- Email: [email protected]
Singular Optimal Control and the Identically Non-Regular Problem in the Calculus of Variations
CopHight © IFA C 9t h Triennial World Congress Bud apest, I-Illn gan. 19H4
SINGULAR OPTIMAL CONTROL AND THE IDENTICALLY NON-REGULAR PROBLEM IN THE CALCULUS OF VARIATIONS P. K. A. Menon, H.
J.
Kelley and E. M. Cliff
Al'I'w/Jon' amI Ou'all 1:'IIKillt'l'rfllg D I'I)(Irlllll'lIl. \ 'ilgill;o I Joly"'("";( I n\I;lull' (Iml Slo/(' L'lIil'I'nil.", H lfI(/dwrg. \';rgillia. L'SA
Abstract . A small but int eresting class of optimal control problems featuring a scalar co nt rol appearing linearly is equiva lent to the c lass of identically nonregular problems in the Calculus of Va riat ions. It is shown that a condition due to Mancill (1950) is eq uivale nt to the generalized Legendre-Clebsch condition fo r this narrow class of problems. Keywords .
Opt imal co nt rol , singular op timal cont r ol .
INTRODUCTION
with
In optimal - control problems featuring scalar con trol appearing linearly in the system diffe r ential
(2)
Note that
equations , singular subarcs can sometimes a ris e .
Along singular subarcs which are minimizing, the Generalized Legendre-Clebsch necessary condi ti on should hold [Kelley, Kopp & Mojer , 1967, Bell & Jacobson , 1975J. A class of such op timal - con tr ol problems can be re cast as identica ll y non-regular problems in the c lassi ca l Calculus of Va riati ons if the d imens i on is low. Specifica l ly ,this transforma ti on appears feas ib le if there a r e a t most two- non - ignorable state va ri ables and one control variable . In general, the procedure involves a change i n the in dependent variable under appropriate smoothness and monoton i c it y assumptions . (The oh r ase " classical Calculus of Varia ti ons" employed
[P(t,x) + Q(t,x)xJ
x* =
(3)
0
It is kn own that the Euler ' s equation for this problem is either an identity o r a finite equation [Bo lza, 1904, Leitmann. 1981. Courant, 1945 - 46 J. If it is an identity, the integral is independent of the :>ath joining two fixed points and no oroo e r mInImum exists. On the other hand, if it is a finite eq uation, the Euler's equation is satisfied on l y along ce rtain paths which in ge neral do not pass through the specified end ooints . These functionals are s ometimes ca lled "de generate " because the Euler equati on for such functionals is not a differential equation, but a fini te equation without any deri va ti ves of the unkn own f un c ti on [Pett:ov, 1968J.
here refers to unconstrained problems, i. e ., not
to Lagrange - '!ayer- Bolza nroblems . ) For this class of :>roblems, ,iancill [195J J has ob tained conditions for a minimizing singular arc .
In this research, Mancill made use of Green 's theorem on line integrals to establish conditions for a strong relative minimum . :!iele [19 50- 51 , 1962J used the Green ' s theorem apDroach for pro blems with control bounds, extended the technique to handle isoperimetric constraints and carried out appl i cations to several fl i gh t problems . Goh [1966J examined the singular Bolza Dr oblem and noted the connection between !Ii ele ' s work and the identically non -regular problem in the Calculus of Variations .
Two the o rems by :1ancill [1950) give necessary and sufficient con ditions fo~ a str ong local minimum i n these problems. These are presented i n the following. Theorem 1.
If E12 is of c lass D' and minimizes the
integral J in the class of admissible cu r ves JOIning 1 and 2, where P(t,x) and Q( t,x) are of class 2n C in a closed re g ion R of x, t space, then
This paper deals with an evaluation of Mancill ' s work and its relation to the Generalized Legendre Clebsch necessa r y condition . A critique on the nature of transversality conditions for this class of problems is presented . Three illustrative ex amples are also given .
if akp/axk = akQ/a t axk- l, k = 1,2,3, ....... 2n - 2 , along a r cs interior t o R, in clud ing all isolated points in common with the boundary of ~:
IDENTICALLY NON- REGULAR PROBLEM
(I ) B
The identically non - regular problem with fixed endpoints in the Calculus of Variat i ons [~Iancill. 1950, Bolza , 1904J is the minimization of an integral of the fo rm t J -_ J 2 [P(t,x ) + Q(t ,x )xJdt (1)
if akp/axk = akQ/a t axk- l, k = 1,2,3 .... r-l, along arcs in commo n with the b ounda ry of R. Let (I') and
(I~)
represent conditions (!) and (I ) B ~ repl aced by
respectively with the inequalities
tl
43
P. K. A. Menon, H. J. Ke11ey and E.
44
the strict inequality>. This is a familiar notati on in the classical Calculus of Variations and it will be employed in this work. The firs t part of (I) with n = I, is the Eu1er's necessary condition fo r thi s problem . The inequality in (I) with n = I , is derived from the second variation. For n > 1 the condi tions (I) are ob t ained from higher var i ations.
