- Email: [email protected]
On the minimization problem of reachable set estimation of control system
Cop)'riabt
~IPAC
GENERALIZED SOLUTIONS IN CONTROL PROBLEMS. 2004.
GENERAL THEORY, LUMPED PARAMETER SYSTEMS. P. 212-'\7
ON THE MINIMIZATION PROBLEM OF REACHABLE SET ESTIMATION OF CONTROL SYSTEM S.Otakulov Samarkand State University, Chair of mathematical modelling, University boulevard, 15, '/03004, Samarkand, Uzbekistan E-mail: [email protected]
Abstract: In this paper considered delay controllable differential inclusions. For these control systems researched the minimization problem of diversion reachable set from terminal set. Differential inclusion, control system, reachable set, terminal set, diversion, minimal estemation, optimal control, saddle point.
Keywords:
1. INTRODUCTION
Deferential inclusions are used widely in solving important problems theory of optimal control. They used as a good mathematical apparatus for describing of conduct controlling objects which is in condition of uncertainty and in some information constraints. It is very necessary to note,that in researching such kind of control systems one must pay attention to different properties of trajectory ensemble, and solutions sets, methods of prognosis and estimating of phase state, minima.x syntheses [1,2].
With the actual problem of control in condition lating information and condition of uncertainly there will be a question of researching controllable of differential inclusions with dealing arguments:
XEF(t,x,x(t-h),u),
t~to,
(1)
i,
where x = x is n-vector state, u is m-vector control, h > 0, F(t, x, y, u) c Er', Er' is n-dimensional Euclidean space of vectors x = (Xl, x2, ..., xn) with norm Hxll = .,j(x, x), (x,() is scalar product of vectors x, ~ E Er'. Let V -convex compact set in Er'. As an admissible control we shell choose every mea-
surable bounded m-vector-function u = u(t), t E T = [to, tIl, such that u(t) E V almost everywhere on T. We denote by V (T) the set of all admissible controls.
By the quantity d(X(tl, u, 'Po), Y) we shall define diversion of reachable set
X (t l , u, 'Po) from terminal set Y. This quantity (3) depends on the parameter of control, i.e. we will obtain the functional
By admissible trajectories we shall call every continuous on T 1 = [to - h, td and absolutely continuous on T n-vector-function x = x(t), considering differential inquisition (1) and initial condition
x(t)
= 'Po(t),
t E To
= [to -
h, to],
J(u)=d(X(t.,u,'Po),Y)'
Let us consider a minimization problem of estimate (4), i.e. we must find control u' E V(T), such that
(2)
where 'Po E Cn(To), cn(To) is space all continuous on To n-vector-functions.
-+ min, u E U(T). (6)
The control u· E U(T), satisfying an equality (5), we shall call optimal control in problem (6). If u· E U(T)- optimal control, then d(X(t l , u', 'Po)) is called minimal estimate for diversion the reachable set X(tl,u, 'Po) from terminal set Y.
In theory of differential inclusions it is very important property of controllability ensemble trajectories (3).
We use the following notation: ntH!') totality all now-empty compacts in H!'; nO(H!') is totality all convex compact in H!';
Let Y is closed convex set in H!'.
Definition. If there exist t l > to and control u E U(T), T = [to, tIl, such those X(t!> u, 'Po) c Y, then be system (1),(2) we call that ('Po, Y) - controllable.
P'(QI,Q2)
=
max{max p(~, Q2), max p(~, QI)} (EQ,
(EQ2
is the Hausdorff distance between QI, Q2 E n(H!'); IIQII = sup II~!I is the norm of set
The property ('Po, Y)- controllability of system (1) researched in [4).
(EQ
Q E n(H!'). System (1) we will research in following conditions: a) F(t,x,y,u) E nO(H!'), V'(t,x,y,u) E T x H!' x H!' x V; b) multi-valued map
With the question of controllability of system (1) there will appear a problem of estimation of diversion reachable set X(t l , u, 'Po) from terminal set Y.
