Nonlinear Analysis of Dynamics and Control of Interconnected Systems with Distributed Parameters

Co p \l i ~h l

©

Parame ter

S\'SIl' lll S.

I F..I.t: COIII ICI I or \l isl r ihllll'd l't'rp i ~ lI ~ 1I 1. Frallce . 1 ~IX~1

NONLINEAR ANALYSIS OF DYNAMICS AND CONTROL OF INTERCONNECTED SYSTEMS WITH DISTRIBUTED PARAMETERS V. M. Matrosov Irlillt.\/: CIJIII/JIIlillg' CI'IIIII' of Ihl' ,\illI'rillll Ihllllrh [ '.'iSH

of

Ihl' l 'SS H ,~ullll'lllr of Sril'lI({',\,

Abstract. Th e pa per pr es ent s some r es ult s on deve lopment a nd a ppli ca ti on of the me th od of vec t o r Lya punov fun c ti on s to th e nonlinea r a n a l ys i s o f dif fe r e nt dynami ca l prope rtie s o f int e r connec t e d contro l sys t ems with di s t r ibut e d pa r ame t e r s . The se r es ult s we re obt a ined by th e a utho r a nd hi s cowo rke r s a t th e Irkutsk Computin g Ce ntr e o f th e Sib e ri a n Branch of th e USSR Aca demy o f Sc i e n ces .

Ke ywo rd s . Int e r connec t e d sys t e m; vec t o r Ly apunov f un c ti o n; compa ris on me thod; o ptima l co ntr o l; di s tribut e d pa r ame t e r; no nlin e ar dy nami cs .

I NTRODUCTION

I . I NTERCONNECTED CONTROL SYSTEM WITH DISTRIBUTED PARAMETE RS .

Vec t o r Lyapunov f un c ti on s( VLF) (R . Be llma n ( 196 2 ); V.M.Ma tr osov ( 196 2»pr opose d t o b e u se d fo r th e anal ysi s of s t a bilit y a nd o th e r dynami cal pr ope rti es of int e r connec ted nonlin ea r sys t ems with di st r i bu t e d pa r ame t e r s in ( I), whe r e compa ri so n th eo r ems a nd the t h e or em on e xp on e nti al s t abilit y we r e obtain e d .

Le t I CSDP b e con s tr uc t ed o f m in t e r connec t e d s ub sys t ems of diffe r e nt n a tur e with contro l s U , • .. , I

Pl " ' " Pm'

U rn' pe rtur ba ti ons

A m' ( with s t a t es x I " " , x

pr ocesse s~

I" ... ,

), whi c h a r e ~th e

m f un c ti on s with va lues belon g in g t o th e se t s Xl" ",

Th e deve l opmen t of Lya punov fun c ti on s me thod, th e c omp a ri son me tho d a nd the VLF-me t hod f or th e sy s t ems with di s tribut e d pa r ame t e r s is des cr i be d in monog r a ph s of V. I.Zub ov( 1957), N. N.Mo i sejev , V. V.Rumjant zev( 196 5), V.La k s hmik a ntham, S.Lee l a ( 1969 ) , T. K.Siraze tdinov( 19 7 1, 1987) , A. A. Ma rt i njuk ( 197 5) , A.N.Mi c he l, R. K.M i l l e r( 19 78) , V.M.Ma t rosov , L.Ju. Anapo l s ki, S . N. Vass i ljev( 1980), E.F. Sa baev( 1980), A. I . Mos ka l e nko( 198 3). Th e comp a ri son prin c ip le with VLF in algo rithmi cal f or m for the de ri vatio n o f th e compa ri son t h eo r e ms and t h e ir pr oofs was a l so fo rmu la t ed i n (2 ) . Th e develo pment of the asso ci a t e d sof twa r e gave th e poss ib i lit y to obt a in two hundr e d comp a ri son theo r ems fo r vario u s dy na mi ca l pr o pe rti es . We have a pp roac he d th e a r ea o f Art if i c i al I nt el li ge nce ( 3) .

Xm res p ec tively . Th e s t a t e space

E~

of eve ry sub -

sys t ems i s s uppose d t o b e th e Ba n ac h s pace (r ea l ), i n whi ch the se t x~ i s u s ual ly den se i n Some open doma in H s E.. !"

r

; :i::.

~

(t)~

x. ~

The dynami cs of s u c h a ICSD P i s desc rib e d by the sys t em of dif fe r en t i a l e qu a ti on s

with non l in ea r ( gen era l ly unb ounde d di s ~ntinu o u s ) ope r a t ors F~: T, x. U~ ' P~-+ E ~ , whe r e' tE L3 "' [ O,'t ) , O < 't~+o<>, x = (x l' ... , Xm) E X"' XI .... ·xms S E =:: E ,x . . . " E , m

UJJ:Tx XIt PJ.j-U , JJ

( U I' . . . , U m) "" U E

Th e decompos iti on- agg r ega ti on ap pr oac h (F. N. Ba il ey ) ( 1965), A. A.Pio ntk ovski(196 7) , V.M. Ma t rosov( 19 7 1) , A. S . Zeml yakov( 1972 , 1976 ) , D.D. Si l j ak( 19 72 , 1978), L. Gruji c( 19 72 ,19 75 ) t o th e p r ocess of const ru c ti on of VLF s a nd compa ri so n sys t ems i s mo d ifie d f o r th e nonlin ea r in t e r con nected systems wi t h d i s tribut ed pa r ame t e r s on t he bas i s of r es ult s of ( 4 ,5,6 ) . I n (7 ) a no th er app r oac h was develo ped and gave a ppl i ca ti ons t o solv in g the opti mal co nt rol p r obl em fo r t h e case of dis tribut e d ecolog i ca l sys t ems .

