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A non-linear extremal problem
A NON-LINEAR
EXTREMAL PROBLEM*
V. F. DEWYANOV Leningrad 23
(Received
THEfollowing
1965)
November
problem is discussed.
Given the non-linear
controlled
Ji = f(X, and the integral
system X(0) =x0
4 t>,
functional
I(t,
u) =
s
F(X(z),
u(z), z)dz.
0
For any t, we can find 0((t)=
inf st F(X(T),u(z),z)&, wzJ 0
where Cl is a given set of control vector functions d~(ktO) / dt.
(controls).
To find
Apart from finding uYD(tf0) /dt, for different classes of control, the application of the results to solving time-optimal problems will be indicated.
1. Differentiation of the minimum function Given the non-linear
*
Zh.
vphisl.
Mat.
mat.
system of ordinary differential
Fit.
7,
1, 41
33 - 51,
1967.
equations
V.F.
42
Dem’yanov
dx(t)
(1.1) X(0) = x0.
U.2)
Here, X(b)= (t*(t), .... P(t)), f(t)= (fi, .... f") are n-dimensional real vector functions and u(t) = (ui(t), . . . , u’(t)) is the real r-dimensional controlling vector function (control), belonging to a class 11 of controls to be defined below, The functions on the right-hand side of (1.1) are assumed cant inuoualy differentiable with respect to xi, and continuous in uj, k (i = 1, . . . , n; j = 1, . . . , r) in the region of permissible values of xj, uj. determined by the class li of controls, the system (1.1). the initial conditions (1.21, and continuous with respect to t in [O,T], 0 < T < co. The class of permissible controls ii will be the set of r-dimensional vector functions snare-su~able on [O, Tl, where, for each t
and i?(t) is a convex closed bounded set of r-dimensional Euclidean space E,. Suppose there exists a finite closed sphere ? C E, such that f?(t) c S for all t E LO, Tl, and that the set C!(t) varies piecewise continuously with t, so that we can put
As usual,
Q(tk~) -+Q(f f 0) denotes here that p(z)=
max
min
(Y-Z)2z~0.
z~Q(fz!cO) Y~Q(fzk@
For instance, li may be the set of vector functions u(t) = (u’(t), square-summable on [O, ~1 and satisfying one of the relations
. . . , u’(t))
where w(t),
B(t),
Xi(t) are piecewise continuous functions bounded on
A non-linear
extremal
problem
43
Lo, z-3 I
We denote by .Y(t, U) the solution ditions (1.2) for a given u E U. Consider
of system (1.1)
with initial
con-
the functional IP,n)={
P(X(V),
u(z),z)dt,
(1.5)
0
where the scalar function F(Y, IL, t) is assumed, unless otherwise stipulated, uniformly continuous in its variables in the region of their permissible values (the region of permissible values of t is the closed interval LO, 7’1, the region of permisslbe values of II is Q(t), and of Y is determined by system (1.11, the initial condition (1.2), the class of controls fl and the interval LO, ~1). In particular, is a closed t G Lo,
[O,
if F is continuous
sohere 1)(Y) c
~1),then
on
En such that
F is in fact
D(T)
X S X [0,T] (where D(7’)
X(t, u) CD(T)
uniformly
continuous
for on D(T)
all
u 5
I,’
X 5’ X
z-1.
The case will sometimes be considered when F is piecewise continuous, bounded in t, continuous in X and u on D?(T) x S. It will be specifically mentioiPed if this type of F is in question. Let = infI(t,.u). UUJ
a(t) Clearly, We fix vided
is continuous
O(t) t E
there
on [O, Tl .
[O, ~1 and consider
exists
a 11E
(4.6)
the set
Me(t) c En (XE&(~),
pro-
U su6h that X(t, u) = X,
ZQ, u) <
inf z(t, u) + e,
e > 0).
VEU
with Ed < Em, we have i%,(t)
~M.s(~).
We form the set M*(t) c En, such that .Y E sequence ‘I~, ~2, . . . , such that USE u,
x, =
X(4 4) E Mt?8(q
,
,1*(t) e,-+O; d-ww
if there
exists
x,-+x.
a
V.F.
44
Dem’yanov
A sequence of controls of this type will be termed a p(t) sequence. The set ?(t) is closed and bounded (in particular. it may consist of a finite or denumerable number of points), and ME(t) -( M*(t), i.e. p(e)=
min (Y -Z)2+O. max e-s4 ZEM*(t)YEM, (0
Notice that M’(t-A)---t~(t-O)cY*(t). A-+-W
.~-(t+A)---+~(t+O)cM’(t), b-M
It can be shown here that @(t + 0) # Mft - 0), M’(t) M(t - 0) # A. In many applications of Q(t) given by (1.6).
\a(t
+ 0) U
it is useful to determine the differentiability
In fact,
W+A)-Q(t) .dQ=w+o~=lim _ dt
r?otice first
fm(t + dt
that,
0)
A
A++0
*
from (l.‘V, t+A
= lim inf -! ~++ou~A
=
b
;
F(X(z,u),4w)d~-
WI]
=
lim 1 inf iI?( ~++o~ucsu~
where
‘+p
R(u, A) =
F(X(z,u),u(z),z)dT-@Q,(1)= 0
=
~r(x(~,.),.(3,~)a_m(t)+(rP(X(~~~).,~~~),.h),.)6n t t = I(t,U)-Q)(t)+ r F‘(X(t,~u),u(z),2)dt. t
Let {rnk) be a p(t) sequence. Now, t-M infR(u,A)~I(t,un)--(t)+C F(X(z,uk),u~@),r)dt UETI
t
4
non-
1inear
1 prob
extrena
45
lem
(where uk(s) is any vector function on (t, t + A] such that Uk(T) E The left-hand side of the last inequality is independent of k, so that
?(T)).
=f~_[l(t, llR)_-Q)(t)+t{AF(X(T, Uk)., Uk(T),T)dT]. t Since X(t, uk) *X E M’(t), k for T E (t, t + A], then
while
ILL(T)
is
arbitrary
and independent of
t+A
infR(u, A)< ILal where k(z)
= /(X(z), U(T), T),
j F(X(z), t X(t)
= X,
u(z), z)dz,
X E M’(2).
Since F(X, u, t) is uniformly continuous,FCY(-r), U(T), 11 < F(X, u(-r), t) + E(A), where E(A) 2 Cl is independent of u E S, E(A) + 0. Since Q(t + T) Q(t + Oj, there must exist, for u(z) E Q(t + T) a T-S+0 u E :!(t + 0) such that &i(A) 0, (1.7) A-+0. and is independent of U(T); for any v i ‘?(t t 0) there exists at least one such vector function U(T) E ,>(T) (given on [t, t + A] 1, for which (1.7) holds. J’(-Y U(T), t)GF(X,
u, t)+si(A),
Since inf H(u, A) < AfF(X, u, t)+‘e(A)+
si(h)]
UEU
for any u E Qlt + 01, then infR(u, A)
V, ~>+E(A)+Q(A)I-
UEU
Hence
dQ(t+o) dt
=~~o~i$R(zz,
-.
A)<
u
+jz[e(A)+ei(A)]=
min F(X, u, t)+
uEQ(t+W
min F(X,u,t).
v=QV+O)
V.F.
46
Dem’yanov
This last inequality holds for auy X E -P(t),
‘Ott + O) at
so that
min F (X, u, t) Xcaf*(t) uEQ(t+o) min
(
.
(we have min rnin here instead of inf inf, ‘j(t + 0) are closed).
since the sets i!*(t) and
The reverse inequality also holds. For,
=
lim -1sup{W(t+A)-I(t,a)]= ~-+I A u
Gm&p
[@@+A)-I(t+A,u)+
+tyF(X(r), u(t),t)dT]=
frnM{stpR(., *
0
~@,A~=@(ti-A)-W+A,
where kfz, 2.z)= f(x(%
u)+
u), a(+)lT),
xtt)
f F(X(z,
u), u(z),~)d’c,
= x(t9 @)*
Let {uk) be a p(t + A) sequence; since supR(u,
A) 2 R(wsr A) = @(t + A)-t-f-A
we
now have t+A
sup R (u, A) >
lim R (uk, A) = lim
k-+oo
k-+m
1 F(X( % uk) d
, uk
(z) , 7)
&.
Since WA
WA f
t we
have
F(X(t,Uk),Uk(Z),Z)dZ~
1 u@tEQ(~t min F(x(~,uk),u(~),~)d~,
t
A)..
A non-linear
where
extremal
problem
X(z)= lim X(z, zzk) (by the Arzela - Ascoli theoreln,
ill,
Y(7)
k-e0
exists
and is a cant inuous vector
function).
