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On finite-dimensional parametrizations of attainability sets
NOIlTit- ~
On Finite-Dimensional Parametrizations of Attainability Sets Valerii I. H e y m a n n
Institute of Mathematics and Mechanics Urals Branch, A cad. Sci. of Russia Kovalevskoi 16 620219 Ekaterinburg, Russia and A r k a d i i V. K r y a z h i m s k i i
International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria Transmitted by F. E. Udwadia
ABSTRACT The attainability set is defined to be a finite-dimensional integral-type image of tile set of all absolutely continuous scalar functions of time whose derivatives take wdues in a given interval. For a class of control systems with scalar controls restricted to the above interval (the class comprises, in particular, some bilinear systems), the attainability set has the traditional meaning. A method of finitedimensional paramctrization of the attainability set is described. The parametrization is universal, i.e., the same for all attainability sets of a fixed dimension. For the case of control systems, the result provides an upper estimate on the number of switchings sufficient to bring the system to an arbitrary reachable state at a prescribed time.
1.
PROBLEM
STATEMENT:
MOTIVATIONS
We start with the accurate problem formulation. F i x reals a a n d b > a. D e n o t e by Z t h e set of all a b s o l u t e l y c o n t i n u o u s f u n c t i o n s z ( ' ) : [ 0 , 1] ~-) [a, b] s u c h t h a t z(1) = 0 a n d - ~ ( t ) ~ [a, b] for
APPLIED MA THEMATICS AND COMPUTATION 78:137-151 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0096-3003/96/$15.00 PII S0096-3003(96)00004-X
138
V. HEYMANN AND A. KRYAZHIMSKII
almost all t ~ [ 0 , 1 ] . For every y ~ [ a , b] denote by Z(y) the set of all z(') ~ Z such that z(0) = y. Fix a natural n. Take continuous mappings dp(.): [ a, b] × [ a, b] ~ R '~, ~ ( ' ) : [ a, b] ~ R'~, and define functions F(.) and F*(.) on Z by
F(z(.)) = ~ l ~ ( z( t), z(O)) dt,
r*(4))
= *(40))
+ r(z(.)).
Put H = { F * ( z ( - ) ) : z ( ' ) ~ Z},
H ( y ) = { F * ( z ( ' ) ) : z ( ' ) ~ Z( y)} (y~
[a,b]).
Call H the attainability set and H ( y ) the y-attainability set (for the mappings O(.), ~(.)). Define a Z-map to be a triple (k, S, z('l ")), where k is a natural number, S is a closed and bounded subset of R ~, and z('l "): s ~ z ( ' l s ) is a continuous mapping of S into Z; continuity is understood with respect to the sup-norIn in Z. Call an above Z-map a parametrization of the attainability set (respectively, a parametrization of the y-attainability set) if the set F*(z(-I s)): s ~ S} coincides with H (respectively, coincides with /4(y)) for arbitrary continuous mappings O ( ' ) , T ( - ) . The problem in question consists in building constructive parametrizations of the attainability and y-attainability sets. The next remark says that a parametrization of the attainability set could be found through a family of parametrizations of y-attainability sets.
REMARK 1. Let for every y ~ [ - b , - a ] , a Z - m a p ( k , S(y), z('l "; y ) ) b e a parametrization of the y-attainability set. Then (k + 1, S, z(-t ")), where S-={(y,s): y~[-b,-a],s~ S(y)}, and z ( ' l ( y , s ) ) = z(-I s; y), is a parametrization of the attainability set. The above problem has a strong link to the theory of control. Recall that for a control system, the attainability set is understood as the set of all states that could be reached by the system at a prescribed time, starting from a given initial state. For some classes of n-dimensional control systems, the sets H and H ( y ) have precisely this meaning. These are the systems operating on the time interval [0, 1], having scalar, Lebesgue measurable controls u(') with values in [ a, b] and characterized by the following: there
Attainability Sets
139
are mappings q~(.), ~ ( - ) such that, for the final point x(1) of system's trajectory z(.) developing from z(0) = z 0 under control ~L(.), one has x(1) = F * ( z ( . ) ) , where z ( t ) = f / is the set of all state x(0) = x 0 ing f(l u ( t ) d t = x(t)
(1.1)
u ( r ) d r . For a system of t h a t kind, H (respectively, H ( y ) ) states z(1) attainable by the system at time 1 from initial using all controls u(.) (respectively, all controls u(-) satisfyy). Provide an example. Consider the control system = (u(t)A
q- B ) x l ( t ) -f- h u ( t )
+ d(x2(t))
,
(1.2)
(1.3)
z 2 ( t ) = g ( z 2 ( t ) ) u ( t);
here x 1 ~ R '*~ x 2 ~ R"~; n 1 q- n 2 = n; x = (Xl, 2;2): A and B are n 1 x ~'zlmatrixes; h ~ R'*~; d(.) is a continuous function froln R "~ to R " ; flmction 9('): R "'~ ~ R'"- is Lipschitz continuous. It is assumed t h a t matrixes A and B 2 commute: A B = BA. T h e n the representation (1.1) holds with
*(y)
=exp(Az+
B):, .... + E ,=(I ( i +
Aiexp( B ) h , 1)!
and ¢P(a, z 0) having, generally, the nonstationary form ~ ( t , z, z0): oo
¢P(t, z, z0) = • i= 0
zi+ 1
-( i +- 1)' A' - e x p ( B ( 1 - t)) Bh
+ exp( A z + B(1 - t ) ) d ( m 2 ( z , ) - z, '%.2)). Here ( Xo. 1, Zo, 2) = xo, and z2(- , xll 2) is the solution to the C a u c h y problem ~(t) = g(s~(t)), ~(0) = Xo,2. This result can be deduced from the trajectory representation given in [1]; one can also verify it explicitly. The formula for ¢P(t, z, z o) takes the stationary form oc
Zi+ l - A~exp( A z ) d( z2( z 0 - z, z~,.2)) ~=o ( i + 1)!
140
V. HEYMANN AND A. KRYAZHIMSKII
provided we have B 2 h = 0, and Bd( x 2) = 0 ( z 2 ~ R"2). Assuming these two conditions we obtain the representation (1.1). Note t h a t (1.2), (1.3) covers, partially, the i m p o r t a n t class of bilinear systems; for a description of their implementations, see the m o n o g r a p h [2]. T h e structure of the reachable sets for bilinear systems was studied in [3]. Our parametrization s t a t e m e n t is relevant to the question on characterization of bang-bang reachable sets for nonlinear control systems (see [4]). Let us give several c o m m e n t s on the above formulated problem of parametrization. Note, first, t h a t the method proposed below results in a constructive solution for any abstract continuous mappings q)(.), T(.); t h a t is why we do not restrict ourselves to the control systems only. Behind the problem formulation, lies the idea of dimension reduction. By definition, the attainability set H is the image, under the m a p p i n g F*(.), of the infinitedimensional (functional) set Z. A parametrization (k, S, z('l ")) of the attainability set represents H as the image of the k-dimensional set S. Besides it gives a rule to reconstruct a z(') ~ Z satisfying F*(z(-)) = x 1, given an x 1 ~ H; the rule is, obviously, as follows: find s ~ S such t h a t F * ( z ( - I s ) = z 1, and put z(-) = z(. Is). This rule, in case H is the attainability set for a control system of the above type, automatically provides us with a control u(.) bringing the system from x 0 to x,; obviously, u ( t ) = - $(t). As a corollary, we obtain a finite-dimensional reduction of (infinitedimensional) optimal control problems with terminal functionals. Namely, the problem of finding a control minimizing to(z(1)) (to(.) is a continuous scalar function) is reduced to the finite-dimensional problem of nfinimizing the superposition ~o(F*( z(" I s))) in s E S; a solution s() of the last problem produces an optimal control u0(.)by uo(t) = - ~ 0 ( t ) , where z 0 ( . ) = z(" Is0). Note t h a t by definition each parametrization works for all mappings (I)(-), q~(-). All above applies to the parametrizations of the y-attainability sets H ( y ) , as well. We solve the problem for the case a < 0 < b. In what follows, these inequalities are assumed to be satisfied. 2.
