Rend. Sem. Mat. Univ. Pol. Torino Vol. 48°, 3 (1990)

Dynamical Systems and O.D.E.

A. Halanay

A D V A N C E S IN L I N E A R C O N T R O L T H E O R Y A N D RICCATI EQUATIONS

A b s t r a c t . Results concerning Riccati equations and their role in the control

of linear time-varying systems are reported: complements to the old Kalman

results, optimal stabilization and input-output properties, solution of the two-

block problem, inner-outer factorizations, best stabilizing compensators for

stochastic systems.

The results reported here may be considered as an addendum to the nonograph of Roberto Conti "Linear Differential Equations and Control", [5]. All of them have been stimulated and motivated by new ideas in linear control theory essentially connected with so called /Z"°°-control.',.

1. A n old result revisited

We shall start with a result of Rudolf Kalman published 30 years ago [17], in a most stimulating paper.

THEOREM 1. Consider the Riccati equation

W + WA(t) + A*(t)W - WB(t)B*(t)W + C*(t)C(t) = 0 .

Assume A,B,C are continuous and bounded on 1R (continuity might be

252

relaxed but this is not relevant for our discussion). Assume there exists 0 > 0,7 > 0,tf > 0 such that for allteWL

XA(s,t)C-(s)C(S)XA(s,t)ds > 7 /

/

t+8 XA(t,s)B(s)B*(s)XA(t,s)ds > pi

(uniform observability and uniform controllability). Then there exists a symmetric, bounded on III solution W+ such that W+ (t) > kl(k > 0) for all t E IR, and A — J?J3*W/+ defines an exponentially stable evolution. This solution is maximal with respect to all symmetric solutions of the Riccati equation defined on III and it is unique with respect to the properties above.

Proof. The assumptions concerning A and B imply existence of a symmetric solution W bounded on IR, W(t) > 0 for all t 6 HI, maximal with respect to symmetric solutions of the Riccati equation defined on IR (see for example W.A.Coppel [6].

If we write

W'(t) = -W(t)[A(t) - B(t)B*(t)W(t)]

~-[A- B(t)B*(t)W(t)]*W{t) - W(t)B(t)B*(t)W(t) - C*(t)C(t)

we have the representation

W(t) = XA_BB.w(t0,t)W(t0)XA_BB.w(t0,t)

- [tXA_BB.w(S,t)W(s)B(s)B*(s)W(s)XA_BB.w(s,t)ds

~ I XA-BB>w(S^)Cm(S)C(S)XA-BB^w(s^)ds Jt0

from here we deduce

W(t) = X*A_BB.w(t0,t)W(to)XA_BB.w(i0,t)

- J ° XA_BB.w(s,t)W(S)B(s)B*(s)W(s)XA_BB,w(s,t)ds

XA-BB*w(s^t)C*(s)C(s)XA-BB*w(sii)ds •

We prove now that there exists a > 0 such that for all / € HI

rt+6

equivalent to

> a

for all f. with | ( | = 1

If the property were not true then for every a > 0, f''wou^. exist with |f | = 1 and

rto+6 I eX*A_BB.w(syt0)[W(s)B(s)B*{s)W(s) Jtn

>to+6

'to

+ C*(s)C(a))XA_BB*w{s,t)Zds< a .

Denote XA-BB*W(SI>^O)( — x(s) a n d remark that

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x'(s) = [A(s) - B(s)B*(s)W(s)]x(s), hence

x(a) = XA(s, t0)( - [' XA(s, o)B{

255

We have

/ XA(&, a)B(a)B*(a)W{a)x((T)dt Jin

v l / 2 <

1/2

(^J'\XA(s9a)B(a)^y (J° \B*(v)W(a)x(*)?dS)

= ( [* \XA(s, a)B(a)\2da) ( / * x*(a)W(a)B'(a)B*:(a)W((r)x(

( / \XA{3,0)B(°)\ d 7 - 26zl2n2cliBe^a

ll2 - 62^Be2^a

hence

7 < a(l + * 2 /&V*«) + 263'2fiyBe^Sa1/2

a contradiction, since a is arbitrary.

256

Since W(t -f S) > 0 we deduce now

^ ( 0 = X\_BB.w{t + 6,t)W(t + 6)XA_BB.w(t+6it)

>t+6 XA_BB.w(s,t)[W(s)B(s)B*(s)W(s) f C*{s)C(s)]

XA-BB*w{s^)ds > a*

+

We shall prove next that A — BB*W defines an exponentially stable evolution.

Let x(t) = XA_BB.w(t,t0)x(t0); we have

x*{t)W{t)x{t) = x'(t0)X*A -BB'W -BB*W

(t, t0)W(t0)XA

/

'o

•[W(s)B(s)B%s)W(s) + C'(a)C(s)]XA_BB.w(»,t)X.A_BB.w(t,ip)x(ta)da'

= ** (< 0 )W(< 0 ) z (*o ) + y " * * ( < o ) ^ - B B . H r ( « , *o)

•[W(«)B(«)B*(«)W(«) + C>(«)C(«)]Xil_BJ,.H,(«,lb)x(*0)J«-.

It follows that

x*(t)W(t)x(t) - x*(t0)W(tQ)x(t0)

= - f x*(s)[W(s)B(s)B*(s)W(s) + C*(s)C(s)]x(s)ds

Jto

hence a|x(2)|2 < x*(t)W(t)x(t) < x*(tQ)W(t0)x(t0) for * > t0 and ^ - BB*W defines an uniformly stable evolution.

257

We have further for tk = /Q + kS

**(*J6+i)Wr*(*Jb+i)*(*ft+i) " **(*k)W*(tk)x(tk)

= - / *+1 x*(tk)X*A_BB,w(s,tk)[W(s)B(s)B*(s)W(s) + C * ( * ) C ( J ) ] Jtk

• ̂ -AB 'W^MfcM**)*** < —«|a?(*ib)|2*.

Denote V* = a?*(*it)W(*ib)a;(^ifc)» w e n a v e Vfc < /Xtu|#(Jfc)|2 and we

deduce that Vfc+1 - V* < —,Vjfc < ( l - — ) V 0 , —

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The controllability and observability assumptions in Theorem 1 may be written, with a change of variables as

J-t-6

/

-t XA(t,-

259

B(t)B*(t)P(t) defines an exponentially stable evolution we have

l^-gS*p(''r)l^*c"flf(

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and in the same way

J X2(t,3)B(3)B*(8)X2(t,8)ds

rt+6 = T(t) / XA(t,s)B(s)B*(s)XA*(t,s)dsT*(t)

>*ndn(T(t)T*(t))I = PI.

We have uniform controllability and uniform observability and from Theorem 1 existence of W and W will follow with W(t) > aI,W(t) < -al.

Let us remark now that

W(t) = T*(t)W(t)T(t) and

W(t) = T*(t)W(t)T(t)

are solutions for the Riccati equation associated to (-A,B,C).

We have indeed:

w'(t)= [T+(/)]/i^(or(o + T*(/)iy(or,(0

+T*(t)[-W(t)A(t) - A*(t)W(t) + W(t)B(t)B*(t)W(t)

-C*(t)C(j)]T(t) = \T*^ T*(t) [ r* (0 ] _ 1

W(t) T~l(i) T'(t)

-r*(/)[r*(0]"1^(/)2i-1(0[71(0A(0T-1(/) + r'(/)r-1(0]T(o

+r*(0]-M^(0]l[^*(0]"1^(0r"1(0}r(0

+T\t)[T*(t)}-1W(t)T-l(t)T(t)B{^

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= (T-\t)T*(t))*W(t) + WftT-WT'it)

-W(t)A(t) - W^T-^^T'it) - A*(t)W(t)

-[T-1(t)T'(t)]*W(t) + W(t)B(t)B*(t)W(t) - C*(t)C(t)

hence W'(t) + W(t)A(t) + A*(t)W(t) - W(t)B(t)B*(t)W(t) + C*(t)C(t) = 0.

The same is true for W. We have next

A{t) - B(t)B*(t)W(t) = T(0 i4( t ) r - 1 (0

-r(0#(0£*(0r*(0[r*(0]_1 ̂ ( 0 ^ ( 0 + T'i^T-^t)

= T(

262

and Yl solves the dual Riccati equation.

In this way invariants of Jonckheere-Silverman are defined as eigenvalues

of n(0^{0-

THEOREM 3. (Bucy, [4]). A - BB*W and A + WC*C are equivalent through the Liapunov transformation I — WW.

Proof. The Jonckheere-Silverman invariants are positive. Let indeed X(t) be an eigenvalue for — W~1(t)W(t), x(t) a corresponding eigenvector, \x(t)\ = 1, -W-\t)W(t)x(t) = X(t)x(t).

Denote z(t) = W(t)x(t); we deduce that -W~-l{t)z(t) = X(t)x(t)

-{W~\t)z{t),z(t)) = (X(t)x(t),z(t)) = X(t)(x(t),z(t))

= X(t)(x(t),W(t)x(t))

where (x,z) is the usual inner product in IRn. Since W(t) < —SI, we have

(W(t)y(t),y(t))

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It is also clear that T is difFerentiable and T1 is bounded. We have

T'(t) = U'(t)W(t) + U(tW(t)

= [A(t) n(o+nc)^*(o - m)c*(t)c(t) n c o + J K O ^ O W ) - II(0[W(

264

We are looking for informations concerning Ker W(t). If A — BB*W defines an exponentially stable evolution, we may represent W as

W(t) = jtC°X*A_BB.w(T,t)[C'(T)C(T) + W(T)B(T)B*(T)W(T)]-

• XA_BB.w(r,t)dT.