PJJ:Tx X - P , JJ

U :; U I " .. . • U rn'

( PI " " , P m) '" PE'? '" ~ I .. .. ' <;1' m whe r e U , P a r e t he given c l asses of a dmi ss i b l e con tr ols a nd pe r turb a ti on s , ) ", x , ;x:(t)", X, ( u l , .. . , umh"u e. U ", m "" UI '" , ' Um, (PI' · ·· ' Pm) "' p e P =PI·····P m·

( x

1

, .. . , x

Fo r insta nce , ti on s

We desc ri be he r e t h e pr ese nt s t a t e of th e VLF - me thod - one of th e most effect i ve me thod s fo r nonl i nea r analys i s of dy nam i ca l p r ope r t i es of i n t e rco nn ec t e d contr o l sys t ems with d i st ribut ed pa r ame t e r s ( IC SDP). Some o th e r app r oac hes descri bed in (8) f o r f i~, it ed i mens i rnal sys t ems a nd a ppl ied to t he pa r ame t r i c sy nt hes i s of the co nt rol and ob serve r s fo r elas ti c s pacec r afts , to th e a nalys i s of s t ab i li t y of large - scale sys t ems , elec tr o- e ne r ge t ic sys t ems , e t c . ca n also b e ge ne r a li zed t o th e d i s t r i bu ted pa r ame t e r systems .

tl

is the c l ass of meas ur a b le f un c -

~

T. x . P .... u p

~

Sys t ems whi c h ca n be r e du ced t o thi s fo r m a r e : ( i ) ICSD P with out pe rturb at i on s , wh ic h a r e desc ri b ed i n ( I ) ; ( i i) in t e r co nn ec t ed co ntr olle d sys t ems with con t inuou s time f r om (2) , wh e r e th e f ir s t s ub system i s desc rib ed by the o r d in a r y d if f e r e nti a l eq ua ti on s i n Rn , , th e second one by int eg r odi ffe r e nt ial eq uat i on , a nd th e r es t s ub sys t ems are des cr ib ed by t he var i ou s classes of evo l u ti ona l pa r t i al di f f e r ent i al eq ua ti on s with the co rres pondin g initial a nd bo und ary cond i tion s , i f we intr odu ce th e co rr es pondi ng f unc ti on al s t a t e s paces fo r th e

4 19

420

V. 1\1. I\latrosO\'

q subsystems ( C(Q), L (Q), W , ... ) (4). p

lieU, peg> ,

If

(for FO=F ) : (\'tOET03uEU3""

p

toETO,=T, xOEHO'=X a re fix e d,

and , if the initia l co nditi on x(t ) = Xo is g i ven, O then it is possible t o con s id e r the so luti on X of (1.1) in the Caratheodory sense (C- so luti on s), the ex ist ence of thi s solution i s s upposed on some T (X ) '= [to' 'l: (X to < 'L ( X ) ~
»;

lutions is denoted by r(to'x ' ll, O state i s x"" x(t), 5'2=. T. X.

p),

the c urr e nt

There are intr od u ce d the r ea l n onnegat i ve functio 1 1 t 1 1 1 na l s .? : Q-.R , ... ,f : ,Q- R , PO:TO'HO-R , ... ,

IS

=

tE T,

.,pl ), p~Q(5)={

... ,p;O).

XO EHO:Po(to'x O) <

O}, Po:;(P~,

p(t,x)=~~>~(t,x),

1

p;,
(cL'.f'\,tO):{pEP

:

, ...

o + )311 P O( to' x ) 11 exp [- .,L(t-t ) ] O O If

,pt P ),

).

0 - expone nti a l invariance, if

J' = 1/1,

0=0,

then - the expone ntia l s t a bilit y. Let ~ ={1t, 3}. For th e purpose of r educ ti on the following notation of t ype quantifiers with small Gothic letters a r e used f r om now on: to'" (.>I toE TO)' t =.()lt ET 1), XO=. ( >l xO E 1 1 O Ep ( 0 \X"(>!UEU),:m =- OIP (.L,~ ,to)E to P ET), 'P:(>l-PEP (oL,ll,t X=. (>l xE

»,

»'

O

er(to'xO'u,p»,

t=(>ltET(X», XOx",Olx=

= :c(t», eX: (>i.LE R1),

F

),

e=.

'6 '" (~J"e

R!),


( >lCE

Y

O/E;E R+), .. .

Here, for exampl e, the definition of

PER;P

0' +

0' E. R! i s the given a nd f i xe d , the n we have

PP

e R/

Let P be the linear normalized space,

tp

P

1 p=(p, ..

t

P (E:)={X€x:p(t,x)~E},

tOETO )

R13J1E R!

EO

F F 0 3rE R 30E R VXOEP (5) 't/pe'.i'\txE + + to Er( to ,x ' U, P ) 'vi t e T( X ) V x= x( t» (p (t ,x) ~

PPO-

stab ili-

zabil it y for perturbations i s written in the form of a word

p(p,.L,to) < p}·

The nonnegative functionals are introduced for any .LE R;,

[gt

pl ( pl,cL ,to ) = sup ( t, .L): _ 2 2 t+l _ 2

P (p

,.L,tO)= s up[S

.p' 1'~( 'p ip:~

g ( s, .L)d s: t E.T

t

,

:0 ; ='

2 . VECTOR LYAPUNOV FUNCTION AND COMPARISON SYSTEM.

tETto]' to

J,

'1£~( :, ~ ;d~

(t, .L) '" s up

[IJ p~

(t, x) 11

x E pt (at )

J

The definitions of th e dynamical properties ( DP) of I CS DP are represented in th e form of t y pe q ua nt if i e r fo rmu lae ~=. ID l ' . . ID n Jt , or

mR 1· .. :m iJ o + . iJ1 A. 1 &m iJ 1 + 1· .. n 2)' o i a r e the universal type quantifiers

.n:;Jl)I .. · 11i iJ T wh ere

ID

m

(m

. =('t/ y. E MYi ) or the existen tial ones

1

1

ID."" 1

'" (3 Y· E MY') for the correspondin g va ri ables 1

Yi EMYi;

:R, R 1 ,.:R 2

out q u ant i f i ers, like

are th e fina l formulae with -

.R. "" (x E pt)

£

£

Let T*sT, t S RO"' {C E R: C ~O }. SO we have the following definitions of DP of ICSDP without perturbations.

.p T*

- controllability :

(tI

tOE TO It XO E HO

O

(xept(e ». ( In part i c u lar, i n Kalman definition .. 1/ x - x 11 , c = {O} , T*=T ).