PARAMETRIZATION
DESIGN
Below, Z I [ t 1, t 2 ] stands for the set of all restrictions of functions from Z to interval [tl, t2] c [0, 1]. Let S be the set of all finite sequences S=(~,TI,~],...,Tk,~k),
~
[a,b], 0~< T1 ~< ~1 < "'" ~< T k < ~k~< 1.
(2.1)
Attainability Sets 141 For such s, let z:[.I s] (respectively, zz[.I s]) be the z(.) ~ Z l[rl, ~.] defined as follows: z(t) = ~ for t ~ [Ta, ~:], k(t) = 0 for t ~ ] ~ , ~:i[, and +(t) = - b (respectively, ~(t) = - a ) for t ~ ] ~ , r~+l[. For every y ~ [0, b] (respectively, y ~ [a, 0]) and s (2.1), define z(. I s; y) to be the continuous extension of z:[.I s] (respectively, zz[.[ s]) to [0, 1] such that P.(t) = - a (respectively, ~(t) = - b) for t ~]0, ~':[ and t ~]~k, 1[. Denote by S ( y ) the set of all s ~ S such thut z(-] s; y) ~ Z(y).
REMARK 1. Clearly, s (2.1) belongs to S ( y ) for y ~ [0, b] if and only if = y - a~-1 and (, -- a(1 - ~'i), where
(2.2) the similar criterion holds for y ~ [ a, 0]. One can easily calculate t h a t for s (2.1) fi'om S(y), one has F ( z ( - I s; y)) = fo(s, y) + f : ( s , y) + f ( s , y),
(2.3)
where; for y ~ [0, b],
afo L( s, y) =
I Jo~Y¢( z' y) dz,
f( s, V) = --~i f r~ko ( z ,
~< Y,
z, y)dz,
~k < 0
z, y) dz,
gk > O,
y) dz + E ( ~ - ~)¢(~, y); i=2
for brevity, we omit the (symmetrical) expressions for f0(s, y), fl(s, y), and jr(s, y) for the case y ~ [ a, 0].
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V. HEYMANN AND A. KRYAZHIMSKII
Let So(y) be the set of all s (2.1) from S ( y ) with k = n + 3. Our main results is as follows.
THEOREM 1. For every y ~ [ a, b], the Z-map ( n + 3, So(y), z(. I s; y)) is a parametrization of the y-attainability set. Specifying Theorem 1 with the help of (2.3), we obtain an explicit representation of the y-attainability set /4(y).
COROLLARY 1. Let y ~ [a, b]. A point x ~ R ~ belongs to the yattainability set H ( y ) if and only if there is an s ~ So(y) such that
fo(s, y) + f l ( s ,
y) + f ( s ,
y) + ~ ( y )
=x;
(2.4)
for every above s, we have F*(z(. I s; y)) = x. Theorem 1, Remark 1, and Corollary 1 yield
COROLLARY 2. The Z-map [a, b ] , s ~ S(y)}, z ( ' l ( y , s ) ) attainability set. A point x ~ only if there is a ( y, s) ~ S o F * ( z ( ' l s; y)) = x,
( n + 4, S(~, z(" I ")), where S o = {( y, s): y = z('l s; y), is a parametrization of the R" belongs to the attainability set H if and satisfying (2.4); for every above s, we have
REMARK 2. Corollary 1 implies t h a t for every control system of the type described in Section 1, each state reachable at time 1 from a prescribed initial state can be attained through a bang-bang control with no more than n + 3 switchings.
PROOF OF THEOREM 1. Fix mappings q)(-), ~ ( - ) (see Section 1), a y~ [a,b]anda z * ( - ) ~ Z ( y ) . L e t Zo(y) b e t h e s e t of all z , ( . ) ~ Z s u c h that ~ , ( t ) ~ { - a , 0, - b } for all t ~ [0, 1], with the exception of a finite number of t's. Clearly, for every s > 0 there is a z , ( . ) ~ Z0(y) such t h a t IF(z*(-)) - F ( z , ( - ) ) l < s. Fix a z , ( - ) ~ Z0(y). It is sufficient to show t h a t
Attainability Sets
143
there exists an s o • S o ( y ) such that qF(z('l So; y)) - F ( z , ( . ) ) ] < s. This fact is obviously ensured by the two lemmas:
LEMMA 1.
There exists an s • S ( y ) such that f o r z(.) = z(. I s; y) one
has
Ir(z(.)) - r( z,(.))l <
(2.5)
LEMMA 2. L e t s • S ( y ) and z(" ) = z(" J s; y). There exists an s o • S o ( y ) such that f o r z0(-) = z(" I So; y) one has
r(z0(.)) = r(z(.)).
(2.6)
PROOF OF LEMMA 1. Introduce several notations for a z(') • Z0(y). Let N(z(.)) be the set of all t • ]0, 1[, where ~(t) does not exist, united with {0} and {1}. Denote by Int(z(.) I do), where d o c [0, 1], the set of all closed intervals [tl, t 2] c d o such that tl, t 2 • N(z(.)), t I > t2, and ~(.) is constant on ]tl, t2[. Denote by t .... (z(.)) (respectively, train(Z('))) the minimum of all t where z(.) takes its m a x i m u m value (respectively, the m a x i m u m of all t where z(.) takes its minimum value. The set of all [tl, t 2] • Int(z(.) I [0, t ..... (z(.))]) (respectively, all It1, t 2] • I n t ( z ( ' ) l [ t m i n ( Z ( ' ) ) , 1])) will be denoted by Int (z(.))(respectively, by Int+(z(.))). Let y >/ 0 (the case y < 0 is treated similarly). We make use of the following fact deduced easily from the definition of the set S( y): for a z(.) • Z0(y) one has z(') = z(. I s; y) with s • S ( y ) if and only if (i) (ii) (iii) (iv)
tm~x(Z(')) ~< tram(Z(')), Z(') is decreasing on [ t .... (z(-)), tmin( Z('))], IInt_(z('))] ~< 1, IInt+(z('))l ~< 1.
Here and in what follows, [ El stands for the number of elements of a finite set E. We define a transformation T 1 which associates to an arbitrary ~(') • Z0(y) a z(.) • Z0(y) which is " n e a r e r " to the class of functions described by (i) and (iii), and, besides, satisfies
= r(;(.)).
(2.7)
144
V. HEYMANN AND A. KRYAZHIMSKII
Repeating this transformation starting with 5(') = z . ( . ) , we build a z(.) satisfying (i), (iii) and (2.5) (which takes the form of the equality, with s replaced by 0). In a similar manner, using other transformations, T 2 and T3, we find a z(.) satisfying (i)-(iv) and (2.5). Define z(') = T 1 5(') for 5(') ~ Z0(Y) as follows. Let t 1 = t ..... (5(')). If IInt_(5('))l ~< 1, put z(') = 5 ( ' ) - L e t us have [ sc,~, S~l] ~ Int_(5(-)), sc, < t 1. Assume, with no loss of generality, that [~:,, t 11 ~ Int(5(-) I[0, tl]), ~(t) = - a > 0 for t ~]SCl, t,[, and ~(t) ~ {0, - b } for t ~]sc,), SOl[. Then there is a ~: ~ [s~,, t 1] such that 5(~,,)
= 5(¢)
= c.
(2.8)
Let c > 0. Since 5 ( t 1) >/ c and 5(1) = 0, one can find a r ~ [tl, 1] such t h a t 5 ( r ) = c. Define z(.) by p e r m u t i n g the graphics of 5(') on [~:0, ~:] and [ ~, r]:
z(t) = 5 ( t +
~ - ~o)
(t~
[~o,~o + r [~o + r -
z(t)
=5(t-r+~)
(te
z(t)
= 5(t)
( t ff [~o, r ] ) .
~]),
~,r]),
(2.9) (2.10) (2.11)
Obviously, we have z ( . ) ~ Z0(y); also t 0 = t ......( z ( ' ) ) = so0 + t 1 - sc, and ~(t) = - a for t ~1~0, t0[- Consequently, Int (z(-)) = I n t ( 5 ( ' ) ) \ {[ ~:0, ~11}, and
lint ( z ( ' ) ) l < lint ( z ( ' ) ) l - 1.