In view of the uni q*nrmcontinuity we have F(X(z), zz,7) 2 F(X, u, t) x=
and is independent
of F(Y, :t, T) for T s E(A), where
lim X(%)EJZ(t+O), %-wiI
e(A)-+O, A4
[t,
t + Al,
e(A) >, 0
of LLE S;
lv(t + 0) = lim ~(t+A)cM’(t). A++0
Since
Q(t + A) AzoQ’t
+ 0) 9 and F&Y, CL, t) is uniformly
continuous
in
any u E Q(t i-T), ‘c E [0, A], there exists a v E q(t t 0) such that F(X, % t) 2 F(X, V, t) *- Q(A), where EI( A) - 0 as A * 0 and is independent of ,Y and U. Hence
A[F(X,u,t)--(A)--i(A)I:
s~pR(u,A)>~~~~~,
u
Thus dQ,‘f + O’ =
lim
~supR(u,A)> a+-++A
6%
lim min [F(X,u, t)-e(A)-ei(A)]= A-c+0uG!W@ (1.8)
=
min
F(X, u, t)-
(1.8)
holds
d@tt + 0) > dt
Fran (1.8)
cl(A)] =
A++0
-~EQ(tSO)
Since
lim [e(A)+
min
F(X, u, t).
uEQWk0)
for any Y E %(t f 01, and @(t + A) c M’(t),
min min F(X, EP (t+o) =Q (f+@
II, t)>
min F(X, min XEIW co WZQ tt+O)
we
have
24, t). (1.9)
- (1.9).
dQ,@+ 0) at
=
min XEM*(f)
min u-sQ(t+W
F (X, 24,t).
(1.10)
V.F. Den’yanov
48
We now
find
fwt-0)
= ;l+KD(t)-
dt
a,(t - *)I=
inf[I(t,U)-@(t-A)]=
m,(t - A)], (1.11)
I(t-A,u)-@(t-A)+
inf
L
UEU
+
*iL+_[Z@,“‘-
(U~(x(~,u),.(T),T)dTJ
=
t-A
inf
=
I(t-A+)-O(t-
A)+
U”A
inf
i F(X(r,u),u(z),z)dr],
vBv t-_h
where u E ~JAif u is given in the range [O, t - A] ; v E V if u(7) is given in (t - A, tl, and u E .?(T) for T E (t - A, tl. I.e. the interior inf in the second term of (1.11) can be taken with control U(T) fixed in [O, t - Al. Continuing (1. ll), inf [I&u)-@(t-A)]< UEU
‘inf [I(t-A+)--(t-A)]+ U@A t +
max
inf
XEM*(t-A)
VEV
s Wq~),u(z),z)dz=
t_A
max inf s p(X(4, -=~*@-A) VEVt_A
S
i WW,~W,$dz<
5 W(t-A),
W,+z;
u(r),t)dz+e(A),
t-A
t-A
where dX(T) / dr =f(X(z),u(z),z),
X(t-A)
=XEM*(~-A);
E(A) + 0 as A + 0 and is independent of X E M’(t - A), u E V. Since Q(t - 0), for any v E Q(t - T) there exists u E g(t - 0) Q(t - 4 7-d-o
such that
F(X(t
-
A), u, t) < F(X(t
-A),
u, t) + Q(A);
Q(A) ~0, +
and is independent of X =M*(t - 0), u E Q(t - z), z E [0, A], where each u E ‘.?(t .- 0) corresponds to at least one v E 3(t - T) for T E CO, Al Now, t
illf s F(X(z),u(r),z)dzGA[ uEv t-A
min.
UEW-0)
F(X(t-A),u,t)+e(A)+ei(A)],
A non-linear
extremal
49
problem
i.e. inf [I(t,u)-@(t-A)]<
<
AE min F(X,u,~)+s(A)+e~(A)l
max
UEQtf-0)
XEM*(f-A)
UUI
min F (X, u, t) T max Ed tt-~~=Q u--o)
max
XEM*
min
F(X,
(f) UEQ +-o)
2.4,t).
where
iii@-O)=
E(t) c M=(t).
lim M(t-A), A-r+0
Thus
dW--0) dE On the other
=
max
(
-
mill
XGiwff
(1.12)
F(X,u,t).
ws2ff--O)
hand,
sF(X(z,u),u(T),Z)dT=
Ab++ f$D(t)-Z(t,u)l+
t-ii
=jilU
1 sup
A--WA
UEU
Iz(u,A),
where R(u,A)=
@((t)-zwq+
F(X(w),44,+~. t-A
Let {uk) be a cl(t) sequence; SUP R(u,
since
A) 2 R(m, A) =(f)(t)-
I(4Uk)$
UEU
1 F(X(z,Uk),Uk(7),Z)dZ, t-A
we have sup R (u, A) 2 lim R ( UR,A) = lim UEU
h-+ca
S F(X( z, Uk) ; 4t (z) I q h.
k+m t-A
50
V.F.
Dem’yanov
Hence
sup
sup R(u, A) a WEU where (ak)
E
w(t),
urn f
(1.13)
p(x(z,uk),uk(2),Z)dZ,
{Uk)-?f) b+w t__-~
iff(t,
&k)k_,m
iIlf 1(2, U). -lsKJ
From (I, 131, it follows that da+-0) --rl=
at
lim
-!-sui
A-MA
R(u,A)> lim
A-d-0
We
A
wk~ww}
k+-
5
F(X(z,Uk),Uk(Zf,Z)dlt.
t_A
have
the Arzela - Ascoli theorem [I], there exists a vector function
By
s-4
k-m
X cz M”(t), then, for k large, j F~XtWh’(~),&G3 t-A
j [F(X,U(Z),Z)-S(A)]&; t-A
~(5) -* 0 and is independent of X ~2 if*(t),
X = T+fi_x
vE
2 and
k.
Here,
(%uk) = M’ (t ) .
Q(t - z) -+ Q(t - 0) as T 3 0, a > 0, whatever the v EZ ?(t - T) with T 6% LO, A], there exists u E Q(t - 0) such that F(X, V, t) 2 ei(A) > 0, Q(A) -+ 0, A-t +O, ~1(A) is independent of FK a, t) -&(A), Since
X EZ V*(t)
and u E V. Then,
k-+ar ‘=Vt__A Hence
min F(X,v,t)----e(A)--es(~)]. uGi(t-a)
ertrema
A non- 1 inear
m(t-0)
lim
->
at
max
min
A.-++0 XEM*(t)
[F(X,u,t)-e(A)-eei(A)l= F (X, u, t) .
min
max
uEQ(t-0)
X=M*(t)
and (1.14),
(1.14)
uEQ(t-0) =
From (1.12)
51
1 problem
we have finally
aw--~=
(1.15)
F (X, u, t) .
min
max
at
xczr+f’(t) uEQ(t--O)
Votes. 1. Let F be uniformly continuous in all its arguments. Now, if the set :4*(t) consists of a single point (as is the case e.g. if (1.1) r(t, u) is strictly is a linear system in X and u, and the functional the function convex), Q(t + 0) = n(t - O), then, from (1.10) and (1.X5), s(t) is differentiable at the point t, and
da@-0)
da -= (4 dt
dQ(t+o)
dt
=
,=
min F(X,u,t), UEQ(U
dt
where Y = i!*(t).
it
2. If F is piecewise continuously differentiable with respect can be shown that (1.10) and (1.15) become respectively
d@,(f+ 0)
min F(X, a, t +O),
min
--------)=
dt
XEM*(t) uEwt+o)
m(t
- 0)
3. Recalling
dt
that Y(t) max
XEN*(t)
=
F(x,u,t-6).
.
dt
dY (t + 0) ___z
to t,
=x,:yj
supZ(t,u)
=
u,~;:)
we get
-inf[-Z(t,u)],
dY(t - 0)
max F,(X, u, t),
dt
uEQ(t+O)
min =
XEN*(t)
max F(X, U, t), uEQ(t-0)
where K’(t) = limNd(t),
XE
N,(t)c
E,,
if
x
=
X(t,
u),
u E
u,
z!t,
u)
e-0
supqt,
v)-
WEV
The above expressions (1.10) and (1.15) handed derivatives of the function O(t). A function
>
h(t)
is
obtained
below,
represent
the first
(left-
left-
and right-
and right-handed)
E.
V.F.
52
derivatives derivatives
Den’yanov
of which are the same at the point t 1 as the corresponding of Q(t) at t 1.
It can prove advantageous to consider h(t) ing the extrema of the latter. Let Q(ti +x) trol
instead of O(t) when find-
= Q
for z E [ti, ti + a], a > 0. We specify IL(T) in the range [O, t], t > t 1
the con-
where U(T) and ul are such that Xi = X(ti, u),
F (Xi, ui, h) =
mln XEbP(lr)
min F(X,UJi).
(1.16)
UEQ
Now.
-Zr(t,,u)+~ F(X(4,WJ).h
h(t) = z(t, u) -
t1
X(z) = fV(4
ah(t)
-=F(X(O,ui,t), at
9
w, 4,
Wti) dt=
X(h) =
xi,
F(Xl, Ul, ti) =
dQ,(+I- 0) dt
.