~.
Cliff
Differentiating this with respect to time, substituting ~ U and using (6) for A, one finds E
(8) Differentiating with respect to time again, while using = u, leads to
x
2n Theorem 2. If P(t,x) and Q(t,x) a r e of c lass C in R and the conditions (I') and (I~) are satis-
(9)
fied along an admissible cu rve E12 joining 1 and Hence the Generalized Legendre-Clebsch necessary condition for first-order singular arc is
2 , then El2 furnishes a strong proper relative minimum for the integral J in the class of admissible c urves joining I and 2 . It is implied in Theorem 2 that the Euler equation is not an identity. This Theorem i s proved using Green ' s the o rem on line integrals. Hancill has given two additional theorems on the necessary and sufficient condi tions for the identically nonregular prob Lem with variable end points. However, the interpretation of these in the light of modern optimal - con trol theory indi ca tes thei r inapplicability owing to the violation of the smoo thness assumption essential t o th e results in Uan c illts work . A detailed discussion of this is presented in section 4 .
At this point, it is perhaps interesting t o compare the results obtained by l1anci11 with those of ;.liele [1950-51,19621. The fi r st oa rt of con dition (I) i n Theo r em I with n= L is termed the" fundamental function " 'JI (t,x) in ~aele ' s work .
il
dU
Q
_ P
tx
< 0
xx
specification on the direction of
traverse along the boundary of the adm issible region, applicable whenever the arcs interior to
that is (ll) The inequality (11) is the same as that in co nddition (l) of lIanc111. One notes that the inequality (I) of Man c ill for o > 1 is no t equivalent to the Generalized LeKendre-Clebsch ne c essary condition but is s omething more general. (See Example 1 b to folloW.)
TRANSFORHATION TO CANONICAL FOR.'1 To investigate the situations in whi c h specified boundary conditions are off the oath defined by the conditions (I), and the variable-endpoint pr oblem, a t rans format ion ap?roach ,I"iscussed in [Kelley,1964 ) is next employed. The identically non-regular problem is first brought into the llayer format:
the admissible region are non - optimal.
P(t,X) + Q(t,x)u
y
With a shllrt develooment it will be sh own that with 11 = I, the inequality in (I) is the Generalized Legendre - Clebs c h necessary condi tion fo r q:l. Consider the optima l control problem t f
r
[P(t,x) + Q(t , x)u1dt
(4)
J
t
with tl' t , x(t ) = Xl' x(t ) = x s!,ecified. 2 l 2 2 minimum of y(t ) is sought with y( tl) = 0 2
Next. a transformation of state variables will be so that the state system has a special form. Th~ new state variables are z and x and th e system is to have the control variable u appea rin g in only one of the state equati ons, the one for x.
p~rform~d
The system is P(t x) + aR(t,x) , at
x
H( A,X,t,U) = P(t,x) + Q( t,x)u + AU
x
-
Q
x
= y
(6)
From the expression (5) for H, one has that along a singular subarc
o
(15)
+ R (t,x)
(16)
where x
R(t,x)
u
Q(t,x {t ) ) + A(t)
z
(5)
and forms the adjoint equation
- P
=u
(14)
and the choice of z leading to it is
To proceed via the "modern" approach one def ines the variational Hamiltonian
u
A
o
subject to the differential constrain t x = u. It is appa rent that this pr oblem is equivalen t t o the identically non - regular problem in the Ca l c ulus o f Variations . Note that the cont r ol u is unbounded.
H
(12) (13)
x = u
THE Pl\l)BLEtl IN AN OPTIMAL - CONTROL FORNIIT
lh n
(10)
The inequality
in ( I ) appears as a specification on the direction of traverse along the extremal . Similarly, the <"ll nclition (Is) of llancill also appea rs in IHele's work as
i £ .2.J H) t = l dt ~
.L
fQ(t,Od
(17)
[Kelley, Kopp & ~oyer, 1967, Kelley, 1964). The end conditions are t , t , x(t ) = Xl' x(t 2 ) = x 2 2 1 1 specified as before. The initial value of z is z(t ) = R(t , Xl) and a ~inimu~ of z(t ) is sou ghL 2 1 1
(7)
Since there are no bounds on the control u, it can
45
Singular Optimal Control Problem behave impulsively and x(t) can jump. If the equation (15) is discarded and a solution sought in the class of functions x(t) piecewise-continuous, x becomes control-like [Kelley, Ko!,!, & 140yer 1%7, Kelley, 1964]. At I)oints t < t < t , x ",inl 2 imizes the right member of equation (14). x = Arg min [P(t,x) + aR (t,x)]
at
x
Such motions have no effect on the performance index. The next example is chosen to illustrate the necessary conditions of Mancill for n > 1.