If x*W{t)x - 0 then C(r)XA_BB.w(T,t)x = 0

B*(T)W(r)XA_BB.w(T,t)x = 0 .

Assume now air functions are C°°, take in the first identity above the derivative with respect to r and use the second identity to obtain [C'(r + C(T)A(T)]XA_BB.W(T, t)x = 0.

Denote TA(C)(T) = C'(T) + C(T)A(T) and write

rA(C)(-r)XA.BB.w(r,t)x = 0.

We may take again derivatives and for r = f we deduce

•'..-. r*(c)(0* = o.

We have thus proved

PROPOSITION'2;• Ker W{t) Cf]k Ker FkA(C)(t).

We shall try to get more precise information in the case A defines an exponentially dichotomic evolution. To this end let us start with some preliminaries.

It is an old result of Perron [18] that a linear system with variable coefficients may be transformed in triangular form by means of a Liapunov transformation. In 1954 R. Vinograd [21] gave a new proof, which we shall reproduce here with some comments.

Let X be a smooth matrix-valued function, detX(/) ^ 0 for all t\ let X j , . . . ,x n be the columns of X. The Gram-Schmidt orthogonalization

265

procedure leads to vectors z\ = xi, z j).

We write further X = UY, Y = i 2 - 1 ; Y is again smooth and triangular. if x'(t) = >i(0*(0»then y ,M = A(*)K( — oo. If we start our constructions above with X having as first k columns the solutions which exponentially decreases for / —• -f , the same structure will remain

valid for Y. We deduce that A = ( ** ,12 I with An defining an

exponentially stable evolution and A22 defining an antistable evolution.

Let us state the final result as

THEOREM 4. (Perron-Vinograd). Let A be a bounded, continuous, matrix valued function defining an exponentially dichotomic evolution. Then there exists an orthogonal Liapunov transformation U such that A =

266

U-^AU - U-W is triangular, A = (A™ ^ 1 2 J , with An,~A*2 defining

exponentially stable evolution.

We may use this theorem to study properties of the stabilizing solution of the Bernoulli equation obtained from the Riccati equation if C = 0.

After the Perron-Vinograd transformation the Bernoulli equation is

*"u + ^11^11 + WnAn - {WUB1 + W12B2){WnBl + W12B2)* = 0

W[2 + A'nWl2 + WnAn - {WxlB, + W12B2){B\W12 + B'2W22) = 0

W22 + A'l2W12 + A22W22 + W?2Al2 + W22A22

-(W^B, + W^B^W^B, + W22B2y = 0

This equation has a solution with Wn = 0, W12 = 0, W22 given by

W

267

where A,B,C are as in Theorem 1.

Assume that A defines an exponentially stable evolution. For every u £ L2{—00,00) there exists a unique solution in L2{—00,00) defined by the formula

x (t) = / XA(t,s)B(s)u(s)ds. J—00

The fact that x as defined is in L (—00, oo,IRn) follows by using directly estimates (see for instance the proof of Lemma 1 in this section). It is useful to sketch a proof for uniqueness in the more general situation of an exponentially dichotomic evolution.

We start by writing for a solution of the linear system

r r

x(t) = XA{t,T) ft X(T) + XA(tyT)(I ~ Y[)XA(T,t)x(t) , A A

Then for / > t > r , we have the estimate

/ \x(t)\2dt > —\X(T)\2 + 2f}2e-as / |z(tf)|2

268

Denote by G the input-output operator u •-> y from Z2(—00,00 ;IItn) to X2(-oo,oo;IRp). Then ||£r|| < 1 if and only if the Riccati equation

W' + A*W + WA + WBB*W + C*C = 0

has a bounded on aR stabilizing solution (A+BB*W defines an exponentially stable.evolution). In one sense the result is easy. We have

/

+00 /-+oo

(u*u-y*y)dt = / (u*u-x*C'Cx)dt -00 J—00

/

+ 0 0 [u*u + x*(Wf + A*W + WA + jyJ91?*W0a;]

269

and for some a > 0, /3 > 1,

\xk(t)\2 < [ Pe-o^ds I e-«

270

THEOREM 5. Assume AyByK,LyM be continuous and bounded on IR,K = K*,M = M*,K(t) > KI,K > 0, is invertible with bounded inverse. Let F(t,x,u) = u*K(t)u+ u*L*(t)x -f x*L(t)u + x*M(t)x. Assume that A defines an exponentially stable evolution and that there exists S > 0 such that for all to and all u 6 L2(to,oo]]Rm) we have

roo rOO

/ F(t,x(i),u(t)dt >S u*(t)u(t)dt,

where x'(t) = A(t)x(t) + B(t)u(t),x(to) ~ 0 .

Then there exists a solution N bounded on JR. for the Riccati equation

that A = A — BK~1(B*N -f L*) defines an exponentially stable evolution.

COROLLARY. If there exists 6 > 0 such that for all u e £2(-oo,oo;IRm) we have

/

+oo r-foo

y*ydt S j _ ^ u*udt and we have seen it leads to

r+oo r+oo / f(t,x(t),u(t))dt >6 u*udt

Jto Jto for all u € I2(*0 ,oo;lR

m), x'(t) = A(t)x(t) + B(t)u(t) x(t0) = 0. From Theorem 5 there exists a bounded on lit solution N with

• N'(t) + A*(t)N.(t) + N(t)A(t)

- N(t)B(t)B*(t)N(t) - C*(t)C(t) = 0

271

Take W(t) = -N(t) and deduce that

W\t) + A\t)W(t) + W(t)A(t)

+ W(t)B(t)B*(t)W(t) + C*{t)C(t) = 0

and A -f BJ3*VT defines an exponentially stable evolution. •

Remark also that as a solution to a Liapunov equation (or inequality) we have W(t) > 0. We shall see in the proof of Theorem 5 that XQN(t)x0 is the optimal value for ff°[u*(s)u(s) - y*(s)y(s)]ds hence

/•OO

XQN^XQ = min / [u*(s)ii(,s) — y*(s)y(s)]ds u Jt

XQW(t)x0 = —min / [u*(s)u(s) - y* (s)y(s)]ds u Jt rOO

= max / [y*(s)y(s) — u* (s)u(s)]ds . u Jt

To prove Theorem 5 we shall use some lemmata.

LEMMA l. Ifu£ Z2(£0,oo;IRm) then every solution x to x1 = A(t)x + B(t)u

is in Z2(

272

we see that u i-> Jt XJ\(t1s)B(s)u(s)ds = (CtQu)(t) is linear and continuous

from Z2(*0,oo;IRm) to L2(t0, oo;IR

n). •

Consider now

J(u) = J™f(t,x(t),u(t))dt

= < M(Ctou + Z(i0,X0)),£

+ < Ku,u > + < LuyCt0u + Z(

+

= < [£*hM£i0 + £?0I + L*Ch + # ] « , * >

+ < (M£to + Z)ti,*(|0>jto) > +•< 2( / (ha;o),(M£

+ Z(t0,Z0)>

= < nt0U,U> + < U,l>{iQtSQ) > + < ^(|0|JB0),M > +/>(

273

Denote by xu(to,*o) > + 2 < u(to}x0)^(i0}xo) >

It is seen that J{u(t0x0)) ls a quadratic form in XQ with coefficients

depending smoothly upon to.

We shall now repeat the dynamic programming argument of Yakubovich. Let r be such that xtiQ x \ is differentiable at r , and r is a Lebesgue point for t » / ^ 3 ( * 0 , * 0 ) ( 0 » M ( « 0 , * O ) ( 0 ) ; denote z 0 = a>g0,*0)(r); consider the solution 3 defined on [r, r -f 6] by the initial value a?o a n d a

constant control v £ l l l m . Denote xi = x(r -f £) and consider the control u defined by

( W(t0,Z0)W , U

274

Then we may write

roo

/ ^('»*(r+*,*i)(0> V+*,*l)(Orf< = x\N(T + 6)xi J T-\-6 W+6

Remark also that we have

Jto

/

oo

Let now u be an arbitrary control and x be the solution obtained by starting at r with the value XQ = xuQ Xo)(

T) a n ^ u&ing the control u; choose a control u equal to U(tQ Xo) on [toyr) and to t; on [r, oo); starting at /o with the initial value XQ under such control we shall have

j(fi>= /V(«,x((0iIo)(o,)rf<

/

oo

/"(«, x(t; r, x0; 5), 5( A^o.xo))

hence

/"(/, x(t; r, x0; w), u(t))dt

/

oo

^(^(/o,*o)W^0,s0)(0)*-

Since w was arbitrary we deduce that

/

oo

H^x(t0txo)(t)^{t0txo)(

t))dt = XON(T)X0 .

Now since J(u) > J(uttQ x \) for u defined above, we deduce that

rr+8 / f(i, x(t), v)dt + x\N(r + 6)> X*0N(T)X0

275

that is

?{t, £(*), v)dt + z*(r + S)N(T + $)s(r + 0 .

From here for S —• 0 we deduce that

^ tJ*(0^( ' )2(0] | l=r+0 + ^ ( ' : . * W . « ' ) > 0 - .

Since x'(t) = A(t)x(t}+ B(t)v we deduce that

[r(r)A\r)i-v*B*(T)]N(T^T)i-r(T)N'(T)x(T) .