(2.12)
By the definition of r, we have 5 ( r ) >~ 5(1) = 0 yielding r ~< tnlin(5(')). The latter, together with (2.11), imply t h a t %re(z(')) = t,,i,,(5(.)) and Int+(z(.))
= Int+(5(-)).
Due to (2.9) and (2.10),
f£°+~-~dP(z(t), y) dt= f ~ ( C(t), y) dt, f~
¢P(z(t), y) dt = frO(~(t), y) dt,
which, together with (2.11), yield (2.7).
(2.13)
Attainability Sets
145
Let c < 0 (see (2.8)). So far as {(0) = y >~ 0, there is a st0o ~ [0, se0] such that ~(seo) = 0. Furthermore, since ~(t 1) /> 0 and ~'(1) = 0, one can find st, ~ [st, tl] , 7 ~ [tl,1] such that {(~) = ~'(~:,) = 0. Define z(.) by permuting the graphics of ~'(') on [st0o, st,] and [se,, r], i.e., by (2.9)-(2.11), where sto, st are replaced by se0o, se,. Then z(-) satisfies, as above, (2.12), (2.13), and (2.7). We have defined the transformation T 1 and shown that for any g'(') Zo(y), the function z(-) = T 1 ~'(-) satisfies (2.12) (whenever lint ({('))l > 1), (2.13), and (2.7). Hence zl(-)= Tl"Z,(.), where m = lint (z,('))l, satisfies the conditions lint ( zl(-))] 4 1,
F( 2:1(")) = F( z• ( ' ) ) .
(2.14)
Similarly, the transformation T 2 (intended to meet (iv)) is defined. For z(-) = Ts~'(-), we have [Int+(z(.))[ ~< [ I n t + ( z ( ' ) ) ] - 1 (whenever Ilnt+(~'('))[ > 1) and the analogue of (2.13): Int (z(')) = Int (~'(-)). Let zs(') = TZ~Zl(-), where m = IInt+(zl(.))l. Taking into account (2.14), we get IInt_(zs(.))l ~< 1,
IInt+(zs(.))] ~ 1,
r(z:(.)) = r(
(.)).
(2.15) (2.16)
Note that (2.15) implies 71 ~ 72, where 71 = t ...... ( Z 2 ( " ) ) ,
72 = tinin( Z2( "))"
Therefore, for z(') = zs('), conditions (i), (iii), and (iv) are satisfied. Now let Z , be the set of all z(-) ~ Z(y) satisfying (i), (iii), and (iv), such that t.... (Z('))
= 71,
train(Z('))
= 72"
(2.17)
Introduce transformation Ts of Z , such that z(.) = Ta {(.) is "closer" than {(-) to the class of functions satisfying (ii) (i.e., decreasing on [rh, 72]), and the inequality
IF(4-))
- r ( C(-))l
s
(2.is)
holds for a sufficiently small 8. We need some notation. For ~'(-) ~ Z , , denote by st(st(-)) the maximum of t ~ [rh, rhl such that ~'(-) is decreasing on [71, t] and by B(~'(-) I st ) the
146
V. HEYMANN AND A. KRYAZHIMSKII
set of all intervals [tl, t 2] ~ I n t ( 5 ( ' ) t E]tl, t2[. P u t also A ( 5 ( ' ) ] ~:) = B ( ~ ( ' ) I sc) is nonempty. For ~'(.) ~ intervals [vii, t2] c [vii, vi2] such that
I]~:, vi2]) such that ~(t) = - b < 0 for rain{t2 - tl:[tx, t2] ~ B ( 5 ( ' ) I ~:)} if Z . , denote by I(5(')) the set of all the following conditions (v), (vi) hold:
(v) for certain ~:1 E]VIl, t2[ and ~2 ~]sci, t2[, function 5(') is decreasing on each of the intervals [vii, ~:1], [ ~ , t2], is increasing on [~1, so2], and satisfies
5(~2) > 5(t2) ~ 5(~1);
(2.19)
(vi) either t 2 ~ N(5(')), or 5,~(t2) = 5(~1); note that ~ = ~(5(.))
(2.20)
and B ( 5 ( - ) I vii) is nonempty provided I ( 5 ( 0 ) is nonempty. It is clear that if 5(') nonempty. For ~'(.) ~ Z . , define z(.) = 5('). Let it not be like in (v), (vi). Take sc*
~ Z . is not decreasing on [vii, vi2], then I(~(.)) is z(-) = T35(-). If 5(') is decreasing on [vii, vi2], put so. Let [vii, t2] ~ / ( 5 ( ' ) ) and ~:1, ~:2 be determined ~ [vii, ~:l] and z* ~ [~:1, ~:2] so that 5 ( s¢*) = 5 ( z * ) = 5 ( t 2 )
(2.21)
(they exist due to (2.19) and the inequality 5(vii) >~ 5(t2)); if 5(t2) = 5(sc~), put sc* = z * = ~ . Define 5 * ( 0 by permuting the graphics of 5(') on
[SOl*, r*] and ['r~', t21: 5"(t)
= 5( t + ~-7 - ~:~*)
(re [~*, ~:~* + t~- ~-*]), (2.22)
5*(t)
= 5(t-
(t e [ ~,* + t~ - ~-*, t~]),
(2.23)
5"(t)
= 5(t)
(t ~ [ ~:a*, t2]).
(2.24)
t~ + ~-*)
Make several observations on 5"(')- From (2.22)-(2.24), the equality V(5*(.))
= V(5(-))
(2.25)
is easily deduced. Note that for ~ = ~:2 - ~'* + ~:1 and t~' = t 2 - ~-* + ~:l*, we have ~:*, t* ~ [sc*, sc* + t 2 - z*]. Hence in view of (2.21), (2.22),
Attainability Sets ~" ~:2 ) = ~'(~:z), ~'*(~:*) = ~'(r*) = ~'(t~) = ~" ( t 2 ); therefore (2.19)) it holds t h a t
t'*(n~) >/t'*(~::) > t'*(t~') = t'*(~*),
(due
147 to
(2.26)
and, besides, ~" *(-) is decreasing on [7~1, ~:t ] and [ s ~ , t~' ] and increasing on [~:*, ~:*]. F u r t h e r m o r e , if t 2 ~ N(~'(-)) (see (vi)), p u t t*3 = ~:1 + t2 - r*; then (see (2.23)) ~(t)
~< 0
( a l m o s t all t ~ ]t~, t~[),
(2.27)
~(t)
~ 0
( a l m o s t all t ~ ]t~, tgD,
(2.28)
t:T - ~1 >I t2 - r * + ~:* - ~:1 >~ t2 - s¢2 >~ A ( ~ ' [ ~ : ( ' ) ) . F r o m (2.27), (2.28), and (2.24), we o b t a i n
B(~*(')lt~)
c B(~'*(.)
I t2) c .B(~*(.)
[ ~1 ).
(2.29)
If t 2 ~ N(~'(-)), then by (vi) ~(t 2) = ~'(~:1) i m p l y i n g s~* = r* and (see (2.23), (2.24)) ~'*(.) = ~'(-). In this case we p u t t~ = m i n N ( g ( . ) ) n [t2, 1] and o b t a i n the inequalities t* - ~:~ > t~ - ~:9 > A(£'(.) I S~l), the equality t* = t2, and the inclusions (2.29). Thus, in b o t h cases we have the inequalities
t~ - ~, >/A(C(.)