(t, X, y, u) -+ F(t, X, y, u)
2. STATEMENT OF THE PROBLEM
is measurable with respect to t E T for V'(x,y,u) E H!'xH!'xVandsatisfiesaLipschitz condition with respect to (x, y, u) for almost all t ET:
= 1'EY inf iI{ - vii is the distance
from point ~ E H!' to set Y; sup
p(~, Y)
sup (EX(I, ,U,'Po)
For fixed t = t l the set X(t., u, 'Po) is reachable set system (1),(2).
=
(5)
Thus, we pose the following minimax control problem:
Let X (t, u, 'Po) = {~ EH!': ~ = x(~), xC) E H(u,'Po)},t E T. By multivalued map t -+ X(t, u, 'Po) we call ensemble trajectories of system (1),(2).
d(X(t l , U, 'Po), Y)
= uEU(T) min J(u).
J(u')
We denote H(u, 'Po) the set of all admissible trajectories x(·, u, 'Po) of the differential inclusion (1) with initial condition (2).
Let: p(~, Y)
(4)
p({, Y).
p'(F(t, x, y, u), F(t, ~,17, v)) :S
(EX(I.,u,'Po)
l(t)[Ilx - ~II
(3) 213
+ Ily -1711 + Ilu - viII,
V(x, y, u), (~, 1), v) E Jr' x Jr' x V, 1(-) E L 2 (T).
Lemma4. Following equality is true:
d(X(t" u, 11'0), Y) = max [c(X(t l , u,/Po), 1/1) - c(Y,l/J)]. (7)
~
ldt), 11 (-) E L1(T), (~O,1)°,VO) E Jr' X R" X Vi d) support functionC(F(t, x, y, u), 1/1) sup{(J,1/I):! E (F(t,x,y,u)} is concave with respect to (x, y) E R" X R" for all t E T, u E V.
c) IIF(t,fl,1)°,vO)ll
!I~I!=I
Proof. Accounting that II~II = J(~,~) max (~, 1/1,) we have:
=
1I~119
p(~,Y) =infll~-yll JIIY
According to results in [5] in conditions a)-c) reachable set X(t., u, /Po) is convex compact in R". It following that for all (u, /Po) E U(T) x C"(To) the estimate (4) exists. More over, inf J(u) > -00.
= inf max(~-y,1/I). I/EYII~1I9
Using the theorem about minimax [1,6]' we obtain p(~, Y)
= max [(~, 1/1) - sup(y, 1/1)] = 111/111:51
..EU(Tl
l/EY
max [(~, 1/1) - C(Y, 1/1)].
1I~1I9
Therefore, 3. SOME PROPERTIES OF SOLUTION SET AND ENSEMBLE TRAJECTORlES
d(X(t" u, 11'0), Y) = max [
sup
II~H9 (EX(t, ....'I'o)
(~,,p)
p(~, Y)
=
- C(Y, 1/1)],
i.e. formula (7) is true.
According to results in [5] the following assertions are true:
It follows from formula (7) and results of convex analysis [6], that this formula is true:
Lemmal. Assume that the conditions a)d) hold. Then for all u E U(T), /Po E cn(To) and t E T the sets
d(X(t" u, 11'0), Y) = max ~[C(X(t"U,lI'o),1/I)
H(u, /Po), X(t, u, /Po)
H;.II:5 1
are convex compacts accordingly Cn(T1 ) and R".
- C(Y, ,p)], (8) where
cone!-
closed concave of function
!(,p).