PP O-

3 uE11

'r;j XE r(to'x ' u ) 3 tE T*n T( X ) ] x= x ( t »

'tIE.EC

g

[ ( tlE'E R1' +

p(t, x) '"

(V' tOE TO 3 UE U

stabil i zabili t y :

3S'E Rl'°tl xO E PO U5')\7'XEr(t 'X '1.U +

o

to

O

t!te:T ( X ) 't/x=x (t»(x E pt( e ' »&( 35 " E R:O tlxO E

o

£3 t

e P to(o") 'v'x.er(t o'xo' u)'t/£ "E1\+ If t e[t1,'f: ) 'Y x=

o H s

E

qc

=Rk~,

TL

v

X ,U s R 9-, P sR'l q =I, 2) qc qc qc qc be give n (na tur al l y partially o r dered) and also th e classes 11 , 'Jl of superpositionally mea qc qc s ur ab l e adm i ss i ble cont r ols U qc and perturbations

where the s ummable functionlfs

g~

xqc SO

Let the fini t edimensional sets

1E.T(X)

X(t »(x e p~( 8" ) 1 .

For DP of ICSDP with per tur bat i ons peP we h ave definitions of P Po - expo n en ti al boundedness

P qc

of compariso n U qc:Tx Xqc~Uqc'

Pq c:T - P qc '

The r es t values of compariso n are derived sim ilarly and de n o t ed by th e index c . The main vector comparison funct i ons are conside k 1 red: V =(V , ... , V q.) :dom v - X (domvl;TxE q q q q qc q i s the open domain) and the auxiliary compariso n fu n c tion s Sq: TO' HO '11.-U qc ' 'I.q:TO' HO'U'P~~ (q = 1, 2 ) sat i sfy i ng th e n ex t cond iti ons. qc

I. There exist the majorizing f un c ti ons k f :T.X 'U , p -R q. s u ch that q qc qc qc

tx. 0

AA

0

-'tx:p ( Vu qc =s q (to'x O' U ) VP qc = 7. q (to'xO 'U'p» i t D+ V q (t,x) ~ ~ fq(t, V q(t, :r: (t», Uqc(t, V q(t, X(t»), Pqc(t» ( q = 1, 2)

(almost for al l t ) .

11 . There exist the ope n domai n A c Tx Rk~; qT " XqcS Aq and the continuation f qup (o n Aq ) of (· , . ), P ( · » b ounded on the func ti on f ( .,., U qc qc q A , superpositionally measurab l e

e~ch compac t

Bc

with respect to

q

t

and quasimonoto ne on Xc

fS (t,x )~fs (t,;;) for x qup c qup c c

S

-s Xc

~ Xc

-~

~ Xc ( ~ #s

( t, x ),( t ,;; ) E A . q c c Ill. For any func ti on x : [t l' t' lJ- X wh i ch is ab solutely continu ous on a ny segment [t , 1

't'J c

T and

)

~2l

Allahsis of Ihllamics alld COlltrol of Illte rco llllectcd Svstcms whose graph i s contained in dom tion

V~(.,Xl'»

for

this

are

sufficient

conand the

t he continuity

one - side Lipschitz condition for to (t , x)) , q

,t he superposi -

mation of Lemma I is executed for it .

is absolutely upper con tinuous in

the sense of V.M.Alexeyev( di ti ous

v~

"~

with respect

~

' ~ lC

wi th some formulas McLe

1,2.

=

The assert i on about CS, whose formula ~ is deri vecl. from the formula ~ by replacing va~iables with sim i lar ones possessing the index c, and al so by replacing type and final formu l as M'\ :R.

(.in par t'lCU 1ar,

<: cc)) is called the comparison system dynamical property, which is associated with DP ;p of the system (1.1). For example, te ' eoc u 'P0~; u , ~c=lVto€To3UcEl1c) [ ('IIEe ER + 3bc ER+ vXoc E te( clvXc E

}tc'" (X c

Wi th the help of f~ the comparison equa ti ons K dr (tl ~ =f..(t ,x c,tic(T ,X c )' Pelt)), XcEXq.c ~ R ~ (2 .1 ) are formed.

t

ER;c According to (2), for any toE To ' xoc Ho c ' UcE- Uc E

there exists the upper On-solution

PcEPc

Xc of

and this solution is continuable to the right(on T( xe ))up to the boundary of any B.The set

{:Cc} is

called the comparison system(CS) (rqc(t o ,x ' ue oc

E Tp:c )\JtE[\c''tc ) 'v'V~ xclt)J(Xc:::

rq,e

c

is one- point set).

(VC!

'

m par i

0

fun c t ion and

son

toxa u. 13 (\ixoc==vq(\, X/"t/u c = Sq,cto'xo'U)

~=1

~

(t,

real measurable functions on

, ... , £~q. )

x

xl,

stabilizability of( 1.1)

COMPARISON PRINCIPLE. DERIVATION OF THEOREMS ON DYNAMICAL PROPERTIES.

The notations are introduced: T ti.. Tq

(t 1

)

S T

t1

, . ..

'"

A

,Ha

formed domains !O~, ~

mf<2.

XO!"z

K

To

t'"

T, V,!

0, rq.:

v

lllJ'lnJ",c

mJl<-.}.;jf"

/' V _ llJJI
v

1)) /"'-

eq, = (f~

~ ~ P ,.....pc ' PelT) == &1 '3 (t) ;

the comparison equations a re chosen to be of the form : '("ct X 1-)(

uq.

dt L e m m a 2 . Under the conditi ons I,ll for the equation (2.3) for p (t,x) = 0 for the vector compari son function (VCF) (v

q , Sq ,7.'J,) (and f q,

=

f~

),

these conditions will take place also f or the sys tem (2.3) with p-J

(t , x)

'!-

0; (V'I ' s'l ,'~) is the

VCF for the comparison equation (2.4), and the esti-

oq

~

.

Tt

DC

), ..

are some how trans1D~1 precedes

.fJa",:ll\ ·· ·:mM nJ M ••.Jt1. J~ 1 v_ ~ re J-q,j''"' }(q, ~

are Lipscitzean

(2 . 4)

i

,

C VtE'Tt

is the type quantifier {~

~

T :· -t ", tj;

Eo

A , Ho ' K,P , ...•.. ... . ,

in the subformula CJ't=

{t

A'

P**, ... .

'.It-

le

=

, T'j. ( x ) = T (X)IIT'J,(t,), ... ,ll=

(2.3)

with the constants

s

e'l ~

PPo-

the controls do not enter under the universal quan tifier, and perturbations do not enter under the existential quantifier.

