.
x(T)
=
x T'
Symbolically, we denote this as (i,f) = 0.
Using known procedures, for each
c1 =
l,...,k we determine the
optimal control
for each scalar functional I (u). c1
Here the vector u(~) has components u1(a),...,u(") and is the m optimal control vector under which the scalar functional I (u) = c1
Qa(x(t) ,u(t) ,t) assumes its optimal value along the trajectory of the system (2.2) which passes between the boundary points (2.5). associated k when
Naturally, there are different periods of time T1,...,T with the control vectors u'")
.
(a=l,. .. ,k)
Moreover ,
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
32
optimizing each scalar functional, additional conditions may be taken into account such as construction of the region N
2 0, spe-
cific boundary conditions and also the class V ( c L ) of admissible controls. We compute the value of the vector
Its components are known numbers. We consider the euclidean norm
-
where the vector I(u)
I*(u) is defined for all admissible controls
u E V.
Obviously, the set of admissible controls V must be the maximal restriction of all the admissible V (1),...,V (k). 0
Definition. ~ -
We will say that the control u (t,x ,x ) optimizes O T the vector functional ( 2 . 3 ) if the inequality
is satisfied for any admissible control u.
We call such a control
optimal relative to the vector functional. Problcm. ~-
Given the equation (2.21, the boundary conditions (2.5),
the vector functional (2.3) and the class V of admissible controls, 0
determine a control u (t,xO,xT) optimizing the vector functional. In the language of the preceding section, here the control vector u(t) plays the role of z and determines the motion of the system (2.2).
As a criterion of approximation R(Z) we choose the
expression ( 2 . 8 ) , while the meaning of the equality (2.1) is embodied in the inequality (2.9).
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Remark 1.
33
It is assumed that in the original problem all the
variables and functionals are reduced to dimensionless form.
If
not, then instead of minimizing the expression (2.8) we must consider minimizing the sum of squares of the relative deviation of the functionals (2.4) from their nominal optimal values (2.71, i.e. we minimize the function I(u)-I* (u)
(2.10)
R(u) =
A generalized form of the approximation function R(S) is to take
the norm (2.11) which, for L
=
1, reduces to a linear combination of the components
of the vector I(u) ean norm R,(u)
-
I*(u), for L = 2 it coincides with the euclid-
I I I (u)-I* (u)I I
and for L
=
m
= max {Ia(u)-Ia(u(a)) la=l,
a
Remark 2.
the supremum norm
...,kl.
(2.12)
It is natural to let T be either free or fixed.
In
do not differ too greatly from particular, if the numbers Tl,...,T k each other, then the time T must also not d.eviate too greatly from these values.
Moreover, it makes sense to consider the case when k 1 Ta 01 T is fixed and equals, for instance, T = a=1
T
5 max{T a Ia=l,...,k}. A geometrical interpretation of the problem.
euclidean space for the vector I(u).
We consider the
The sum (2.8) is the square
of the distance from the point corresponding to the control u to
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
34
the point having coordinates I ( u ( l )) ,I2 (u(~) ) ,. ..,Ik (u(~) ) . We 1 call this latter point the ideal (or utopian) point. The problem consists in choosing a control u which minimizes this distance over a time T, which does not deviate too much from the numbers T
1'""
TkA physical interpretation of the problem.
If the vector
0
u (t,x ,x ) minimizes (2.8), then it will be the vector under which O T 0 the functionals I (u ) ( a = l , ...,k) assume values as near as possible a
(2) (k) to the numbers I ( u ' l ) ) , I 2(u 1 , . ..,Ik(u 1. 1 The sense of this approximation is as follows: we assume that
we have chosen some scalar functional I
a'
class of admissible controls control u'~).
determined the desired
for it and found the optimal
The number I a ( u ( a ) ) we take as the index of quality
of the system, which may be achieved in an ideal case.
The ideal
case is that in which the total control resources are used for achieving the optimal value for only one of the criteria. Moreover, in this ideal case it is possible, in general, to follow two different directions characterized by the choice of the class of admissible controls and also by the choice of boundary conditions for optimization of each scalar functional taken individually. In actuality, we may not ignore the other performance indices and often, in order to achieve an improvement in the system relative to the set of all these indices, we must operate in a more restricted class of admissible controls. 0
For the chosen control u
,
there takes place some deterioration
of each performance index I (u) if taken separately (in comparison a
with that value obtained if we would optimize only with respect to that particular criterion); however, this deterioration is spread throughout the whole set of criteria Ia (u) and is as small as possible.
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Remark 3 .
35
The usual approach to the problem of optimizing k
functionals is to minimize the sum C X I (u). Here it is assumed c1 act that the weighting coefficients are given exactly, i.e. the weights of the functionals I (u) (a=l,...lk) are known exactly. c1
However,
this is possible only in a few cases. Let us now assume that the statement of the vector optimization problem is freed from the constraint of knowing the weights of the functionals. We seek a control which uniformly approximates the 0 values of the functionals I (u ) to their optimal values I (u‘“)). c1
c1
There are now opportunities for other forms of approximation. Remark 4. Here we have formulated the problem of programming optimal trajectories. Usually, it is also possible to formulate 0
the problem of analytic construction of a regulator u (x,t), which optimizes the vector functional. Again it is relevant to emphasize the essential difference of the given problem statement from that which was given in section 6 of Chapter I.
1. The choice of the numbers I1 (u(1)) ,I2 ( u ” ) 1 , . stipulated in a unique fashion.
. . ,Ik(U(k)) is
In the space of the vector
1 this means the following (see Fig. 4 for a simple k illustration of the case k = 2). Let the point I* have coordinates 1{11,1 2,...,I
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
36
... ,Ik (u(~)) .