Lemma2. Suppose that the conditions a)c)hold. More over, let UI E U(T), U2 E U(T). Then for each Xl(-) E H(Ul,II'O) there exits X2(-) E H(U2, 11'0)' such that
Ilxd') -
sup
{EX(t....'I'.l
Accounting the equality (7) for lower bound of diversion (4) the following estimate is true:
x2(-)i1C"(TIl ~
inf d(X(t" u, /Po), Y) ~
uEU(T)
exp (21.1(t)dt) Ill(')I!L,(Tl
max[ inf C(X(t l ,u,/Po),1/I)-
1I~1I:51
IluI - u21I L 2'(T)'
.. EU(T)
- C(Y, tP)]·
The following lemma follows from lemmas 1,2:
4. MAIN RESULTS
Lemma3. Assume that the conditions a)c) hold. Then multi-valued maps u -+ H(u,/po' (t,u) -+ X(t,u,/po) are continuous accordingly on U(T) and T x U(T).
We research conditions of optimality in problem (6). These conditions lets find minimal estimate diversion of reachable set system (1) from terminal set. 214
Let in system (1)
F(t,x,y,u)
I,
!
(9)
+ C(~(tlJ t)b(t, u(t)), ,p)dt.
where A(t), Al (t)- are n x n- matrixes, elements of which summable on T, multivalued map (t, u) -+ b(t, u) measurable with respect to t E T and satisfies a Lipschitz conditions with respect to u E V. Then for multi-valued map (9) wile be true conditions a)-b).
uEU
(11)
to
Let
= C(X(tl, u, <'oo),,p) -
J.I.(u,,p)
C(Y, ,p). (12)
Theorem 1. Let control the condition:
As we remained for estimate of diversion (4) is important properties of convexity and compacts reachable set X (t l , u, <,00)' Be found with these property will have reachable set X(t l , u, <,00), if matrices A(t), AI(t) satisfy above mentioned conditions and multi-valued map (t, u) -+ b(t, u) inside of Lipschitz condition satisfies conditions of continuouty with respect to u E V, moreover sup Ilb(t, ulll ~
= (S(tl, <,00)' ,p)+
C(X(t IJ u, '1'0), Y)
= A(t)x+AI(t)y+b(t,u),
minC(~(tl' t)b(t, ueU
C(~(tl, t)b(t,
UO
E U(T) satisfies
u), ,pO) =
UO(t)), ,pO), t E T, (13)
where ,po EEl", ,po #: 0, is point of maximum of function J.I.(UO,,p) on 8 1 = {t/I E El" :, 11,p11 ~ I}. Then reachable set X (tl, u, '1'0) will have minimal estimate, i.e.
.8 1 (t): lull + .82(t),
d(X(tl> uO, <,00), Y) = min d(X(t l , u, '1'0)' Y). (14)
.810, .82(-) E £1 (T).
ueU(T)
This fact follows from this representation
[4] Proof. Let ,po EEl", ,po #: 0, is point of global maximum of function J.I.( uO, ,p) on B I = {,p EEl":, ll,pll ~ I}. Then, using (7),(11),(12),(13), we have:
X(t1,u,<,Oo) = S(tlJ<,Oo)+ I,
!
+ ~(tl, t)b(t, u(t))dt,
(10)
I,
= max J.I.(UO,,p) = II"II~I J.I.(UO, ,pO) = (S(t l , '1'0), ,p0)+
where ~(t,r) is n x n-matrix function, satisfying an equation
d(X(tl>uo,<,Oo),Y)
~~r) = -~(t,r)A(r)-
~(t,
~(t,t-O)=E,
S(t 1 , 'Po) =
~(t,
r)A I (r
+ h), r
~(t,r)=O,
~
I,
+! uev C(~(tl' t)b(t,
t,
min
r~t+O,
u), ,p°)dt-
10
-C(Y,,p)
to)<'oo(to)+
to+h
+
! ~(tlJ r)A (r)<'oo(r - h)dr, I
< max [(S(tl, <,00), ,pO) + - il"'ll~l
to
E- unity n x n-matrix.
I,
+
Using formula (10) and properties of integral of multi-valued map, we obtain, that for support function C(X(t IJ u, <'oo),,p) of set X (t l , u, r,?o) the following equality is true:
! to
215
min C(~(tl, t)b(t, u), ,p°)dtuev
~
for system (17) find this formula inf max [(S(t h 'Po), !J>0)
ueu(TJ:I"II'S\
+
C(X(t h u, 'Po),!J» = (S(tl' 'Po), t,!J)+ I,
I,
+ /(~(tl:t)B(t)U(t),t/J)dt+
+ / C(~(th t)b(t, u(t)),!J>°)dt-
to
to
I,
+ / C(~(tl, t)G(t), tP)dt.