<),
(2.30)
(2.27), and (2.29). W e pass to the final stage of our definition of z(') = T a ~'('). Fix a 6 > 0. C h o o s e % = s~ ' < , . . . , < %+1 = t'e so t h a t t~+ 1 - r i = A, and ~:
f ' : ¢,( c * ( t ) , v) dt - E ¢( ~?,
y)±
< ,s,
(2.31)
i= 0
) , ~" * ( x22* ) ] c where ~'i* = ~" *(ri)" In view of (2.26), we have ~'~* ~ [g" * ( *SOl [ ( *(~1" ), ~ *0ql)]. R e o r d e r points ~'i* ( w i t h o u t using new notations) so t h a t ~'i* > ~'i* 1- Since sr *(') is decreasing on [7/1, s~* ], there are t% > "'" > IXk b e k m g i n g to [rtl, s~l*] such t h a t g *(/x,) = ~'**. Define z(') replacing points (/xi, ~'i*) on the graphic of r *(.) b y tile s e g m e n t s {(t, ~'i*): v~ ~< t ~< v~ + A},
V. HEYMANN AND A. KRYAZHIMSKII
148
where v i = txi + (i - 1)A. More accurately, we put
z( t) = C*( t)
(t E [711 , Vl]),
z(t)
(t
=
z ( t ) = ~ * ( t - iA)
+ a]),
( t E [Pi -~ A, "i+1])
with uk+ 1 = t*2, and
z( t) = ~*( t)
(2.32)
( t ~ [771, t ~ ] ) .
Function z(-) is defined. W e have
(
alP( z(t), y) dt=
s" + f:2-¢P(C*(t),
k
alP( C *(t), y) dt + E
rh
kA
':
i=0
s,< alp( C*(t), ~£i+1
y) dt
,
k
y) dt+ ~_, ¢P(~*, y)A. (2.33) i=0
So far as kA = t 2 - ~*, the sum of the first three items on the right of (2.33) is the integral of ¢P(~('), y) over [711, ~=1"]. Thus by virtue of (2.31)
This, together with (2.32) and (2.25), provides (2.18). Furthermore, by the definition of z(.), this function is decreasing on [~71, t~ ]. Then by (2.27) and (2.32) we have so(z(.)) >~ t~ ; hence in view of (2.30) and (2.20)
~(z(.)) - ~(~(-)) >/A(~'(.) I ~:(~'(.))) >/h(~'(') I ~1). (2.34) Using (2.32) and (2.29), we get
B( z( . ) l ~ ( z( . ) ) ) c B( z( . ) l t~ ) c B ( z ( - ) l ~ : l )
cB(z(-)l-ql)
implying
A ( Z ( ' ) I ~=( Z ( " ) ) ) ~ /~(Z(') I 'g~l)"
(2.35)
Attainability Sets 149 Consequently, if ~ ' ( . ) ~ Z , is not decreasing o n [771 , g~2], then z ( . ) = T 3 ~'(') ~ Z , satisfies (2.18), (2.34), and (2.35). Pass to the final stage in proving the inequality (2.5). Let z2(-) be not decreasing on [~71, ~72]. Denoting z(°)(.) = z2(.) , z(i+l)(.) = Ttz(i)(.) and assuming A = A( z 2 [ ~1 ), w e get
IF(z(~+l'('))
~ : ( z ( ' + " ( ' ) ) > ~:(z(°(.)) + A,
-
r(~")(.))l <
provided z(°(') is not decreasing on [~71, ~72]- Take m >t A -1 and put z(.) = z(m)(.). Then z(-) is decreasing on [~?a, 7/2], i.e., satisfies (ii), and
IF(4))
- r ( ~ : ( . ) ) l < .~a;
recall that z(.) satisfies (i), (iii), and (iv) by the definition of Z . . Taking into account (2.16) and letting 6 < e / m , we obtain the inequality (2.5). If z2(.) is decreasing on [r/l, r/2], then we have the same with m = 1. Lemma 1 is proved.
PROOF OF LEMMA 2. Let z(') = z(-I s; y), where s ~ 5"(Y) has the form (2.1). With no loss of generality assume k >_- n + 3. Consider the case Y > 0 ( Y < 0 is treated identically). We have z(t) = z i for t c= [%, ~i], ~,(t) = - b for t ~]~:,, ri+l], and z I = ~'. P u t also z~ = ~'~ where ~', is defined by (2.2). The following relations hold:
r(z(.)) = f)oTl ¢P(~'*(t), y) dt +
f l Op( { *( t), 8) dt + A + IX, (2.36)
where k
i=1
,
k
k
= E ¢(z~, y)(~,-
~)=¢
i=1
E ¢(~,
81¢~,
(2.37/
i--1
k
13= Y'~ (s~ - ri) ,
/3~=(s~i - r~)//3,
(2.38)
i=1
= Ek fff+, ,t,( z ( t ) , 8/dt = - -1,, ~ i=1
_
u
1 f(dp( z, y) dz.
i=1
f < * ( z , y) dz Zi+l
(2.39)
150
V. HEYMANN AND A. KRYAZHIMSKII
Note t h a t E~=l /3i = 1 and z i e [~'k, ~']. T h e n by the t h e o r e m of C a r a t h e o d o r y there exist Zo, 2 . . . . . Zo,,, e [~'k, ~'], zo,, > zo,,+l with m = n + 2, such t h a t E CO( z~, y)/3~ = * ( Zo.,, y)/30 ~ i=l i=1 with certain /3o, i >/0,
Ei"k i
(2.40)
rio, ~ = 1. T a k e an
80 = ( ~ ' , T 0 . ] , a 0 , , . . . .
,'T0, m + l ,
~o,,,+t) e S
(2.41)
such that, first, s o e So(y), second ~0,1 = TO, ] = T I '
(2.42)
s~o, ,,+, = to, ,,,+~ = sck,
(2.43)
i)/J ~
(2.44)
J~0, i = ( ~0, i -- T0,
(2 ~< i < m ) ,
~'o, ,,, +, = {k,
(2.45)
~'0, m+l ~---~ - - b ~ (To, i+ 1 - ~o,i), i=1
(2.46)
where
and, third, Zo(-) - z(. I So; y) satisfies Zo(ro,1) = ~,
zo(s¢o,,~+,) = ~o,,n+],
z o ( t ) = ~'o,i
( t e [%, i, ~:o, i], 2 ~< i ~< m)
(2.47) (2.48)
(the existence of s o is justified below). F r o m (2.40), (2.44), we deduce (like in
(2.37)) rn + 2
* =
y)dt;
E i=1
"To, i
f r o m (2.42), (2.48), (2.46), w e get ( l i k e i n (2.39)) m+
]
i~ J~r°"+' [o, ( Zo(t), y) dt.
Attainability Sets
151
These equalities together with (2.39), yield the desired equality (2.6), and finalize the proof. It remains to show the existence of element s o. Assume (2.42), (2.44), and put Zo, m + l = ~k, (2.49)
%.2 = G o , 1 - ( Zo,2 - ~ ) / b , T0, i+ 1 = ~0,1 -- (Zo, i+l -- Zo, i ) / b
(2 < i •
m),
(2.50)
(2.51)
G0, m + l = TO, re+l"
Then (to, i+, - Go,,) = ( ( -
~k)/b,
(2.52)
i=1
which is equivalent to (2.45) (see (2.46)). S u m m i n g up (2.51) and G0, ~ - ~0, (2 4 i ~< m), and using (2.44) and (2.42), we obtain
the last equality follows from (2.38) and (2.2). Thus, (2.43) is also satisfied. Introduce s o (2.41). T h e equalities (2.45), (2.42), (2.43), and the inclusion s ~ So(y) imply s o ~ So(y) by R e m a r k 1. Finally, (2.48) and (2.47) for z0(') = z('l so; y) follow from the definition of z0(.) and (2.49)-(2.52). L e m m a is proved. REFERENCES 1 R . W . Brockett, System theory on group manifolds and coset spaces, SIAM J. Control 10:265-284 (1972). 2 R. R. Mohler, Bilinear Control Processes, with Appliances to Engineering, Ecology, and Medicine, Academic Press, New York, 1973, p. 224. 3 R.W. Brockett, On the reachable set for bilinear systems, in Variable Structure Systems with Application to Economics and Biology (A. J. Krener, Ed.), Lecture Notes in Econom. and Mathem. Systems, Vol. 3, Springer, Berlin, 1975. 4 A.A. Agrachev and S. A. Vakhrameev, Nonlinear control systems of constant rank and bang-bang conditions for extremal controls, Soviet Math. Dokl. 30:620-624 (1984).