The right-handed derivatives of h(t) and O(t) are thus identical. Since h(t) can easily be obtained, we select X, and u1 satisfying (1.16) so that h(t) decreases at the fastest possible rate. Continuous differentiability with respect to A’ and t has to be required of F(X, u, t) at the point (‘Y,, t 1):
where 3F / 8X = duct. Since h(tt + A) =
(aF 1 dxi,. . . , 8F / ax,),
Z(ti + A, u) = Z(ti, u) + A dhj;)
and * indicates the scalar pro-
;
;
A2@;lf” +
o(A2),
A non-
61, iii) (1.16)
CM be selected such that
ertrema
1 inear
1 prob
53
lem
from the set of points (X1, ul) satisfying
Higher order derivatives of h(t) at the point t 1 can be similarly determined; we just have to require that F and f be differentiable with respect to X and t at the point (X,, t 1) the relevant number of times. In particular, if there is more than one pair of points (.y,, ii,) satisfying (1.18), we csn select from them the pair (x1*, ul*) such that
dsh(h, XI*, UI*) = at3
dYh(h,
min
(%,*G)
21,;I)
dt3
?
The higher order derivatives of h(t) can also be minimized (until the set on which the minimumis sought consists of a single pair of Points 01, u1); of course, it can happen that the set contains more than one pair of points whatever the order of the derivative). NOWlet Q(t) E Q h(t)
= T(t,
.X1E ‘d*(tl),
u) on [tl
for
a, h], a > 0. Consider the function where u E I/ is such that .X(t I, u) =
t E [ti -
- a, t ,I,
i.e. 1 (k U) = minI(ti, u), VEV
F (Xi, ui, ti) =
wt4
dt
_
;i$
+
(1.19)
max XfziW(t)
s
min F (X, u, t) ,
uEQ
F(X(z,u),u(z),z)d~.
(1.301
t,-A
It will be shown that
Wi) -
dt
= min F (Xi, u, ti) = F (X,, vi, tt).
(1.21)
UEQ
For, by the mean value theorem, whatever the summablefunctions .X(r), U(T) 0 where X(T) E D(A), U(T) E Q, D(A) is the convex linear envelope fi(~), T E [tl - A, t,]),
54
V.F.
Dem’yanov
t*
s
NW(A),
F(X(T?),U(Z),T)az=
@(A),WI.
tr-8 where Z(a)
ED(A),
E(A) E Q, 5(A) E [k From y(A)
----+ X, T(A) + ti.
We have x(lh)
A, tt]. we choose
some convergent
A-M
sequence ii(
G(&),
. . . , E(AJ -
ZJE Q, A8 -
-
ah(h)
ah(h)
-=F(X,u,t~), at
-=
0;
then,
8--roe
- 0)
dh(ti
at
at
=F(X,u,t1).
Clearly, F(X, u, ti) 2
minF(X, u, ti) = F(X, ui, ti). UEQ
If
it happened that
UA (.c) =
and thus obtain
I(&,
we could take the control
F(X, m, t!) < P(X, II, tf),
uA) <
u(z)9
r
ui,
‘t E
I(ti,
E
u) for
Al, A, tJ,
(0, ti (ti
-
small A, which contradicts
(1.19).
Thus (1.21) is proved, i.e. the left-handed derivatives of h(t), are identical at the point t 1; h(t) can be found without difficulty, provided a control II E II satisfying (1.191 is known. It is easily that
Q(t) shown
(1.22) where
ax(h) /
at= m,
From the pairs of points be chosen so that
(-Vi, ui)
&% (ti, Xi, ,&)
dt2
F(h) = F(Xl, m, b).
ui, h),
a=
satisfying
min (Xl; UI)
(1.29),
(fl,
2,)
a% (h, xi, Ui)
dt2
If there is more than one pair (.fi, ii,) satisfying particular pair (Y,*, ul*) can be taken such that
must
(1.23)
’ (1.231,
the
A non-linear
and so on (the max occurs
h(h
-A)=
Vi)
extremal
problem
55
here because + o(A3),
- A
and in general, (.Y,, 1~~) have to be taken so as to minimize the even, and maximize the odd, derivatives of h(t)). 4. It was assumed above that ?(t) is constant in the neighbourhood of the point tl. If u(t) = u(t)w(t), where a(t) is a continuously differentiable bounded scalar real function, w(t) E ;? (in which case we write y(t) = a(t)!?), then (1.17), (1.22) become
2. Controls with integral hounds Returning to system (1.1) under initial conditions (1.2), the same conditions as before are imposed on the functions on the right-hand side of the system. The control vector function u(-r) on [O, tl is assumed to belong to the set U, of functions measurable on LO, tl, satisfying t s
uy+z
& ci,
i = I,...,?.,
O
(2.1)
0
or t
s7.4.(%I (4 CL
24(‘d)d-f<
c,
(2.2)
0
where P(T) is a symmetric real r x r matrix, with continuous elements bounded on [O, tl. Consider
the function (P(t)=
min I(t,u), u=t
positive
definite-on
[O, tl,
V.F.
56
where T(t,
u) is the functional
Dpm’yanov
(1.5).
Thus, (P(t)==
min Z (t, 2.4)= min s P(X(z, U&J
lAe
u), W),W.
0
Consider T(t, u). We fix a u G 51 (a vector [O, tl and find the variation of the functional
SZ(t* 24)=
function in the range at the point (u, t):
W(t, u),~(t)dt)bt + [ G&W% 0
where
G&z) = (Gt(t,z),
(-$)‘Y,(t,r)+ a:uy),
Gui(t,7) = mu&T) dz
Gu2(t,~), . . . , W(t,z)),
= -
(~)‘Y,(t,+)-
fu(4 = f(Xh
‘F;r)
m ax
dF -ax’
aF
(
afl
af
.
afl
.
J-g,
If ut E fJ is such that
YY,(t,t)=
0,
4axn 1 af'
ax,
p’““---
.
afn
(2.3)
aF
ax2’“”
a’
ax=
,
1,. . . ,r,
Fub) = F(X(T 4, u(z),z),
u), u(494, -=
i=
.
.
.
.
.
.
.
.
.
.
afn axn
afn azp”“‘-
Z(t, ZQ) = min Z(t, u), then UUI
t
s
Gu(t,+Ldz
2
0
0
for any permissible (u+&) E 4.
variations
For A > 0, we have
(a variation
6, is permissible
if
(2.4)
(2.5)
A non-linear
extremal
57
prob Een
t+A
s
F (x(7,
@@+A)-O(t)=
Ut+A) , Ut+A (7)) T)&
-
0
t
Whatever
A) > 3.
A > 0, q(t,
We want to find dO(t)/dt tion (2.1).
for the class of controls
satisfying
condi-
First consider the functional I(t,u)=
i F(u(T),T)& 0
which differs nates.
from (1.5)
We denote by ut E
.!J
in that it is independent of the phase coordi-
the control
for which t
@(t)=I(t,ut)=
min s F(u(z),~)&. *0
We shall sometimes write q(s) = u(t, T) for convenience, thereby emphasizing the dependence of the optimal control nt (7) on t. In this case the vector function Q(T) s u(t,v) is continuously differentiable with respect to t almost everywhere. Then,
suppose that t s 0
U”2(t,‘F)dr=Cc
(i=
L..J).
58
V.F.
Dem’yanov
Lagrange’s method is used to form the functional rlh(4 u) = Equating the variation
of this functional
“‘,z) “) + 2hiQ i.e.
M(zQ(T), 7) / ihi = -2Afui(~).
to zero,
(I) = 0,
(2.6)
Since
s
Ui2(t, T) dlJ = Ci,
0
we have
i=i
From (2.V,
0
we have
hiUi(t,T)=-_Z
1 afl(W,‘6),4 dui
,
?LiUi(t, t) =
-
1 aF(u(t, 09 t)
-g
aui
1
so that
ui(t, t) . An expression is easily obtained for the derivative
when, for some i,
f pi' 0t, T dr < Ci. 0 Returning to our problem, let I(t, u) be the functional let IJ~E !I be such that CD’(t) = min I (t, u) UEU
=
I&u,)=
jF(X(r,nl),2li(z),7)d~. 0
(2.7 )
(1. f;), and
A non-linear
I prob
eztrema
59
lem
As previously, we shall sometimes write IQ(T) = U(&T). In view of the conditions imposed on F and system (1.11, the vector function u(t, T) is continuously differentiable with respect to t almost everywhere. Also, whatever i = 1, . . . , r, either t zP(t,T)&=
s
(2.8)
ci,
0
or t
s
lP(t,Z)dz
< Ct.
(2.9)
0
Suppose (2.8)
is true for all i = 1, . . . . r, min I(& 7.4)< UEW
If
min I(t, u), u-t
where %’ is the set of vector functions square summablein the range LO, tl (there are no other restrictions on II E ‘U, then we also have 11
s
UytiJ)dT
=
ci.
0
for ti E [t - a, t + a],
where a > 0, Then
cm(0
5
(2.10) an*;;r)d~.
f Gu; (t,z) 0 i=i
-=W+,Ut),w(t),t)+ at hit t
s
Ui2 (t, 7) Oh = Ci.
0
Differentiating
with respect to t, we obtain
(2.11)
We now find the minimumof l(t, the functional )?a(C u) =
u) (with fixed t).
s [F(X(t,u),u(z),z)+ 0
i i=i
To
W(T)]
this end, we write
d-c.
V.F.
60
where hi = hi(t),
Dem’yanou
i = 1, . . . . r.
The relat ionshlps G,,‘(t,a)+2hiut’(z)=O must be satisfied Hence G,,i(t,z) Substituting
(i=
i,...,r)
on the optimal control ut for almost all -r E = -PA&
(2.12)
= -2A&(t,z)
[O, tl.