(rz"n),

P

A/'A_

:J'~c is the set in the space of

then we assume that

wi t h respect to

stab i lizabil it y of

It is supposed, as a rule that in the formulae ;/} and;jJc

To

depend only on t and are summable,

+L

.....

is associated with under perturbations .

..

VteT('X)nT(x )w (t,X(t))"" i: ct). c q. c If in (I.I)perturbations additively enter in (1.1)

ep

V"

\ipc:::

X(,v'XcE r'lC (To'X oc ' tic' Pc)

d :x:(t)

......

=(:>It,ET(+o)) · A - is an empty f ormula,

Then

- - - = F (t, x, u) dt

£: )]

.j}e:: toc'Uc ec o:.c bcXoc "Pc Xc tc :lJJ xe 1t c

v e c t -

s'l ' rq, )be the

be the corresponding subsystem of CS(for(I.I).

="lq,(\'Xo'U,P))

PPo-

is associated with

3.

tor

te 'tf Xoc E Pi-DC Cbe lV Xc E re (to' x oc ' tic) \.J" v ec E R+ j t1cE

"""""

(2.2)

L e m m a l. Let

"

(1. 1 ) ,

the in itial problem

Pc ) = {ic }

I;J X== :Xect))(Xc< £~ 1~(3o~E

E rclto' Xoc ,U c) VtETCrc)

0'

0

oc

422

\',

~!.

I\!at roS()\' ..,.. e RK

i st the vector

, VLF (v,s, 't), constant

+

De

~

~

k • k -ma tri ces P,A,B ,the functions f : T • R~ RI< , 1": T" RK;",V_RK wh i ch are con tinuous and bounded on eve r y cyl inJer T. O~ such that for any tEOT, x e X, lieU , p eep the follow in g co' nditions are sa ti sfied: ~ m,Q,x (pt(t,xll2... m~ V S (t,xl (0< ~< 00).

Jii<+

S::.1,K

C::·1,£

- Up• lto,Xol 11 2. for II'lr(i:o,x.,lU:s:f'I

(to,xoleTo'Ho' v ' (t,x,U,P) ..

~ACvtt,xl-rc) + B'S(U)(t,V(t,xll+ f '(t,V(t, X)-oc,S(Ul(t,V(t,X)ll (3Uc£Uc'v'(t,xc)€T.XclA(Xc-oc1+ BUc(t,~I+f'(t, Xc-

Dc' Uc(t,Xc))":!O P(X c- ocJ+f tt, Xc-Qc)'

f(t,xc-Oc)=;;O for X .... Ps,~O(j'i'S).(_1)S·I~1~,:,·p!~I>O 11 Xc- Qc U tE;:T C J Ps""PSS ""'S I -S (5::1,KJ, f (t,Y.-Oc)'!!Ej (t,Y-Qc) for y'S:.y~ y'j~yJ(j*S), ,{,y'ERI< (S=:1,"'i
oc.

l1c ~ slU), (VtoETo)'1: (to' 'P) c;; ':Pc'

Then the system(I.I) ei s PPo-exponentially bounded. Besides , if re: R+, ~c e: R: are preliminarily (the condi ti on of l ower boundeness of some type of vec tor-func ti on V~).

fixed and mg

The vector compa ri son func ti on (Vq , Sq , r q ) (q = 1,2)satisfying the conditions II 0,:IJ~ ,:IJ x,
so

p

S=1,K

tern (1.l) is

r S ~ ~.

(a

l )2., then the sysmin l:U exponehtially invariant. If a l-

C

PPo -

Qc = 0, then (1. l) is ppo- exponent i ally stable ( 0' = 0) and respectively (1.l) i s pp" expon,!;,.nt i al -

ly

stable in the whole ( 0=0, b=+oo) for f = O.

XoCj.

nov function wi th respect t o

DP~

for the system

(1. 1)

Vnder the condition of absence of independent exis tential quantifiers and under some natural assumption abo ut the order of entering of connected variables in the formu l a ~ and in the condition of domain's variation under the typical conditions etc. (see(2»the corn p a r i son p r n c i p 1 e is pro v e d for d e r i v n g cap r 0 h e 0 rem s o n d y n a m p e r tie e s . Let exist VLF (Irq , s q, , rq,) (q=1,2) with respect to DP of the system (LI)and certain conditions qc wh ich are sufficient for the existence of the comparison pro perty 11 in the associated CS,be sa-

The 0 rem 2. Let for the system (2.3) there exist VLF (V, s , <.), where IT satisfies the one s id e Lipschizean condition with respect t o (t,x) and the Lipschitzean condition with respect to x with the constant f feRK, s :U-U, 't:

-req.

tisfied. Then the system (1.1) has the dynamical prope rt y Shorter: (3Vq , Sq' r q , f q,) I,g,

1".

.1lxq '

Ill,

llxoq,lltq,lluq.,.1lpq

,:J'q,c

(q = 1,2)

,~.f',

., Th1S 1S

, expl1c1t

,

the slmplest form of the compar1 so n principle, representing the scheme of theorems on dynamical properties of th e system(I.I). The comparison principle is established with the complete foundation in the general form ; it is in tended for arb itr ary dynamical properties and mathe matical descriptions of control systems (2). It defi nes th e algor ithms of derivation of comparison theorems and theorems on dynamica l propert i es , which are implemented in the form of the software(prog ram package VLF-I) a nd gave the possibility to obtain hundr eds of th eorems compat ib le with new resul ts published in per i odicals (3). It is done for the first time, if one compares it with the known results in Art i ficial Intelligence.

-t

C

(t) j

'V''(t, X, tL)":!O t (t, V(t,Xl, s (U) (t, V(t, X)) for (t,x,u) e: T ~ X. V , a nd f up be quasimono t one and have continuous bounded derivatlves with r es peet to t,

j

x c;

(t,x c ' Ucl=:P(t, Xc )x-tQtt,X c )t.I. c ' r 'I. "c

UcE

where P(t , x

R,

0

5 Xc 5

R

is the k. k matrix and Q(t,x

c

) is

c

the k ~ r matrix, which both have continuous derivatives on T • R"up to (k-1) -t h order bounded in the ; ('/toETo lio sR)!8eR:)V(to , c O pOt (bJ),"Pt (b l={X ER~oe:xoc
cylinders T • 0'1.