On Finite-Dimensional Parametrizations of Attainability Sets Valerii I. H e y m a n n
Institute of Mathematics and Mechanics Urals Branch, A cad. Sci. of Russia Kovalevskoi 16 620219 Ekaterinburg, Russia and A r k a d i i V. K r y a z h i m s k i i
International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria Transmitted by F. E. Udwadia
ABSTRACT The attainability set is defined to be a finite-dimensional integral-type image of tile set of all absolutely continuous scalar functions of time whose derivatives take wdues in a given interval. For a class of control systems with scalar controls restricted to the above interval (the class comprises, in particular, some bilinear systems), the attainability set has the traditional meaning. A method of finitedimensional paramctrization of the attainability set is described. The parametrization is universal, i.e., the same for all attainability sets of a fixed dimension. For the case of control systems, the result provides an upper estimate on the number of switchings sufficient to bring the system to an arbitrary reachable state at a prescribed time.
1.
PROBLEM
STATEMENT:
MOTIVATIONS
We start with the accurate problem formulation. F i x reals a a n d b > a. D e n o t e by Z t h e set of all a b s o l u t e l y c o n t i n u o u s f u n c t i o n s z ( ' ) : [ 0 , 1] ~-) [a, b] s u c h t h a t z(1) = 0 a n d - ~ ( t ) ~ [a, b] for
APPLIED MA THEMATICS AND COMPUTATION 78:137-151 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0096-3003/96/$15.00 PII S0096-3003(96)00004-X
138
V. HEYMANN AND A. KRYAZHIMSKII
almost all t ~ [ 0 , 1 ] . For every y ~ [ a , b] denote by Z(y) the set of all z(') ~ Z such that z(0) = y. Fix a natural n. Take continuous mappings dp(.): [ a, b] × [ a, b] ~ R '~, ~ ( ' ) : [ a, b] ~ R'~, and define functions F(.) and F*(.) on Z by
F(z(.)) = ~ l ~ ( z( t), z(O)) dt,
r*(4))
= *(40))
+ r(z(.)).
Put H = { F * ( z ( - ) ) : z ( ' ) ~ Z},
H ( y ) = { F * ( z ( ' ) ) : z ( ' ) ~ Z( y)} (y~
[a,b]).
Call H the attainability set and H ( y ) the y-attainability set (for the mappings O(.), ~(.)). Define a Z-map to be a triple (k, S, z('l ")), where k is a natural number, S is a closed and bounded subset of R ~, and z('l "): s ~ z ( ' l s ) is a continuous mapping of S into Z; continuity is understood with respect to the sup-norIn in Z. Call an above Z-map a parametrization of the attainability set (respectively, a parametrization of the y-attainability set) if the set F*(z(-I s)): s ~ S} coincides with H (respectively, coincides with /4(y)) for arbitrary continuous mappings O ( ' ) , T ( - ) . The problem in question consists in building constructive parametrizations of the attainability and y-attainability sets. The next remark says that a parametrization of the attainability set could be found through a family of parametrizations of y-attainability sets.
REMARK 1. Let for every y ~ [ - b , - a ] , a Z - m a p ( k , S(y), z('l "; y ) ) b e a parametrization of the y-attainability set. Then (k + 1, S, z(-t ")), where S-={(y,s): y~[-b,-a],s~ S(y)}, and z ( ' l ( y , s ) ) = z(-I s; y), is a parametrization of the attainability set. The above problem has a strong link to the theory of control. Recall that for a control system, the attainability set is understood as the set of all states that could be reached by the system at a prescribed time, starting from a given initial state. For some classes of n-dimensional control systems, the sets H and H ( y ) have precisely this meaning. These are the systems operating on the time interval [0, 1], having scalar, Lebesgue measurable controls u(') with values in [ a, b] and characterized by the following: there
Attainability Sets
139
are mappings q~(.), ~ ( - ) such that, for the final point x(1) of system's trajectory z(.) developing from z(0) = z 0 under control ~L(.), one has x(1) = F * ( z ( . ) ) , where z ( t ) = f / is the set of all state x(0) = x 0 ing f(l u ( t ) d t = x(t)
(1.1)
u ( r ) d r . For a system of t h a t kind, H (respectively, H ( y ) ) states z(1) attainable by the system at time 1 from initial using all controls u(.) (respectively, all controls u(-) satisfyy). Provide an example. Consider the control system = (u(t)A
q- B ) x l ( t ) -f- h u ( t )
+ d(x2(t))
,
(1.2)
(1.3)
z 2 ( t ) = g ( z 2 ( t ) ) u ( t);
here x 1 ~ R '*~ x 2 ~ R"~; n 1 q- n 2 = n; x = (Xl, 2;2): A and B are n 1 x ~'zlmatrixes; h ~ R'*~; d(.) is a continuous function froln R "~ to R " ; flmction 9('): R "'~ ~ R'"- is Lipschitz continuous. It is assumed t h a t matrixes A and B 2 commute: A B = BA. T h e n the representation (1.1) holds with
*(y)
=exp(Az+
B):, .... + E ,=(I ( i +
Aiexp( B ) h , 1)!
and ¢P(a, z 0) having, generally, the nonstationary form ~ ( t , z, z0): oo
¢P(t, z, z0) = • i= 0
zi+ 1
-( i +- 1)' A' - e x p ( B ( 1 - t)) Bh
+ exp( A z + B(1 - t ) ) d ( m 2 ( z , ) - z, '%.2)). Here ( Xo. 1, Zo, 2) = xo, and z2(- , xll 2) is the solution to the C a u c h y problem ~(t) = g(s~(t)), ~(0) = Xo,2. This result can be deduced from the trajectory representation given in [1]; one can also verify it explicitly. The formula for ¢P(t, z, z o) takes the stationary form oc
Zi+ l - A~exp( A z ) d( z2( z 0 - z, z~,.2)) ~=o ( i + 1)!
140
V. HEYMANN AND A. KRYAZHIMSKII
provided we have B 2 h = 0, and Bd( x 2) = 0 ( z 2 ~ R"2). Assuming these two conditions we obtain the representation (1.1). Note t h a t (1.2), (1.3) covers, partially, the i m p o r t a n t class of bilinear systems; for a description of their implementations, see the m o n o g r a p h [2]. T h e structure of the reachable sets for bilinear systems was studied in [3]. Our parametrization s t a t e m e n t is relevant to the question on characterization of bang-bang reachable sets for nonlinear control systems (see [4]). Let us give several c o m m e n t s on the above formulated problem of parametrization. Note, first, t h a t the method proposed below results in a constructive solution for any abstract continuous mappings q)(.), T(.); t h a t is why we do not restrict ourselves to the control systems only. Behind the problem formulation, lies the idea of dimension reduction. By definition, the attainability set H is the image, under the m a p p i n g F*(.), of the infinitedimensional (functional) set Z. A parametrization (k, S, z('l ")) of the attainability set represents H as the image of the k-dimensional set S. Besides it gives a rule to reconstruct a z(') ~ Z satisfying F*(z(-)) = x 1, given an x 1 ~ H; the rule is, obviously, as follows: find s ~ S such t h a t F * ( z ( - I s ) = z 1, and put z(-) = z(. Is). This rule, in case H is the attainability set for a control system of the above type, automatically provides us with a control u(.) bringing the system from x 0 to x,; obviously, u ( t ) = - $(t). As a corollary, we obtain a finite-dimensional reduction of (infinitedimensional) optimal control problems with terminal functionals. Namely, the problem of finding a control minimizing to(z(1)) (to(.) is a continuous scalar function) is reduced to the finite-dimensional problem of nfinimizing the superposition ~o(F*( z(" I s))) in s E S; a solution s() of the last problem produces an optimal control u0(.)by uo(t) = - ~ 0 ( t ) , where z 0 ( . ) = z(" Is0). Note t h a t by definition each parametrization works for all mappings (I)(-), q~(-). All above applies to the parametrizations of the y-attainability sets H ( y ) , as well. We solve the problem for the case a < 0 < b. In what follows, these inequalities are assumed to be satisfied. 2.