(1)) ,I2 (u(~) ), I1 (u
Then it is clear that optimization
of the vector functionals (2.3) is possible only in the shaded positive octant Q.
But this means that as a measure of approxi-
mation of the vector I(u) to the point I*, we may choose not only the euclidean norm (2.8) or (2.10), but also (2.11) for any L
2
1
and the linear function (2.13) for constants c
c1
> 0 given in advance.
We can also use any con-
tinuous positive function R(u) defined on Q. 2.
Assignment of the boundary values (2.5) may be carried out
in a different manner for each optimization taking the functionals I (u) separately, but then for optimization of the function R(u) a these conditions must reflect a basic and not auxiliary aspect of the problem. 3.
In each particular optimization problem, the set of admis-
sible controls is particular to that problem.
Since the statement
of the vector optimization problem is developed for the set of (1),v(2) admissible controls V, which is itself restricted by V , . . . I V(k), the position of the point I* depends upon the structures of
,.. .,V(k).
The value min R(u) will be minimal uev in the case when the point I* is also defined for u E V.
the sets V(l) ,V(2)
3.
Flight at a Prescribed Altitude
We illustrate the given problem statement by an example from the book (10).
The equations of flight of an aircraft at a given
altitude under reasonable assumptions may be expressed in the form (1.11)
.
31
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
We introduce the boundary conditions: (i):
v
x =0, i
= vo,
mi = m0' (2.14)
(f):. xf = xT,
vf = vTI
mf = mTl
mo, vT, mT are given numbers since the values x
Here v
T
T are free.
and
As a functional we take
AG = -X TI
(2.15)
the minimum of which corresponds to maximizing the distance x
T'
Now among the admissible x, v, m, f3 we find that curve which minimizes the functional (2.15)
.
According to the maximum principle ( 6 ) , we form the function H = v\Y1
-
\Y2
+
(:
)@
and write the equation for the co-state vector Y i Y Y
Y
Here
1 2
(2.16)
'4 -\Y 2 3
Y ,Y 1 1' 2 3
= 0,
=-Y
QV
1
+-Y
m
2'
(2.17)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
38
A first integral and the transversality conditions are written in the form
-
vY1
Y2 +
(z
Y2-Y3)B
=
(2.18)
c,
(2.19) respectively. By virtue of the boundary conditions (2.14), condition (2.19) gives
YIT
c = 0.
= 1,
(2.20)
For the optimal control, the maximum principle defines the solution
B
=
P
for far
= 0
C Y m 2
Ey2
Y
3
> 0,
(2.21)
- Y3 < o .
The singular solution 0
5 B 5
appears on an interval
where we have the two integrals VY
1
- m8 ,2 = o ,
;
- Y2
-
Y3
(2.22)
= 0.
By Poisson's theorem (c.f. (18), section 181, we obtain a third integral
cmY 1
+
(mQm-cQV-Q)Y2 = 0.
(2.23)
39
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Determination of the three relations (2.221, (2.23) requires
(2.24)
This relation defines the manifold S,(m,v)
=
cQ
+
v(mQ -cQ -Q) = 0. m v
(2.25)
In the particular case of a parabolic polar, where m Q = Av2'+ B V
(2.26)
2
with A, B given positive constants, we will have 2 S (m,v) = Bm (v+3c) 1
-
4
Av (v+c) = 0.
(2.27)
It is not difficult to establish that here Poisson's theorem does not give any other linearly independent integrals. According to (18) (cf. 518), the form of the sinqular control is written as QSlv B 1= cSlv - mS
(2.28) lm
which, in the case of the parabolic polar (2.26), will have the form
B, =
4 2 [Av (3v+2c) + Bm (v+6c)]Q 4 2 2 2 ' Acv (3v+2c) + Bm v(2v+7c) + 6Bc m
(2.29)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
40
Thus, the optimal control of the thrust is represented, generally speaking, as a piecewise-continous function consisting of the three arcs (2.211,
(2.28).
As was already noted in section 3 of
the preceeding chapter, the optimal combination of these arcs may be the curve impuf depicted in Fig. 5. From the construction, it follows that in any case the control
8. Actually, since the 1dm manifold ( 2 . 2 5 ) passes through the origin, - > 0 and, consequently, dv for the control (2.28) cB < Q(m,v) while at the time when B = B (2.28)
satisfies the condition 0 5 B
we have cB > Q(m,v). Now we define a program thrust which will be optimal in the minimum time sense, i.e. for the system performance index we take the functional T
I2 =
I0 dt = T.
(2.30)
We describe the problem in new variables. pendent variable. dt _ _ -1 dx
dv -
v
=
cB
dx
We let x be the inde-
Then Eq. (1.11) takes the form
I
-
Q(m,v)
mv
(2.31)
I
dm _ _ - BV dx The functional ( 2 . 3 0 ) G=t,
is written in the Mayer form as
A G = tf = Tmin'
(2.32)
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
41
m
S,(m.
vl
=
0
S, (m. V ) = o
7
C
"
Fig. 5 We now apply the maximum principle.
We have the function (2.33)
V
the auxiliary equations
-d y-1 dx
- 0,
2 A + vQv ax
ax
(2.34)
mv
V
-dy3 - -
- Q
mQm
mv
- Q Y Y - CB Y
m v
m2v
V
VECTOR-VALUED OPTlMlZATlON PROBLEMS IN CONTROL THEORY
42
and the form of the optimal thrust a s 6 = 0 f o r L ~- - Y3 < 0, mv 2 v
(2.35)
-
fo
1 for%^ - - Y >o. m v 2 v 3 = -1 and C = 0. 1T Since x is free, we have
The transversality conditions determine Y We construct the singular control. the first integral P
+
MB = 0 ,
where P and M are yl
C
M = - Y
P = v - % Y z ,
mv
On the interval
2
[ ,t21, the
- -vy 3.
