: v e l 6 ~ t. U~mber 2 " ~tST~t~ & CO.'¢IX 0L L ~ ,~ug~,t Z9St

Fin i t e e scape t ime for Ricca t i differential r eqhat ions

~ydc ItIARTl N *

Rm+~'d 11 ~-l. l 1981

1o + p + ~ me ~ + c m ~ ot ~ m + t i d i [ t + ~ mmgon is stmim and {I ~ m + ~ mat + t a l c ml,gom if th O obL + tiC, ls ldwl~, l'wam or ~l~ys ~=rwJ t L

K~,m'~: pdmti dll tmt la] eqmdm, modic mtutl m F~ nlt: +

I. Inlrodlt~:tion

Consider the qmdrat ic functional

d (u ) = f f ( y ' y + Wu) dt+W(r)S~(T) (1+1)

snbj~t to the ~ns t ra in t

J = A x + B u , 5=Cx (12 )

where the system is ~nt ro l lable and observable. This is. of cour.~e, ode of the most studied objO~L~ in control theorY- Kalman'$ ~eminal paper of 1960 a~x~'ered the f ~ d m ~ t u i quc~tio~ ,:onc~ning the r~ul t l t |g opthr:d syst crfl hut raised ~verui intriguing po~ i b i f i t i ~ In ~ m = sense Kniman's recognition that the matrix R i ~ t i equat{on replaced the Hamil lon-Jaeobi equations ~ ~ d is. the moat fundamental r~t t i t in the hnear -quad~6e optimization l i t~ ture~ Retail that the Pdccati equation ~,.~=i~ led with (1.1). (1.2) is

~ O ) = - a ' ~ ' ( O - e { O a - P U ) a ~ ' ~ ( O + c ' c . e ( r ) = s . ( i .3)

In [51 it was shown that if S = 0 mad Tgoe~ to infinity. P(t) converges to a positive definite equilibrium ~:nt ion. P. of (I .3) and that this mlution genemt~ the optimal control, namely

u ( r ) = -- B'ex(,).

Furthermore i t w ~ sl',6wn that the Solution of the RJccati equation exists for all t < 0 and i t is ea~d to s ~ that in ~act l lm ,_±=P( tp =P. H o w l e r , the t e ~ k n ! q ~ t~exl seemed to imply that thor= might ~ t a 7- >. 0 such that P(3 '~) did not exist, the finite ~ p e time ph~omenm

It i:* t~efni to e.r, aqfin¢ a special ~ namely when the u - j s t~ (1.2) is ~cui~ and 2 = u. The t~ul t lng Pd ~ a t i eq~tion is

p = l , p a . p(O) = O.

T h ~ ~ two equilibrium so lu t ion p = ~ I. SO ~ ante that any solution mJth initial condition irx the i n t ~ a l {-- 1.1) dots not ~ t p e but is b o n d e d for all time L In part imlar the P d m t i eqmtioa of optimal control is bounded mr al l time. On the othe~ hand any solution out~de of that i n t ~ ' u i ~ p ~ in finite

• S~m~ m m by N ~ G ~ N ~ ; 3~ ~ ~ A O 2 - ~ ONR e ~ a t ~ Nm N ~ I * m C * l ~ , ~ ~ ~ N = I~ACm.~0rtt~L~S.

0167-6911]81/NXI0-0000/SOZS0 ~ 19~1 No.'t/a-Holl~d IX7

podfive t~me ,X ~ rm[ e n~ a ,~e time. In fact if ~ lln~ is pro~f i f icd . ~ have t~e fol]o,~ing plcturc:

,C> It is not t~ tu l for the on . ,~Sm~iona l case to be ind~cad~ of the sltuati6n for .1Afh dlmcr, sions but "~ the c~c of the t]c<ad ~ u t i o n it win be s.~own dzat the ol;c-dimc~[ona] ~ is typic~.

Z Cano,-Jcel c h t m and ftnlL. ~ p e time " . '

In t~ s ~c~an ~ ~il! ~ the r e p ~ t a ~ o ~ of th= I ~ d e £ ~ d o n ~ a ~ght i n v ~ a n t ~ 1 o r ~ d on t~© man;foId of La£rangi~ ~bsp~ccs of R 2~. ~ vicwl:*oint h ~ bcc~ u.scd ~ t ~ [ v e I y in [6,7,.?.,3,4 ~-

lta~c~x[ly, the i d ~ of Grassmann and L.~grangi~ G ~ s r n a n n manifold is that w~ ~ thL, tk of one-parameter I 7"oups of m a t H ~ ~,.~n g on the set of subspa~ s by l l n ~ ~ f o r m a t i o n s . If we t~cst ~.~.t ~ ¢ s't;bspact~ t(* be of f'~cd dlmen~on they ~ be patumet,'rized Ioctlly by ~ t r i ~ ~ d L ~ . l ~ a l paramet erlzl tion~ form a set (3f charts for the ma~fold. ]n fa¢ t the charts so obtained are open ~ d d e ~ e m d so the O r m ' ~ m n m,anlfo!d ~ be Lhousb t of ~ a ~m~'l~fi~fou of a ~ t of ~ t ~ For o~ timaI control ~-c are hb:~csted in the ~ of n-dim~sioual ~b~?ac¢~ of R 2. and the G ~ manifold is the compactifi~ tim| of the n X n ~ t r i ~ The Lat f a n , a n m a ~ fold ~ be thought of ~ the ~rres ix .ndlng compactificati0;I of ~'nxmeh~ matric~. In beth ~ th~ gicc~ti equa~ons arls¢ by con~de/~g the n~resentaticr~ of the action of the ode-paramet .-r groups in the local coordinnte~