-C(Y,!J»]. (15)
(18)
to
Accounting (12),(18), we have:
Therefore, equality (14) is true.
conc,plJ(u,!J» = (S(th 'Po),!J»+ Remark. It follows easily from these relations (15), that each point of global maximum function IJ( uO,!J» on B l is also point of global maximum of function
II
+ / (~(th t)B(t)u(t),!J»dt+ to I,
+ COfle.,lj C(~(t\, t)G(t),!J»dt10
I,
+/ min C(~(th t)b(t, u),!J»dt-C(Y,!J». to
ueV
- C(Y, !J»]. (19) Accounting to fonnula (8)
(16)
d(X(t\,u,'Po), Y) where~lJ(u,!J»
It follows, that obtaining the minimal estimate of reachable set has closely connected with the problems of existing saddle point offunctional IJ( u, !J».
= 11,,119 max con41J(u,!J», is defined by (19).
It is seen from (19) that function a(u,!J» = COflc,pIJ( u, tP) is linearly with respect to u E U(t) and concave with respect to tP E R". That is way for function a(u,!J» we may use theorem about minimax [1,6]. Therefore for system (17) following assertion is true.
Now we consider line armodel of control system (1) i.e. system:
x E A(t)x+A\(t)x(t-h)+B(t)u(t)+G(t), (17)
Theorem 2. In order to, that reachable set X(thuO,'Po) for system (17) possess minimal estimate it is necessary and sufficient of existing vector t/J0 R", which is a point of global maximum of function
We research system (17) in following conditions:
ail
elements of matrixes A(t), Al(t) and B(t) are surnmable on T = [ta, td;
rJ(1P)
= S(tl, 'Po), tP)+ I,
+ / min(~(tl, t)B(t)u, !J»dt+ ueV to
b\) for each t E T,
G(t) E O(R");
h
+COfle.,lj C(~(th t)G(t), !J»dt-C(Y, tP)] to
cIl multi-valued map t --. G(t) mesurable on T, moreover !IG(t)11 ~ g(t), t E T, g(.) E LdT)
and holding following condition
=
min(~(tl, t)B(t)u, !J>0) uev (~(th t)B(t)uO(t),!J>°),
Then support function of the set
t E T.
Assertion of theorem shows, that control UO E U(T) gives minimal estimate of
X(t h u, 'Po) 216
diversion set X(t" u, 'Po) from terminal set Y, if and only if when there exists a vector 'P" ER", such that pair (u", 'P") is saddle point of functional u(u, t/J) on U(T) x B l •
REFERENCES [I} A.B.Kurzbanskii. Control and observation under conditions of indeterminacy, "Nauka",Moscow, 1977.392p.(Russian) [2] F.L.Chernous'ko and A.A. Melikyan, Game problems of control and search, " Nauka" ,Moscow, 1978.27Op.(Russian) [3] S.Otalrulov, On the conditions controUability of diHerential inclusions. lzvestiya, Russian Academy of Sciences , ser, texn, kibemitika. 1992, no 2, pp. 57-62(Russian) [4J I.Israilov and S.Otakulov, On the one property of trajectory ensembles of delay differential inclusion. Proceedings of Inter.Conf., "Modern problems of mathematical physics and infonnational technology". vo!. 2. Tashkent 2003. pp. 213-215(Russian) [5] I.Israilov, B. E.Davronov , About properties of compact and convexity family trajectory of delay controliable differential inclusions. Vestnik Samarkandskogo Unversiteta, Samarkand.2003,no 3, pp.6-9 (Russian) {6J B.N.Pshenkhnyi, Convex analysis and extremal problems, "Nauka" , Moscow, 1980. 320 p. (Russian)
217
~IPAC
GENERALIZED SOLUTIONS IN CONTROL PROBLEMS. 2004.