+ z*(T)N(T)[A(T)X(T) + J?(r)v] + ^"(r, s ( r ) , v) > 0 .

On the other hand

* x0 N(to)xQ= J ^(/,a;(id|a.o)(0,t*(«0,xo)(0)^ + *o^(r)*o

x(t0tzo)(

3)>u(to;*o)(3))d3 J to

By differentiating with respect to t at r we deduce

•^>*(«„,*o)('-)>'Wo)(r)) + I'M.W^W

+ '«(«., . .)( r)^^O^W»(«o,«.)( r .) + *(l.,..)('-)^

276

Since this equality holds for every r where x/tQXQ\ is differentiable and since #(Xo) is absolutely continuous, the equality holds everywhere.

We know that £(

277

and finally we see that V{t,x) < 7|z|2; by using the uniform boundedness of the coefficients we see also that V(t,x) > a |x|2 .

We have

|x(*)| > ^V{t,x(t))

±V(t,x(t))

278

Since A defines an exponentially stable evolution u = 0 is admisible and

for all (/(b#o) w e have

x*QN(t0)x0< / x*0X*A(t,t0)M(t)XA(t,t0)x0dt Jto

and N(to) is bounded. •

We shall develop further the result in Theorem 5 by considering a new input-output operator.

THEOREM 6. Consider matrix-valued functions A,B\,B2, C, continuous and bounded on HI, assume A defines an exponentially stable evolution and (A,(B[B2)) is uniformly controllable. Define the input-output operator G as

CJ) — y

x'(t) = A(t)x(t) + fli(0t*i(0 + B2(t)u2(t),

til € £ 2 ( -oo ,oo ; I f t m i ) , u2 G L2(-00,00;HI™2)

x e L2(-oo,oo;lRn), y(t) = C(t)x(t) ,

Let W* = {(J£) eL2(-oo,oo',mmi+m2),u1(t) = 0fort > * 0 } .

Define Tto = IJ+ G\wt0, where [ ]+ is the projection of L2(-oo, 00; JRP) onto

L2(t0,00; 1RP). If there exists q < 1 sucii that | | r< 0 | | < q for all tQ, then the

Riccati equation

W' + A*W + WA + WB2B^W + C*C = 0 (x)

has a bounded on III solution W such that A+BiBl^W defines an exponentially stable evolution and moreover the spectral radius ofW(t)P(t) is strictly less than 1, for all t £ HI, P(*) being the bounded on HI solution of the Liapunov equation

P' = AP + PA* + BB\ B = (Bi B2).

Conversely, if there exists a solution of the Riccati equation (x) with the above stated properties, then for all tQ we have \\Tto\\ < 1.

279

/•CO

Proof. If ||r*°11 < q < 1, then for all u G Wto we have / y*ydt <

/

-foo roo

u*udt\ we choose u G Wto with u\ = 0 and deduce that / y*ydt < -co JIQ

/

-foo /"-foo r+oo

^21*2^ f° r a ^ 0̂? hence / y*ydt < q / u^^dt. -00 J—00 J—00

It follows that the input-output operator associated to (/i,Z?2>C) has norm strictly less than 1, hence from Theorem 5 we deduce existence of the stabilizing solution of the corresponding Riccati equation.

Existence and properties of P follow from exponential stability and from the controllability of (A,B). We may write further for any u G Wto

r + C O / " f e d /«oo rOO r / o

/ y*ydt — / u*udt = / y*ydt — / u^^dt — / u*udt to J—00 Jto JtQ J—00

Let XQ ^ 0 be arbitrary; for such XQ there exists an optimal U2 such

that maxW2 J^°(y*y — ̂ 2 ^ 2 ) ^ = /t°°(2/*3/ ~ ^ 2 ^ 2 ) ^ = ^ o ^ ^ o ^ o - For the

same #0 there exists an optimal u^u = (-1) defined on (—00,^0] such that

/ • 0 0

and •*o rh

a j JP - (^o)^o = m m / u*udt= / u*udt . u J-00 J-00

Remark first that A -f BB* P~l defines an antistable evolution since

A + BB*P~1 = P(-A*)P-1 + P'P-1 and A+BB*P~X

is similar to (—A*) via the Liapunov transformation defined by P.

Consider the solution X(T) — XA+BB*F-I(T^)X0) w e have

x G L ( — 0 0 , / ) ,

x'(r) = A(T)X(T) + B(T)B\T)P-1(T)X{T)

= A(T)X(T) + J9( r )u( r ) , u(r) = J 5 * ( r ) P " 1 ( r ) 5 ( r ) ,

280

hence u G Z 2 ( -oo , t ) ,x ( t ) = XQ.

f u*(T)u{r)dr = I x\T)p-l(T)B\r)B(T)p-l{T)x{T)dr J—oo J—oo

= • / l^*(r)p-1(T)Pl(r)P'1{T)x(T) - 5 * ( r ) ^ ~ V ^ M * " ) J—oo

- X\T)A\T)P-1(T)X{T)](IT

=L{-r{T\jtp~l{T¥T)

- X\T)P-1{T)[X\T) - B(T)B\T)P-1{T)X(T)} -

- [X*(T))' - 5 * ( r ) p - 1 ( r ) 5 ( r ) i ? * ( r ) ] p - 1 ( r ) ^ ( r ) | ( / r

rt

( 0 i , " 1 ( 0 « ( 0 + 2 / ^ ( r ) p - 1 ( r ) i ? ( r ) Z ? * ( r ) p - 1 ( r ) 5 ( r ) ^ ./—oo

= —a;

= -x*(t)p-\t)x(t) + 2 / «*(r)ii(r)rfi . / — CO

hence

J—oo

On the other hand for an arbitrary admissible couple (u,x) we have

jt[x*(T)p-\r)x{T)] = X\T)P-\T)[A{T)X{T) + B(r )u( r ) ]

+ [a?*(r)i4*(r) + n*(r) i7*(r)]p- 1(r)a:(r) - X*{T)P-1{T)P9(T)P-1{T)X(T)

= t**(r) t t ( r ) . - . [ t i ( r ) , - . . i?*(r)p- 1 ( r )x(r) ]*Kr) - ^ ( ^ " ^ ^ ( r ) ] .

281

Since x G L2{—oo,t),x' (E L2(—oo,/), we have l im r_+_0 0 x{r) = 0 and

/ u*(T)u{r)dT = x*(t)p-1(t)x(t) J—oo

+ / \u(r) - B\T)P-1(T)X(T)\2(1T > x*(t)P~l(t)x(t). J—oo

Consider now the control equal to u in (—00^0) a n d to ( ) in (/u?oo);

this control belongs to Wto and

/ y*ydt- / u*udt = XQW(t0)xo ~ XQP^^XQ . Jto J—oo

Since | | r*° | | < q < 1 we deduce that

'OO r + O O

y*ydt < q / 'to J—oo

•+00

roo r+00 / y*ydt < q / u*udt,

Jtn J—oo

/

i-oo u*udt .

-00

Let p{to) be the largest eigenvalue of the matrix W(to)P(to)y xt0 a corresponding eigenvector; we have W(to)P(to)xt0 = p{to)xtQ. Denote

_ W(tQ)zh = pmP~lW^z^{t0)zt0

= p(to)zJ0P'\to)ziofzi0W{t0)zio - ztQP-\t0)ziQ

= [p(t0)-l]z;0P-l(t0)zi0.

Since in the inequality above XQ was arbitrary we may replace it by zt0 to deduce that

/

+ 0 0 u*toutodt < 0 .

-OO

282

It follows that p(to) - 1 < 0 for all to.

Assume now there exists IQ such that | | r< 0 | | > 1; then there exists u e W*M|t2|| = 1 such that J^Tydt > 1; define

/

to XA(t0,s)B(s)u(s)ds .

-oo

For this XQ we have

XoW(to)xo

rOO rOO

> / y*ydt - / u^dt ;

— x^P (^o)^o ^ ~ / u*udt,

J—oo rOO f+OO

x*0W(t0)x0 - £oP- 1(*o)£o ^ / V*ydt - I u*udt > 0 .

On the other hand

0 < x*0W(t0)x0 - xlp-^to^o < [p(t0) - l]xuP~1(/0)a:o

hence p(to) > 1, a contradiction. •

In the stationary case it is true that if p( kl,k > 0, M(t) = M*(t).

283

Associate the Riccati equation

N' + A*(t)N + NA(t) + M(t)

- [L(ij+ NB^K-^lLit) + NB(t)]* = 0

(the same as in Theorem 5). Assume that this equation has an antistabilizing solution, that is

A — BK"l{L + NB)* defines as antistable evolution. Then the minimum of /JJ^-F (/,#(£), u(t))dt subject to x(t) = f_

Xj[(t,s) B(s) u(s) ds, x(to) — XQ equals — XQN(to)xo and is associated to the feedback control.

u(t}= -K-l(t)[L(t) + N(t)B(t)]*x(t) .