Let

be the ~f'mh cslz3~y syt?plecdc m t l ~ r '~so~at c~ ~ d l (:.3) ~ d let e m b= the ~ , ~ p , ~ d i n g o n ~ - p ~ e ~ ~'oup. ~*Vith cv¢~ symr~etri¢ ~ t z ' i x S is assOC~led the L a g r a n ~ sub~;p~

[(~xx) . . . . . ~ p . . . . . . tesp~) T h ~ ~ t h t~c [n~ti~ ~ n c ~ t ~ J; for (1-3) ~ ~s,>date the ~ r v c

~ ' 1 ( 5 : ) . . . . "1 ia the Lagra=~ian w~ifold. Ev~T finite point of tAe solu~it.~ of ~1~) is associated vAth a subspace of the above form. So t h ~ is a f'mitc escape tirn~ for t~¢ (~bh with in~d~J posldo~ S if[ there is a nonzero x ~ R" s ~ch that

for sor~. ~ ~ CcState. FoUowlng [8] we c~u wrY= (~ .I) ~n the con~x~ient form

We ~ the ~f .U~¢ u~fo~d~

Velm© 1. N~mbc¢ 2~ SY ST~'.~ & 0. '~ FROL ~ ^~.~t 19SI

~: ,i(~ ° , o ~ , , , , 0 , ,)( F ,) (, , ) ( , ~)(~:) (o ~) where P is the urdqo¢ posltivc d efmil¢ solut ion of the Ric~.ari ¢qual ion; thus we have (2.2) ~'rl t tcn

- - B B " x 0 ,0 ,~p(:-"'P -~ . . . . ".,')'(~S-F,~)=(0) ~2.3, where t~ "- BB'P is a sgahl¢ mat r ix (all e i g ~ i n ~ in the ] e f t - h ~ d p I ~ ) Let A d ~ o t e A -- BB'P.

W e can eva;uaze

~(~ - 7 ) , by }hi•grating the co r r~pond lng d i [ f~ raz ia l equat ions to have i~ closed forr~

o _ . , ) l ( s _ e ) x j = t o ) . (2.4)

F rom {2.4) ~ then have the following p~pos i t i on :

Prog43slt'ion 2.1. The Ri~ali equation ( | ,3) n ~ ;,~itc p ~ i r i ~ ~eape time iff there ~i~ tx ~ x E R " cad a t ~> 0 such tkat

[ , -12 e-*'BB" ©-';" d s ( g - - P ) l . . . .

The fol inwlng t e~Janical I ~ wil l be ~ f u l i n the sequel.

L ¢ ~ 2.2. Let x ~ R" ~ c h d~r x" x = 1.7~en gieen any T > 0 she~ is a finite r ~ c h that

x" f r e -2"BB" • -2", ds x > y.

Using co~pactne~x there ix o finite t ~ e h that for ¢11 z o f n o ~ 1 the inequ~di o" ix zolisfied.

Remark. M ~ y ve~'~io~ of t t ~ lemm~ ~ knov~n in the f i t ~ t u r e ~ see [.~ .~.'d [11 for ey, za~pl ~ The proof is ~ y ¢~sy ~ n s e q u e J 3 ~ of ~ n t r o l l a h i l l t y and the stabBity of -~ ~ d is Ieft z-. the ~ d ~ .

3 . M a l n t h e o ~

In thLs section a complele classlfication of those orbits wi th f in i te ~ c a p e t ime ~ U be g i ~ and se-,eral consequ e.nc~s of the cla~sLqc~tion ~ J I be considered.

Le t

" ( [ ) : I + ( I ; ~ - - ~ l " ' ~ l ~ d ' ) ( " - - S ) .