PARAMETRIZATION
DESIGN
Below, Z I [ t 1, t 2 ] stands for the set of all restrictions of functions from Z to interval [tl, t2] c [0, 1]. Let S be the set of all finite sequences S=(~,TI,~],...,Tk,~k),
~
[a,b], 0~< T1 ~< ~1 < "'" ~< T k < ~k~< 1.
(2.1)
Attainability Sets 141 For such s, let z:[.I s] (respectively, zz[.I s]) be the z(.) ~ Z l[rl, ~.] defined as follows: z(t) = ~ for t ~ [Ta, ~:], k(t) = 0 for t ~ ] ~ , ~:i[, and +(t) = - b (respectively, ~(t) = - a ) for t ~ ] ~ , r~+l[. For every y ~ [0, b] (respectively, y ~ [a, 0]) and s (2.1), define z(. I s; y) to be the continuous extension of z:[.I s] (respectively, zz[.[ s]) to [0, 1] such that P.(t) = - a (respectively, ~(t) = - b) for t ~]0, ~':[ and t ~]~k, 1[. Denote by S ( y ) the set of all s ~ S such thut z(-] s; y) ~ Z(y).
REMARK 1. Clearly, s (2.1) belongs to S ( y ) for y ~ [0, b] if and only if = y - a~-1 and (, -- a(1 - ~'i), where
(2.2) the similar criterion holds for y ~ [ a, 0]. One can easily calculate t h a t for s (2.1) fi'om S(y), one has F ( z ( - I s; y)) = fo(s, y) + f : ( s , y) + f ( s , y),
(2.3)
where; for y ~ [0, b],
afo L( s, y) =
I Jo~Y¢( z' y) dz,
f( s, V) = --~i f r~ko ( z ,
~< Y,
z, y)dz,
~k < 0
z, y) dz,
gk > O,
y) dz + E ( ~ - ~)¢(~, y); i=2
for brevity, we omit the (symmetrical) expressions for f0(s, y), fl(s, y), and jr(s, y) for the case y ~ [ a, 0].
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V. HEYMANN AND A. KRYAZHIMSKII
Let So(y) be the set of all s (2.1) from S ( y ) with k = n + 3. Our main results is as follows.
THEOREM 1. For every y ~ [ a, b], the Z-map ( n + 3, So(y), z(. I s; y)) is a parametrization of the y-attainability set. Specifying Theorem 1 with the help of (2.3), we obtain an explicit representation of the y-attainability set /4(y).
COROLLARY 1. Let y ~ [a, b]. A point x ~ R ~ belongs to the yattainability set H ( y ) if and only if there is an s ~ So(y) such that
fo(s, y) + f l ( s ,
y) + f ( s ,
y) + ~ ( y )
=x;
(2.4)
for every above s, we have F*(z(. I s; y)) = x. Theorem 1, Remark 1, and Corollary 1 yield
COROLLARY 2. The Z-map [a, b ] , s ~ S(y)}, z ( ' l ( y , s ) ) attainability set. A point x ~ only if there is a ( y, s) ~ S o F * ( z ( ' l s; y)) = x,
( n + 4, S(~, z(" I ")), where S o = {( y, s): y = z('l s; y), is a parametrization of the R" belongs to the attainability set H if and satisfying (2.4); for every above s, we have
REMARK 2. Corollary 1 implies t h a t for every control system of the type described in Section 1, each state reachable at time 1 from a prescribed initial state can be attained through a bang-bang control with no more than n + 3 switchings.
PROOF OF THEOREM 1. Fix mappings q)(-), ~ ( - ) (see Section 1), a y~ [a,b]anda z * ( - ) ~ Z ( y ) . L e t Zo(y) b e t h e s e t of all z , ( . ) ~ Z s u c h that ~ , ( t ) ~ { - a , 0, - b } for all t ~ [0, 1], with the exception of a finite number of t's. Clearly, for every s > 0 there is a z , ( . ) ~ Z0(y) such t h a t IF(z*(-)) - F ( z , ( - ) ) l < s. Fix a z , ( - ) ~ Z0(y). It is sufficient to show t h a t
Attainability Sets
143
there exists an s o • S o ( y ) such that qF(z('l So; y)) - F ( z , ( . ) ) ] < s. This fact is obviously ensured by the two lemmas:
LEMMA 1.
There exists an s • S ( y ) such that f o r z(.) = z(. I s; y) one
has
Ir(z(.)) - r( z,(.))l <
(2.5)
LEMMA 2. L e t s • S ( y ) and z(" ) = z(" J s; y). There exists an s o • S o ( y ) such that f o r z0(-) = z(" I So; y) one has
r(z0(.)) = r(z(.)).
(2.6)
PROOF OF LEMMA 1. Introduce several notations for a z(') • Z0(y). Let N(z(.)) be the set of all t • ]0, 1[, where ~(t) does not exist, united with {0} and {1}. Denote by Int(z(.) I do), where d o c [0, 1], the set of all closed intervals [tl, t 2] c d o such that tl, t 2 • N(z(.)), t I > t2, and ~(.) is constant on ]tl, t2[. Denote by t .... (z(.)) (respectively, train(Z('))) the minimum of all t where z(.) takes its m a x i m u m value (respectively, the m a x i m u m of all t where z(.) takes its minimum value. The set of all [tl, t 2] • Int(z(.) I [0, t ..... (z(.))]) (respectively, all It1, t 2] • I n t ( z ( ' ) l [ t m i n ( Z ( ' ) ) , 1])) will be denoted by Int (z(.))(respectively, by Int+(z(.))). Let y >/ 0 (the case y < 0 is treated similarly). We make use of the following fact deduced easily from the definition of the set S( y): for a z(.) • Z0(y) one has z(') = z(. I s; y) with s • S ( y ) if and only if (i) (ii) (iii) (iv)
tm~x(Z(')) ~< tram(Z(')), Z(') is decreasing on [ t .... (z(-)), tmin( Z('))], IInt_(z('))] ~< 1, IInt+(z('))l ~< 1.
Here and in what follows, [ El stands for the number of elements of a finite set E. We define a transformation T 1 which associates to an arbitrary ~(') • Z0(y) a z(.) • Z0(y) which is " n e a r e r " to the class of functions described by (i) and (iii), and, besides, satisfies
= r(;(.)).
(2.7)
144
V. HEYMANN AND A. KRYAZHIMSKII
Repeating this transformation starting with 5(') = z . ( . ) , we build a z(.) satisfying (i), (iii) and (2.5) (which takes the form of the equality, with s replaced by 0). In a similar manner, using other transformations, T 2 and T3, we find a z(.) satisfying (i)-(iv) and (2.5). Define z(') = T 1 5(') for 5(') ~ Z0(Y) as follows. Let t 1 = t ..... (5(')). If IInt_(5('))l ~< 1, put z(') = 5 ( ' ) - L e t us have [ sc,~, S~l] ~ Int_(5(-)), sc, < t 1. Assume, with no loss of generality, that [~:,, t 11 ~ Int(5(-) I[0, tl]), ~(t) = - a > 0 for t ~]SCl, t,[, and ~(t) ~ {0, - b } for t ~]sc,), SOl[. Then there is a ~: ~ [s~,, t 1] such that 5(~,,)
= 5(¢)
= c.
(2.8)
Let c > 0. Since 5 ( t 1) >/ c and 5(1) = 0, one can find a r ~ [tl, 1] such t h a t 5 ( r ) = c. Define z(.) by p e r m u t i n g the graphics of 5(') on [~:0, ~:] and [ ~, r]:
z(t) = 5 ( t +
~ - ~o)
(t~
[~o,~o + r [~o + r -
z(t)
=5(t-r+~)
(te
z(t)
= 5(t)
( t ff [~o, r ] ) .
~]),
~,r]),
(2.9) (2.10) (2.11)
Obviously, we have z ( . ) ~ Z0(y); also t 0 = t ......( z ( ' ) ) = so0 + t 1 - sc, and ~(t) = - a for t ~1~0, t0[- Consequently, Int (z(-)) = I n t ( 5 ( ' ) ) \ {[ ~:0, ~11}, and
lint ( z ( ' ) ) l < lint ( z ( ' ) ) l - 1.