(2.36)
singular solution satisfies the
two integrals
=o,
p = -yl- L y
v
mv
M = % Y
mv
2
2
-_ y3 v
(2.37) 0.
The Poisson bracket gives the new first integral mQm V
mv
mv
- Q
(2.38)
m v
or, what is equivalent (2.39)
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
43
The integral manifold of the singular solutions must satisfy the equation
- _1
- -C
0
mv
-cm
= 0.
V
V(mQm-cQv-Q)
(2.40)
mQ
Thus, as follows from (2.43), we have the manifold of singular solutions s2(mtV) = mQm
- cQV
-
(2.41)
Q = 0,
which, in the case of the parabolic polar (2.261, is represented by the expression s2(m,v)
-
=
*v4
(2.42)
= 0.
Analagously, by the formula (2.281, the form of the singular control is written as
8,
=
QS2V cs - mS2m 2v
1
(
'2v
= -as2
av
'2rn
1
(2.43)
In the case of the parabolic polar (2.26), the singular control may be written as 3
8,
=
2Av Q 2Acv3
-
(2.44)
2' Bm
Turning to graphics, the manifold
S
2
(m,v) = 0 will be located
to the left of the manifold S (m,v) = 0, as is shown in Fig. 5 . 1 In order to convince ourselves of this, from formulas (2.25) and (2.41) we determine Q.
In the case S (m,v) = 0, we have 1
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
44
Q
V
= Ql(m,v) = v - c (m&-cQv)
(2.45)
I
while in the case of the manifold S2 (m,v)
Q
= Q2(m,V) = mQm
-
=
0,
(2.46)
CQ,.
Since the strength of the aerodynamic drag Q(m,v) is positive and c > 0 , the comparison of (2.45) with (2.46) gives
Q, (mlv)> Q2 (m,v).
(2.47)
Since the mass only decreases, for a fixed mass, equality of Q, and Q can only be achieved under the condition v > v2. The 2 1 validity of this is shown by the arrangement of the curves S = 0 1
and S2
= 0
in Fig. 5.
Such an ordering becomes evident for determination of Q in the form of the parabolic polar (2.26).
Describing the manifold (2.27)
in the form
and the manifold (2.42) in the form
it is evident that for a fixed value of the velocity we will always have m 1 < m2' In the first problem for the functional (2.15), the optimal trajectory, as was noted, corresponds to the curve impnf, while in the second problem for the functional (2.32), the optimal curve is im'p'n'f (Fig. 5).
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
4s
Now we consider the motion which is optimal, say, for the criterion
(2.47') in which xo is the maximal achieved distance in the first optimization problem, while to is the minimal time in the second problem. The optimal control will be composed of f3 = 0, f3 singular arcs.
=
and
In fact, the Pontryagin function has the same form as in (2.16); the auxiliary equations also coincide with (2.17) and the first integral remains in the form (2.18).
This insures that the form of
the optimal control is as defined in (2.21); however, for determination of the vector Y we use the following transversality
[(-
condition
+
( + x y
. ) & t + y 2 6 v + y 3 6 1 j
f
= 0.
(2.48)
i According to the boundary conditions (2.14), the condition (2.48) splits into the relations (2.49)
(2.50) Now the first integral is written as 0
T - t P-Mf3=-1 (to)
(2.51)
in which T is the duration of the process, while P and M have the forms
46
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
P=vY
C
M = - Y
- Q
m
2
(2.52)
- '3.
, the singular solution will satisfy the
On the interval following integrals
(2.53) M = C Y - Y =o. m 2 3
(2.54)
We calculate the first Poisson bracket R = (P,M) =
cmYl m
+ NY2
2
(2.55)
= 0,
in which N(m,v)
=
Qmm
-
cQv
-
Q
=
(2.56)
S2 (m,v).
Now we calculate the second Poisson bracket (R,P+MB)
= (R,P)
+
(R,M)B = 0 ,
(2.57)
which, as a result, defines the new integral -rn(cB+N)Yl
+ [(cNv-mm)B + NQv-NVQ]Y2
= 0.
(2.58)
From the integrals (2.54), (2.55) and (2.58), we determine the manifold of singular solutions.
We have the determinant
0
C -. m
1
cm
N
0
= 0,
(2.59)
2.
OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
which after expansion gives S =
-N2
-
c (NQ -QN )
v
v
-
c
c N+ (cNv-mNm)] 8 = 0 .
41
(2.60)
From the last expression, we determine the singular control as
'
N~ + C(NQ~-QN,) =
(2.61)
c (mm-cNv-N)
Taking into consideration (2.56), the last expression may be written in the form
'
=
s2 +
QS2V c~~~ - mS
+ 2m
S2
CQ-mQm)
(2.62)
c ( s2+cS 2v-mS2m)
We note that if in (2.52), S2 = 0, then it is converted into (2.43), i.e. into the expression which was defined for the solution of the minimum time problem. The relation S1 = CQ + vS2, obtained by comparing (2.25) and (2.41), gives the possibility to determine the dependencies
s2v
- _1 (Slv+Q-mQm)r v
substitution of which into (2.62) determines the singular control 3, in the following form:
8
=
cs lv
QS1" - mSlm
+ s1 +
s1(Q-mQ,)
-
c ( c ~ ~ +s ~ )- m ~ ~ ~
(2.63)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
48
=O
V
Fig. 6 If we suppose that
S
1
=
0 in (2.63), then it is converted into
(2.28), i.e. into the expression which was defined for the solution of the maximal distance problem. The location of the curve (2.60) in the (m,v) plane depends on the form of the function Q(m,v), characterizing the aerodynamic
drag.