From P ~ p o 6 6 o n 2.1 wc have that z( t ) is ~ n g a l a r a t t o i f f tbe orbi t o r the PA~at i e~aati,~n (1.3) ;'.as f ini te ~ p o r ime t = t o. Let ¥ ( t ) = ( P -- S)=(t). Y(t) is symmetr ic ~ d f rom L ~ a m a 2.2 we have thaz Y( t ) is evc~tu.zIly n" ~negat ive defini te ~ d Y(0) = P - - S. S i n ~ f~ e -/hBj~" e -'; '* d$ is pos i s i~ deft'mit e ~ Ires• the fo l l a~ ing l- .mma:

Lemma 3.1. T,~ere exist x and t o ~ that Y ( t ) x = 0 at t = to if]" x ~ Ker(~ -- S ) ~ P -- S has a negati~ ~genoalue.

t ~

I , ~ i . ~ 2 ., s-~ E r ~ M s A c o l r r R e L L E r r I ~ . S - ^ u ~ t 19al-

PleoL ~ P - - S has a ~ ' ~ . l i ~ ¢ig~valuc then Y{O) d ~ ~ On the o t h ~ b ~ t h ~ is a t o sucl! that Y(~o) has no n e ~ t i ~ ~ $ ~ v a h ~ k ~-~1~ Y{t) is ~ T , ~ e ~ ¢ its ~ s ~ v ~ l u ~ ~ real ~a~ since th¢ r ~ g ~ ' a l u ~ arc coathmous f ~ c ~ o r ~ o f t ~ ! ~ m ~ t be a ¢." s[k~l: d ~ t d im Ker Y( te ) int~ases.. - : ,

compl~liel~; cthar~.ct erlaes t h ~ e orblts thaf have finite e ~ p e tim ~. Note that w ~ t The foHo~-:ng thcorcal ~ really counting is t he tumbcu" of limes mt osbit intersec~ the ~ o r a ~ h3~ersurface in the

L a g r ~ Grassm ~ manirohL A.,.~o see [lO] for related derivations.

:3,2. T~e O~l t o f the ~ f ~ a t i eq~ti~ (13) ~ ¢ $ infinite'in f l~i te p ~ i t i ~ t ime I f f P - - S h~" a negative ¢igenl~lue. Tht.numb2~ o f poi~ta at which d~e ~ be t imes infinite is finite.

Proof. F~r~ ~.t~ame ¥ -- S has no z~o ~geav~ ~e~ ~ e n

is a ~ ~=mcu~c ~ t t i n ~ d for large t is i ~ t h ~ definite by L e m ~ 2.2. If t = 0 it has caczcfl~ th= numbe~ of negative e~gen~'alu~ as do¢~ P -- S. "~['int$ there ~ at ]eax[ that number of sEnguhris[ ~ ( b ~ a u s : ~h~: ~ l u ~ of .*(tX, ~ -- $ ) = z ~ ~ ~ l ) . O a th~ oLh~ hand b~ v [ ~ 2.2 z ( t X P -- S ) - t is ~ l t u a l ~ p~titix~: dea'm[ re. Thus Ihefe are a~ mo~t finally many ~

If P -- S is s ingubr :he ~ u u l e ~ t is more t ed~out. One proceeds by co~ide t ing

( e - s)~( , ) = ( ~ - s ) + t e - s)J.' S )

which is ~mmeu'ic emd eventually uonnegaf i~ One a rgu~ by diagonalizing ( P -- 5 ) and ~ , m p ~ n $ r~ull sT~a~ f ~ P - S ~ d ( ~ - - ~:):(t), ~ l e detai:s a ~ Ir~t to the ~ d ~ .

. ~ ( t ) = - - ~ ' P ( t ) - P ( t ) A - - P ( t ) B B ' P ( t ) + C C ' , P(O)=O

Proof. It ,,wa~ origlnaly s a ~ in [>] 21at the ~ lu t ion ex~:~ts for all t < : 0 ae_:l T h ~ 3.2 prOveS the o,lb:t h a l l

(~roU~r~ 3 ~ ~'he peno~?c ~ a i ~ o f ".~e R i ~ t i e~uatl~l a ~ e i ~ " alwa),s flnJte or ~ a l~) ,~ contained !n the ~ ) ~ r p i ~ e ~[ infini~F. :

Proof. By petlod~c~ ty, ~ .~,. c~,capcs onc~ i t ~ p e ~ inrmi:ely ~ n y t i m ~

Remm~ This was in ¢ ~ n c ~ p:~nted ~ t to me by J ~ 9till~x~, {9] in the ~ that n = 2 ~ d in ~xne ~ l ~ the argnm~lt h¢~'e is j ~ t a g~erallzat/o n of bis p ~ L :' " ':-" "

R e f ~

v ,a~== i . Nu=~ z s v s l " ~ , i s & COntrOL LEtt~ ^ . ~ t i~st

+ Eds~ c~,~naea/~fmhoda l ~ a ~ ~ r o e . O " / J ~ S ~ , ~ (RdacL D o r a l ~ . ~ t 9~0) pp. 195-21Z

18} ' I ; S u t g a w ~ A nc~k~auy mJ~d ~l~ci= =d d~ Jar d= ~a= o [ L~- ~J=d c q ~ t i ~ to Ix: .etiodi¢. IEEE T = A ~

: ( ~ a ~ tt9to) s * 4 - ~

It0] LC wnlem~ l .co t z ~ m r ~ opdm~l ~ n l m l ~ me ~ $ & ~ ¢ m ~ lI ~ d ~ IEEE ~%InL A ~ C ~ l M 16 ( t ~ l } 621-634,

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