(2.12)
By the definition of r, we have 5 ( r ) >~ 5(1) = 0 yielding r ~< tnlin(5(')). The latter, together with (2.11), imply t h a t %re(z(')) = t,,i,,(5(.)) and Int+(z(.))
= Int+(5(-)).
Due to (2.9) and (2.10),
f£°+~-~dP(z(t), y) dt= f ~ ( C(t), y) dt, f~
¢P(z(t), y) dt = frO(~(t), y) dt,
which, together with (2.11), yield (2.7).
(2.13)
Attainability Sets
145
Let c < 0 (see (2.8)). So far as {(0) = y >~ 0, there is a st0o ~ [0, se0] such that ~(seo) = 0. Furthermore, since ~(t 1) /> 0 and ~'(1) = 0, one can find st, ~ [st, tl] , 7 ~ [tl,1] such that {(~) = ~'(~:,) = 0. Define z(.) by permuting the graphics of ~'(') on [st0o, st,] and [se,, r], i.e., by (2.9)-(2.11), where sto, st are replaced by se0o, se,. Then z(-) satisfies, as above, (2.12), (2.13), and (2.7). We have defined the transformation T 1 and shown that for any g'(') Zo(y), the function z(-) = T 1 ~'(-) satisfies (2.12) (whenever lint ({('))l > 1), (2.13), and (2.7). Hence zl(-)= Tl"Z,(.), where m = lint (z,('))l, satisfies the conditions lint ( zl(-))] 4 1,
F( 2:1(")) = F( z• ( ' ) ) .
(2.14)
Similarly, the transformation T 2 (intended to meet (iv)) is defined. For z(-) = Ts~'(-), we have [Int+(z(.))[ ~< [ I n t + ( z ( ' ) ) ] - 1 (whenever Ilnt+(~'('))[ > 1) and the analogue of (2.13): Int (z(')) = Int (~'(-)). Let zs(') = TZ~Zl(-), where m = IInt+(zl(.))l. Taking into account (2.14), we get IInt_(zs(.))l ~< 1,
IInt+(zs(.))] ~ 1,
r(z:(.)) = r(
(.)).
(2.15) (2.16)
Note that (2.15) implies 71 ~ 72, where 71 = t ...... ( Z 2 ( " ) ) ,
72 = tinin( Z2( "))"
Therefore, for z(') = zs('), conditions (i), (iii), and (iv) are satisfied. Now let Z , be the set of all z(-) ~ Z(y) satisfying (i), (iii), and (iv), such that t.... (Z('))
= 71,
train(Z('))
= 72"
(2.17)
Introduce transformation Ts of Z , such that z(.) = Ta {(.) is "closer" than {(-) to the class of functions satisfying (ii) (i.e., decreasing on [rh, 72]), and the inequality
IF(4-))
- r ( C(-))l
s
(2.is)
holds for a sufficiently small 8. We need some notation. For ~'(-) ~ Z , , denote by st(st(-)) the maximum of t ~ [rh, rhl such that ~'(-) is decreasing on [71, t] and by B(~'(-) I st ) the
146
V. HEYMANN AND A. KRYAZHIMSKII
set of all intervals [tl, t 2] ~ I n t ( 5 ( ' ) t E]tl, t2[. P u t also A ( 5 ( ' ) ] ~:) = B ( ~ ( ' ) I sc) is nonempty. For ~'(.) ~ intervals [vii, t2] c [vii, vi2] such that
I]~:, vi2]) such that ~(t) = - b < 0 for rain{t2 - tl:[tx, t2] ~ B ( 5 ( ' ) I ~:)} if Z . , denote by I(5(')) the set of all the following conditions (v), (vi) hold:
(v) for certain ~:1 E]VIl, t2[ and ~2 ~]sci, t2[, function 5(') is decreasing on each of the intervals [vii, ~:1], [ ~ , t2], is increasing on [~1, so2], and satisfies
5(~2) > 5(t2) ~ 5(~1);
(2.19)
(vi) either t 2 ~ N(5(')), or 5,~(t2) = 5(~1); note that ~ = ~(5(.))
(2.20)
and B ( 5 ( - ) I vii) is nonempty provided I ( 5 ( 0 ) is nonempty. It is clear that if 5(') nonempty. For ~'(.) ~ Z . , define z(.) = 5('). Let it not be like in (v), (vi). Take sc*
~ Z . is not decreasing on [vii, vi2], then I(~(.)) is z(-) = T35(-). If 5(') is decreasing on [vii, vi2], put so. Let [vii, t2] ~ / ( 5 ( ' ) ) and ~:1, ~:2 be determined ~ [vii, ~:l] and z* ~ [~:1, ~:2] so that 5 ( s¢*) = 5 ( z * ) = 5 ( t 2 )
(2.21)
(they exist due to (2.19) and the inequality 5(vii) >~ 5(t2)); if 5(t2) = 5(sc~), put sc* = z * = ~ . Define 5 * ( 0 by permuting the graphics of 5(') on
[SOl*, r*] and ['r~', t21: 5"(t)
= 5( t + ~-7 - ~:~*)
(re [~*, ~:~* + t~- ~-*]), (2.22)
5*(t)
= 5(t-
(t e [ ~,* + t~ - ~-*, t~]),
(2.23)
5"(t)
= 5(t)
(t ~ [ ~:a*, t2]).
(2.24)
t~ + ~-*)
Make several observations on 5"(')- From (2.22)-(2.24), the equality V(5*(.))
= V(5(-))
(2.25)
is easily deduced. Note that for ~ = ~:2 - ~'* + ~:1 and t~' = t 2 - ~-* + ~:l*, we have ~:*, t* ~ [sc*, sc* + t 2 - z*]. Hence in view of (2.21), (2.22),
Attainability Sets ~" ~:2 ) = ~'(~:z), ~'*(~:*) = ~'(r*) = ~'(t~) = ~" ( t 2 ); therefore (2.19)) it holds t h a t
t'*(n~) >/t'*(~::) > t'*(t~') = t'*(~*),
(due
147 to
(2.26)
and, besides, ~" *(-) is decreasing on [7~1, ~:t ] and [ s ~ , t~' ] and increasing on [~:*, ~:*]. F u r t h e r m o r e , if t 2 ~ N(~'(-)) (see (vi)), p u t t*3 = ~:1 + t2 - r*; then (see (2.23)) ~(t)
~< 0
( a l m o s t all t ~ ]t~, t~[),
(2.27)
~(t)
~ 0
( a l m o s t all t ~ ]t~, tgD,
(2.28)
t:T - ~1 >I t2 - r * + ~:* - ~:1 >~ t2 - s¢2 >~ A ( ~ ' [ ~ : ( ' ) ) . F r o m (2.27), (2.28), and (2.24), we o b t a i n
B(~*(')lt~)
c B(~'*(.)
I t2) c .B(~*(.)
[ ~1 ).
(2.29)
If t 2 ~ N(~'(-)), then by (vi) ~(t 2) = ~'(~:1) i m p l y i n g s~* = r* and (see (2.23), (2.24)) ~'*(.) = ~'(-). In this case we p u t t~ = m i n N ( g ( . ) ) n [t2, 1] and o b t a i n the inequalities t* - ~:~ > t~ - ~:9 > A(£'(.) I S~l), the equality t* = t2, and the inclusions (2.29). Thus, in b o t h cases we have the inequalities
t~ - ~, >/A(C(.)