x4 dt, subject to x
Kin
(b)
z
U
(18)
possibly exhibiting jum!, discontinuities in the interior of the interval depending on the nature of the time de!,endence of equation (14). The variable x will gene rally jump at the initial and final times to satisfy the end conditions unless the value emerging from expression (18) happens fortuitously to satisfy them.
The conditions x(t ) - Xo and x(t ) a x specified. f o f Since there are no bounds on the control variable, the problem in the Calculus of Variations format is (26)
Min
The situation with endooint freedom is interesting. Consider for example, tl and t2 fixed as before, but x(t ) unspecified. To minimize y(t ), x 2 2 should jump at the final time t2 to the value x(t ) = Arg max rr(t ,x) 2 2
(19 )
x
The necessary conditions for a minimum are 4x
3
(1)
Two elementary examples: t f 2 (a) lIin x dt, subject to x = u )t 0
24x
x(t ) 0
x
0
(20) o
P(t,x)
x
Q( t,x) _ 0
(21)
V Siny
Min
0
(32)
(23)
J
Changing the independent variable from time to altitude,
(24)
dV dh
g{T- D2 W V Siny
is met in the strengthened form along the arc xzO and hence, the trajectory x =0 affords a strong relative minimum. The result (25) was obtained in [Kelley, Kopp & Moyer, 1967] via the Generalized Legendre-Clebsch necessary conditicn. If the initial and final conditions are off the x = 0 path, jumps in x are required at the end points.
- £ V
(34)
(V f ,h ) f Hin
(25)
(33)
dt
The sufficient condition 2 > 0
(31)
(Vi ,hi)
V' >
Siny)
Differential equations for range rate and fuelflow rate have been dropped from the system, since they are ignorable in this problem. The optimal-control problem is the minimization of time required to fly from an initial ' (V,h) pair to a final (V,h) pair , viz.
(22)
and 2
-
(Vf·h f )
The necessary conditions of Uancill [1950] become, 2x = 0
(30)
g[ {(T-D) /W}
z
nz
With the identification of 2
0
Minimum-time aircraft climb
v
Since there are no bounds on the control, the differential constraint is inactive. Hence, the problem in classical Calculus of Variations format is
t
>
(29)
Following Kiele [1950-51), a model of aircraft in symmetric flight under the assumptions of constant weight and thrust, T, and drag D, functions of altitude, h, and airspeed, V, only, is:
and x(t ) = x specified. f f
Hin
0
a
Note that the sufficient condition (I') in Theorem 1, (30) with strengthened inequality, is met for n = 4. Just as in the previous example, jumps in x must be permitted at the endpoints if the specified conditions are off the x = 0 path.
(2)
r
Further, (28)
24
To co nvey an impression of Mancill's work, three examples are given in the following.
(27)
Hence x • 0 is the extremal.
This seems to be the nearest thing to a transversality condition that one can have with x controllike.
ILLUSTRATIVE EXAMPLES
• 0
J
~
V Siny
(35)
(V ,h ) i 1 Substituting next for Siny in (35) from (34), the !,roblem in the classical Calculus-of-Variations format is
P. K. A. Menon, H. J. Kelley and E. M. Cliff
46
W V'
+ - - - dh
Hin
g(T-D)
(36)
In this development, the monotonicity of the altitude variable has been tacitly assumed. If des ired, Siny may be constrained by defining an ad· missible region in the V-h space as suggested in [ltiele,1962j, however,this falls outside the Manci)] model. Empioying conditions (I) in Theorem 1, the necessary conditions for a minimum for arcs interior t o the admissible region, are
.L
av
W g(T-D)
I
=.L ah
!
\
W I.g (T-D)
\ W I V(T-D)
I
I
I ? Lah aV I W I I V(T-D) I
!