(i = 1,. . .,r).
(2.12)
in (2.10)
a@ tt) - at = P(x(t,w)tw(t),q-
1 22 0
i=1
&,(t,r) yy)
a%.
Using (2. ll),
a@(t) -=F(X(t&),ut(t),t)+~ Afqt,t). at i=i
Again applying (2.12)) aa (t) -=~(x(t,u,),.,(r),t)-f~ at Recalling
(2.3)
- (2.5).
G,$(t,t)d(t,t). r=t
the ffnal result
aF(x(t,w),w(t),t
a@(t)
-=F(X(t,ut),Ilt(t),t)-~
u(t, t). (2.13)
du
at
This is the same as X(t, u) depends on the form of the derivative (1, 2, . . . , r-1. It can
Is
(2.7). i.e. the fact that the vector function phase coordinates does not affect the general of Q(t). Now let (2.9) hold for certain i E Y c be assumed without loss of generality that
t s qt,
Z) az = ci,
i=
1,2, . . . , r,
ri <
r;
0
t s d2 p, T) a%< cr,
i=ri+1,
ri+2,...,r.
0
We
again assume the existence
t +al.
of a > 0 such that,
for tl
E [t - a,
A non-linear
extremal
81
problem
11
s
ui2 (ti, z) dz = Ci,
i=
l,...,r.
0
Let
be
Er_rl
(r
-
rl)-dimensional
Euclidean space. It is easily shown
that
d@(t) _ dt
my,,,
F(X(t,ut),Uti(t),....
(,rl+i
,...,
(2.14)
r--r,
. . . . W’ (q ) u’r+‘,. . . , ut)- -!- 2r’ ~Jqx(wt),~t(q,q &d 2 i=i It may happen that,
ui(t,t).
for all or only some i , t
s
u’2(t, z) dz = Ci,
0
where for all
ti E [ti - U,t) or ti E (t, & + ~1 r s uia(ti,z)dz < Cr. 0
Since in this case Gui (t,z) = 0, we have aF (t) / dui = 0, minF(X(t,Ut),Ui(t)
, . . . , zP(t),
zd, .i+qq,.
. . , u’(t),
t) =
u =
and (2.13).
(2.14)
ww&),qq,.
. . , u’(t))
retain the same form.
In particular, it can happen that point t for the control ut for all i term is missing in (2.14).
rl =
= 0, i.e. 1, . . . , r;
(2.9) holds at the in this case the second
If the function on the right-hand side of system (1.1) or the function F have discontinuities of the 1st kind, we have to distinguish between the left- and right-handed derivatives of Q(t) as before, and also &(t, t f O);F(X(t,ut), u(t f O), t rt 0); W(x(t,w),u(t
etc.
The following
is similarly
zk 0), tztO)/dui
obtained for the class of controls
V.F.
62
satisfying
condition
Den’yanov
(2.2) :
if
ut*(T)p((7)Ut(T)dt=
.f
c,
(2.15)
0
then d@(t) i dt = F(X’(t, ut), ut(t), derivative
is evaluated
in this
t) .- ~/zC;,~*(t, t)u(t,
t),
i.e.
the
case from (2.13);
if t &*(.t)p(Z)w(T)dt
s 0
<
c,
then
d@(t)
-
It turns out that,
=
dt
for
min P(X(t, =-?
the class
ut), u, t).
of controls
(2.11,
(2.2),
d@(t)
(2.16)
dt~17(X(t,ut),ut(t),t), if all t (i.e. Votes.
the “energy” is utilized on the optimal (2.8) or (2.15) is satisfied). 5.
Let the conditions
s
ziiz(z)dt
(2.1)
< C,(t),
control
ut at the instant
be i = 1, . . . . r:
tE
to,Tl,
0
where Ci(t) is a function, positive for all t, and continuously entiable over the necessary time interval. Now, if
s
&Z(T)& = c<(t)%
for
all
i = 1, ,.,,
r,
it can be shown, exactly
differ-
(2.17) as above,
that (2.18)
A non-
linear
extrema
1 prob
Zen
63
If (2.17) is not satisfied for all i, instead of (2.18) we get an analogue of (2.14). The last term in (2.18) may be
(2.19)
where -ri is a point of LO, tl. If Gi (t,t) = 0, some other Feint I FE [O, t) has to be taken in (2.19, at whici?G,:(t, Q) # 0. If ~u:(t, z) s 0 for some i and almost all T E [O, tl , terms corresponding to this i will be missing (zero) in (2.18). 6. The second derivat Ives of @(t ) are not easy to evaluate in the case of integral constraints. The following procedure is best for finding dO/dt. Since the control (2.20)
iS
permissible,
we have o(t) < F(X(t, at), 0, t). On the other hand, D(2) >
minF(X(t, ut), U, t) = F(X(t, w), li, t). rJ E E,
In other words, the upper and lower bounds of en(t) sre known. The left- and right-handed derivatives of the function 1(t, at the point t, in the case of control (2.20), are dh(t + 0) 4 =
at
Recalling
cih(fF(X(4 ut), 0, q,
(2. lS), we now have dh(1- 0) dt
0)
= F(X(h ut), Q(t), 9,
02
d
u) = h(t)
a (t + 0) dt
*
3. Application to the solution of time-optimal problems
Returning to system (1.1) under initial conditions (1.2), to find, from a given class u of controls (u is the set of functions square summable in LO, 7’1, satisfying one of relations (1.3). (1.4), (2.1), (2.2)),
64
V.F.
D em’yanov
the control u G U such that: (1) Y(T, u) = 0 for some T > 0; (2) the T in (1) is the least possible. This is the time-optimal problem. Various closely related effective methods of successive approximations [2 - 61 have been developed for the case when system (1.1) is linear. The following
approach may be used to solve the non-linear problem.
Let R(t) be the region of admissibility of system (1.1) - (1.2) at the instant t; then Z E R(t) if there exists a permissible control u E 11such that .Y(t, U) = Z. In the case of a linear system (when (1.1) has the form X = A(t)X(t) + B(t)u(t) + F(t), where A(t) is an R x h matrix, B(t) is n x r, F(t) is an n-dimensional vector, whose components, like the elements of A, B, are assumed piecewise continuous bounded functions of time), the set R(t) is convex, closed, and bounded for any t (0 < t
= min$Xx(t,u). UUJ
Clearly, cp(t ) is a continuous non-negative funct Ion. The optimal time T is the least root of the equation q(t) = 0, i.e. T is the first instant at which q(T) = 0 [VI. method for finding q(t) at any instant t, 0 < t < co Is proposed In for non-linear systems, while methods of successive approximations are proposed in U.91 for finding various values, including q(t), in the case of non-linear systems, though here the successive approximations may lead to a local minimumof the function Y Z2 on the set R(t). A
M
apart from a constant, is the function 6(t) with F(X(x, u), u(x), 7) = X0(x, u)f(X( z, u), u(z)., z), it Is sufficient find the least root of the equation q(t) = 0. Since q+(t),
to
Since the derivatives of q(t) are known, this problem can be solved by the usual methods, e.g. Newton’s, the chord or secant methods, etc. The same approach can also be useful
for numerical solution
of certain
A non-
other
problems,
as outlined
linear
extrera
e.g.
in [lo].
1 prob
65
lem
Translated
by D.E.
Brown
REFERENCES 1.
KANTOROVICH, L.V. and AKILOV, G.?. Functional analysis in normed spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh) Fiamatgiz, Moscow, 1959.
2.
NEUSTADT, L.W. Synthesizing time-optimal Applic. 1, 3 - 4, 484 - 493, 1960.
3.
F4TON.
J.H. An iterative Analysis Applic. 5, 2,
solution 329
- 344,
J.
systems.
to time-optimal 1982.
Math.
Control,
Analysis
J. Math.
4.
KIRIN. N.E. A numerical method for linear time-optimal problems, In Symp. Computational methods (MetodY vychislenii), No. II, Leningrad State University, Leningrad, 67 - 74. 1963.
5.
PSHENICHNYI. B.N. A numerical control of a linear system, 60. 1964.
6.
BABUNASHVILI, T.G. Nauk
7.
SSSR,
155,
Synthesis 2,
295
method Zh.
for
vlchis
finding 1.
of linear 1964.
the
Mat.
optimal
time-optimal 4r 1, 52 -
mat.
Fiz.
systems
Dokl.
Akad.
- 298,
HO YU CHI, A successive approximation systems subject to input saturation, D-84, 1, 33 - .37, 1962.
technique Trans.
for Am.
optimal
Sot.
nech.
control Engrs,
a.
DEM’YANOV, V.F. Optimal program design for matika Tclenekh. 25, 1, 3 - 11, 1964.
9.
DEM’YANOV, V.F. Solution of some optimal mekh. 26. 7, 1153 - 1160, 1965.
10.
PONTRYAGIN. L.S., BOLTYANSKII. V.G., GAMKRELIDZE, R.V. and MISHCHENKO, E.F. Mathematical theory of optimal processes (Matemat ichesksya teoriya optimal’ nykh protsessov) , Fizmatgiz, Moscow, 1961.
a linear
problems,
system,
Avtomatika
Avto-
Tele-
EXTREMAL PROBLEM*
V. F. DEWYANOV Leningrad 23
(Received
THEfollowing
1965)
November
problem is discussed.