It may be situated to the right of the curve S (m,v) = 0 1 or to the left of the curve S (m,v) = 0, or between them. The 2 motion of the point from i to f in this plane will be the curves im p n f, im p n f, im3p3n3f, respectively (Fig. 6). 111 2 2 2 We hope that this example has completely cleared up the expediency of the approach to the vector optimization problem in the spirit of section 1 of this chapter.
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
4.
49
Analytic Construction of Optimal Regulators
The problem of analytic construction of optimal regulators (the ACOR problem) has the same statement as the problem of programming optimal trajectories. However, within it are specifications which demand its separate consideration. First of all, we assume that we know the programmed control
which represents the exact solution of the vector equation (2.2) with boundary conditions(2.5). The ACOR problem is posed relative to the perturbed dynamics described by the equation (18)
where
q
is the deviation of the vector x from the programmed value
x*, generated by means of the disturbance $(t); 5 is the increment in the control vector used to bring the actual motion nearer to the programmed motion, i.e. rli(t)
=
x
p
- xZ(t) ,
(i=l,.. .,n; j=1,.
5 . (t)
=
u . (t)
af(x,u,t)
,
=
3
3
-
u3 (t) 7
..,m).
Here the matrices B
=
ax
af (xrurt) are known, timeau
varying functions defined on the motion (2.2) and calculated for the given program (2.64); by o(n,(,t)
we denote a power series
convergent for all q r 5 . We impose a series of requirements on the perturbed motion given by Eq. (2.65).
These requirements are based upon physical
aspects of the controlled system (cf. (39)) .
50
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
Let (2.66) be functionals reflecting these requirements.
This collection of
functionals forms a vector functional.
( n , < ) ,...,w k ( n , < ) are assumed to be 1 positive-definite and continuously-differentiable in their arguThe integrand functions w
ments. Let
I Ino/ I
5 A be a neighborhood of the origin in which the
perturbed motion can begin. Let (2.67) be a feedback law lying in the class of admissible controls V, yielding the optimum of the index I for the closed-loop system a (2.68) The law (2.67) may be synthesized for any a = l,...fkl using known methods of solution of the ACOR problem for a single functional (18). We note that the class of admissible controls V consists only of those functions S(q,t) for which the system of equations (2.65) is asymptotically stable. It is clear that for each fixed a , the law (2.67) may ignore all requirements contained in the other functionals of (2.66). We assume that all the functions (2.67) are known and we compute the numerical value of all functionals Ia(<(@")) = Ia0' We form the euclidean norm in the space of vectors 1(11,1 2,..., Ikl:
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
k
51
k (2.69)
defined for all admissible controls 5
E
V.
Definition. We will say that the feedback law C0(n,t) optimizes the vector functional with components (2.66)
if the inequality (2.70)
is satisfied for any admissible control 5
E
V.
We will say that
such a feedback law is optimal relative to the vector functional. Problem.
Given Eq. (2.65)
,
the vector functional (2.66)
and
the class of admissible controls V, it is required to determine a 0
feedback law 5 (n,t) optimizing the vector functional in the above sense.
This problem is studied in (143,148).
The mathematical formulation of the given problem is no different than the formulation of the problem of programming optimal trajectories considered in section 2 of this chapter. The distinction lies only in the given boundary conditions:
in the ACOR problem
all components of the state vector at the endpoints of the trajectory are assumed to be free and only in the special case of the infinite-interval problem must we impose the condition
~ ( m )=
0.
However, there is a real difference in the form of the solutions to these two problems.
In the problem of determining optimal pro-
grammed trajectories, the solution is sought in the form of a timevarying function of the boundary conditions, while in the ACOR problem the feedback law is sought in the form of a vector function of the state and time. It is clear that all the remarks made relative to the choice of the function R(u) in section 2 of this chapter retain their meaning and value for the synthesis problem.
1.
The General Mathematical Formulation of the Problem
We formulate a new statement of the problem of optimizing a vector functional. The idea of such a formulation was first presented in the author's works.(142,143). [ Z ] , Let each element of some space Z define a system S characterized by some vector performance index I
=
which is
-
I(I l,. . ,Ik).
In such a situation each element of the space Z, defining the system S [ Z ] ,
will have associated with it a definite point with
.,Ik (Z) in the space of performance
Coordinates I (Z) I (Z),.. 1
indices I(Z).
2
So, 'Eor example, for the element z E Z, the system
will have the performance indices I1(z),. .. ,I (2) dek fining a point I(z) E I ( Z ) .
Sz
E
S [ Z ] ,
We let z a
E
Z be an element of the space 2 for which
with a = 1, ...,k. This means that for the element za
E
Z, only the performance
index I ( z ) assumes its optimal value. a We consider the euclidean space I(2) with coordinates I (Z),..., 1 Ik(Z). Let the values I ( z ) (a=l,...,k) be given. It is always a a possible to say that there exists an element z* E 2, forming a system
Sz*
E
S [ Z ]
which, in the space I (Z), the values of the
functionals I (z*) 1
.,Ik ( z * )
will be the best (in some agreed
29
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
30
upon sense) approximation to the true optimal values of these functionals I ( z ),I ( z ),...,I ( z ) . As a criterion for such an 1 1 2 2 k k approximation, it is possible to take any positive-definite function
of the variables I (Z),I (Z),... ,Ik(Z). We let R(Z) denote such a 1 2 function which forms one or another metric in the space I(Z). Definition. A n element z*
E
Z optimizes the system S in the
R(Z) -sense if R(z*) = inf R(2). ZEZ
Consequently, the problem of optimizing the system S with a vector criterion may be formulated as: find z*
E
given S [ Z ] ,
I(Z), R(Z),
Z.