(37)
(38)
The expressions (37) and (38) may be put in the followi ng form
.L [V(T-D)] ah
I
0
array of results of seemingly enormous power (e. g., sufficiency by strengthening inequalities), which are in fact of extremely limited applicability because of their smoothness hypotheses. An unwelcome complication of the Mancill theory is the incorporation of state-inequality constraints, a relic of his earlier work on this special type of .problem [Mancill, 1947], which do not alleviate the smoothness difficulties. Treatment of variational problems with x(t) piecewise continuous only has been given by V.F. Krotov [19611. (See also Petrov,[1968].) Bounded-control probleQs approached by Green's Theorem have been studied by lIiele [1950-51, 1962]. CONCLUDING REMARKS Mancill's two Theorems given in the present work are of interest and seem to have been ahead of their time. For the narrow class of problems considered by Mancill, the inequality (I) with n = 1 is equivalent to the generalized Legendre-Clebsch condition. Perhaps equally important was l1ancill's introduction of the Green's Theorem device for the study of problems of small dimension .
(39i
E = Constant
a.=-[V(T-D)] ah2
ACKNOWLEDGI1ENT
I E = Constant
<
0
(40)
The necessary conditi on for a strong relative
The presently reported resear ch was suppo rted by NASA Langley Research Center, Hampton, Virginia, U.S.A., under Grant NAG 1-203, Dr. Christopher Gracey serving as Technical Monitor.
minimum, then, is
[V(T-D)]
I E
Constant ~ 0
(41)
This result was obtained in [Kelley, Kopp & Moyer, )967] using the Generalized Legendre-Glebsch necessary condition . The expression (39) corresponds to stationary points of excess power V(T-D) along contours of co nstant energy E " h + V2 /2g. Inequality (41) implies that the stationary points of excess ~ower d
along cons tant-energy contours must be maxima,
result in accord with engineering intuition. If the endpoints are off the path defined by (39) jumps in airspeed and al titude must be permitted to meet the boundary condition. With bounds on control, on the o ther hand, operation at one of the control limits is indicated. SMOOTHNESS DIFFICULTIES AND THEIR IlIPACT Tn Mancill [1950], and in classical Calculus-ofVariations treatments generally (e.g. [Bolza, 1904, Lietmann, 1981, Courant, 1945-46], the function x(t), which appears along with its derivative, x(t), as an argument of the integrand, is assumed to possess a first derivative which is at least piecewise co ntinuous. The various theorems of [lIancill, 1950] do not apply to discontinu~us solutions of the type examined in the preced1ng sections. In the classical setting one would say that no minimum exists in the class of admissible function s, but only a lower bound. Indeed the classica l treatment [Bol7.a, 1904,Leitmann,198l, Courant, 1945-46] focus entirely on the degenerate case in which the integral is independent of the path. One is fac ed with the choice between extending the theory to admissible x(t) piecewise continuous, or the introduction of bounds on the control u(t). Unfortunately l~ncill did neither and produced an
REFERENCES Bell, D. J. and Jacobson, D. H., (1975). 'Singular Optimal Control Problems', Academic Press, New York. Bolza, 0., (1904). 'Lectures on the Calculus of Variations', G. E. Stechert and Co., New York. Courant, R., (1945-46). 'Cal cul us of Variations and Supplementary Notes and Exercises', mimeographed class notes, New Yo rk Unive rsit y . Goh, B. S., (1966). 'The Second Variation for the Singular Bolza Problem', SIAM Journal of Control, Vol. 4 , pp. 309-325. Kelley, H. J., (1964). 'A Transformation Approach to Singular ~ubarcs in Optimal Trajectory and Control Problems', SIAM Journal of Control, Vol.2, pp. 234-240. Kelley, n. J., Kopp, R. E. and Moyer, G. H., (1967). 'Singular Extremals', in Topics in Optimization, Ed. G. Leitmann, Chapter 3, Academic Press, New York. Krotov, F. F., (1961). 'The Principal Problem of the Calculus of Variations for the Simplest Functional on a set of Discontinuous Functions', Soviet lIathematics Doklady, Vol. 2, DD. 231-234. Leitmann, G., (1981). 'The Calculus of Va riati ons and OPtimal Control', Plenum Press, New York. !lancill, J'. D., (1947). 'Unilateral Variations with Variable Endpoints', American Journal of Mathematics, Vol. LXIX, pp. 121-138. Mancill, J . D., (1950). 'Identically Non-Regular Problems in the Calculus of Variations', Mathematica Y Fisica Teorica, Universidad Nacional del Tucuman, Rep~blica Argentina, Vol. 7, No. 2. Miele, A., (1950-51). 'Problemi di Minimo Tempo Nel Vola Non-Stazionario degli Aeroplani', Atti della Accademia delle Scienze di Torino, Vol. 85, pp. 41-42.
Singular Optimal Control Problem Miele, A., (1962). 'Extremization of Linear Integrals by Green's Theorem', in Optimization Techniques, Ed. G. Leitmann, Chapter 3, Academic Press, New York. Petrov, I. P., (1968). 'Variational Methods in Optimum Control Theory', Academic Press, llew York.