The conclusion follows from a direct calculation

f° [u*(t)K(t)u(t) + u*(t)L*(t)x(t) + x*(t)L(t)u(t) J—oo

+x*(t)M(i)x(t)]dt = f° {u*(t)K(t)u(t) + u*(t)L*(t)x(t) J—oo

+x*(t)L(t)u(t) + x*(t)[L(t) + ^( .0J?(0]A'" 1(0[ i(0 + ^ ( O ^ W l ^ W

-:x*(t)N\t)x(t)-x*(t)A*(t)N(t)x(t).-^

,/o d f° ±[x*(t)N(t)x(t)]dt

J-oo dt

•OO

/ ° {«*(*) + x*(t)[L(t) + JV(t)5(i)]tf_1 (

284

We had in the proof of Theorem 6

K(t) = / , L(t) = 0, M(t) = 0 ,

hence

N'(t) + A*(t)N(t) + N(t)A(t) - N(t)B(t)B*(t)N(t) = 0

and

p(t) = -mr1 •

This remark suggests

THEOREM 7. Assume A,B,K,L,M be continuous and bounded on IR,K = K*,K(t) > kl, for all t G 1R, k > 0, M = M\ Let F(t,x,u) = u*K(t)u + u*L(t)x + x*L*(t)u + x*M(t)x. Assume that A defines an exponentially stable evolution and that (A,B) is uniformly controllable.

Assume also that there exists S > 0 such that for all /Q

/

to fio

T{t,x,{t),u{i))dt >S u*(t)u(t)dt -oo J—oo

where x'{i) = A(t)x(t) + B(t)u(t)

u G L2(-oo,^0;iRm), x e L2(-,t0',m

n).

Then there exists a solution N bounded on IR to the Riccati equation above such that A — BK~*(L -f NB)* defines an antistable evolution.

Proof. We may try the same procedure as for Theorem 5. Consider the problem

F{t, x{t), u(t))dt, x(t) = / XA(t, s)B((s)u(s)ds , . -oo J—oo

x(to) = xo\.

Start by computing

F{t,x(t),u(t))dt -oo

= f° t0> S < uyu >t0

and it follows that TZt0 > 6Iy7Z7 exists. We have further

-to r r* f° I / u*(s)B*(s)X*i(t1s)dsL(t)u(t) dt

J—oolJ—oo

= f° u%s)B*(s)\[ ° XX(t,s)L(t)u(t)dt ds

= f° u*(t)B*(t)\ [ ° X*A(s,t)L(s)u(s)ds]dt]

/ °. [ / u*(s)B*(s)X\(t,s)ds]M(t)\ f XA(tya)B(a)u(a)da dt J—oolJ—oo J \.J—oo .

= f° u\s)B%s)\ f ° X*A(t,s)M(t)( f XA{t,a)B(cr)u(

286

= lQ u\s)B\s)\ I I I"X\{t,s)M{t)Xji(t,o)dl\B(

287

We may write the operator in a different way by using the Liapunov equation

P' + A*(t)P + PA(t) + M(t) = 0 .

We have

/ ° B*(t)X\(s, t)M(s)XA(sy a)B((r)ds Jma.x(t,(T)

= - / ° B*(t)XA(a,t)[A*(s)P(s) + P(s)A(s) + P'(s)]XA(s,a)B{

288

with

(7j0tO(0 = K(t)u(t) + / [L*(t) + B\t)P{t)]XA{t,a)B(a)u(

289

We deduce that

(Bio«)(0 = (7io«)(0 - (A*toP(t0)Atou)(t).

We have seen that

/•(*,x(t) , t t(to , -00

hence •«b

^•(t,a;(t),.M(*))rf/=< M,7i0t* >t0 = to

- (*At0u,P(to)Atou) =< u,TtQ >( 6 < u,u>tQ.

We shall deduce from here that

< u,7J0 > | 0 > - < u,u >to

for all ô a n d all u, and it will follows that Tt0 is invertible.

As for many other results in this section we follow here closely reasoning of Jonckheere and Silverman [13], [14], [15].

(7i0)(i) = K(t)u(t)

+ /• [L* (t) + B*(t)P{t)]XA{t,a)B(

290

and

We have

< u,Kt0 >Tt0u> >t0 + < u,HtQu >|0

- < u,Htou >to

= J*^u\t)B\t)X\{tQ,t)P{tAj ° J T ^ o . i r ^ ^ M a J r f a l A

XA{t0,o)B{a)u{(j)d(T\ P(t0)l XA(t0ja)B((j)u(a)da

M*(t)A'(0t*(0^ -oo

J° [J u*(t)[L*(t) + B*(t)P(t)]XA{t,a)B(a)u(a)da\dt

j ° I / ° u*(t)B*(t)X*A(

We shall have

- < v, HTv > T

= (J_ XA{T,a)B{a)v{a)da\ P(T)(J XA(T,

292

We deduce that

< v,1Zjv > j = < v>,Tto,u >to

+ < v,Hjv >j> 6 < v,v > j = 6 < u,u >t0 ,

< u,Tto >to> 6 < u,u >tQ - < v,Hjv>T

and for T large enough we shall have < u,Tt0u >t0> - < u,u >(0

=< A^AtoT^A^xo-AlPMxo^A^AtoT-'At^xo >«0

= ( U J O ^ 1 ^ ) - 1 ^ - PMxo^toT^ADiAtJ^AlX^o)

= ({AtoT^A^xo^o) - {P(to)x0,xo);

< K«.(«-^1^M«o.^r1-*;0)'1*o),«-7S'X(^o^rM*to)"1*o >

293

=< Kt0u,u >tQ -((At0Ti-1AtQ)-

lxo,xo) + (P(t0)xQ,xo)

- ( ( A ) ^ ~ X ) ~ l x o , z o ) + (P(to)xo,x0)

HiAtoT-U^x^xo) - (P(t0)x0,x0)

=< ntou,u>to -((AtoT^Al^xo.xo)

+(P{to}xo,xo).

We see from here that

to= ((AtoT^Al^xo^xo) - (P(to)*p,*o)+

+ t0 .

We deduce from here that < 1ZtQu,u >t0 has a minimal value equal to {[(AtQTt~

lA*iQ)~X - P(to)]x0.x0) obtained for

u = ^1AtMto^1^or1xo-It remains to repeat the dynamic programming reasoning to show that

the symmetric matrix giving the optimum — N(to) = (AtQTt~ AtQ) — P(to) is in fact an antistabilizing solution for the Riccati equation.

From this formula we remark that N is differentiable (we have explicit formulae for At0 and Tt0 and P is a solution for a differential equation).

Let r be such that the optimal solution above, denoted xrtQXQ\ is dif-ferentiable at r, and r is a Lebesgue point fori *-*• F(trX(t0iXQy(t),U(tQXQ\(t)). Consider an arbitrary v £ Rm denote x the solution starting at xrtQXQ\(r) and under the constant control v evoluating back on [r — #,r]; let #/ = X(T — 6). Consider the control u defined by

( 'tt(«o,*o)(*)- ' ' r

The corresponding solution is

x(t) = i

X(0.

295

It follows that

d dt[x(t)N(t)x(t)]t=T_0 + HT,*(T),V) > 0

that is [x*(r)A*(r) + V*B*(T)]N(T)X(T)

+ X*(T)N{T)[A(T)X(T) + # ( T > ]

+ X*(T)N*(T)X(T) + ^ ( r , z(r) , w) > 0 .

On the other hand it is seen that

K « o , * o ) M ^ ) + « (< .« )«^( r ) ] JV( r )* ( l 0 i ^ ) ( r )

296

Denote

V(t,x) = / x*X~(8,t)xda, J-oo A

A = A-BK'^B+N + L*).

For x solution of x' = A(t)x we have

V(t,x(t)) = J_ \x(s)\2ds , jV(t,x{t)) = |5(0| :

From the general formulae we started with, we have \\uuQ x \\\ < ^|«ol>

hence ||^(foXo)|| < «|^ol» hence V(2o?#o) ^ 7l#o| f°r a ^ (*0>#o)j we see also

that V(t,x > a\x\z (since coefficients are bounded!). ft 1

We deduce —V(t,x(t)) V(t,x(t)) > 0 and from here

e - l /T(«-«o)K ( t > J ( i > ) < K(f0,a:o) < *

297

Tf1 = -(7, - A*tP{t)At)-\AtP{t)AtTt-1 - I]

AtT^Al = -At(Tt - AlPMAtr^AtPWAtTf1 - I]A*t

= -AlTlTl{A*tP{t){AtTl^A''t)-A\)

= -AtKT'AUPitHAtT^A*,) - I).

We deduce that

i = ~AinTlA\[P{i)-{Atri-

lA\rl]

= -Atn^lAlN{t).

It follows that (Tt is invertible) N(t) is nonsingular and N(t) = -(Atnj

lA\)-1 < 0.

4. Jonckheere-Juang-Silverman solution for the two-block problem

The last questions we have discussed are essentially time-varying analogues of results obtained by Jonckheere, Silverman and Juang [13].

We shall follow again these authors to show the solution of the two-block problem.

THEOREM 8. Consider the nodes

H = (A,B,CHlDH), V = (A,B\Cv,Dv).

Assume A defines an exponentially stable evolution and that there exists 72 such that *)2I > DyDy and the Riccati equation

S' + SA + A*S + C*HCH

-[(P - S)B + C*VDV](72I - D*VDV)-

1[B*(P - S) 4- D*VCV] = 0

298

where P is the bounded on lit solution of the Liapunov equation

P' + A*p + PA + C*HCH + C^rCv = 0),

lias a solution on IR and such that there exists a > 0 with S(t) < —aI. Define the node

Q = [-(A + S-lC*HCH)*,PB + CVDV,CHS-\DH]

Then Q has an antistahle evolution and

||(//-QHI2 + ||iHI2

299

300

Again a direct calculation shows that

'-foo /--f-oo

dt | | ( / / - Q)u\\2dt + / ||Ku|| -OO J — OO

/

+CO

-oo

/

-foo K ( f t 0 + C*£>)[7

2/ - T)*^]-1 - u*}( 72 / - £>*£>){

-oo

/

+oo u*udt.