'*

the admissible controls U e and the perturbations p of the comparison are continuous ,11.c s s ( U ), c 1 e e

Ct7\ETo VPcER+V£ER+3~ER:l'l(to,'P(E,p,i;,»s.'J'c (}3e J j

there exist continuous positively defined (for all t E T) symmetric (k. k)-matrices A(t),B(t)such that A is bounded, continuously differentiable on T and . T T x;CA(t)+A (t)p(t,x l+ P (t,X )A(t1+ B(t)J)(c~ 0

c

for

4. THEOREMS ON DYNAMICAL PROPERTIES .

"

for the equation (2.3) without perturbations

.ts,

r

~

'f- 'fc' Pcl't)=: ~1 ~

c

XERK:xTA(t) Q(tx 1==0

cc'

C

for t e T; fup

is

First of all we shall consider the theorems for DP wi th ( IT1 ' s 1 ' r 1 ) '" ( Vz. ' Sz ' r 2. ) = (1[', s , r)

quasimonotone for the stabilizing comparison cont rol U c ( . . ) . Then the system (2.3) is

(3) , where V

stabilizable under perturbations. If in this case the system (2 . 3) is considered without per turb ati ons, then fJP o- stabilizability up to asymptot i c stability takes place . Let there be defined xGeX,

are locally-lipschitzean with respect

to x, for wh i ch by (1.l) 'j)+V(t,

U,p)= Rim

At-O+

The

0

xlt))

~V'(t,x,

A\ [vtt +~t, X+ lltF(t,x, tl,P»)-V(t, xl}.

rem I. Let for the system (I.I)there ex-

PPo-

y (t,x) = II x

-xG

II ,

the sets

T"

=

T and

8 = {O},

423

Analysis o f Dynamics and Co ntro l of Inte rcon nected Systems ti be th e class of measurable functions of time p(t,x) = o.

Th e 0 rem 3. If the r e ex i st th e re a l k ~ k -ma trix P , (k. r ) - mat r ix Q, VLF (IT, s )for the sy s t em (I.I)without perturbations,satisfying the conditi ons: ,

,

[

,

1

V(t,x )-1r(t,X)~ Q t-t+lI)( -xII

]

'

•

t~ t,x,X€X,

lER:, V(t,X6)~O, vtTo,HokHoc' VlTo,H o )

is

bo-

und e d se t, (\ftoETo 'o'tETt V'xeX\{Xsr) V(t,X),* 0, , 0 (V'toETo) lie £ 5 (to,11 ), 1T (t .;,U)~ Pv(t.X)+ QSlto,U)(t)

( 55, t :6\

for t"o€To, 'tlEU,tt,x)eT 'X, Ps~~ 0 ("*S), rank [Q, ~Q,p2Q, ... ,l>I<-lQJ:: K all the solutions of the system (I.I)and the CS XC = PX c + QUe (t), licE ti,l' where tic is the set of ex -

tJ

ponen tia l functions of time,defined on T:= [to' (~

i s the time of occ ur e nce a t th e tar ge t poin t

x

),

GC

J ,

a re continu able on T"= [ to' t 1

the sys tem (1. 1) without per turb a ti ons is trollable .

then

p T* - con-

Another su ffi cient co nditi ons on different t ype of control l ab ilit y in terms of VCF and CS a re obtai ned in ( 14,15). For example , bo undary ( toU Ho (t , G, )) - contr ollabili t y

(3VEU\iXo EH 0 '1XE'l.(to,Xo,u))(t1 Elt 0(X.l~X(tlEG,cX) 1 of the sys t em (I . I/ with finitedimensional contro l u: T - X_ U s R for re striction (t. , x , u ) e D £: T ~ X x R~ wi thout perturbations lS'= c:p) (he r e X i s a n open set , a na l ogo us restrictions a r e intr oduce d fo r CS).Th e following s uffic ient conditions are obtained III

(11

*

1

(311E CRK (T.X13 JEC R" lR ) 3 U Ell 'It E'T'o'XEX.)

.

V

l1

,. (t,)<.))~f(t) t1

S

XE 'Z.(t o

'" X •

,x o ' u* )are continuable.

For the sys t em (I.I)without perturbations und er the conditi on s of ex i s t e nce of C-solutions continua b il ity on Tt = [t o' +09 the condi t ions of eq ui co trollability °wi th target se t G a r e obt a in ed (3UE'U 'v'cER~o3d.->O VXoE-P~ le) 'o'):'E«t ,x ,11) 3 t E [to ,1:0+.L] Vx = (t))(XE G).

is ope n, V

dUoEU \fteTt

o

RI(:

is con-

XC ~

s:'-+R~

a.:T)( Y--+R

K

,

V:

In solvi ng ap plied problems,the it ems conce rnin g the con struc ti on of vecto r-func ti ons and comparison systems are the most difficult. The decomposition aggregation technique for the analys i s of exponential s t abilit y was extended to int e r connecte d systems de scribed by the differential eq uation s in Banach space in (4) , and elaborated for some i nfini t edimensional systems ,incl uding also hybrid systems in (6, 10). I n thi s case , havi ng selected the isolated sub system$ with distributed parameters,the quadratic Lyapunov functionals a r e constructed by Siraze tdinov ' s method(5)or ot hers ;the co nditions of so l vab ility of t he linear operato r Lyapunov ' s eq ua tion in Gilbert space with unbounded(differential)operator are obt ained in (17,9) . Let the interconnected sys tem(I.I)b e represented in the form ,. • x. = F.0 (t,x. ,u.

t

L

)

l

1.

~( t,x,u,p ), .l=l,k +F.

,

()

5 .1

l

•

T.

xoc T,

vc'1.1 )It,x,Uo(t, Xl) ~ t(1f(t,X)), flO)

=0,

t(Xc);to for

fceXc)
Xc-:f.O,

f i s quasimonotone by Xc

0

L

ic

i ng the es t ima t es •

0

t

.

L

U

l

•

l

·0

l ]

'Si\Ul)(t,VL(t,xlll (t,X)€S{ li.=1"--;-K". = s · (1./,. ) are measura bl e r eal functions 'l/, : RZ-+ le l L LC _ Ri .