47
SINGULAR OPTIMAL CONTROL AND THE IDENTICALLY NON-REGULAR PROBLEM IN THE CALCULUS OF VARIATIONS P. K. A. Menon, H.
J.
Kelley and E. M. Cliff
Al'I'w/Jon' amI Ou'all 1:'IIKillt'l'rfllg D I'I)(Irlllll'lIl. \ 'ilgill;o I Joly"'("";( I n\I;lull' (Iml Slo/(' L'lIil'I'nil.", H lfI(/dwrg. \';rgillia. L'SA
Abstract . A small but int eresting class of optimal control problems featuring a scalar co nt rol appearing linearly is equiva lent to the c lass of identically nonregular problems in the Calculus of Va riat ions. It is shown that a condition due to Mancill (1950) is eq uivale nt to the generalized Legendre-Clebsch condition fo r this narrow class of problems. Keywords .
Opt imal co nt rol , singular op timal cont r ol .
INTRODUCTION
with
In optimal - control problems featuring scalar con trol appearing linearly in the system diffe r ential
(2)
Note that
equations , singular subarcs can sometimes a ris e .
Along singular subarcs which are minimizing, the Generalized Legendre-Clebsch necessary condi ti on should hold [Kelley, Kopp & Mojer , 1967, Bell & Jacobson , 1975J. A class of such op timal - con tr ol problems can be re cast as identica ll y non-regular problems in the c lassi ca l Calculus of Va riati ons if the d imens i on is low. Specifica l ly ,this transforma ti on appears feas ib le if there a r e a t most two- non - ignorable state va ri ables and one control variable . In general, the procedure involves a change i n the in dependent variable under appropriate smoothness and monoton i c it y assumptions . (The oh r ase " classical Calculus of Varia ti ons" employed
[P(t,x) + Q(t,x)xJ
x* =
(3)
0
It is kn own that the Euler ' s equation for this problem is either an identity o r a finite equation [Bo lza, 1904, Leitmann. 1981. Courant, 1945 - 46 J. If it is an identity, the integral is independent of the :>ath joining two fixed points and no oroo e r mInImum exists. On the other hand, if it is a finite eq uation, the Euler's equation is satisfied on l y along ce rtain paths which in ge neral do not pass through the specified end ooints . These functionals are s ometimes ca lled "de generate " because the Euler equati on for such functionals is not a differential equation, but a fini te equation without any deri va ti ves of the unkn own f un c ti on [Pett:ov, 1968J.
here refers to unconstrained problems, i. e ., not
to Lagrange - '!ayer- Bolza nroblems . ) For this class of :>roblems, ,iancill [195J J has ob tained conditions for a minimizing singular arc .
In this research, Mancill made use of Green 's theorem on line integrals to establish conditions for a strong relative minimum . :!iele [19 50- 51 , 1962J used the Green ' s theorem apDroach for pro blems with control bounds, extended the technique to handle isoperimetric constraints and carried out appl i cations to several fl i gh t problems . Goh [1966J examined the singular Bolza Dr oblem and noted the connection between !Ii ele ' s work and the identically non -regular problem in the Calculus of Variations .
Two the o rems by :1ancill [1950) give necessary and sufficient con ditions fo~ a str ong local minimum i n these problems. These are presented i n the following. Theorem 1.
If E12 is of c lass D' and minimizes the
integral J in the class of admissible cu r ves JOIning 1 and 2, where P(t,x) and Q( t,x) are of class 2n C in a closed re g ion R of x, t space, then
This paper deals with an evaluation of Mancill ' s work and its relation to the Generalized Legendre Clebsch necessa r y condition . A critique on the nature of transversality conditions for this class of problems is presented . Three illustrative ex amples are also given .
if akp/axk = akQ/a t axk- l, k = 1,2,3, ....... 2n - 2 , along a r cs interior t o R, in clud ing all isolated points in common with the boundary of ~:
IDENTICALLY NON- REGULAR PROBLEM
(I ) B
The identically non - regular problem with fixed endpoints in the Calculus of Variat i ons [~Iancill. 1950, Bolza , 1904J is the minimization of an integral of the fo rm t J -_ J 2 [P(t,x ) + Q(t ,x )xJdt (1)
if akp/axk = akQ/a t axk- l, k = 1,2,3 .... r-l, along arcs in commo n with the b ounda ry of R. Let (I') and
(I~)
represent conditions (!) and (I ) B ~ repl aced by
respectively with the inequalities
tl
43
P. K. A. Menon, H. J. Ke11ey and E.
44
the strict inequality>. This is a familiar notati on in the classical Calculus of Variations and it will be employed in this work. The firs t part of (I) with n = I, is the Eu1er's necessary condition fo r thi s problem . The inequality in (I) with n = I , is derived from the second variation. For n > 1 the condi tions (I) are ob t ained from higher var i ations.