GENERAL THEORY, LUMPED PARAMETER SYSTEMS. P. 212-'\7
ON THE MINIMIZATION PROBLEM OF REACHABLE SET ESTIMATION OF CONTROL SYSTEM S.Otakulov Samarkand State University, Chair of mathematical modelling, University boulevard, 15, '/03004, Samarkand, Uzbekistan E-mail: [email protected]
Abstract: In this paper considered delay controllable differential inclusions. For these control systems researched the minimization problem of diversion reachable set from terminal set. Differential inclusion, control system, reachable set, terminal set, diversion, minimal estemation, optimal control, saddle point.
Keywords:
1. INTRODUCTION
Deferential inclusions are used widely in solving important problems theory of optimal control. They used as a good mathematical apparatus for describing of conduct controlling objects which is in condition of uncertainty and in some information constraints. It is very necessary to note,that in researching such kind of control systems one must pay attention to different properties of trajectory ensemble, and solutions sets, methods of prognosis and estimating of phase state, minima.x syntheses [1,2].
With the actual problem of control in condition lating information and condition of uncertainly there will be a question of researching controllable of differential inclusions with dealing arguments:
XEF(t,x,x(t-h),u),
t~to,
(1)
i,
where x = x is n-vector state, u is m-vector control, h > 0, F(t, x, y, u) c Er', Er' is n-dimensional Euclidean space of vectors x = (Xl, x2, ..., xn) with norm Hxll = .,j(x, x), (x,() is scalar product of vectors x, ~ E Er'. Let V -convex compact set in Er'. As an admissible control we shell choose every mea-
surable bounded m-vector-function u = u(t), t E T = [to, tIl, such that u(t) E V almost everywhere on T. We denote by V (T) the set of all admissible controls.
By the quantity d(X(tl, u, 'Po), Y) we shall define diversion of reachable set
X (t l , u, 'Po) from terminal set Y. This quantity (3) depends on the parameter of control, i.e. we will obtain the functional
By admissible trajectories we shall call every continuous on T 1 = [to - h, td and absolutely continuous on T n-vector-function x = x(t), considering differential inquisition (1) and initial condition
x(t)
= 'Po(t),
t E To
= [to -
h, to],
J(u)=d(X(t.,u,'Po),Y)'
Let us consider a minimization problem of estimate (4), i.e. we must find control u' E V(T), such that
(2)
where 'Po E Cn(To), cn(To) is space all continuous on To n-vector-functions.
-+ min, u E U(T). (6)
The control u· E U(T), satisfying an equality (5), we shall call optimal control in problem (6). If u· E U(T)- optimal control, then d(X(t l , u', 'Po)) is called minimal estimate for diversion the reachable set X(tl,u, 'Po) from terminal set Y.
In theory of differential inclusions it is very important property of controllability ensemble trajectories (3).
We use the following notation: ntH!') totality all now-empty compacts in H!'; nO(H!') is totality all convex compact in H!';
Let Y is closed convex set in H!'.
Definition. If there exist t l > to and control u E U(T), T = [to, tIl, such those X(t!> u, 'Po) c Y, then be system (1),(2) we call that ('Po, Y) - controllable.
P'(QI,Q2)
=
max{max p(~, Q2), max p(~, QI)} (EQ,
(EQ2
is the Hausdorff distance between QI, Q2 E n(H!'); IIQII = sup II~!I is the norm of set
The property ('Po, Y)- controllability of system (1) researched in [4).
(EQ
Q E n(H!'). System (1) we will research in following conditions: a) F(t,x,y,u) E nO(H!'), V'(t,x,y,u) E T x H!' x H!' x V; b) multi-valued map
With the question of controllability of system (1) there will appear a problem of estimation of diversion reachable set X(t l , u, 'Po) from terminal set Y.