Then the vecto r functio n V= ( V 1, •.• on acco unt of the interconnec ti ons

V-1'-Y :T' X- RI(

i.L

,due t o the

who l e sys t em (5. l)h as the total derivative with respecttot.

...

,positively heterogene -

ous to th e power me (O ,ll, i.e. , f( A x) =fi.mf (x) for i\.> O, X E R~. Then the system (I.r) i s eq ui controllable and the time of the solution hit ( uO)wl' t h X'~"t

V et o , x Q

)

~

t o

x oc in G not more b

'

11

Xo

x o~

b is the cons t a nt depending only on f. yev) .

'

1-m

>

= Ft (t,x i ,u,t (t , x, )), X,EO Xl" Ei (5.2) there exist the solu ti ons xL of th e initial value probl em (5 .2 ), xl(t o ) = xiofor any x l~HlO' 1,I,i"~i on T~ ( Xi ~ and differentiable fu nctiona ls Vl: T' x ~i- ~ R and funct i ons s (. : 1tl -+U- sa ti sfy X

't/XEX \G)

.. RK (3XcE +)

Tx

no: To""

Y -+ RK •

l

(X '\ G )

{Xc €

U'

V Ct,X.,U.(t,X . ))': a L · lt,v (t,X . l)v (t,"'.t;.}.B.,(t).

tinuous by t and l ocally-lip schi t zean by x on

(\1'5 E R!.o 3XocE R:)1T (to ;p;o lO')) £:

11.

d.

The 0 rem 4. Let exis t nonnegative vector - f unction v- : Tx( X\ G ) -+ RK, continuo u s f uncti on f : Ror< .... R" s uch that the se t X\ G

Toy, Ho

5. VLF CONSTRUCTION

&

o o

)(

For " isolated s ub systems"

(VxeX\G1)V(t"xl~ ~ f(t)dt+M,(VxoEHohr(\,Xol~M, the so luti ons

ditions was obtained und e r vec t or f un ct i ons

E;. are Gilbe rt spaces with the scala r produc t <",'

.,)lt,X,U

o

suit - evasion problems in the differen tia l games and on o ther int e r es tin g dynami c properties for bo th non-l inear int erconnec ted co nt ro l sys tem s with di st ribut ed parame t ers an d di ffe r e ntial inclusions were obt a in ed . These theo r ems give suffic ient condi ti ons of presence of these dynamical properties(DP) in t e rms of VLF(2,3, 10, 16). The problem of the co nve r s i on of theo r em on dynam ic a l properties is import a nt. The conversion of the compari son principl e, the possibi lit y of u sage of di ffe r e nti al equa lit ies for VLFs we re studied in (12,14). In (3) th e deriva ti on of theorems on universal dynamical properties E~ )with necessa r y and suffi cien t con-

, where

(A.V.Lake-

Similarly,some new theorems on solvabilit y of pur -

I f it is possible to constr uct the es timat i on

424

V. M. Matrosov

air' ~ \{ ~ j I< ~ <-ax.L,F.l (t,x,U(t,x),plt,X)"".L:a. .. (t,lflt,x))v (t,X .) + ;LB (t, l . K J' 1 lJ J" l)

v\T,X)l'lLctt,t/(t, x.ll+~ d o, P (tl J . J" j~, II Je

i..=-1,i<,(t,x)e:-T.X (d.~constl lJ

then we ob taln

V- (t , x, u,p ) ,;: A(t,

V (t,x»

·u.c (t, V(t, x»

V- (t,x) + B( t, V (t,x» .

+ D

lization y (';!: ) t, X ( ~1 ) ' u

Pc (t )

=

-\r(2?, t ,X(,?,», w(J;;)

(?,»

belong to

Yt ,W t

=

Ul ( ~ ,

' re spec -

tive l y . The function -\ri s supe r positionally differentiable with respect to t , i f XE)[ the supe rposi tion 1)'(;;: , t ) = -Ir(,? , t , :x(~" t » E c""(Q T l

where

a lj (t,x c ) = aij (t, Xc ) (jh) ,aLL (t,x c )

and there exis ts the function

o

xc) + a , (t,x c )'

:e(~,t':X('?l),j!(>l»; Q,' (j f.

Xt'

C(2)-Y

such that the equal lt y

Hence,the equation of comparison fo ll ows: Xc =

Suppose they are correlated wi t h the sys t em of comparison, if (VteT V :x: (~, ) EXi: VU(~,)E UJre a -

holds. The expressio n written as

A(t , x c )x c + B(t,x c ) lLe (t,x c ) + D Pc (t)

with additively entering comparison pe rturb ation s of the type (2.4) , bu t wi th p : T - - Rk . c For nonlinear interconnec t ed systems,described by ordinary differential equations in R~ and functio nal differential equations,the decomposition '- aggregation method for construct in g vector functions and comparison systems is implemented in the form of the program package VLF-2 on IBM compatible software. Th is package is used for solving a number of concrete stability problems for mechanical and ele ctromechanical systems, control l ed multiconnected systems, for analyzing immunological models,etc; (8,15) .

.:to:,t, X(!;:,) ,

tC;;:l' t,X(>,l ut?;;,)))

and deno ted by d A

1)' ( ~,t

,:x: (2?, ,

cal le d the total derivative of

\r

due to the system

The 0 rem 5. Let {~~, w~}be the mi nim i zing sequ ence for le on ~, and let there exist functions ~, ~ of

the above structure such that

1) dJl V'C.,;,t::x:\;;:,11Idt=

~(!;;,t,"'(~"t, :r.(~z », W\~"t,X(~z)' 'U.(>z)),

2)

Je (l;,t, ~ (l;"t,X(~z)l, W(>;:l't, X (?;;'Z)' U(?;;'z mEQc(~'1') J(~,t,xl~,), U(j;:,l18Q(~,t). l,<;:;t)€

Let us consider the problem of minimization of the functional I = F( xl);, t K ), u (~,tK» on the

is-

A.

6.0PTlMAL CONTROL. The i deology of the VLF method is extended to the property of optimal control of systems with distributed parameters(7 ,1 3)etc .). There is the association of this approach with V.F.Krotov ' s theory of sufficient conditions of op t imality,which are of the s i milar significance in the optimal control theory as A.M.Lyapunov ' s theorem in s t abi lit y theory.