Given the non-linear
controlled
Ji = f(X, and the integral
system X(0) =x0
4 t>,
functional
I(t,
u) =
s
F(X(z),
u(z), z)dz.
0
For any t, we can find 0((t)=
inf st F(X(T),u(z),z)&, wzJ 0
where Cl is a given set of control vector functions d~(ktO) / dt.
(controls).
To find
Apart from finding uYD(tf0) /dt, for different classes of control, the application of the results to solving time-optimal problems will be indicated.
1. Differentiation of the minimum function Given the non-linear
*
Zh.
vphisl.
Mat.
mat.
system of ordinary differential
Fit.
7,
1, 41
33 - 51,
1967.
equations
V.F.
42
Dem’yanov
dx(t)
(1.1) X(0) = x0.
U.2)
Here, X(b)= (t*(t), .... P(t)), f(t)= (fi, .... f") are n-dimensional real vector functions and u(t) = (ui(t), . . . , u’(t)) is the real r-dimensional controlling vector function (control), belonging to a class 11 of controls to be defined below, The functions on the right-hand side of (1.1) are assumed cant inuoualy differentiable with respect to xi, and continuous in uj, k (i = 1, . . . , n; j = 1, . . . , r) in the region of permissible values of xj, uj. determined by the class li of controls, the system (1.1). the initial conditions (1.21, and continuous with respect to t in [O,T], 0 < T < co. The class of permissible controls ii will be the set of r-dimensional vector functions snare-su~able on [O, Tl, where, for each t
and i?(t) is a convex closed bounded set of r-dimensional Euclidean space E,. Suppose there exists a finite closed sphere ? C E, such that f?(t) c S for all t E LO, Tl, and that the set C!(t) varies piecewise continuously with t, so that we can put
As usual,
Q(tk~) -+Q(f f 0) denotes here that p(z)=
max
min
(Y-Z)2z~0.
z~Q(fz!cO) Y~Q(fzk@
For instance, li may be the set of vector functions u(t) = (u’(t), square-summable on [O, ~1 and satisfying one of the relations
. . . , u’(t))
where w(t),
B(t),
Xi(t) are piecewise continuous functions bounded on
A non-linear
extremal
problem
43
Lo, z-3 I
We denote by .Y(t, U) the solution ditions (1.2) for a given u E U. Consider
of system (1.1)
with initial
con-
the functional IP,n)={
P(X(V),
u(z),z)dt,
(1.5)
0
where the scalar function F(Y, IL, t) is assumed, unless otherwise stipulated, uniformly continuous in its variables in the region of their permissible values (the region of permissible values of t is the closed interval LO, 7’1, the region of permisslbe values of II is Q(t), and of Y is determined by system (1.11, the initial condition (1.2), the class of controls fl and the interval LO, ~1). In particular, is a closed t G Lo,
[O,
if F is continuous
sohere 1)(Y) c
~1),then
on
En such that
F is in fact
D(T)
X S X [0,T] (where D(7’)
X(t, u) CD(T)
uniformly
continuous
for on D(T)
all
u 5
I,’
X 5’ X
z-1.
The case will sometimes be considered when F is piecewise continuous, bounded in t, continuous in X and u on D?(T) x S. It will be specifically mentioiPed if this type of F is in question. Let = infI(t,.u). UUJ
a(t) Clearly, We fix vided
is continuous
O(t) t E
there
on [O, Tl .
[O, ~1 and consider
exists
a 11E
(4.6)
the set
Me(t) c En (XE&(~),
pro-
U su6h that X(t, u) = X,
ZQ, u) <
inf z(t, u) + e,
e > 0).
VEU
with Ed < Em, we have i%,(t)
~M.s(~).
We form the set M*(t) c En, such that .Y E sequence ‘I~, ~2, . . . , such that USE u,
x, =
X(4 4) E Mt?8(q
,
,1*(t) e,-+O; d-ww
if there
exists
x,-+x.
a
V.F.
44
Dem’yanov
A sequence of controls of this type will be termed a p(t) sequence. The set ?(t) is closed and bounded (in particular. it may consist of a finite or denumerable number of points), and ME(t) -( M*(t), i.e. p(e)=
min (Y -Z)2+O. max e-s4 ZEM*(t)YEM, (0
Notice that M’(t-A)---t~(t-O)cY*(t). A-+-W
.~-(t+A)---+~(t+O)cM’(t), b-M
It can be shown here that @(t + 0) # Mft - 0), M’(t) M(t - 0) # A. In many applications of Q(t) given by (1.6).
\a(t
+ 0) U
it is useful to determine the differentiability
In fact,
W+A)-Q(t) .dQ=w+o~=lim _ dt
r?otice first
fm(t + dt
that,
0)
A
A++0
*
from (l.‘V, t+A
= lim inf -! ~++ou~A
=
b
;
F(X(z,u),4w)d~-
WI]
=
lim 1 inf iI?( ~++o~ucsu~
where
‘+p
R(u, A) =
F(X(z,u),u(z),z)dT-@Q,(1)= 0
=
~r(x(~,.),.(3,~)a_m(t)+(rP(X(~~~).,~~~),.h),.)6n t t = I(t,U)-Q)(t)+ r F‘(X(t,~u),u(z),2)dt. t
Let {rnk) be a p(t) sequence. Now, t-M infR(u,A)~I(t,un)--(t)+C F(X(z,uk),u~@),r)dt UETI
t
4
non-
1inear
1 prob
extrena
45
lem
(where uk(s) is any vector function on (t, t + A] such that Uk(T) E The left-hand side of the last inequality is independent of k, so that
?(T)).
=f~_[l(t, llR)_-Q)(t)+t{AF(X(T, Uk)., Uk(T),T)dT]. t Since X(t, uk) *X E M’(t), k for T E (t, t + A], then
while
ILL(T)
is
arbitrary
and independent of
t+A
infR(u, A)< ILal where k(z)
= /(X(z), U(T), T),
j F(X(z), t X(t)
= X,
u(z), z)dz,
X E M’(2).
Since F(X, u, t) is uniformly continuous,FCY(-r), U(T), 11 < F(X, u(-r), t) + E(A), where E(A) 2 Cl is independent of u E S, E(A) + 0. Since Q(t + T) Q(t + Oj, there must exist, for u(z) E Q(t + T) a T-S+0 u E :!(t + 0) such that &i(A) 0, (1.7) A-+0. and is independent of U(T); for any v i ‘?(t t 0) there exists at least one such vector function U(T) E ,>(T) (given on [t, t + A] 1, for which (1.7) holds. J’(-Y U(T), t)GF(X,
u, t)+si(A),
Since inf H(u, A) < AfF(X, u, t)+‘e(A)+
si(h)]
UEU
for any u E Qlt + 01, then infR(u, A)
V, ~>+E(A)+Q(A)I-
UEU
Hence
dQ(t+o) dt
=~~o~i$R(zz,
-.
A)<
u
+jz[e(A)+ei(A)]=
min F(X, u, t)+
uEQ(t+W
min F(X,u,t).
v=QV+O)
V.F.
46
Dem’yanov
This last inequality holds for auy X E -P(t),
‘Ott + O) at
so that
min F (X, u, t) Xcaf*(t) uEQ(t+o) min
(
.
(we have min rnin here instead of inf inf, ‘j(t + 0) are closed).
since the sets i!*(t) and
The reverse inequality also holds. For,
=
lim -1sup{W(t+A)-I(t,a)]= ~-+I A u
Gm&p
[@@+A)-I(t+A,u)+
+tyF(X(r), u(t),t)dT]=
frnM{stpR(., *
0
~@,A~=@(ti-A)-W+A,
where kfz, 2.z)= f(x(%
u)+
u), a(+)lT),
xtt)
f F(X(z,
u), u(z),~)d’c,
= x(t9 @)*
Let {uk) be a p(t + A) sequence; since supR(u,
A) 2 R(wsr A) = @(t + A)-t-f-A
we
now have t+A
sup R (u, A) >
lim R (uk, A) = lim
k-+oo
k-+m
1 F(X( % uk) d
, uk
(z) , 7)
&.
Since WA
WA f
t we
have
F(X(t,Uk),Uk(Z),Z)dZ~
1 u@tEQ(~t min F(x(~,uk),u(~),~)d~,
t
A)..
A non-linear
where
extremal
problem
X(z)= lim X(z, zzk) (by the Arzela - Ascoli theoreln,
ill,
Y(7)
k-e0
exists
and is a cant inuous vector
function).