(t, X, y, u) -+ F(t, X, y, u)
2. STATEMENT OF THE PROBLEM
is measurable with respect to t E T for V'(x,y,u) E H!'xH!'xVandsatisfiesaLipschitz condition with respect to (x, y, u) for almost all t ET:
= 1'EY inf iI{ - vii is the distance
from point ~ E H!' to set Y; sup
p(~, Y)
sup (EX(I, ,U,'Po)
For fixed t = t l the set X(t., u, 'Po) is reachable set system (1),(2).
=
(5)
Thus, we pose the following minimax control problem:
Let X (t, u, 'Po) = {~ EH!': ~ = x(~), xC) E H(u,'Po)},t E T. By multivalued map t -+ X(t, u, 'Po) we call ensemble trajectories of system (1),(2).
d(X(t l , U, 'Po), Y)
= uEU(T) min J(u).
J(u')
We denote H(u, 'Po) the set of all admissible trajectories x(·, u, 'Po) of the differential inclusion (1) with initial condition (2).
Let: p(~, Y)
(4)
p({, Y).
p'(F(t, x, y, u), F(t, ~,17, v)) :S
(EX(I.,u,'Po)
l(t)[Ilx - ~II
(3) 213
+ Ily -1711 + Ilu - viII,
V(x, y, u), (~, 1), v) E Jr' x Jr' x V, 1(-) E L 2 (T).
Lemma4. Following equality is true:
d(X(t" u, 11'0), Y) = max [c(X(t l , u,/Po), 1/1) - c(Y,l/J)]. (7)
~
ldt), 11 (-) E L1(T), (~O,1)°,VO) E Jr' X R" X Vi d) support functionC(F(t, x, y, u), 1/1) sup{(J,1/I):! E (F(t,x,y,u)} is concave with respect to (x, y) E R" X R" for all t E T, u E V.
c) IIF(t,fl,1)°,vO)ll
!I~I!=I
Proof. Accounting that II~II = J(~,~) max (~, 1/1,) we have:
=
1I~119
p(~,Y) =infll~-yll JIIY
According to results in [5] in conditions a)-c) reachable set X(t., u, /Po) is convex compact in R". It following that for all (u, /Po) E U(T) x C"(To) the estimate (4) exists. More over, inf J(u) > -00.
= inf max(~-y,1/I). I/EYII~1I9
Using the theorem about minimax [1,6]' we obtain p(~, Y)
= max [(~, 1/1) - sup(y, 1/1)] = 111/111:51
..EU(Tl
l/EY
max [(~, 1/1) - C(Y, 1/1)].
1I~1I9
Therefore, 3. SOME PROPERTIES OF SOLUTION SET AND ENSEMBLE TRAJECTORlES
d(X(t" u, 11'0), Y) = max [
sup
II~H9 (EX(t, ....'I'o)
(~,,p)
p(~, Y)
=
- C(Y, 1/1)],
i.e. formula (7) is true.
According to results in [5] the following assertions are true:
It follows from formula (7) and results of convex analysis [6], that this formula is true:
Lemmal. Assume that the conditions a)d) hold. Then for all u E U(T), /Po E cn(To) and t E T the sets
d(X(t" u, 11'0), Y) = max ~[C(X(t"U,lI'o),1/I)
H(u, /Po), X(t, u, /Po)
H;.II:5 1
are convex compacts accordingly Cn(T1 ) and R".
- C(Y, ,p)], (8) where
cone!-
closed concave of function
!(,p).
Lemma2. Suppose that the conditions a)c)hold. More over, let UI E U(T), U2 E U(T). Then for each Xl(-) E H(Ul,II'O) there exits X2(-) E H(U2, 11'0)' such that
Ilxd') -
sup
{EX(t....'I'.l
Accounting the equality (7) for lower bound of diversion (4) the following estimate is true:
x2(-)i1C"(TIl ~
inf d(X(t" u, /Po), Y) ~
uEU(T)
exp (21.1(t)dt) Ill(')I!L,(Tl
max[ inf C(X(t l ,u,/Po),1/I)-
1I~1I:51
IluI - u21I L 2'(T)'
.. EU(T)
- C(Y, tP)]·
The following lemma follows from lemmas 1,2:
4. MAIN RESULTS
Lemma3. Assume that the conditions a)c) hold. Then multi-valued maps u -+ H(u,/po' (t,u) -+ X(t,u,/po) are continuous accordingly on U(T) and T x U(T).