~.
Cliff
Differentiating this with respect to time, substituting ~ U and using (6) for A, one finds E
(8) Differentiating with respect to time again, while using = u, leads to
x
2n Theorem 2. If P(t,x) and Q(t,x) a r e of c lass C in R and the conditions (I') and (I~) are satis-
(9)
fied along an admissible cu rve E12 joining 1 and Hence the Generalized Legendre-Clebsch necessary condition for first-order singular arc is
2 , then El2 furnishes a strong proper relative minimum for the integral J in the class of admissible c urves joining I and 2 . It is implied in Theorem 2 that the Euler equation is not an identity. This Theorem i s proved using Green ' s the o rem on line integrals. Hancill has given two additional theorems on the necessary and sufficient condi tions for the identically nonregular prob Lem with variable end points. However, the interpretation of these in the light of modern optimal - con trol theory indi ca tes thei r inapplicability owing to the violation of the smoo thness assumption essential t o th e results in Uan c illts work . A detailed discussion of this is presented in section 4 .
At this point, it is perhaps interesting t o compare the results obtained by l1anci11 with those of ;.liele [1950-51,19621. The fi r st oa rt of con dition (I) i n Theo r em I with n= L is termed the" fundamental function " 'JI (t,x) in ~aele ' s work .
il
dU
Q
_ P
tx
< 0
xx
specification on the direction of
traverse along the boundary of the adm issible region, applicable whenever the arcs interior to
that is (ll) The inequality (11) is the same as that in co nddition (l) of lIanc111. One notes that the inequality (I) of Man c ill for o > 1 is no t equivalent to the Generalized LeKendre-Clebsch ne c essary condition but is s omething more general. (See Example 1 b to folloW.)
TRANSFORHATION TO CANONICAL FOR.'1 To investigate the situations in whi c h specified boundary conditions are off the oath defined by the conditions (I), and the variable-endpoint pr oblem, a t rans format ion ap?roach ,I"iscussed in [Kelley,1964 ) is next employed. The identically non-regular problem is first brought into the llayer format:
the admissible region are non - optimal.
P(t,X) + Q(t,x)u
y
With a shllrt develooment it will be sh own that with 11 = I, the inequality in (I) is the Generalized Legendre - Clebs c h necessary condi tion fo r q:l. Consider the optima l control problem t f
r
[P(t,x) + Q(t , x)u1dt
(4)
J
t
with tl' t , x(t ) = Xl' x(t ) = x s!,ecified. 2 l 2 2 minimum of y(t ) is sought with y( tl) = 0 2
Next. a transformation of state variables will be so that the state system has a special form. Th~ new state variables are z and x and th e system is to have the control variable u appea rin g in only one of the state equati ons, the one for x.
p~rform~d
The system is P(t x) + aR(t,x) , at
x
H( A,X,t,U) = P(t,x) + Q( t,x)u + AU
x
-
Q
x
= y
(6)
From the expression (5) for H, one has that along a singular subarc
o
(15)
+ R (t,x)
(16)
where x
R(t,x)
u
Q(t,x {t ) ) + A(t)
z
(5)
and forms the adjoint equation
- P
=u
(14)
and the choice of z leading to it is
To proceed via the "modern" approach one def ines the variational Hamiltonian
u
A
o
subject to the differential constrain t x = u. It is appa rent that this pr oblem is equivalen t t o the identically non - regular problem in the Ca l c ulus o f Variations . Note that the cont r ol u is unbounded.
H
(12) (13)
x = u
THE Pl\l)BLEtl IN AN OPTIMAL - CONTROL FORNIIT
lh n
(10)
The inequality
in ( I ) appears as a specification on the direction of traverse along the extremal . Similarly, the <"ll nclition (Is) of llancill also appea rs in IHele's work as
i £ .2.J H) t = l dt ~
.L
fQ(t,Od
(17)
[Kelley, Kopp & ~oyer, 1967, Kelley, 1964). The end conditions are t , t , x(t ) = Xl' x(t 2 ) = x 2 2 1 1 specified as before. The initial value of z is z(t ) = R(t , Xl) and a ~inimu~ of z(t ) is sou ghL 2 1 1
(7)
Since there are no bounds on the control u, it can
45
Singular Optimal Control Problem behave impulsively and x(t) can jump. If the equation (15) is discarded and a solution sought in the class of functions x(t) piecewise-continuous, x becomes control-like [Kelley, Ko!,!, & 140yer 1%7, Kelley, 1964]. At I)oints t < t < t , x ",inl 2 imizes the right member of equation (14). x = Arg min [P(t,x) + aR (t,x)]
at
x
Such motions have no effect on the performance index. The next example is chosen to illustrate the necessary conditions of Mancill for n > 1.