In view of the uni q*nrmcontinuity we have F(X(z), zz,7) 2 F(X, u, t) x=
and is independent
of F(Y, :t, T) for T s E(A), where
lim X(%)EJZ(t+O), %-wiI
e(A)-+O, A4
[t,
t + Al,
e(A) >, 0
of LLE S;
lv(t + 0) = lim ~(t+A)cM’(t). A++0
Since
Q(t + A) AzoQ’t
+ 0) 9 and F&Y, CL, t) is uniformly
continuous
in
any u E Q(t i-T), ‘c E [0, A], there exists a v E q(t t 0) such that F(X, % t) 2 F(X, V, t) *- Q(A), where EI( A) - 0 as A * 0 and is independent of ,Y and U. Hence
A[F(X,u,t)--(A)--i(A)I:
s~pR(u,A)>~~~~~,
u
Thus dQ,‘f + O’ =
lim
~supR(u,A)> a+-++A
6%
lim min [F(X,u, t)-e(A)-ei(A)]= A-c+0uG!W@ (1.8)
=
min
F(X, u, t)-
(1.8)
holds
d@tt + 0) > dt
Fran (1.8)
cl(A)] =
A++0
-~EQ(tSO)
Since
lim [e(A)+
min
F(X, u, t).
uEQWk0)
for any Y E %(t f 01, and @(t + A) c M’(t),
min min F(X, EP (t+o) =Q (f+@
II, t)>
min F(X, min XEIW co WZQ tt+O)
we
have
24, t). (1.9)
- (1.9).
dQ,@+ 0) at
=
min XEM*(f)
min u-sQ(t+W
F (X, 24,t).
(1.10)
V.F. Den’yanov
48
We now
find
fwt-0)
= ;l+KD(t)-
dt
a,(t - *)I=
inf[I(t,U)-@(t-A)]=
m,(t - A)], (1.11)
I(t-A,u)-@(t-A)+
inf
L
UEU
+
*iL+_[Z@,“‘-
(U~(x(~,u),.(T),T)dTJ
=
t-A
inf
=
I(t-A+)-O(t-
A)+
U”A
inf
i F(X(r,u),u(z),z)dr],
vBv t-_h
where u E ~JAif u is given in the range [O, t - A] ; v E V if u(7) is given in (t - A, tl, and u E .?(T) for T E (t - A, tl. I.e. the interior inf in the second term of (1.11) can be taken with control U(T) fixed in [O, t - Al. Continuing (1. ll), inf [I&u)-@(t-A)]< UEU
‘inf [I(t-A+)--(t-A)]+ U@A t +
max
inf
XEM*(t-A)
VEV
s Wq~),u(z),z)dz=
t_A
max inf s p(X(4, -=~*@-A) VEVt_A
S
i WW,~W,$dz<
5 W(t-A),
W,+z;
u(r),t)dz+e(A),
t-A
t-A
where dX(T) / dr =f(X(z),u(z),z),
X(t-A)
=XEM*(~-A);
E(A) + 0 as A + 0 and is independent of X E M’(t - A), u E V. Since Q(t - 0), for any v E Q(t - T) there exists u E g(t - 0) Q(t - 4 7-d-o
such that
F(X(t
-
A), u, t) < F(X(t
-A),
u, t) + Q(A);
Q(A) ~0, +
and is independent of X =M*(t - 0), u E Q(t - z), z E [0, A], where each u E ‘.?(t .- 0) corresponds to at least one v E 3(t - T) for T E CO, Al Now, t
illf s F(X(z),u(r),z)dzGA[ uEv t-A
min.
UEW-0)
F(X(t-A),u,t)+e(A)+ei(A)],
A non-linear
extremal
49
problem
i.e. inf [I(t,u)-@(t-A)]<
<
AE min F(X,u,~)+s(A)+e~(A)l
max
UEQtf-0)
XEM*(f-A)
UUI
min F (X, u, t) T max Ed tt-~~=Q u--o)
max
XEM*
min
F(X,
(f) UEQ +-o)
2.4,t).
where
iii@-O)=
E(t) c M=(t).
lim M(t-A), A-r+0
Thus
dW--0) dE On the other
=
max
(
-
mill
XGiwff
(1.12)
F(X,u,t).
ws2ff--O)
hand,
sF(X(z,u),u(T),Z)dT=
Ab++ f$D(t)-Z(t,u)l+
t-ii
=jilU
1 sup
A--WA
UEU
Iz(u,A),
where R(u,A)=
@((t)-zwq+
F(X(w),44,+~. t-A
Let {uk) be a cl(t) sequence; SUP R(u,
since
A) 2 R(m, A) =(f)(t)-
I(4Uk)$
UEU
1 F(X(z,Uk),Uk(7),Z)dZ, t-A
we have sup R (u, A) 2 lim R ( UR,A) = lim UEU
h-+ca
S F(X( z, Uk) ; 4t (z) I q h.
k+m t-A
50
V.F.
Dem’yanov
Hence
sup
sup R(u, A) a WEU where (ak)
E
w(t),
urn f
(1.13)
p(x(z,uk),uk(2),Z)dZ,
{Uk)-?f) b+w t__-~
iff(t,
&k)k_,m
iIlf 1(2, U). -lsKJ
From (I, 131, it follows that da+-0) --rl=
at
lim
-!-sui
A-MA
R(u,A)> lim
A-d-0
We
A
wk~ww}
k+-
5
F(X(z,Uk),Uk(Zf,Z)dlt.
t_A
have
the Arzela - Ascoli theorem [I], there exists a vector function
By
s-4
k-m
X cz M”(t), then, for k large, j F~XtWh’(~),&G3 t-A
j [F(X,U(Z),Z)-S(A)]&; t-A
~(5) -* 0 and is independent of X ~2 if*(t),
X = T+fi_x
vE
2 and
k.
Here,
(%uk) = M’ (t ) .
Q(t - z) -+ Q(t - 0) as T 3 0, a > 0, whatever the v EZ ?(t - T) with T 6% LO, A], there exists u E Q(t - 0) such that F(X, V, t) 2 ei(A) > 0, Q(A) -+ 0, A-t +O, ~1(A) is independent of FK a, t) -&(A), Since
X EZ V*(t)
and u E V. Then,
k-+ar ‘=Vt__A Hence
min F(X,v,t)----e(A)--es(~)]. uGi(t-a)
ertrema
A non- 1 inear
m(t-0)
lim
->
at
max
min
A.-++0 XEM*(t)
[F(X,u,t)-e(A)-eei(A)l= F (X, u, t) .
min
max
uEQ(t-0)
X=M*(t)
and (1.14),
(1.14)
uEQ(t-0) =
From (1.12)
51
1 problem
we have finally
aw--~=
(1.15)
F (X, u, t) .
min
max
at
xczr+f’(t) uEQ(t--O)
Votes. 1. Let F be uniformly continuous in all its arguments. Now, if the set :4*(t) consists of a single point (as is the case e.g. if (1.1) r(t, u) is strictly is a linear system in X and u, and the functional the function convex), Q(t + 0) = n(t - O), then, from (1.10) and (1.X5), s(t) is differentiable at the point t, and
da@-0)
da -= (4 dt
dQ(t+o)
dt
=
,=
min F(X,u,t), UEQ(U
dt
where Y = i!*(t).
it
2. If F is piecewise continuously differentiable with respect can be shown that (1.10) and (1.15) become respectively
d@,(f+ 0)
min F(X, a, t +O),
min
--------)=
dt
XEM*(t) uEwt+o)
m(t
- 0)
3. Recalling
dt
that Y(t) max
XEN*(t)
=
F(x,u,t-6).
.
dt
dY (t + 0) ___z
to t,
=x,:yj
supZ(t,u)
=
u,~;:)
we get
-inf[-Z(t,u)],
dY(t - 0)
max F,(X, u, t),
dt
uEQ(t+O)
min =
XEN*(t)
max F(X, U, t), uEQ(t-0)
where K’(t) = limNd(t),
XE
N,(t)c
E,,
if
x
=
X(t,
u),
u E
u,
z!t,
u)
e-0
supqt,
v)-
WEV
The above expressions (1.10) and (1.15) handed derivatives of the function O(t). A function
>
h(t)
is
obtained
below,
represent
the first
(left-
left-
and right-
and right-handed)
E.
V.F.
52
derivatives derivatives
Den’yanov
of which are the same at the point t 1 as the corresponding of Q(t) at t 1.
It can prove advantageous to consider h(t) ing the extrema of the latter. Let Q(ti +x) trol
instead of O(t) when find-
= Q
for z E [ti, ti + a], a > 0. We specify IL(T) in the range [O, t], t > t 1
the con-
where U(T) and ul are such that Xi = X(ti, u),
F (Xi, ui, h) =
mln XEbP(lr)
min F(X,UJi).
(1.16)
UEQ
Now.
-Zr(t,,u)+~ F(X(4,WJ).h
h(t) = z(t, u) -
t1
X(z) = fV(4
ah(t)
-=F(X(O,ui,t), at
9
w, 4,
Wti) dt=
X(h) =
xi,
F(Xl, Ul, ti) =
dQ,(+I- 0) dt
.
The right-handed derivatives of h(t) and O(t) are thus identical. Since h(t) can easily be obtained, we select X, and u1 satisfying (1.16) so that h(t) decreases at the fastest possible rate. Continuous differentiability with respect to A’ and t has to be required of F(X, u, t) at the point (‘Y,, t 1):
where 3F / 8X = duct. Since h(tt + A) =
(aF 1 dxi,. . . , 8F / ax,),
Z(ti + A, u) = Z(ti, u) + A dhj;)
and * indicates the scalar pro-
;
;
A2@;lf” +
o(A2),
A non-
61, iii) (1.16)
CM be selected such that
ertrema
1 inear
1 prob
53
lem
from the set of points (X1, ul) satisfying
Higher order derivatives of h(t) at the point t 1 can be similarly determined; we just have to require that F and f be differentiable with respect to X and t at the point (X,, t 1) the relevant number of times. In particular, if there is more than one pair of points (.y,, ii,) satisfying (1.18), we csn select from them the pair (x1*, ul*) such that
dsh(h, XI*, UI*) = at3
dYh(h,
min
(%,*G)
21,;I)
dt3
?