-oo

It follows that

•-foo r+oo

/

i-oo r-t-oo

y*ydt

/•oo

/ x*(t)C*H(t)CH(t)x(t)dt Jtn 'to

' + CO

[u\t)D*v{t) + x*(t)Ct(t)][Cv(t)x(t) + ZV(*M0]^ -co

u*(*)u(*)d*. -co

Use the Liapunov equations

P'H+.PHA + A*PH + C*HCH-=O

P'v +' P K ^ + A*PV + Cf CV = 0

to write the inequality as

72 / u*(t)u(t)dt >x*(t0)PH(to)x(t0)

J—oo

f*o ( 0 PvB + ClDv\ fx\

7_ooV }\B*PH+D*VCV D*VDV ) \uj

On the other hand

,*o /-C*„CH P'HB\ fxX

/ - J ' ^ U V o ) ( > = *>W^W) and we deduce that for all to

^ . . " . . I -C*HCH PB + C^Dv , ,x B*P + D*VCV D*VDV I U , /

to (x* „*)

-CO

dt

< 72 / u* J — OO

t0

udt

302

hence

fl\ + , J C%CH -(PB + C*VDV)\ / ^ / (x*u*)[ _ ) ( )dt>0.

y_oov \-(PB + CvDv)* j*I-D*vDv ) W

Now from Theorem 7 it follows existence of the negative definite solution to the Riccati equation that has been assumed in Theorem 8 (with a small change in 7) .

5. Riccati equations and inner-outer factorizations

A nonstationary node [1] is a system (A^B^C^D^X, U,Y) consisting of spaces X, £/, Y and operator-valued functions A,B,C,D with A(i) : X —• X, B(t) : U -* X, C(t) : X -> Y, D(t) : U -+ Y, alf operators being assumed linear and bounded and all functions defined on IR, continuous and bounded. To the node we associate the system

x1 = A(t)x + B(t)u , y = C(t)x + D{t)u #

We shall assume that A defines an exponential dichotomy and we shall associate the solution defined by

x(t)= f XA(t,t0)HxA(tQ,s)B(s)u(s)ds J—00 J to

XA(t,t0)(I-l[)XA(to,8)B(s)u(3)d3

to

and the corresponding output

y{t) = C{t)x(t) + D(t)u(t).

Here Ylt is the projection defining the exponential dichotomy.

We have defined in this way an operator from £2(—00,00; U) into 2 / 2 ( -oo ,oo ;y ) .

303

This operator is preserved by node similarity defined by a Liapunov transformation (corresponding to a change of state variables).

Let the node similarity be defined by

Ai(*) = S-*(t)A2(t)S(t) - S-Ht)S'(t)

B^t) = S-*(t)Ba(t), C,(«) = C2(t)S(t).

It follows that XA2(t,s) = S(t)XAl(t,s)S~l(s) and

C ,J(0^,( ' .*O)II2.I.^( 'O.^2(*)'

= c3(t)S(t)xA, (*, t0)s-Ht0)U,,h S(tv)xAi (t0, *)s-H«)na(»)

= Cx{t)XM (t, t0) n i i ( 0 XAl (t0,3)3,(8)

where we have denoted ITi.to = ^" ' ( 'o ) IL,

304

If x'. = Aji^Xj + B.(t)Uj, yj = Ci{t)xi + DJWUJ, j = 1,2 are the corresponding control systems, node multiplication corresponds to u = u2, ui — 2/2' V = î» leading to a system in a product space

4 = i42(J)a?2 + B2{t)u

x[ = ^ ( O ^ + £i(0[C2(0*2 + ^2(0«]

2/ = C1(t)xl + /?1(«)[C2(0a?2 + ^ ( ' M •

Remark that the multiplication defined above corresponds to the product of the associated input-output operators.

These input-output operators may be written as

-bo.

Ay

CAt,8)= <

Gj(t)XAi(t,t0)l[xAj(ti,3)B(8), t > s to

- GjWX^to) (i - j } ) XAi(t»8)B{8)y t

305

The above operations make sense also in the case C = 0 when we have direct connection between the input and the output. Given the node G = (A,B,C, D,X, U,Y) we define the adjoint node G by

G* = (~A*,-C*,B*iD*iX*,Y+,U*).

The adjoint node corresponds to the adjoint of the input-output operator with respect to the L2 scalar product.

We have

/

+oo r + o o / r-\-oo \

(y(t),z(t))dt = ( / G(t,s)u(s)ds +D{t)u(t),z(t)]dt -co J—oo \J—oo /

/

+ o o /• + o o / r - foo V

(u(t), D*(t)z(t))dt + / (t*(5), / G*(s to

A

-C(t)XA(t,t0)(i -l])XA(t0,s)B(s),t < s to

we have

G*(t,8)= <

B*(s)XA(tQys)(]JyXA(t,t0)C^t) ,t>* t0

A

- B*(s)X*A(t0,s)(I - l[)*X*A(tt0)C*(t) ,t.

306

Remark that X^(s,t) = X_A.(tys) and

x_A.(5,g[/-(nyjx^.(

307

and consider the system

x'(t) = A(t)x(t) - B(t)[D*(t)D(t)]-iD*(t)C(t)x(t)

- B(t)[D*(t)D(t)]-iB*(t)x(t) + B(t)[D*(t)D(t)]-iy(t)

x'(t) = -A*(t)x(t) - C*(t)C(t)x(t)

+ C*(0^(0[^*(0^(01"1^*(0*(0 + C*{t)D{t)[D*{t)D(t)]-iD*{t)C{t)x{t)

-C*(t)D(t)[D*(t)D(t)]-iy{t).

For this system y is the input and u is the output. The node is described by

A(t) - B{t)[D*(t)D(t)]^D*(t)C(t) -fl(0'[I>*(

308

The Iliccati equation is written as

W' + W(A - DF) + {A- BF)* W + (C - DF)*{C - DF) = 0 .

Let us mention in this connection a recent result [11].

Assume A,B,C are ^-periodic; assume also that for every XQ £ IRn

there exists u G Z2(0,oo;IRm) such that for x(0) = XQ the solution x of x' = A(t)x + B(t)u(t) belongs to / ,2(0,oo;IRn) ; assume finally that for the solution matrix Z of the Hamiltonian system with Z(0) = / we have det[Z(u>) — pi) ^ 0 for all p with \p\ = 1. Then, the corresponding Riccati equation has a stabilizing periodic solution. The condition concerning the Hamiltonian corresponds to the necessary and sufficient condition in the stationary case (no eigenvalues on the imaginary axis for the Hamiltonian matrix). The genmeral result of Yakubovich [23] and the above mentioned theorem (as well as Theorems 6 and 7) relate properties of the Riccati equation to linear-quadratic optimization problems in infinite time.

Concerning periodic Riccati equations results of S. Bittanti and his colleagues should also be mentioned [2], [3].

Under our fundamental assumption and with the notation above, define the nodes

Gi = (A - BF,B(D*Dy/2,C- DF,D(D*D)~ll2,X, U,Y)

Go = (A,B,(D*D)l/2F,(D*D)l/2tX,U,U).

The product G{GQ corresponds to the system

x' = Ax + Bu

x' = (A - BF)x + B(D*D)~lf2[(D*D)l'2Fx + (D*D)ll2u]

y = (C- DF)x + D{D*D)-ll2[{D*D)ll2Fx + (D*D)^2u]

hence

x' = Ax -f Bu

x' = (A-BF)x-\-BFx-\-Bu

y = (C~DF)x + DFx + Du.

309

If we take x = x - x we obtain a similar node

x' = Ax -f # M

5' = (A - £ F ) x

2/ = Cx + £ u - (C - .DF)£ .

This node is the sum of the node

x' = Ax + # u , 2/ = Cx + ^ ^

with the node

x' = (A - B F ) 5 , 2/ = - ( C - DF)x .

Since A—BF defines an exponentially stable evolution, the only solution of x' = (A — BF)x belonging to X2( —oo,oo;IRn) is the zero one and the corresponding input-output operator is zero. We deduce that the node defined by GJGQ is equivalent to the given node G. The node GQ will be called outer: it has a right inverse with exponentially stable evolution; the node GQ is written as x' = Ax + Bu,y = {D*D)

x/2u +D*D)ll2Fx and the inverse, obtained by taking u + (D*D)ll2y — Fxy has the evolution defined by x'= (A-BF)x-\-B(D*D)-1f2y.

The node G,- will be called inner; it has the property that G^G, is the identitiy. We have indeed Gt- associated to the system

x' = (A - BF)x + B{D*D)~ll2u

y = (C-DF)x + D(D*D)-l/2u

and G* associated to

x' = -(A-BF)*x-(C-DF)*u

y = {D*D)-*l2B*x + (D*D)-ll2D*u.

310

We deduce that G^G{ is associated to

x' = (A - BF)x + B(D*D)-l/2u

x' = - (A - 2?F)*x - (C - DF)*[(C - DF)x + D(D*D)~ll2u]

y = (D*D)-V2B*x + (D*D)"1/2D*[{C - £ F ) z + D(D*Dyl/2u].

We see that y = (JD*!?)-1/2^** + £*(C - £>/>] + ti. Recall now that F = (D*D)~l(D*C + £* W) to deduce that

2/ = u + (D*D)~ll2B*{x - Wx).