We research conditions of optimality in problem (6). These conditions lets find minimal estimate diversion of reachable set system (1) from terminal set. 214
Let in system (1)
F(t,x,y,u)
I,
!
(9)
+ C(~(tlJ t)b(t, u(t)), ,p)dt.
where A(t), Al (t)- are n x n- matrixes, elements of which summable on T, multivalued map (t, u) -+ b(t, u) measurable with respect to t E T and satisfies a Lipschitz conditions with respect to u E V. Then for multi-valued map (9) wile be true conditions a)-b).
uEU
(11)
to
Let
= C(X(tl, u, <'oo),,p) -
J.I.(u,,p)
C(Y, ,p). (12)
Theorem 1. Let control the condition:
As we remained for estimate of diversion (4) is important properties of convexity and compacts reachable set X (t l , u, <,00)' Be found with these property will have reachable set X(t l , u, <,00), if matrices A(t), AI(t) satisfy above mentioned conditions and multi-valued map (t, u) -+ b(t, u) inside of Lipschitz condition satisfies conditions of continuouty with respect to u E V, moreover sup Ilb(t, ulll ~
= (S(tl, <,00)' ,p)+
C(X(t IJ u, '1'0), Y)
= A(t)x+AI(t)y+b(t,u),
minC(~(tl' t)b(t, ueU
C(~(tl, t)b(t,
UO
E U(T) satisfies
u), ,pO) =
UO(t)), ,pO), t E T, (13)
where ,po EEl", ,po #: 0, is point of maximum of function J.I.(UO,,p) on 8 1 = {t/I E El" :, 11,p11 ~ I}. Then reachable set X (tl, u, '1'0) will have minimal estimate, i.e.
.8 1 (t): lull + .82(t),
d(X(tl> uO, <,00), Y) = min d(X(t l , u, '1'0)' Y). (14)
.810, .82(-) E £1 (T).
ueU(T)
This fact follows from this representation
[4] Proof. Let ,po EEl", ,po #: 0, is point of global maximum of function J.I.( uO, ,p) on B I = {,p EEl":, ll,pll ~ I}. Then, using (7),(11),(12),(13), we have:
X(t1,u,<,Oo) = S(tlJ<,Oo)+ I,
!
+ ~(tl, t)b(t, u(t))dt,
(10)
I,
= max J.I.(UO,,p) = II"II~I J.I.(UO, ,pO) = (S(t l , '1'0), ,p0)+
where ~(t,r) is n x n-matrix function, satisfying an equation
d(X(tl>uo,<,Oo),Y)
~~r) = -~(t,r)A(r)-
~(t,
~(t,t-O)=E,
S(t 1 , 'Po) =
~(t,
r)A I (r
+ h), r
~(t,r)=O,
~
I,
+! uev C(~(tl' t)b(t,
t,
min
r~t+O,
u), ,p°)dt-
10
-C(Y,,p)
to)<'oo(to)+
to+h
+
! ~(tlJ r)A (r)<'oo(r - h)dr, I
< max [(S(tl, <,00), ,pO) + - il"'ll~l
to
E- unity n x n-matrix.
I,
+
Using formula (10) and properties of integral of multi-valued map, we obtain, that for support function C(X(t IJ u, <'oo),,p) of set X (t l , u, r,?o) the following equality is true:
! to
215
min C(~(tl, t)b(t, u), ,p°)dtuev
~
for system (17) find this formula inf max [(S(t h 'Po), !J>0)
ueu(TJ:I"II'S\
+
C(X(t h u, 'Po),!J» = (S(tl' 'Po), t,!J)+ I,
I,
+ /(~(tl:t)B(t)U(t),t/J)dt+
+ / C(~(th t)b(t, u(t)),!J>°)dt-
to
to
I,
+ / C(~(tl, t)G(t), tP)dt.