As noted in the last chapter, the vector optimization problem was considered in much the same spirit in ( 8 9 ) ; however, it is important to emphasize the essential difference between the problem statement given above and that in (89).
Recently, these ideas have
been developed in the works (91-99).
2.
A Statement of Optimal Trajectory Programming Problems with a Vector-Valued Criterion
We present a statement of the problem in the same form as it was given in (142). Let the system dynamics be given by the differential equations
which are defined in some region N(x(t),U(t))
x(xll...,xn and control space u(ul ,...,um 1 , t
0 of state space E
T, T =
Po
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
31
The right side of the expression (2.2) is a vector with coordinates
...,fn (x,u,t),
fl(xru,t),f2(x,urt),
which are assumed to be con-
tinuously differentiable functions in the variables x, u and t. Let some admissible class of controls U be given where u assumes values in the region N
2 0.
E
U
Also, let there be given a
vector functional
with components
It is assumed that the functions @[email protected] have continuous n first partial derivatives with respect to all arguments in the region N
20
for t
E
T.
Let the boundary values for the vector x(t) be x(t 0) = xor where T is free.
x(T)
=
x T'
Symbolically, we denote this as (i,f) = 0.
Using known procedures, for each
c1 =
l,...,k we determine the
optimal control
for each scalar functional I (u). c1
Here the vector u(~) has components u1(a),...,u(") and is the m optimal control vector under which the scalar functional I (u) = c1
Qa(x(t) ,u(t) ,t) assumes its optimal value along the trajectory of the system (2.2) which passes between the boundary points (2.5). associated k when
Naturally, there are different periods of time T1,...,T with the control vectors u'")
.
(a=l,. .. ,k)
Moreover ,
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
32
optimizing each scalar functional, additional conditions may be taken into account such as construction of the region N
2 0, spe-
cific boundary conditions and also the class V ( c L ) of admissible controls. We compute the value of the vector
Its components are known numbers. We consider the euclidean norm
-
where the vector I(u)
I*(u) is defined for all admissible controls
u E V.
Obviously, the set of admissible controls V must be the maximal restriction of all the admissible V (1),...,V (k). 0
Definition. ~ -
We will say that the control u (t,x ,x ) optimizes O T the vector functional ( 2 . 3 ) if the inequality
is satisfied for any admissible control u.
We call such a control
optimal relative to the vector functional. Problcm. ~-
Given the equation (2.21, the boundary conditions (2.5),
the vector functional (2.3) and the class V of admissible controls, 0
determine a control u (t,xO,xT) optimizing the vector functional. In the language of the preceding section, here the control vector u(t) plays the role of z and determines the motion of the system (2.2).
As a criterion of approximation R(Z) we choose the
expression ( 2 . 8 ) , while the meaning of the equality (2.1) is embodied in the inequality (2.9).
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Remark 1.
33
It is assumed that in the original problem all the
variables and functionals are reduced to dimensionless form.
If
not, then instead of minimizing the expression (2.8) we must consider minimizing the sum of squares of the relative deviation of the functionals (2.4) from their nominal optimal values (2.71, i.e. we minimize the function I(u)-I* (u)
(2.10)
R(u) =
A generalized form of the approximation function R(S) is to take
the norm (2.11) which, for L
=
1, reduces to a linear combination of the components
of the vector I(u) ean norm R,(u)
-
I*(u), for L = 2 it coincides with the euclid-
I I I (u)-I* (u)I I
and for L
=
m
= max {Ia(u)-Ia(u(a)) la=l,
a
Remark 2.
the supremum norm
...,kl.
(2.12)
It is natural to let T be either free or fixed.
In
do not differ too greatly from particular, if the numbers Tl,...,T k each other, then the time T must also not d.eviate too greatly from these values.
Moreover, it makes sense to consider the case when k 1 Ta 01 T is fixed and equals, for instance, T = a=1
T
5 max{T a Ia=l,...,k}. A geometrical interpretation of the problem.
euclidean space for the vector I(u).
We consider the
The sum (2.8) is the square
of the distance from the point corresponding to the control u to
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
34
the point having coordinates I ( u ( l )) ,I2 (u(~) ) ,. ..,Ik (u(~) ) . We 1 call this latter point the ideal (or utopian) point. The problem consists in choosing a control u which minimizes this distance over a time T, which does not deviate too much from the numbers T
1'""
TkA physical interpretation of the problem.
If the vector
0
u (t,x ,x ) minimizes (2.8), then it will be the vector under which O T 0 the functionals I (u ) ( a = l , ...,k) assume values as near as possible a
(2) (k) to the numbers I ( u ' l ) ) , I 2(u 1 , . ..,Ik(u 1. 1 The sense of this approximation is as follows: we assume that
we have chosen some scalar functional I
a'
class of admissible controls control u'~).
determined the desired
for it and found the optimal
The number I a ( u ( a ) ) we take as the index of quality
of the system, which may be achieved in an ideal case.
The ideal
case is that in which the total control resources are used for achieving the optimal value for only one of the criteria. Moreover, in this ideal case it is possible, in general, to follow two different directions characterized by the choice of the class of admissible controls and also by the choice of boundary conditions for optimization of each scalar functional taken individually. In actuality, we may not ignore the other performance indices and often, in order to achieve an improvement in the system relative to the set of all these indices, we must operate in a more restricted class of admissible controls. 0
For the chosen control u
,
there takes place some deterioration
of each performance index I (u) if taken separately (in comparison a
with that value obtained if we would optimize only with respect to that particular criterion); however, this deterioration is spread throughout the whole set of criteria Ia (u) and is as small as possible.