<),
(2.30)
(2.27), and (2.29). W e pass to the final stage of our definition of z(') = T a ~'('). Fix a 6 > 0. C h o o s e % = s~ ' < , . . . , < %+1 = t'e so t h a t t~+ 1 - r i = A, and ~:
f ' : ¢,( c * ( t ) , v) dt - E ¢( ~?,
y)±
< ,s,
(2.31)
i= 0
) , ~" * ( x22* ) ] c where ~'i* = ~" *(ri)" In view of (2.26), we have ~'~* ~ [g" * ( *SOl [ ( *(~1" ), ~ *0ql)]. R e o r d e r points ~'i* ( w i t h o u t using new notations) so t h a t ~'i* > ~'i* 1- Since sr *(') is decreasing on [7/1, s~* ], there are t% > "'" > IXk b e k m g i n g to [rtl, s~l*] such t h a t g *(/x,) = ~'**. Define z(') replacing points (/xi, ~'i*) on the graphic of r *(.) b y tile s e g m e n t s {(t, ~'i*): v~ ~< t ~< v~ + A},
V. HEYMANN AND A. KRYAZHIMSKII
148
where v i = txi + (i - 1)A. More accurately, we put
z( t) = C*( t)
(t E [711 , Vl]),
z(t)
(t
=
z ( t ) = ~ * ( t - iA)
+ a]),
( t E [Pi -~ A, "i+1])
with uk+ 1 = t*2, and
z( t) = ~*( t)
(2.32)
( t ~ [771, t ~ ] ) .
Function z(-) is defined. W e have
(
alP( z(t), y) dt=
s" + f:2-¢P(C*(t),
k
alP( C *(t), y) dt + E
rh
kA
':
i=0
s,< alp( C*(t), ~£i+1
y) dt
,
k
y) dt+ ~_, ¢P(~*, y)A. (2.33) i=0
So far as kA = t 2 - ~*, the sum of the first three items on the right of (2.33) is the integral of ¢P(~('), y) over [711, ~=1"]. Thus by virtue of (2.31)
This, together with (2.32) and (2.25), provides (2.18). Furthermore, by the definition of z(.), this function is decreasing on [~71, t~ ]. Then by (2.27) and (2.32) we have so(z(.)) >~ t~ ; hence in view of (2.30) and (2.20)
~(z(.)) - ~(~(-)) >/A(~'(.) I ~:(~'(.))) >/h(~'(') I ~1). (2.34) Using (2.32) and (2.29), we get
B( z( . ) l ~ ( z( . ) ) ) c B( z( . ) l t~ ) c B ( z ( - ) l ~ : l )
cB(z(-)l-ql)
implying
A ( Z ( ' ) I ~=( Z ( " ) ) ) ~ /~(Z(') I 'g~l)"
(2.35)
Attainability Sets 149 Consequently, if ~ ' ( . ) ~ Z , is not decreasing o n [771 , g~2], then z ( . ) = T 3 ~'(') ~ Z , satisfies (2.18), (2.34), and (2.35). Pass to the final stage in proving the inequality (2.5). Let z2(-) be not decreasing on [~71, ~72]. Denoting z(°)(.) = z2(.) , z(i+l)(.) = Ttz(i)(.) and assuming A = A( z 2 [ ~1 ), w e get
IF(z(~+l'('))
~ : ( z ( ' + " ( ' ) ) > ~:(z(°(.)) + A,
-
r(~")(.))l <
provided z(°(') is not decreasing on [~71, ~72]- Take m >t A -1 and put z(.) = z(m)(.). Then z(-) is decreasing on [~?a, 7/2], i.e., satisfies (ii), and
IF(4))
- r ( ~ : ( . ) ) l < .~a;
recall that z(.) satisfies (i), (iii), and (iv) by the definition of Z . . Taking into account (2.16) and letting 6 < e / m , we obtain the inequality (2.5). If z2(.) is decreasing on [r/l, r/2], then we have the same with m = 1. Lemma 1 is proved.
PROOF OF LEMMA 2. Let z(') = z(-I s; y), where s ~ 5"(Y) has the form (2.1). With no loss of generality assume k >_- n + 3. Consider the case Y > 0 ( Y < 0 is treated identically). We have z(t) = z i for t c= [%, ~i], ~,(t) = - b for t ~]~:,, ri+l], and z I = ~'. P u t also z~ = ~'~ where ~', is defined by (2.2). The following relations hold:
r(z(.)) = f)oTl ¢P(~'*(t), y) dt +
f l Op( { *( t), 8) dt + A + IX, (2.36)
where k
i=1
,
k
k
= E ¢(z~, y)(~,-
~)=¢
i=1
E ¢(~,
81¢~,
(2.37/
i--1
k
13= Y'~ (s~ - ri) ,
/3~=(s~i - r~)//3,
(2.38)
i=1
= Ek fff+, ,t,( z ( t ) , 8/dt = - -1,, ~ i=1
_
u
1 f(dp( z, y) dz.
i=1
f < * ( z , y) dz Zi+l
(2.39)
150
V. HEYMANN AND A. KRYAZHIMSKII
Note t h a t E~=l /3i = 1 and z i e [~'k, ~']. T h e n by the t h e o r e m of C a r a t h e o d o r y there exist Zo, 2 . . . . . Zo,,, e [~'k, ~'], zo,, > zo,,+l with m = n + 2, such t h a t E CO( z~, y)/3~ = * ( Zo.,, y)/30 ~ i=l i=1 with certain /3o, i >/0,
Ei"k i
(2.40)
rio, ~ = 1. T a k e an
80 = ( ~ ' , T 0 . ] , a 0 , , . . . .
,'T0, m + l ,
~o,,,+t) e S
(2.41)
such that, first, s o e So(y), second ~0,1 = TO, ] = T I '
(2.42)
s~o, ,,+, = to, ,,,+~ = sck,
(2.43)
i)/J ~
(2.44)
J~0, i = ( ~0, i -- T0,
(2 ~< i < m ) ,
~'o, ,,, +, = {k,
(2.45)
~'0, m+l ~---~ - - b ~ (To, i+ 1 - ~o,i), i=1
(2.46)
where
and, third, Zo(-) - z(. I So; y) satisfies Zo(ro,1) = ~,
zo(s¢o,,~+,) = ~o,,n+],
z o ( t ) = ~'o,i
( t e [%, i, ~:o, i], 2 ~< i ~< m)
(2.47) (2.48)
(the existence of s o is justified below). F r o m (2.40), (2.44), we deduce (like in
(2.37)) rn + 2
* =
y)dt;
E i=1
"To, i
f r o m (2.42), (2.48), (2.46), w e get ( l i k e i n (2.39)) m+
]
i~ J~r°"+' [o, ( Zo(t), y) dt.
Attainability Sets
151
These equalities together with (2.39), yield the desired equality (2.6), and finalize the proof. It remains to show the existence of element s o. Assume (2.42), (2.44), and put Zo, m + l = ~k, (2.49)
%.2 = G o , 1 - ( Zo,2 - ~ ) / b , T0, i+ 1 = ~0,1 -- (Zo, i+l -- Zo, i ) / b
(2 < i •
m),
(2.50)
(2.51)
G0, m + l = TO, re+l"
Then (to, i+, - Go,,) = ( ( -
~k)/b,
(2.52)
i=1
which is equivalent to (2.45) (see (2.46)). S u m m i n g up (2.51) and G0, ~ - ~0, (2 4 i ~< m), and using (2.44) and (2.42), we obtain
the last equality follows from (2.38) and (2.2). Thus, (2.43) is also satisfied. Introduce s o (2.41). T h e equalities (2.45), (2.42), (2.43), and the inclusion s ~ So(y) imply s o ~ So(y) by R e m a r k 1. Finally, (2.48) and (2.47) for z0(') = z('l so; y) follow from the definition of z0(.) and (2.49)-(2.52). L e m m a is proved. REFERENCES 1 R . W . Brockett, System theory on group manifolds and coset spaces, SIAM J. Control 10:265-284 (1972). 2 R. R. Mohler, Bilinear Control Processes, with Appliances to Engineering, Ecology, and Medicine, Academic Press, New York, 1973, p. 224. 3 R.W. Brockett, On the reachable set for bilinear systems, in Variable Structure Systems with Application to Economics and Biology (A. J. Krener, Ed.), Lecture Notes in Econom. and Mathem. Systems, Vol. 3, Springer, Berlin, 1975. 4 A.A. Agrachev and S. A. Vakhrameev, Nonlinear control systems of constant rank and bang-bang conditions for extremal controls, Soviet Math. Dokl. 30:620-624 (1984).