x4 dt, subject to x
Kin
(b)
z
U
(18)
possibly exhibiting jum!, discontinuities in the interior of the interval depending on the nature of the time de!,endence of equation (14). The variable x will gene rally jump at the initial and final times to satisfy the end conditions unless the value emerging from expression (18) happens fortuitously to satisfy them.
The conditions x(t ) - Xo and x(t ) a x specified. f o f Since there are no bounds on the control variable, the problem in the Calculus of Variations format is (26)
Min
The situation with endooint freedom is interesting. Consider for example, tl and t2 fixed as before, but x(t ) unspecified. To minimize y(t ), x 2 2 should jump at the final time t2 to the value x(t ) = Arg max rr(t ,x) 2 2
(19 )
x
The necessary conditions for a minimum are 4x
3
(1)
Two elementary examples: t f 2 (a) lIin x dt, subject to x = u )t 0
24x
x(t ) 0
x
0
(20) o
P(t,x)
x
Q( t,x) _ 0
(21)
V Siny
Min
0
(32)
(23)
J
Changing the independent variable from time to altitude,
(24)
dV dh
g{T- D2 W V Siny
is met in the strengthened form along the arc xzO and hence, the trajectory x =0 affords a strong relative minimum. The result (25) was obtained in [Kelley, Kopp & Moyer, 1967] via the Generalized Legendre-Clebsch necessary conditicn. If the initial and final conditions are off the x = 0 path, jumps in x are required at the end points.
- £ V
(34)
(V f ,h ) f Hin
(25)
(33)
dt
The sufficient condition 2 > 0
(31)
(Vi ,hi)
V' >
Siny)
Differential equations for range rate and fuelflow rate have been dropped from the system, since they are ignorable in this problem. The optimal-control problem is the minimization of time required to fly from an initial ' (V,h) pair to a final (V,h) pair , viz.
(22)
and 2
-
(Vf·h f )
The necessary conditions of Uancill [1950] become, 2x = 0
(30)
g[ {(T-D) /W}
z
nz
With the identification of 2
0
Minimum-time aircraft climb
v
Since there are no bounds on the control, the differential constraint is inactive. Hence, the problem in classical Calculus of Variations format is
t
>
(29)
Following Kiele [1950-51), a model of aircraft in symmetric flight under the assumptions of constant weight and thrust, T, and drag D, functions of altitude, h, and airspeed, V, only, is:
and x(t ) = x specified. f f
Hin
0
a
Note that the sufficient condition (I') in Theorem 1, (30) with strengthened inequality, is met for n = 4. Just as in the previous example, jumps in x must be permitted at the endpoints if the specified conditions are off the x = 0 path.
(2)
r
Further, (28)
24
To co nvey an impression of Mancill's work, three examples are given in the following.
(27)
Hence x • 0 is the extremal.
This seems to be the nearest thing to a transversality condition that one can have with x controllike.
ILLUSTRATIVE EXAMPLES
• 0
J
~
V Siny
(35)
(V ,h ) i 1 Substituting next for Siny in (35) from (34), the !,roblem in the classical Calculus-of-Variations format is
P. K. A. Menon, H. J. Kelley and E. M. Cliff
46
W V'
+ - - - dh
Hin
g(T-D)
(36)
In this development, the monotonicity of the altitude variable has been tacitly assumed. If des ired, Siny may be constrained by defining an ad· missible region in the V-h space as suggested in [ltiele,1962j, however,this falls outside the Manci)] model. Empioying conditions (I) in Theorem 1, the necessary conditions for a minimum for arcs interior t o the admissible region, are
.L
av
W g(T-D)
I
=.L ah
!
\
W I.g (T-D)
\ W I V(T-D)
I
I
I ? Lah aV I W I I V(T-D) I
!