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Remark 3 .
35
The usual approach to the problem of optimizing k
functionals is to minimize the sum C X I (u). Here it is assumed c1 act that the weighting coefficients are given exactly, i.e. the weights of the functionals I (u) (a=l,...lk) are known exactly. c1
However,
this is possible only in a few cases. Let us now assume that the statement of the vector optimization problem is freed from the constraint of knowing the weights of the functionals. We seek a control which uniformly approximates the 0 values of the functionals I (u ) to their optimal values I (u‘“)). c1
c1
There are now opportunities for other forms of approximation. Remark 4. Here we have formulated the problem of programming optimal trajectories. Usually, it is also possible to formulate 0
the problem of analytic construction of a regulator u (x,t), which optimizes the vector functional. Again it is relevant to emphasize the essential difference of the given problem statement from that which was given in section 6 of Chapter I.
1. The choice of the numbers I1 (u(1)) ,I2 ( u ” ) 1 , . stipulated in a unique fashion.
. . ,Ik(U(k)) is
In the space of the vector
1 this means the following (see Fig. 4 for a simple k illustration of the case k = 2). Let the point I* have coordinates 1{11,1 2,...,I
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
36
... ,Ik (u(~)) .
(1)) ,I2 (u(~) ), I1 (u
Then it is clear that optimization
of the vector functionals (2.3) is possible only in the shaded positive octant Q.
But this means that as a measure of approxi-
mation of the vector I(u) to the point I*, we may choose not only the euclidean norm (2.8) or (2.10), but also (2.11) for any L
2
1
and the linear function (2.13) for constants c
c1
> 0 given in advance.
We can also use any con-
tinuous positive function R(u) defined on Q. 2.
Assignment of the boundary values (2.5) may be carried out
in a different manner for each optimization taking the functionals I (u) separately, but then for optimization of the function R(u) a these conditions must reflect a basic and not auxiliary aspect of the problem. 3.
In each particular optimization problem, the set of admis-
sible controls is particular to that problem.
Since the statement
of the vector optimization problem is developed for the set of (1),v(2) admissible controls V, which is itself restricted by V , . . . I V(k), the position of the point I* depends upon the structures of
,.. .,V(k).
The value min R(u) will be minimal uev in the case when the point I* is also defined for u E V.
the sets V(l) ,V(2)
3.
Flight at a Prescribed Altitude
We illustrate the given problem statement by an example from the book (10).
The equations of flight of an aircraft at a given
altitude under reasonable assumptions may be expressed in the form (1.11)
.
31
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
We introduce the boundary conditions: (i):
v
x =0, i
= vo,
mi = m0' (2.14)
(f):. xf = xT,
vf = vTI
mf = mTl
mo, vT, mT are given numbers since the values x
Here v
T
T are free.
and
As a functional we take
AG = -X TI
(2.15)
the minimum of which corresponds to maximizing the distance x
T'
Now among the admissible x, v, m, f3 we find that curve which minimizes the functional (2.15)
.
According to the maximum principle ( 6 ) , we form the function H = v\Y1
-
\Y2
+
(:
)@
and write the equation for the co-state vector Y i Y Y
Y
Here
1 2
(2.16)
'4 -\Y 2 3
Y ,Y 1 1' 2 3
= 0,
=-Y
QV
1
+-Y
m
2'
(2.17)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
38
A first integral and the transversality conditions are written in the form
-
vY1
Y2 +
(z
Y2-Y3)B
=
(2.18)
c,
(2.19) respectively. By virtue of the boundary conditions (2.14), condition (2.19) gives
YIT
c = 0.
= 1,
(2.20)
For the optimal control, the maximum principle defines the solution
B
=
P
for far
= 0
C Y m 2
Ey2
Y
3
> 0,
(2.21)
- Y3 < o .
The singular solution 0
5 B 5
appears on an interval
where we have the two integrals VY
1
- m8 ,2 = o ,
;
- Y2
-
Y3
(2.22)
= 0.
By Poisson's theorem (c.f. (18), section 181, we obtain a third integral
cmY 1
+
(mQm-cQV-Q)Y2 = 0.
(2.23)
39
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
Determination of the three relations (2.221, (2.23) requires
(2.24)
This relation defines the manifold S,(m,v)
=
cQ
+
v(mQ -cQ -Q) = 0. m v
(2.25)
In the particular case of a parabolic polar, where m Q = Av2'+ B V
(2.26)
2
with A, B given positive constants, we will have 2 S (m,v) = Bm (v+3c) 1
-
4
Av (v+c) = 0.
(2.27)
It is not difficult to establish that here Poisson's theorem does not give any other linearly independent integrals. According to (18) (cf. 518), the form of the sinqular control is written as QSlv B 1= cSlv - mS
(2.28) lm
which, in the case of the parabolic polar (2.26), will have the form
B, =
4 2 [Av (3v+2c) + Bm (v+6c)]Q 4 2 2 2 ' Acv (3v+2c) + Bm v(2v+7c) + 6Bc m
(2.29)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
40
Thus, the optimal control of the thrust is represented, generally speaking, as a piecewise-continous function consisting of the three arcs (2.211,
(2.28).