The higher order derivatives of h(t) can also be minimized (until the set on which the minimumis sought consists of a single pair of Points 01, u1); of course, it can happen that the set contains more than one pair of points whatever the order of the derivative). NOWlet Q(t) E Q h(t)
= T(t,
.X1E ‘d*(tl),
u) on [tl
for
a, h], a > 0. Consider the function where u E I/ is such that .X(t I, u) =
t E [ti -
- a, t ,I,
i.e. 1 (k U) = minI(ti, u), VEV
F (Xi, ui, ti) =
wt4
dt
_
;i$
+
(1.19)
max XfziW(t)
s
min F (X, u, t) ,
uEQ
F(X(z,u),u(z),z)d~.
(1.301
t,-A
It will be shown that
Wi) -
dt
= min F (Xi, u, ti) = F (X,, vi, tt).
(1.21)
UEQ
For, by the mean value theorem, whatever the summablefunctions .X(r), U(T) 0 where X(T) E D(A), U(T) E Q, D(A) is the convex linear envelope fi(~), T E [tl - A, t,]),
54
V.F.
Dem’yanov
t*
s
NW(A),
F(X(T?),U(Z),T)az=
@(A),WI.
tr-8 where Z(a)
ED(A),
E(A) E Q, 5(A) E [k From y(A)
----+ X, T(A) + ti.
We have x(lh)
A, tt]. we choose
some convergent
A-M
sequence ii(
G(&),
. . . , E(AJ -
ZJE Q, A8 -
-
ah(h)
ah(h)
-=F(X,u,t~), at
-=
0;
then,
8--roe
- 0)
dh(ti
at
at
=F(X,u,t1).
Clearly, F(X, u, ti) 2
minF(X, u, ti) = F(X, ui, ti). UEQ
If
it happened that
UA (.c) =
and thus obtain
I(&,
we could take the control
F(X, m, t!) < P(X, II, tf),
uA) <
u(z)9
r
ui,
‘t E
I(ti,
E
u) for
Al, A, tJ,
(0, ti (ti
-
small A, which contradicts
(1.19).
Thus (1.21) is proved, i.e. the left-handed derivatives of h(t), are identical at the point t 1; h(t) can be found without difficulty, provided a control II E II satisfying (1.191 is known. It is easily that
Q(t) shown
(1.22) where
ax(h) /
at= m,
From the pairs of points be chosen so that
(-Vi, ui)
&% (ti, Xi, ,&)
dt2
F(h) = F(Xl, m, b).
ui, h),
a=
satisfying
min (Xl; UI)
(1.29),
(fl,
2,)
a% (h, xi, Ui)
dt2
If there is more than one pair (.fi, ii,) satisfying particular pair (Y,*, ul*) can be taken such that
must
(1.23)
’ (1.231,
the
A non-linear
and so on (the max occurs
h(h
-A)=
Vi)
extremal
problem
55
here because + o(A3),
- A
and in general, (.Y,, 1~~) have to be taken so as to minimize the even, and maximize the odd, derivatives of h(t)). 4. It was assumed above that ?(t) is constant in the neighbourhood of the point tl. If u(t) = u(t)w(t), where a(t) is a continuously differentiable bounded scalar real function, w(t) E ;? (in which case we write y(t) = a(t)!?), then (1.17), (1.22) become
2. Controls with integral hounds Returning to system (1.1) under initial conditions (1.2), the same conditions as before are imposed on the functions on the right-hand side of the system. The control vector function u(-r) on [O, tl is assumed to belong to the set U, of functions measurable on LO, tl, satisfying t s
uy+z
& ci,
i = I,...,?.,
O
(2.1)
0
or t
s7.4.(%I (4 CL
24(‘d)d-f<
c,
(2.2)
0
where P(T) is a symmetric real r x r matrix, with continuous elements bounded on [O, tl. Consider
the function (P(t)=
min I(t,u), u=t
positive
definite-on
[O, tl,
V.F.
56
where T(t,
u) is the functional
Dpm’yanov
(1.5).
Thus, (P(t)==
min Z (t, 2.4)= min s P(X(z, U&J
lAe
u), W),W.
0
Consider T(t, u). We fix a u G 51 (a vector [O, tl and find the variation of the functional
SZ(t* 24)=
function in the range at the point (u, t):
W(t, u),~(t)dt)bt + [ G&W% 0
where
G&z) = (Gt(t,z),
(-$)‘Y,(t,r)+ a:uy),
Gui(t,7) = mu&T) dz
Gu2(t,~), . . . , W(t,z)),
= -
(~)‘Y,(t,+)-
fu(4 = f(Xh
‘F;r)
m ax
dF -ax’
aF
(
afl
af
.
afl
.
J-g,
If ut E fJ is such that
YY,(t,t)=
0,
4axn 1 af'
ax,
p’““---
.
afn
(2.3)
aF
ax2’“”
a’
ax=
,
1,. . . ,r,
Fub) = F(X(T 4, u(z),z),
u), u(494, -=
i=
.
.
.
.
.
.
.
.
.
.
afn axn
afn azp”“‘-
Z(t, ZQ) = min Z(t, u), then UUI
t
s
Gu(t,+Ldz
2
0
0
for any permissible (u+&) E 4.
variations
For A > 0, we have
(a variation
6, is permissible
if
(2.4)
(2.5)
A non-linear
extremal
57
prob Een
t+A
s
F (x(7,
@@+A)-O(t)=
Ut+A) , Ut+A (7)) T)&
-
0
t
Whatever
A) > 3.
A > 0, q(t,
We want to find dO(t)/dt tion (2.1).
for the class of controls
satisfying
condi-
First consider the functional I(t,u)=
i F(u(T),T)& 0
which differs nates.
from (1.5)
We denote by ut E
.!J
in that it is independent of the phase coordi-
the control
for which t
@(t)=I(t,ut)=
min s F(u(z),~)&. *0
We shall sometimes write q(s) = u(t, T) for convenience, thereby emphasizing the dependence of the optimal control nt (7) on t. In this case the vector function Q(T) s u(t,v) is continuously differentiable with respect to t almost everywhere. Then,
suppose that t s 0
U”2(t,‘F)dr=Cc
(i=
L..J).
58
V.F.
Dem’yanov
Lagrange’s method is used to form the functional rlh(4 u) = Equating the variation
of this functional
“‘,z) “) + 2hiQ i.e.
M(zQ(T), 7) / ihi = -2Afui(~).
to zero,
(I) = 0,
(2.6)
Since
s
Ui2(t, T) dlJ = Ci,
0
we have
i=i
From (2.V,
0
we have
hiUi(t,T)=-_Z
1 afl(W,‘6),4 dui
,
?LiUi(t, t) =
-
1 aF(u(t, 09 t)
-g
aui
1
so that
ui(t, t) . An expression is easily obtained for the derivative
when, for some i,
f pi' 0t, T dr < Ci. 0 Returning to our problem, let I(t, u) be the functional let IJ~E !I be such that CD’(t) = min I (t, u) UEU
=
I&u,)=
jF(X(r,nl),2li(z),7)d~. 0
(2.7 )
(1. f;), and
A non-linear
I prob
eztrema
59
lem
As previously, we shall sometimes write IQ(T) = U(&T). In view of the conditions imposed on F and system (1.11, the vector function u(t, T) is continuously differentiable with respect to t almost everywhere. Also, whatever i = 1, . . . , r, either t zP(t,T)&=
s
(2.8)
ci,
0
or t
s
lP(t,Z)dz
< Ct.
(2.9)
0
Suppose (2.8)
is true for all i = 1, . . . . r, min I(& 7.4)< UEW
If
min I(t, u), u-t
where %’ is the set of vector functions square summablein the range LO, tl (there are no other restrictions on II E ‘U, then we also have 11
s
UytiJ)dT
=
ci.
0
for ti E [t - a, t + a],
where a > 0, Then
cm(0
5
(2.10) an*;;r)d~.
f Gu; (t,z) 0 i=i
-=W+,Ut),w(t),t)+ at hit t
s
Ui2 (t, 7) Oh = Ci.
0
Differentiating
with respect to t, we obtain
(2.11)
We now find the minimumof l(t, the functional )?a(C u) =
u) (with fixed t).
s [F(X(t,u),u(z),z)+ 0
i i=i
To
W(T)]
this end, we write
d-c.
V.F.
60
where hi = hi(t),
Dem’yanou
i = 1, . . . . r.
The relat ionshlps G,,‘(t,a)+2hiut’(z)=O must be satisfied Hence G,,i(t,z) Substituting
(i=
i,...,r)
on the optimal control ut for almost all -r E = -PA&
(2.12)
= -2A&(t,z)
[O, tl.