-C(Y,!J»]. (15)
(18)
to
Accounting (12),(18), we have:
Therefore, equality (14) is true.
conc,plJ(u,!J» = (S(th 'Po),!J»+ Remark. It follows easily from these relations (15), that each point of global maximum function IJ( uO,!J» on B l is also point of global maximum of function
II
+ / (~(th t)B(t)u(t),!J»dt+ to I,
+ COfle.,lj C(~(t\, t)G(t),!J»dt10
I,
+/ min C(~(th t)b(t, u),!J»dt-C(Y,!J». to
ueV
- C(Y, !J»]. (19) Accounting to fonnula (8)
(16)
d(X(t\,u,'Po), Y) where~lJ(u,!J»
It follows, that obtaining the minimal estimate of reachable set has closely connected with the problems of existing saddle point offunctional IJ( u, !J».
= 11,,119 max con41J(u,!J», is defined by (19).
It is seen from (19) that function a(u,!J» = COflc,pIJ( u, tP) is linearly with respect to u E U(t) and concave with respect to tP E R". That is way for function a(u,!J» we may use theorem about minimax [1,6]. Therefore for system (17) following assertion is true.
Now we consider line armodel of control system (1) i.e. system:
x E A(t)x+A\(t)x(t-h)+B(t)u(t)+G(t), (17)
Theorem 2. In order to, that reachable set X(thuO,'Po) for system (17) possess minimal estimate it is necessary and sufficient of existing vector t/J0 R", which is a point of global maximum of function
We research system (17) in following conditions:
ail
elements of matrixes A(t), Al(t) and B(t) are surnmable on T = [ta, td;
rJ(1P)
= S(tl, 'Po), tP)+ I,
+ / min(~(tl, t)B(t)u, !J»dt+ ueV to
b\) for each t E T,
G(t) E O(R");
h
+COfle.,lj C(~(th t)G(t), !J»dt-C(Y, tP)] to
cIl multi-valued map t --. G(t) mesurable on T, moreover !IG(t)11 ~ g(t), t E T, g(.) E LdT)
and holding following condition
=
min(~(tl, t)B(t)u, !J>0) uev (~(th t)B(t)uO(t),!J>°),
Then support function of the set
t E T.
Assertion of theorem shows, that control UO E U(T) gives minimal estimate of
X(t h u, 'Po) 216
diversion set X(t" u, 'Po) from terminal set Y, if and only if when there exists a vector 'P" ER", such that pair (u", 'P") is saddle point of functional u(u, t/J) on U(T) x B l •
REFERENCES [I} A.B.Kurzbanskii. Control and observation under conditions of indeterminacy, "Nauka",Moscow, 1977.392p.(Russian) [2] F.L.Chernous'ko and A.A. Melikyan, Game problems of control and search, " Nauka" ,Moscow, 1978.27Op.(Russian) [3] S.Otalrulov, On the conditions controUability of diHerential inclusions. lzvestiya, Russian Academy of Sciences , ser, texn, kibemitika. 1992, no 2, pp. 57-62(Russian) [4J I.Israilov and S.Otakulov, On the one property of trajectory ensembles of delay differential inclusion. Proceedings of Inter.Conf., "Modern problems of mathematical physics and infonnational technology". vo!. 2. Tashkent 2003. pp. 213-215(Russian) [5] I.Israilov, B. E.Davronov , About properties of compact and convexity family trajectory of delay controliable differential inclusions. Vestnik Samarkandskogo Unversiteta, Samarkand.2003,no 3, pp.6-9 (Russian) {6J B.N.Pshenkhnyi, Convex analysis and extremal problems, "Nauka" , Moscow, 1980. 320 p. (Russian)
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