As was already noted in section 3 of
the preceeding chapter, the optimal combination of these arcs may be the curve impuf depicted in Fig. 5. From the construction, it follows that in any case the control
8. Actually, since the 1dm manifold ( 2 . 2 5 ) passes through the origin, - > 0 and, consequently, dv for the control (2.28) cB < Q(m,v) while at the time when B = B (2.28)
satisfies the condition 0 5 B
we have cB > Q(m,v). Now we define a program thrust which will be optimal in the minimum time sense, i.e. for the system performance index we take the functional T
I2 =
I0 dt = T.
(2.30)
We describe the problem in new variables. pendent variable. dt _ _ -1 dx
dv -
v
=
cB
dx
We let x be the inde-
Then Eq. (1.11) takes the form
I
-
Q(m,v)
mv
(2.31)
I
dm _ _ - BV dx The functional ( 2 . 3 0 ) G=t,
is written in the Mayer form as
A G = tf = Tmin'
(2.32)
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
41
m
S,(m.
vl
=
0
S, (m. V ) = o
7
C
"
Fig. 5 We now apply the maximum principle.
We have the function (2.33)
V
the auxiliary equations
-d y-1 dx
- 0,
2 A + vQv ax
ax
(2.34)
mv
V
-dy3 - -
- Q
mQm
mv
- Q Y Y - CB Y
m v
m2v
V
VECTOR-VALUED OPTlMlZATlON PROBLEMS IN CONTROL THEORY
42
and the form of the optimal thrust a s 6 = 0 f o r L ~- - Y3 < 0, mv 2 v
(2.35)
-
fo
1 for%^ - - Y >o. m v 2 v 3 = -1 and C = 0. 1T Since x is free, we have
The transversality conditions determine Y We construct the singular control. the first integral P
+
MB = 0 ,
where P and M are yl
C
M = - Y
P = v - % Y z ,
mv
On the interval
2
[ ,t21, the
- -vy 3.
(2.36)
singular solution satisfies the
two integrals
=o,
p = -yl- L y
v
mv
M = % Y
mv
2
2
-_ y3 v
(2.37) 0.
The Poisson bracket gives the new first integral mQm V
mv
mv
- Q
(2.38)
m v
or, what is equivalent (2.39)
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
43
The integral manifold of the singular solutions must satisfy the equation
- _1
- -C
0
mv
-cm
= 0.
V
V(mQm-cQv-Q)
(2.40)
mQ
Thus, as follows from (2.43), we have the manifold of singular solutions s2(mtV) = mQm
- cQV
-
(2.41)
Q = 0,
which, in the case of the parabolic polar (2.261, is represented by the expression s2(m,v)
-
=
*v4
(2.42)
= 0.
Analagously, by the formula (2.281, the form of the singular control is written as
8,
=
QS2V cs - mS2m 2v
1
(
'2v
= -as2
av
'2rn
1
(2.43)
In the case of the parabolic polar (2.26), the singular control may be written as 3
8,
=
2Av Q 2Acv3
-
(2.44)
2' Bm
Turning to graphics, the manifold
S
2
(m,v) = 0 will be located
to the left of the manifold S (m,v) = 0, as is shown in Fig. 5 . 1 In order to convince ourselves of this, from formulas (2.25) and (2.41) we determine Q.
In the case S (m,v) = 0, we have 1
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
44
Q
V
= Ql(m,v) = v - c (m&-cQv)
(2.45)
I
while in the case of the manifold S2 (m,v)
Q
= Q2(m,V) = mQm
-
=
0,
(2.46)
CQ,.
Since the strength of the aerodynamic drag Q(m,v) is positive and c > 0 , the comparison of (2.45) with (2.46) gives
Q, (mlv)> Q2 (m,v).
(2.47)
Since the mass only decreases, for a fixed mass, equality of Q, and Q can only be achieved under the condition v > v2. The 2 1 validity of this is shown by the arrangement of the curves S = 0 1
and S2
= 0
in Fig. 5.
Such an ordering becomes evident for determination of Q in the form of the parabolic polar (2.26).
Describing the manifold (2.27)
in the form
and the manifold (2.42) in the form
it is evident that for a fixed value of the velocity we will always have m 1 < m2' In the first problem for the functional (2.15), the optimal trajectory, as was noted, corresponds to the curve impnf, while in the second problem for the functional (2.32), the optimal curve is im'p'n'f (Fig. 5).
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
4s
Now we consider the motion which is optimal, say, for the criterion
(2.47') in which xo is the maximal achieved distance in the first optimization problem, while to is the minimal time in the second problem. The optimal control will be composed of f3 = 0, f3 singular arcs.
=
and
In fact, the Pontryagin function has the same form as in (2.16); the auxiliary equations also coincide with (2.17) and the first integral remains in the form (2.18).
This insures that the form of
the optimal control is as defined in (2.21); however, for determination of the vector Y we use the following transversality
[(-
condition
+
( + x y
. ) & t + y 2 6 v + y 3 6 1 j
f
= 0.
(2.48)
i According to the boundary conditions (2.14), the condition (2.48) splits into the relations (2.49)
(2.50) Now the first integral is written as 0
T - t P-Mf3=-1 (to)
(2.51)
in which T is the duration of the process, while P and M have the forms
46
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
P=vY
C
M = - Y
- Q
m
2
(2.52)
- '3.
, the singular solution will satisfy the
On the interval following integrals
(2.53) M = C Y - Y =o. m 2 3
(2.54)
We calculate the first Poisson bracket R = (P,M) =
cmYl m
+ NY2
2
(2.55)
= 0,
in which N(m,v)
=
Qmm
-
cQv
-
Q
=
(2.56)
S2 (m,v).
Now we calculate the second Poisson bracket (R,P+MB)
= (R,P)
+
(R,M)B = 0 ,
(2.57)
which, as a result, defines the new integral -rn(cB+N)Yl
+ [(cNv-mm)B + NQv-NVQ]Y2
= 0.