If we define x — x — Wx, a direct computation using the Riccati equation leads to

x' = ~.(A-BF)*x

and G*Gi is similar to the node associated with the system

x' = (A- BF)x + B(D*DD)~ll2u

x' = -(A-BF)*x

y±u + {D*D)rll2B*x:

Since —(A — BF)* defines an antistable evolution the only solution x belonging to L2(—oo, oo;lltn) is the zero one and the input-output operator corresponding to the node G\G\ is the identity.

Remark that the evolution associated to the node G{ is exponentially stable and that from G = GiGQ we have G*G•='GJGJG'.G'o = G*0G0.

Consider now the "filtering Riccati equation"

S' = S(A- IIC)* + (A- IIC)S + (B - IID)(B - IID)*

II = (BD* + SC*)(DD*)~l .

Assume this equation has a bounded stabilizing solution, that is A — HC defines an exponentially stable evolution.

311

Under the above assumption define the nodes

Gi,co = {A-HC,B- IID, (DD*)-V2C,(DD*)~ll2D).

The node GQ co has the inverse

(A - IIC, H{DD*)~ll2, -(DD*)-1/2^ (DD*)-1/2)

and it is outer.

We have further G: roG*irn = I. We have indeed

GUo = (-(A - IICy,-C*(DD*)-V2,{B - HDy.D^DD*)-1/2).

The product GicoG*co corresponds to the system

x'=-(A-'HC)*x-C*(DD*)-l/2u

x' = (A - HC)x + {B- HD)[(B - HDyx + D*(DD*)-ll2u]

y - (DD*)~ll2Cx + (DD+)-ll2D[(B - HDYx + D^DD*)-1/2*]

= u + (DD*)-ll2[Cx + (DB* - DD*H*)x]

= u + {DD*)-ll2C(x-Sx).

Denote x — x — Sx; we deduce by using the filtering Riccati equation that

P = (A'-RG)x..

Since A — IIC defines an exponentially stable evolution we have x = 0, hence y = u.

The product G0 C0Gi co corresponds to the system

• x' = (A - HC)x + (B - HD)u

x1 = Ax + H{DD*)-ll2[((DD*)-ll2Cx + (DD+y^Du]

y = Cx + (DD*y/2[(DD*)-1/2Cx + (DD*)~1/2Du]

= C(x + x) + Du .

312

Denote x = x -f %', we have

x' = Ax -f i?u

and we see that G0,coGi co ls equivalent to G.

REMARK. Consider the linear-quadratic problem defined by A,B,K,L,M and assume that the corresponding Iticcati equation

W' + {A- BK-lL*)*W + W{A- BK~lL*)

- WBK~lB*W + M - LK-XL* =0

has a bounded stabilizing solution.

The usual computations lead to

/ (u*Ku + u*L*x + x*Lu + x*Mx)dt

= x*(t0)W{tQ)x(io)

roo

+ . / [u + K-\B*W 4- L*)x]*K[u + K~l(B*W + L*)x]dt J to

/•OO

= -x*(to)W{to)x(tQ)+ / (u + Fx)* K(u + Fx)dt. Jtn

with F±K-l(B*W + L*).-

If x(to) = 0 we deduce that

roo rOO

/ Jr(t,x,u)dt = / |j/(0|V< Jto Jto

where y is the output associated to the node 6*0 = (A,B,K1'2F,K1'2).

The property of Go to be invertible with the inverse defining an exponentially stable evolution corresponds to the fact that the system

313

obtained by taking the optimal feedback has stable zeros (in the stationary case).

If we consider inputs in 2>2(—00,00; IRm) we shall have, assuming that A defines an exponentially stable evolution

/

+oo r+oo

.F(*,z,u)cft= / \y(t)\2dt. •oo J — oo

Since G0 is given by x' = Ax + Bu, y = Kll2Fx -f Kll2u the inverse is

obtained with iz = Kll2y — Fxy x' = (A — BF)x 4- BKll2y and has indeed an

exponentially stable evolution.

REMARK. If a node (A,B,C,D) with ;4 defining an exponentially stable evolution satisfies D*(t)D(t) = / and D*(t)C(t) + B*(t)Q(t) = 0 then

y*(t)y(t)dt = / u*(t)u(t)dt; •oo J—oo

here Q is the unique bounded on IR solution of the Liapunov equation

Q''+A*{t)Q+QA(t) + C*{t)C{t) = U.

We shall extend now the nodes G± and

314

""•••' (A~~* o \ *

Define R = -U I ^_ " J u a n d

consider the extended node

G?xt = (4 - BF,{B{D*D)~XI2 RC*D±)yC -DF,(D(D*D)-^2 D±)).

We have ((&D)-*lrD* \ ^p*Dyl/2 D±j = 7 a n d

since VK is a solution of the Liapunov equation

W' + (A - £F)*VF + W(A - BF) + (C - DF)*(C - DF) = 0 .

Let us check that; the above condition is written

(D*D)-^2D*(C - DF) + (D*D)~1/2B*W = 0

D\{C - DF) + D*±CRW = 0 .

Since F = (©*-/7):r"1(J9*VF-+ D*C) the first equality is obvious. The second one reduces to

D*±(C - D?)(I + RW) = 0 .

Using definition of R we have

For a; 6 A'er.W, we have iiVFa; = 0 and (C — DF)x = 0, since Ker W C Ker(C - DF). It follows that for x e Ker W

D*±(C - DF){I + RW)x = 0 .

315

If x G 7Z(W) we may write x = Wv

= '-u(* ®)u*v = -Wv = -x

and (I-\-RW)x = 0. Taking into account that X = 1Z{W) © Ker W, we deduce that

D*±(C - DF)(I + RW) = 0 and the extended node G?x t is an isometry.

To obtain a factorization where G, is replaced by G^xt we have to replace also Go be Ggx t . Since we have added to G, new inputs, we have to add to Go new zero outputs in order to preserve the product node unchanged. We shall take thus

^,(A,s,(^f/a'),(^f/8;

In the same way we may take D± with DD*^ = QyD±D* — / , associate

to S an 5-inm the same way R was associated to W and define

It is seen in the same way as for the control Riccati equation that

KerS CKer(B-HD)* .

The conditions for G^0 to be isometric are

nDD*)ll2& \ DL

and

(B - IID){D*{DD*)-ll2 D*±) + S(C*{DD*)-1/2 - SBD*±) = 0

^(^(DD*)-1/2 Bl) = /

316

that is

(B - HD)D* + SC* = 0,(2? + HD)D\ - SSBD*± = 0 .

The first condition is satisfied from the definition of / / and the second one is written as

(/ - SS)(B - HD)D*± = 0 .

Let

/ - 5 5 = [ / Y Q J J U*; if x*S = Owe have Sx = 0 hence

(B-HD)*x = 0,

x*[(B - HD)D*± - SS(B - HD)D*±] = 0 ;

if x = Sv, x* = v*5,

x * ( / - 5 5 )

- « ( * s)«-"(s ;)«•-"(J !)(!;)»*-• and GJ*^ is isometric.

We define the node G%% = (Ar(;H('DD*)lf*0), £,((£>£>* J1/-2 •())) and

deduce the factorization G = G^G^.

We have used above a condition for G^0 to be isometric. Let us discuss this condition.

317

Let have a node (A,B,C,D) with A defining an exponentially stable evolution and such that DD* = / , BD*+SC* = 0 where S solves the Liapunov equation

S' = SA*(t) + A(t)S + B(t)B+(t). ^ «̂ >» ^ v y±

Associate the adjoint node (A,B,C,D) = (—A*,—C*,B*,D*); we have

D*D = DD* = J, C*S - SB = 0 ,

S' = - 5 2 ( 0 - A*(t)5 + C*(t)C(t)

and a direct computation shows that for x' = Ax -f 2?w, j / = Cx -f DM

£(x*Sx) = (£*£* + £*C*)(i)i2 + Cx) - u*D*Du ILL

= y*y — u*u ,

hence < y,y >=< u,u > and under the conditions above G* = G is isometric. Let us stress that the study of nonstationary nodes has been inspired

by the important book of Bart, Gohberg, Kashoek [1].

6. Best stabilizing compensators for stochastic systems

A. Consider a stochastic system

dx(t) = A(t)x(t)dt + B(t)dw(t)

z(t) = C[t)x(t)

Assume that A defines an exponentially stable evolution. A quality index for this evolution is taken as

•

To see how this index may be expressed we write

x(t) = XA(t,tQ)x(t0)+ f XA(tJs)B(s)dw(s)

(t) = zJt)+ I II(t,s)dw(s)

Jtn ' ' 0

z0(t) = C(i)XA(i,t0)x(i0)

II(t,s) = C(t)XA(t,s)B(s).