(37)
(38)
The expressions (37) and (38) may be put in the followi ng form
.L [V(T-D)] ah
I
0
array of results of seemingly enormous power (e. g., sufficiency by strengthening inequalities), which are in fact of extremely limited applicability because of their smoothness hypotheses. An unwelcome complication of the Mancill theory is the incorporation of state-inequality constraints, a relic of his earlier work on this special type of .problem [Mancill, 1947], which do not alleviate the smoothness difficulties. Treatment of variational problems with x(t) piecewise continuous only has been given by V.F. Krotov [19611. (See also Petrov,[1968].) Bounded-control probleQs approached by Green's Theorem have been studied by lIiele [1950-51, 1962]. CONCLUDING REMARKS Mancill's two Theorems given in the present work are of interest and seem to have been ahead of their time. For the narrow class of problems considered by Mancill, the inequality (I) with n = 1 is equivalent to the generalized Legendre-Clebsch condition. Perhaps equally important was l1ancill's introduction of the Green's Theorem device for the study of problems of small dimension .
(39i
E = Constant
a.=-[V(T-D)] ah2
ACKNOWLEDGI1ENT
I E = Constant
<
0
(40)
The necessary conditi on for a strong relative
The presently reported resear ch was suppo rted by NASA Langley Research Center, Hampton, Virginia, U.S.A., under Grant NAG 1-203, Dr. Christopher Gracey serving as Technical Monitor.
minimum, then, is
[V(T-D)]
I E
Constant ~ 0
(41)
This result was obtained in [Kelley, Kopp & Moyer, )967] using the Generalized Legendre-Glebsch necessary condition . The expression (39) corresponds to stationary points of excess power V(T-D) along contours of co nstant energy E " h + V2 /2g. Inequality (41) implies that the stationary points of excess ~ower d
along cons tant-energy contours must be maxima,
result in accord with engineering intuition. If the endpoints are off the path defined by (39) jumps in airspeed and al titude must be permitted to meet the boundary condition. With bounds on control, on the o ther hand, operation at one of the control limits is indicated. SMOOTHNESS DIFFICULTIES AND THEIR IlIPACT Tn Mancill [1950], and in classical Calculus-ofVariations treatments generally (e.g. [Bolza, 1904, Lietmann, 1981, Courant, 1945-46], the function x(t), which appears along with its derivative, x(t), as an argument of the integrand, is assumed to possess a first derivative which is at least piecewise co ntinuous. The various theorems of [lIancill, 1950] do not apply to discontinu~us solutions of the type examined in the preced1ng sections. In the classical setting one would say that no minimum exists in the class of admissible function s, but only a lower bound. Indeed the classica l treatment [Bol7.a, 1904,Leitmann,198l, Courant, 1945-46] focus entirely on the degenerate case in which the integral is independent of the path. One is fac ed with the choice between extending the theory to admissible x(t) piecewise continuous, or the introduction of bounds on the control u(t). Unfortunately l~ncill did neither and produced an
REFERENCES Bell, D. J. and Jacobson, D. H., (1975). 'Singular Optimal Control Problems', Academic Press, New York. Bolza, 0., (1904). 'Lectures on the Calculus of Variations', G. E. Stechert and Co., New York. Courant, R., (1945-46). 'Cal cul us of Variations and Supplementary Notes and Exercises', mimeographed class notes, New Yo rk Unive rsit y . Goh, B. S., (1966). 'The Second Variation for the Singular Bolza Problem', SIAM Journal of Control, Vol. 4 , pp. 309-325. Kelley, H. J., (1964). 'A Transformation Approach to Singular ~ubarcs in Optimal Trajectory and Control Problems', SIAM Journal of Control, Vol.2, pp. 234-240. Kelley, n. J., Kopp, R. E. and Moyer, G. H., (1967). 'Singular Extremals', in Topics in Optimization, Ed. G. Leitmann, Chapter 3, Academic Press, New York. Krotov, F. F., (1961). 'The Principal Problem of the Calculus of Variations for the Simplest Functional on a set of Discontinuous Functions', Soviet lIathematics Doklady, Vol. 2, DD. 231-234. Leitmann, G., (1981). 'The Calculus of Va riati ons and OPtimal Control', Plenum Press, New York. !lancill, J'. D., (1947). 'Unilateral Variations with Variable Endpoints', American Journal of Mathematics, Vol. LXIX, pp. 121-138. Mancill, J . D., (1950). 'Identically Non-Regular Problems in the Calculus of Variations', Mathematica Y Fisica Teorica, Universidad Nacional del Tucuman, Rep~blica Argentina, Vol. 7, No. 2. Miele, A., (1950-51). 'Problemi di Minimo Tempo Nel Vola Non-Stazionario degli Aeroplani', Atti della Accademia delle Scienze di Torino, Vol. 85, pp. 41-42.
Singular Optimal Control Problem Miele, A., (1962). 'Extremization of Linear Integrals by Green's Theorem', in Optimization Techniques, Ed. G. Leitmann, Chapter 3, Academic Press, New York. Petrov, I. P., (1968). 'Variational Methods in Optimum Control Theory', Academic Press, llew York.
47