(2.58)
From the integrals (2.54), (2.55) and (2.58), we determine the manifold of singular solutions.
We have the determinant
0
C -. m
1
cm
N
0
= 0,
(2.59)
2.
OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
which after expansion gives S =
-N2
-
c (NQ -QN )
v
v
-
c
c N+ (cNv-mNm)] 8 = 0 .
41
(2.60)
From the last expression, we determine the singular control as
'
N~ + C(NQ~-QN,) =
(2.61)
c (mm-cNv-N)
Taking into consideration (2.56), the last expression may be written in the form
'
=
s2 +
QS2V c~~~ - mS
+ 2m
S2
CQ-mQm)
(2.62)
c ( s2+cS 2v-mS2m)
We note that if in (2.52), S2 = 0, then it is converted into (2.43), i.e. into the expression which was defined for the solution of the minimum time problem. The relation S1 = CQ + vS2, obtained by comparing (2.25) and (2.41), gives the possibility to determine the dependencies
s2v
- _1 (Slv+Q-mQm)r v
substitution of which into (2.62) determines the singular control 3, in the following form:
8
=
cs lv
QS1" - mSlm
+ s1 +
s1(Q-mQ,)
-
c ( c ~ ~ +s ~ )- m ~ ~ ~
(2.63)
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
48
=O
V
Fig. 6 If we suppose that
S
1
=
0 in (2.63), then it is converted into
(2.28), i.e. into the expression which was defined for the solution of the maximal distance problem. The location of the curve (2.60) in the (m,v) plane depends on the form of the function Q(m,v), characterizing the aerodynamic
drag.
It may be situated to the right of the curve S (m,v) = 0 1 or to the left of the curve S (m,v) = 0, or between them. The 2 motion of the point from i to f in this plane will be the curves im p n f, im p n f, im3p3n3f, respectively (Fig. 6). 111 2 2 2 We hope that this example has completely cleared up the expediency of the approach to the vector optimization problem in the spirit of section 1 of this chapter.
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
4.
49
Analytic Construction of Optimal Regulators
The problem of analytic construction of optimal regulators (the ACOR problem) has the same statement as the problem of programming optimal trajectories. However, within it are specifications which demand its separate consideration. First of all, we assume that we know the programmed control
which represents the exact solution of the vector equation (2.2) with boundary conditions(2.5). The ACOR problem is posed relative to the perturbed dynamics described by the equation (18)
where
q
is the deviation of the vector x from the programmed value
x*, generated by means of the disturbance $(t); 5 is the increment in the control vector used to bring the actual motion nearer to the programmed motion, i.e. rli(t)
=
x
p
- xZ(t) ,
(i=l,.. .,n; j=1,.
5 . (t)
=
u . (t)
af(x,u,t)
,
=
3
3
-
u3 (t) 7
..,m).
Here the matrices B
=
ax
af (xrurt) are known, timeau
varying functions defined on the motion (2.2) and calculated for the given program (2.64); by o(n,(,t)
we denote a power series
convergent for all q r 5 . We impose a series of requirements on the perturbed motion given by Eq. (2.65).
These requirements are based upon physical
aspects of the controlled system (cf. (39)) .
50
VECTOR-VALUED OPTIMIZATION PROBLEMS IN CONTROL THEORY
Let (2.66) be functionals reflecting these requirements.
This collection of
functionals forms a vector functional.
( n , < ) ,...,w k ( n , < ) are assumed to be 1 positive-definite and continuously-differentiable in their arguThe integrand functions w
ments. Let
I Ino/ I
5 A be a neighborhood of the origin in which the
perturbed motion can begin. Let (2.67) be a feedback law lying in the class of admissible controls V, yielding the optimum of the index I for the closed-loop system a (2.68) The law (2.67) may be synthesized for any a = l,...fkl using known methods of solution of the ACOR problem for a single functional (18). We note that the class of admissible controls V consists only of those functions S(q,t) for which the system of equations (2.65) is asymptotically stable. It is clear that for each fixed a , the law (2.67) may ignore all requirements contained in the other functionals of (2.66). We assume that all the functions (2.67) are known and we compute the numerical value of all functionals Ia(<(@")) = Ia0' We form the euclidean norm in the space of vectors 1(11,1 2,..., Ikl:
2. OPTIMIZATION PROBLEMS WITH VECTOR FUNCTIONALS
k
51
k (2.69)
defined for all admissible controls 5
E
V.
Definition. We will say that the feedback law C0(n,t) optimizes the vector functional with components (2.66)
if the inequality (2.70)
is satisfied for any admissible control 5
E
V.
We will say that
such a feedback law is optimal relative to the vector functional. Problem.
Given Eq. (2.65)
,
the vector functional (2.66)
and
the class of admissible controls V, it is required to determine a 0
feedback law 5 (n,t) optimizing the vector functional in the above sense.
This problem is studied in (143,148).
The mathematical formulation of the given problem is no different than the formulation of the problem of programming optimal trajectories considered in section 2 of this chapter. The distinction lies only in the given boundary conditions:
in the ACOR problem
all components of the state vector at the endpoints of the trajectory are assumed to be free and only in the special case of the infinite-interval problem must we impose the condition
~ ( m )=
0.
However, there is a real difference in the form of the solutions to these two problems.
In the problem of determining optimal pro-
grammed trajectories, the solution is sought in the form of a timevarying function of the boundary conditions, while in the ACOR problem the feedback law is sought in the form of a vector function of the state and time. It is clear that all the remarks made relative to the choice of the function R(u) in section 2 of this chapter retain their meaning and value for the synthesis problem.