We deduce that

E\z(t)\2 = E[z\t)z{t)}

= \z0(t)\2 + z*(t)E f II(t,s)dw(s) + E\ [ II(t,s)dw(s)

Jto Ut0

+ E< f II(t,s)dw(s)] f H(t,s)dw{s)\ I Ut0 1 Jto J

= M 0 l 2 + / Tr[II\t,s)II{t,s)]ds. Jto

Since \zQ(t)\ < /?e-a('_ '°)|a;(*0)|,a > 0,t > tQ , we deduce

lim - / \z0(t)\2dt = 0

T—oo I Jto

lim - / E\z(t)\2dt T-KX> I Jto

— 1 /"to+T = lim - / {Tr[H*(t,s)H(t,s)]ds}dt

T->oo T Jt

Since l i m ^ ^ f_? Tr[H*(t,s)H(t,s)]ds = 0 we may also write

# X T t0 + T t 1 / 'O+T I f f 1

lim - / E\z(i)\2dt= lim - / / Tr[H*(t,s)H(t,s)]d T-+00 T Jto T-KX> T J ./.Qo

and

Let us remark that

Tr[H*(t,s)H(t9a)] = Tr[H(t,s)H*(t,s)]

= Tr[C(t)XA(t,s)B(s)B*(s)XA(t,s)C*(t)]

f Tr[C(t)XA{t,s)B(s)B*(s)XA(t,s)C*(t)}ds J — oo

= Tr[C(t) f XA(t,s)B(s)B*(s)XA(t,S)dsC*(t)] J—oo

Tr[C(*)P(t)C*(t)] where

J—oo IS

-OO

is the unique bounded on IR solution of the Liapunov equation

P1 = A{t)P+.PA*(t) + B{t)B*(t) .

We have proved that

lim" - / ° E\z(t)\2dt =.. lhn" - / rr[C(t)P(/)C*(t)]dt. T-+00 T Jto T—00 T 7

to

In the same way we have by changing the order of integration

•

320

and since I fto+T /-oo

/ ds I H*(t,s)H(t,s)dt < oo \Jto Jt+T

we deduce that

i fto+T i fto+T roo lim - / E\z(t)\2dt = lim - / ds I Tr[H*(t,s)H(t,s)]dt.

T-foo T j t o T->OO T y

In the same way

1 fto + T roo

lim - / ds I Tr[H*(ts)H(t,s)]dt T-oo T JtQ J3

1 fto+T r+oo = lim - / ds Tr[II*(t,s)II{t,s))dt

T-+00 T Jto J_00

hence

i r*o + i /"too lim - • / dt / Tr[H*(t,s)H(t,s)]ds

T-oo T JtQ J_00 i fto+T r+oo

= lim - / ds / Tr[H*(t,s)H(t,s)]dt T-^OO T j t o y . ^

All above formulae make sense also if H(t,s) corresponds to exponentially dichotomic evolution. We have

lim — / dt / H*{t,s)H(t,s)ds

T-*oo T JtQ J_00

i fto+T ( ft0 = lim - / dt\ / H*(t,s)H(i,s)ds

T - o o T Jto (J-oo

+ / H*(t,s)H(t,s)ds\

and from \H(t,s)\ < /3e~al*~sl we deduce that

I fto i roo lim - / H*(t,s)H(t,s)ds= lim - / H*(tys)H(t,s)ds = 0

r-4oo T y . o o T-+00 T JtQ+T

322

hence •«o+T /-+oo

f'-^'oo T _ ^

1 flQ-t I f-tOO lim - / dt I H\t,s)H(t,s)ds "-+00 T Jto J_00

i rto+T /"

Then

•-foo r + o o r + o o f-foo

/ H*(t,s)H(tys)dt= / iq(t,s)H2(t,s)dt. J—oo «/—oo

Proof. We have //(«,«) = D^H^s) + / + ~ H1(t,s)H2(

324

LEMMA 2. Assume S = SiS2,S2S% = / , D*2{t) = [D2(t)]~l a n c f 5 l

corresponds to an integral operktor. Then

/

-hoo r-too

Tr[II*(t,s)H(t,s)]ds= / rr[//f(t,a)jyi(2(5)//2*(r,S)//r(«,r)//1(

325

We use now Tr(AB) - Tr(BA) to deduce

*4-oo r+oo

Tr[H*(t,s)H(t,s)]ds = / Tr[HJ(trs)H.1(t;8)]ds -oo •/—oo

/

+oo r-f-00 ./. ff1*(t,«)/r1(

326

Associate a stabilizing compensator

dxc(t) = Ac(t)xc(t)dt + Dc(t)duc(t)

Vc(t) = Cc(t)xc(t).

The compensator is coupled to the system by taking uc = y,u = yc as in the deterministic case. By such coupling we obtain the system

dx(t) = [A{t)x(t) + B2(t)Cc(t)xc(t)]dt + B\(t)dw(t)

dxc{t) = [Ac(t)xc(t) + Bc(t)C2(t)x{t)]dt + Bc(t)D2\(t)div(t)

z(t) = C i ( 0 * ( 0 + Di2{t)Cc(t)xc(t).

The compensator is stabilizing if the evolution defined by ( r* p \ c

is exponentially stable. The quality of the stabilizing compensator is measured

by the index described above lim - / / 0 + T E\z(t)\2dt, and we may state the T—*oo T °

problem of finding the stabilizing compensators that minimizes the quality index.

We shall describe now the class of stabilizing compensators for which this minimization problem will be considered. Define a system (S) by

dx(t) = {[A(t) - B2(t)F(t)]x{t) + B2{t)F{t)x(t) + B2(t)u(t)}dt

+Bi(t)dw(t)

dx(t) = [A{t)II(t)C2{t)]x{t)]dt + [Bx(t) - H{t)D

327

We couple the "parameter system7' (L) to (S) by taking ue = y2, u2 — Ve

and obtain

:dx(t) = {[A(t) - B2(t)F(t)]x(t) + B2{t)F(t)x(t) +B2WCe(J)xe(J)}dt + BiWdw(r)

dx(t) = [A(t)'- Il(t)C2(t)]x(t)]dt + [Bi(t) - H(t)D21(t)]dw(t)

dxe(t) = [i4c(0*e(0 + Be{t)C2{t)x{t)]dt + Be(t)D2i(t)dw(t)

z(t) = [Ci(0 - £i2(0F(*)M*) + ^12(0^(0*(0;

+£>l2Ce(0*e(*) .

Let us now make the crucial remark that this last system may be also considered as obtained by coupling to (*) the "parametrized family of compensators"

dx(t) - {[A(t) - B2(t)F(t) - H{t)C2(t)]x(t) -B2(t)Ce(t)xe(t)}dt-H(t)duc(t)

dxe(t) = {[Ae(t)xe{t) + Be{t)C2{t)x(t)}dt + Be{t)duc(t)

yc(t)=F(t)x(t) + Ce{t)xe{t).

Coupling this family to (.*) means uc = y,u = yc\ such coupling leads to

dx(t)= [A(i)x(t) + B2(t)F(t)x(t) + B2{t)Ce(t)xe(t)]dt + Bi(t)dw(t)

dx{t) = {[A(t) - B2(t)F(t) - E(t)C2(t)]x(t) - B2(t)Ce(t)xe{t)

-II(t)C2(t)x(t)}dt-II(t)D21(t)dw(t)

dxe(t) = [Ae(t)xe{t) + Be(t)C2(t)x(t) + Be(t)C2(t)x(t)]dt

+Be(t)D21(t)dw(t)

z(t) = C^xit) + Dl2(t)F(t)x(t) + D12(t)Ce(t)xe(t).

328

If we take x(t) = x(t) + x(t) we obtain the same system as the one resulting if coupling (L) to (S).

If (A, B2) is stabilizable and (C2 , A) is detectable we may choose F such that A — B2F defines an exponentially stable evolution and we may choose / / such that A - HC2 defines an exponentially stable evolution. Assume that Ae defines an exponentially stable evolution.

Then we see that the family of compensators parametrized by (L) is stabilizing for the given system (*) and we may ask for the parameter (L) that gives an optimal quality index.

We have seen that the quality index is expressed in terms of Trll*(t,s)II(t,s) where / / is the kernel of the input output operator from the noise to the quality output z. Let us remark now that the same kernel is obtained if we start with the deterministic system

x' = Ax -f B2u2 -f B\u\

y2 = C2x + D2lux

z = C^x -f D12u2

and stabilize it by using the family of compensators

x' = (A - D2F - IIC2)x - B2Cexe - Huc

< = K*e + B e ^ + Be"c

y = Fx -f Cxa Jc e e

(with u2 = yc, uc = y2) or equivalently by coupling to the system (S1)

x' = (A - B2F)x + B2Fx + B2u2 + Blul

x1 = (A - HC2)x -f (Bx - HD21)Ul

y2 = C2x + D2lUl

z = (C1-DuF)x + D12Fx + D12u2

the "parameter system" [L')

x' — Axo -\- Bua , yo = Cxc . e e e e e , ; / e e e

329

If CS1) is written as ( )' = ( „ n } 2 ) ( * ) we see that S„ is \ ' \y2J \S21 S22J \u2J " defined by the node

and S12 is defined by the node

) v(*02) , (C, - D12F Dl2F),Dl2y

A-B2F B2F

0 A-HC2

Since for the node Si2 we have x' = (A — IIC2)x we deduce x = 0 and Su reduces to (A ~ B2FiBvCl - Dl2F,.D12):

We see next that S21 is defined by (A — HC2,B1 — HD21,C2,D21) and that S22 = 0. Coupling (L') to (5

;) means

ue yvu2- ye\ we have u2 - ye = L'ue = Z'?/2 and

* = ^jjt*! + Suu2 = 5-JJMJ + S12L'y2 = ( 5 n + S12L'S21)u .

We have seen that our optimization problem consists in finding 2/ in order to minimize

i /^o+T z'+oo lim - / dt rr[//*(t,a)jy(t,5)]da

T->-oo T 7 fo J -oo

. i rto+T r+oo = lim - / ds I Tr[H\t,s)H{t,s)]ds

where 7/ is the kernel associated to the input-output operator Su + S12L'S21.