)fcit

0., ;

3) (V'.L= 1, ... 3 [xJ..' UJ..1E 1)) .;T(>;:,t, X.I.. (>;:, ;t)lo.~,,(. (~, t),

W(1;,t,X~(~1,t), Ui;:l't))::~(~,t), (~,t)EG.T; 4)

G\"'(~,tK,X(3;1 1), W(>;;,t K' X(~1)' U(~1 ))l-F (x(!71)' U(~,))~ 0

set D, composed of pairs of functions [x(~,t ), u (~, t )] s ubj ec t to condit i ons :x.e:X, ueU A :Xt(~, t) J('!;; ,t,

=

f( S,t,:x:(~"t ),

ut'?"~

t »;

:r.(~" t), u(~"t»E:Q(~,t )CE-t>

5) on th e seq uence

{x...,uJ.-} ( J.,,= 1, ... ) the left

side of the inequality 4) L e ~ds to zero. Then the sequence {XJ.. ' uJ..\i s minimizin~ for I on D.

(>;;"t)EQ1"

Here, P.Levi notations are used,

X

,U

are the

sets of belongness of trajectories and controls; f (~,t, :x.(~, ) , u(~,» (J resp,) is the functional vector acting from

QT x Xt,Ut

in X (in E, resp.),

are the sets of belongness of states and va l ues of co ntr ol; X,U are Eucledean spaces with dimensions

nand r ; the case U = U corresponds to con tr ols, depending only on t . The problem of comparison: Ic = G(y (~, t K ), w( ~ ,t K »),

JJ

= [,/

(~,t), W(~ , tl} : 'It t~,t) 0. ~(~,-!:, Y(~l,t), W(~"t)), J (>,t,

In (7,16) another class of compa r ison theorems i s stated , which is based on usage of t he const r ucti on of t otal derivative due to the comparison sys t em. I n (13),these theorems are extended to important pracricaly optimizing problems of improvement (relaxation). I n (2) comparison theorems in opti mal control, which use the conditions of the type of differential inequalities were also obta in ed. Th i s approac h was used for the purpose of investi gations of 0 p 1 m a 1 c 0 n t r 0 1 0 s p a c e - a g e f 0 r e s t s s t r u c t u r e in (7) .

c

V(>"t), W(~"tl)EQe(>,t), (>,t)EQT' 'fEY,

W

EW.

Let )l:EA be a point of plane,

Int roduce the f unc t ions V'( ,?, t, X (2?, »

:

Q,'X-t-+ Y,

be the characteristic dimension of a tree, n (t, p, ~ ) = (n', ... ,nl m ) be the vector of space -

425

Ana lysis of Dy nam ics and Control of I n tercon nected Systems age density of a stand of timber, i

=

;-:--Im be

the indices of tree breeds . The dynamics of distribut i on with respect to appro priate dimensions the processes of renewal and propag~tion are described by the equations: l i i. i at +ap(1f (t,p, s ', n) · n )=- (t ,?, ~ ; n) ·n -u(t,p, g) (6.2) 1 n(tH,p,~):nH CP , ~), pE [ D. ,

i

an' a

pm ,

. n' (t,A ,"g)= ~

~

p~

A 11

Here ut

1 n) n (t'Pl'~1)dp1d~ (6 . 3)

i.

1\(t;~,Pl '~1 ;

are the rates of fellings(control)fun • ) , v i ( • ), b~ ( • )have the sense

ctions d L

of coefficients of dying, of velocity of growth of trees and renewal,respectively . The struc t ure of these coefficients is the operator one ( there takes place the dependence on integrals over p of nand essentially nonlinear.

terns of ordinary differential equations are appli cable t o it. Numerical calc ulations revealed the fact that the optimal densi t y of a s t and of timber n ,,( t, p , J1; ) gives i ts qualitative structure in the form of two or three layers with t he average diameters of approxima t e l y 14 , 32 and 50 cm. The similar multilayer struct ure was realized in the middle centu r es in Europe for surbur b forests in order to obta i n the maximal volume of wood from regular selec t ed felling. This optimal solution would be expedient to re alize especially, in the regions of exhaus t ed re source of wood . The distributed optimal control of rotations and spinning osc i llations of the flying wing was studied which described by a nonlinear equation and the following conditions:

a

02.9

ae

lo(~) at2.::: ol;; (GI(-s) o~ 1+ mK(~ ,t,e,8t) +ll(~,t) ,

The criterion of optimality characte r izes the s t ate of the resource " forest " at some final time moment. Lm p;' i. . I

~I~ ~ WKCp,lO:ln\ \ .p, ~ldpdJ;:,(sup ,

=

(6 . 4)

A Cl.

t -

t.

~ ~

i

.

i

£ S \WCp,~lu'ct, p , ~)d~d'pdt :: Po ,

t~

~

Cl.

Here

W~

(? , ~ ) ~

0,

w'-

i.::.Dm '

(6.5)

( p,!E ) are weight coef -

ficients(in particular, WC ( p , ;E ) are volumes of wood in one tree of i-th breed , having the dimension p growing at che point jii ). The equations(6 . 5), which have the sense of assigned general volumes of fellings (R " )for each tree breed on the time in terval [ t , t ",] . Thus, the problem is to ful fil the pla~ (6.5) under the condition of maximiza tion of the stock of timber resource I at the fi nite time moment

tK

As a result of successive application of theorems on JOlnt optimality with nonlinear vector mapp ings {Jo , Ul it is found, that the problem solution has the main- l i ne structure . Besides, t he mainline n .. (t, p , "ll> ) (optimal density for t e et H ' tK »

corresponds to maximal productivity of a

stand of timber and is obtained as a result of sol v ing the following auxiliary con c e n t r a t e d optimization problem: to maximize the funct ional . pi Lm

m

'

lt~=~<{~ B\t,p, ~; A

2.

2.

l =~ [.L1(~)l9l~,t)<.)) +.L2.l~)l8tl;;,t,,"n Jd~ --- Lnf. o L

The additional restrictions are: n' ""O,

i

I1)n C t, p;E)dp + i.