(i = 1,. . .,r).
(2.12)
in (2.10)
a@ tt) - at = P(x(t,w)tw(t),q-
1 22 0
i=1
&,(t,r) yy)
a%.
Using (2. ll),
a@(t) -=F(X(t&),ut(t),t)+~ Afqt,t). at i=i
Again applying (2.12)) aa (t) -=~(x(t,u,),.,(r),t)-f~ at Recalling
(2.3)
- (2.5).
G,$(t,t)d(t,t). r=t
the ffnal result
aF(x(t,w),w(t),t
a@(t)
-=F(X(t,ut),Ilt(t),t)-~
u(t, t). (2.13)
du
at
This is the same as X(t, u) depends on the form of the derivative (1, 2, . . . , r-1. It can
Is
(2.7). i.e. the fact that the vector function phase coordinates does not affect the general of Q(t). Now let (2.9) hold for certain i E Y c be assumed without loss of generality that
t s qt,
Z) az = ci,
i=
1,2, . . . , r,
ri <
r;
0
t s d2 p, T) a%< cr,
i=ri+1,
ri+2,...,r.
0
We
again assume the existence
t +al.
of a > 0 such that,
for tl
E [t - a,
A non-linear
extremal
81
problem
11
s
ui2 (ti, z) dz = Ci,
i=
l,...,r.
0
Let
be
Er_rl
(r
-
rl)-dimensional
Euclidean space. It is easily shown
that
d@(t) _ dt
my,,,
F(X(t,ut),Uti(t),....
(,rl+i
,...,
(2.14)
r--r,
. . . . W’ (q ) u’r+‘,. . . , ut)- -!- 2r’ ~Jqx(wt),~t(q,q &d 2 i=i It may happen that,
ui(t,t).
for all or only some i , t
s
u’2(t, z) dz = Ci,
0
where for all
ti E [ti - U,t) or ti E (t, & + ~1 r s uia(ti,z)dz < Cr. 0
Since in this case Gui (t,z) = 0, we have aF (t) / dui = 0, minF(X(t,Ut),Ui(t)
, . . . , zP(t),
zd, .i+qq,.
. . , u’(t),
t) =
u =
and (2.13).
(2.14)
ww&),qq,.
. . , u’(t))
retain the same form.
In particular, it can happen that point t for the control ut for all i term is missing in (2.14).
rl =
= 0, i.e. 1, . . . , r;
(2.9) holds at the in this case the second
If the function on the right-hand side of system (1.1) or the function F have discontinuities of the 1st kind, we have to distinguish between the left- and right-handed derivatives of Q(t) as before, and also &(t, t f O);F(X(t,ut), u(t f O), t rt 0); W(x(t,w),u(t
etc.
The following
is similarly
zk 0), tztO)/dui
obtained for the class of controls
V.F.
62
satisfying
condition
Den’yanov
(2.2) :
if
ut*(T)p((7)Ut(T)dt=
.f
c,
(2.15)
0
then d@(t) i dt = F(X’(t, ut), ut(t), derivative
is evaluated
in this
t) .- ~/zC;,~*(t, t)u(t,
t),
i.e.
the
case from (2.13);
if t &*(.t)p(Z)w(T)dt
s 0
<
c,
then
d@(t)
-
It turns out that,
=
dt
for
min P(X(t, =-?
the class
ut), u, t).
of controls
(2.11,
(2.2),
d@(t)
(2.16)
dt~17(X(t,ut),ut(t),t), if all t (i.e. Votes.
the “energy” is utilized on the optimal (2.8) or (2.15) is satisfied). 5.
Let the conditions
s
ziiz(z)dt
(2.1)
< C,(t),
control
ut at the instant
be i = 1, . . . . r:
tE
to,Tl,
0
where Ci(t) is a function, positive for all t, and continuously entiable over the necessary time interval. Now, if
s
&Z(T)& = c<(t)%
for
all
i = 1, ,.,,
r,
it can be shown, exactly
differ-
(2.17) as above,
that (2.18)
A non-
linear
extrema
1 prob
Zen
63
If (2.17) is not satisfied for all i, instead of (2.18) we get an analogue of (2.14). The last term in (2.18) may be
(2.19)
where -ri is a point of LO, tl. If Gi (t,t) = 0, some other Feint I FE [O, t) has to be taken in (2.19, at whici?G,:(t, Q) # 0. If ~u:(t, z) s 0 for some i and almost all T E [O, tl , terms corresponding to this i will be missing (zero) in (2.18). 6. The second derivat Ives of @(t ) are not easy to evaluate in the case of integral constraints. The following procedure is best for finding dO/dt. Since the control (2.20)
iS
permissible,
we have o(t) < F(X(t, at), 0, t). On the other hand, D(2) >
minF(X(t, ut), U, t) = F(X(t, w), li, t). rJ E E,
In other words, the upper and lower bounds of en(t) sre known. The left- and right-handed derivatives of the function 1(t, at the point t, in the case of control (2.20), are dh(t + 0) 4 =
at
Recalling
cih(fF(X(4 ut), 0, q,
(2. lS), we now have dh(1- 0) dt
0)
= F(X(h ut), Q(t), 9,
02
d
u) = h(t)
a (t + 0) dt
*
3. Application to the solution of time-optimal problems
Returning to system (1.1) under initial conditions (1.2), to find, from a given class u of controls (u is the set of functions square summable in LO, 7’1, satisfying one of relations (1.3). (1.4), (2.1), (2.2)),
64
V.F.
D em’yanov
the control u G U such that: (1) Y(T, u) = 0 for some T > 0; (2) the T in (1) is the least possible. This is the time-optimal problem. Various closely related effective methods of successive approximations [2 - 61 have been developed for the case when system (1.1) is linear. The following
approach may be used to solve the non-linear problem.
Let R(t) be the region of admissibility of system (1.1) - (1.2) at the instant t; then Z E R(t) if there exists a permissible control u E 11such that .Y(t, U) = Z. In the case of a linear system (when (1.1) has the form X = A(t)X(t) + B(t)u(t) + F(t), where A(t) is an R x h matrix, B(t) is n x r, F(t) is an n-dimensional vector, whose components, like the elements of A, B, are assumed piecewise continuous bounded functions of time), the set R(t) is convex, closed, and bounded for any t (0 < t
= min$Xx(t,u). UUJ
Clearly, cp(t ) is a continuous non-negative funct Ion. The optimal time T is the least root of the equation q(t) = 0, i.e. T is the first instant at which q(T) = 0 [VI. method for finding q(t) at any instant t, 0 < t < co Is proposed In for non-linear systems, while methods of successive approximations are proposed in U.91 for finding various values, including q(t), in the case of non-linear systems, though here the successive approximations may lead to a local minimumof the function Y Z2 on the set R(t). A
M
apart from a constant, is the function 6(t) with F(X(x, u), u(x), 7) = X0(x, u)f(X( z, u), u(z)., z), it Is sufficient find the least root of the equation q(t) = 0. Since q+(t),
to
Since the derivatives of q(t) are known, this problem can be solved by the usual methods, e.g. Newton’s, the chord or secant methods, etc. The same approach can also be useful
for numerical solution
of certain
A non-
other
problems,
as outlined
linear
extrera
e.g.
in [lo].
1 prob
65
lem
Translated
by D.E.
Brown
REFERENCES 1.
KANTOROVICH, L.V. and AKILOV, G.?. Functional analysis in normed spaces (Funktsional’nyi analiz v normirovannykh prostranstvakh) Fiamatgiz, Moscow, 1959.
2.
NEUSTADT, L.W. Synthesizing time-optimal Applic. 1, 3 - 4, 484 - 493, 1960.
3.
F4TON.
J.H. An iterative Analysis Applic. 5, 2,
solution 329
- 344,
J.
systems.
to time-optimal 1982.
Math.
Control,
Analysis
J. Math.
4.
KIRIN. N.E. A numerical method for linear time-optimal problems, In Symp. Computational methods (MetodY vychislenii), No. II, Leningrad State University, Leningrad, 67 - 74. 1963.
5.
PSHENICHNYI. B.N. A numerical control of a linear system, 60. 1964.
6.
BABUNASHVILI, T.G. Nauk
7.
SSSR,
155,
Synthesis 2,
295
method Zh.
for
vlchis
finding 1.
of linear 1964.
the
Mat.
optimal
time-optimal 4r 1, 52 -
mat.
Fiz.
systems
Dokl.
Akad.
- 298,
HO YU CHI, A successive approximation systems subject to input saturation, D-84, 1, 33 - .37, 1962.
technique Trans.
for Am.
optimal
Sot.
nech.
control Engrs,
a.
DEM’YANOV, V.F. Optimal program design for matika Tclenekh. 25, 1, 3 - 11, 1964.
9.
DEM’YANOV, V.F. Solution of some optimal mekh. 26. 7, 1153 - 1160, 1965.
10.
PONTRYAGIN. L.S., BOLTYANSKII. V.G., GAMKRELIDZE, R.V. and MISHCHENKO, E.F. Mathematical theory of optimal processes (Matemat ichesksya teoriya optimal’ nykh protsessov) , Fizmatgiz, Moscow, 1961.
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