Jll(S,t)d!;:= UZ;(tl , o where tl( is the finite time moment, uL:(t) is the additi~nal (t) l~

U

control, subject to the restriction !u L . Al l given functions are conti nuous and

.

differentiable the necessary number of t ites . "Mi nimizat i on of the functional" is understood in the sense of finding the minimizing sequence. Application of theorem 5 allowed to reduce the prob lem, to delete the partial derivatives with respect to the dist r ibuted parameter and the integral restriction on the control, so that to get the next essentially simpler problem:

..

I =

L.

1

'Z.

(If

It

l-

a·[(JW-(\)+~ lo (~)1( (S,t~ )d~J + ~

--

inf

o

... { 1

t

1..

at the set D = 'l. (>:;,t),~ (~,tl ' JYI (t) . ULlt) i: l!~(~'~,,) ,

.1 7~(~,t) =I ~~) m K(~,t, ~2+bA(~) -\roJ, ~'(~'+1I) ::: e1(~ ) ; o

r

~ d"'t ::: J I o

f

1

0

(~)7. (I;;,t) ds+f'1 (t), ~*(\l ::: jl (~l·e ~)d~ 0 0

d,M(tl

-

0

2

(it=uL(t) , jlf(tH l::: O, IUL (tlk U ; ~ (~,tHl:::eo(~) '

+ B~(t,~ ; J1~(Pm)) '" (O,t)='" (L ,t)::: O . ~

~

Here

is the scalar continuous and sufficiently smooth function, for which (!: ld ~ = 1, the function

SL.c

o A

y Z (~ , t ) is the control ,o.and r-const . The later problem is substantially simpler than the initia l one. So, after some modifications, the known numer i cal methods of op t imal control of sys -

It is solved numerically on the basis of known me thods.

V. M. Ma trosov

426

The op t im~ trajectory of initial problem is obtained by the formulae e(>;;t)O:W(~ ,tl+8A(~ltro(+ ' ~ , ~ (t ,3:; ,)) , , {'i2(~,tl\tE ltH,t".l , ~ E [0, LJ

w ls t)= ~ .. (tKl[ll!;;l_ Oll t>;;)]+J..

,

r

0

6"lo(~)

8 ol!;)\t=.ttj'

°11(>;;)

6·1

(>;;)

Ito:tK,~E[O,L]

3:;E[o,Ll

On t he basis of VLF method investigated the problems of mUlticrite r i al se l ection and optimal control with the vec t or criterion of quality by finitedimensional control -u,. : T. X U '= R~ without perturbations with the restriction (t , x , u) = (t, x ( t) , UC t , x( t ) »

So

DE T • X' U

Notations are 21? i s the set of solutions ( x ,u) of eq . ( l.l )for t he indicated restrict i on, ~c is a sim ilar se t with CS for the restriction

x/t) , Uc(+, Xc(t))) s; Dc sA '

R~ ~

The terminal criter i a of control of quality are cons i dered: m m J : ;;E - + R , J c: ;;Ec - R , J ( x,U)= J(X(t,)),

tlt(:t ' Ucl e;;e )J(:t ,Ul:J ( X (t )l, c C e r e c c,

dom J c ~(.

:J : X_R

m

,

'J C:(RK)_R

m

V A(t»)'U'c.(t,E T is the fixed time moment)

t "''T

The solvab i lity of the optimal synthesis problem of unimprovable(with the accuracy up to £ '= Rm)control is consisted of the establishment of existence of sequences ( x" u , ), ( Xl,u ), . . . !;; ;;E of the admissible l

solutions of (1. l)which has the following

proper -

ties:

l)For minimally admissible values of quality ind ices kiE R: with respect to c r iteria J L (i = T;ffi) K= =(k 1 , .. .. . ... ,k rn ), ( 3neN:; { ',2 .... }\fseN : s:> n) J(:x: s ' Us) = J(X S (t , J) " 2) (\t EER: \ {0}3ne-NltseN : $::> n lj(:t,U)E~)

K;

J(Xs,Us)+E '" J (:r , U] V (3 i= 1, m) J. (Xs.u s )+£i'" Tt (l:. lA] (i . e . for arbitrary small E and large s(>n)(xs 'u ) are £ - effective(with respect to the criterion J) solutions of the optimal synthesis problem) The analogous properties le)' 2c)are introduced for corresponding sequence of admissible solutions( x ,' c u c , ), ( :x.Cl'u CZ ) ' " .of the comparison equation(2.1) (me = m) Consider the VCFs( v ,s)satisfying the conditions of the I - st,II- nd,III - d types, but with s* :(T . E- ~ K

"R

) _

quence {(x sc ' u )} t he presence of proper t ies I) , sc is de2)for the seque nce i<:x: ' Us ») Vc = I, ~ c

=v

duced (S.N. Vassilyev [1 5J

o I 0 (~) whe r e J. is an arbitrary constant, y =(w,y.. ) ( A. I .Moskalenko , N. A.Ovsyannikova, 1988 ) .

(\;ICX: , U)E;;El

Then from t he satisfac t ion of le ), 2e )of the se-

s

L Z _\ bA(>;;) d 6-J-2-- >;;,

(t ,

for absolutely conti nuous x : T - X admissible UE E U the f unction s* ( ., :d · ), u( - ,:1: ( ' » ,- ) :A-+ - - > R~c belongs to 1ic .

~~c , D(t)" A(t) '= (dom s*) (t)

and

assume that U e (t, V (t,x» = s *(t,x,U( t,x), 1{'( t,x»; a function f( o , • , u e ( · , - »is satisfied the Ca ratheodory condition in A for ~ ~ ~c ' solutions of CS are continuable on [ t o ' t ,] . Corn par i son the 0 rem. Let there exist the seq uence (x ' Us )} ~;:e , VCF ( V ,s* )satisS : fying the conditions (3 neN'tISE N : s > n) Jc(x s /t:,ll~

J (xs(t, 1), l'

(\I(t , X' 'U)€Il\lx EA(t] : \Jlt,x)~ X J(t,xc' S (t,x , U,XC )]E]c' c e

('t,f XE X):J (x) ~ JclVlt, X)),

REFERENCES 1.Ma t rosov , V. M.(1973) . Method of vector Lyapunov f unctions in the analysis of interconnected systems with distrib u ted parameters(Add.preprint on I IFAC symposi um on control of distributed paramete r sys tems, Banf, Canada, 197 1) '_ ~~! 2~~! ~_ ~~~_ ! ~!~~~£b ~~_ ~~

5-22.

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