E. We shall now use the specific assumptions that

D12=(°I),D21 = (0 l)yBl = {Bu 0 ) , ^ = ^ " ) .

330

In this case Sn is defined by

((A-B2 B2F \ (Bn 0 \ (Cn 0\ \ U 0 A-IW2)'\Bn -Il)'\-F FJ'V'

Assume that F , / / have been constructed by using stabilizing solutions to Riccati equations

R' -f A*R + RA- RB2B^R + C{CX = 0

S' = SA* + AS - SC;C2S + BxBl

F = D\2CX + B^R, II = BXD^ + SC*

(D\2CX=V, 2 ^ = 0 ! ) .

For Sl2 = (A - B2F,B2, Cx - D12F, D12) we have S*2 = (-(A - B2F)*,-{Cl ~ DuF)*,B*2,D\2)y the product node S{2Sl2 is written as

x\ = (A - B2F)x1 + B2u

x'2 = -(A - B2FYx2 - (Cx - Dl2FY[{Cl - Dl2F)Xl + D12u)

y = B2>x2 + D*12[(C1-D12F)x + ni2u]

= u + B*x2 +(Di2Cv- F)xl

• = u + B2(x2 — Rxx).

If we denote x3 — x2 — Rxx and use the Riccati equation for R we deduce

^ 3 = v,y = u, »J 12^12 = •*•

We may extend S12 as in the preceding section to a node

W-(A-V.(* t(cn o)Q),(c«) .(}),.(•{))

-(/l-JJ.F.B, "C„),(^).(? „)) ,

331

for which the adjoint is

(st^y = (c(A-B2FYA-cu >,*)-(c7!y.(? £))

In the same way we see that

S21S21 ==••/", and we extend 52 1 to

sir=(A-Hc2,(Bn-H),(^_^s ) , ( ; j )

to obtain an isometric node. We have

Q 'J-QTfQ — Q _|_ Cext ( L " ] Cext ^11 T ^ 1 2 ^ °21 - °11 + ^12 I 0 0 / 2 1

Since SJ? = (512 512),5ff = (|j)

(•?i2 î2)(o o j G " ) - ^12^ / /^21

We deduce

^11 + ^ 1 2 ^ ' ^ 2 1

cext "~ °12

__ Cext — «->i2

5 l l + ( o 0). cext ^21

^11,11 "*" ^ ' ^11,12

^11,21 ^11,22

Cext C _ / C e x t V * c Y C e x t V °21 » ^11 — Wl2 ) °\\\D2\ )

332

By using Lemma 3 we deduce that if H is the kernel corresponding to Su -f- S12L

,S2l there

1 fto+T f+oo lim - / dt Tr[II+(t,s)H{t,s)]ds

T-^OO T j t o j ^

i fto+T f+oo = lim - / ds / Tr[H*(t,s)H(t,s)]dt

T-+oo T JtQ J_00

= lim - / ds / Tr[//1*1(^5)//11(^5)]^

i fto+T f+oo _ _ = lim - / dt TrlH^t^s)!!^^)^

T^oo T Jto J_00

where Hu is the kernel corresponding to the node

^11,11 + L' ^11,12

^11,21 ^11,22

hence has the structure

#11,11 + ^ # i i , 12

#11,21 #11,22

We have further

(#11,11 + W l l , 1 2 + £ ) + ^11,21^11,21 *

irnHu = [ * #11,12#11,12 + #11,22#11,22>

and

rrii^ir.j = rr[(51IiU + £)•(//„,„ + Z>]

+ Tr(Hih2lHn>2l) + Tr(Hlhl2HlhU) + Tr(H'll>22Hll>22).

The only relevant term for our optimization problem is

Tr[(Hn,n + l)*{Hu,u + £)].

Let us consider closely the node 5 n n ; we have

Sn = (SlfySniSff)* described by

x\ = -(A - HC2)*xi - C%ui + S*Bnu2

x2 = (A-B2F)x2 + B2Fx3 + Bn(Bl1x1 + u2)

4 = {A- HC2)x3 + # 1 1 ( ^ * 1 + n2) -IU-IVXX + t»i)

x'A = -(A - B2F)*xA - C{1Cnx2 + F*F(x3 - x2)

2/1 = # 2 * 4 - ^ 2 + ^X3

2/2 = CnRxi +Ci\x2 .

We deduce that 5 n n from u\ to 2/1 is described by

x\ = -(A - HC2Yxi - C^ui

x'2 = (A - B2F)x2 + B2Fx3 + fl-nflf!*!

4 = ( A - 77C2)x3 i-BnB^xi +HH*Xl - Hui

x'A = ~{A- B2F)*x4 - C[xCnx2 "+ F*Ffa.- x2)

Vl = B^Xi +F(xz-x2) .

Write the Riccati equations in Liapunov form

R! + (A - JS2*y # + i?(A - £ 2 F) + F*F + C*nCn = 0

5 ' = S(A - HC2y + (A- HC2)S + IUI* + i^ i i^ i i .

334

We may write then

x'3 = (A- IIC2)x3 + [S' - S(A - HC2y -(A- IIC2)S]x1

- Hui ={A- TIC2)(x3 - Sxi) + S'xx + S(x[ + C2*ui) - II u\

and since / / = SC^ we deduce (2:3 — Sx\)' = (A — IIC2)(x3 — Sx\) hence #3 = Sx\.

In the same way

(x4 - Rx2)' = -(A - B2F)*xA - 01x0x1X2 + F*F(x3 - x2)

- R(A - B2F)x2 - RB2Fx3 - RBx\B\xX\

+ (A-B2F)Rx2 + R{A-B2F)x2HF*F + CnCn)x2

= -{A - B2F)*(xA - Rx2) + (F*F - RB2F)x3

••- RBx\B\xx\ .

We have F* = RB2 hence

(a?4 - -/te2)' = - ( 4 - 52F)*(a;4 - Rx2) - RBx\B\xX\ .

Denote X5 = 0:4—72^2 and consider

* i : =-(A- iyC 2 )*a ; i . - ' e, 2Ui

4 = -(A - B2F)*x5 - tf*n2*Ii*i

2/1 = B\{xh + ^ 2 ) - Fx2 + ^ 3

The node Sxifn is thus described by

335

and corresponds to an antistable evolution while V corresponds to an exponentially stable evolution.

We deduce that

w , #11,11 (M) , 't < ̂ (#11,11+ £ ) ( M ) = {„

£(/,s) t>S

and

Jim - / e/* / Tr{[Hlhll(t,8) + Z M n f f i l , i i ( M ) + L(t,s)]}ds

-- lim - / < ^ / rr[//1*111(a)// l l f l l(*,*)]da + r r . /

(tys)L(t1s)ds > . + L

It is now clear that the minimum will be obtained for L = 0 hence for L = 0.

Let us state the final result

THEOREM 9. Consider the stochastic system

dx(t) = [A(t)x(t) + B2{t)u(t)]dt + Bn(t)dw(t)

dy(t) = C2{t)x(t)dt + dv(t)

z1(t) = Cnx(t) , z2(t) = u(t)

with (A,B2,C2) uniformly controllable and observable. Then the compensator that stabilizes the system and minimizes

•

336

is given by

dx(t) = [A(t) - B2(t)F(t) - H(t)C2(t)]x(t)dt

- H(t)duc(t)

yc(t) = F(t)x(t)

where

F(t) = B;(t)R(t), H(t) = S(t)C;(t)

Ry S, being stabilizing solutions defined on IR for the Riccati equations

R' + A*(t)R + RA(t) - RB2(t)B;(t)R + C*,(t)Cn(t) = 0

S' = SA*(t) + A(t)S - SC*{t)C2{t)S +.Bll(t)Bl1(t).

ADDENDUM

Hankel and Toeplitz Operators. Theorem of Glover

A. Let (A,B,C,0) be a node with A defining an exponentially dichotomic evolution. Associate the operator

G[f'B'C) : L2(-oo,oo;IRm).-• Z2(-oo,oo;IRp)

with the usual formula

A

C(i)XA(t,t0)l[xA(t0,s)B(8),s < t

G^B'c\t,s)=< to

- C(t)XA(t,t0)(I - H)XA(t0,s)B(s),s > t to

337

We may write

Z2(-oo,oo) = Z 2 ( -oo , / 0 )© Z2(*0,oo)

and associate Toeplitz and Henkel operators

j4A,BtC) = p G{AyB,C) to 2 to

n(A,B,C) _ p.-G(A,B,C) to 1 to

L2((

338

(since u = 0 for t < t0 and for t < t0 < s we have

G#'B'C)(M) = -C(t)XA(t,t0)(l- fl\xA(t0,s)B(s)). to

If the node has an antistable evolution then for all /0 we have Y\t0 — 0 and

too

H\*'B'C)u){t) = - C(t)XA(t,s)B(s)u(s)ds. Jt0

If the node has an exponentially stable evolution, then I ~Ylt = 0 and n{A,ByC) = Q

If (i4, J9, C) is the sum of a node with antistable evolution and of a node with exponentially stable evolution, we have

(* * ) • « - ( £ ) • c-'c. c->

n-(!!)- '-ti-('ol) to X '

a node (A, B,C) with exponentially stable evolution we shall have

hence

MA,B,C)

L2((«o,oo):IRm)

A0\ /B

r{AyB,C)

L2((*0,oo):IRm)

L2(( n Ow-*)). to H (A,B,C) to

{A>B,C) to

r(A,B,C)\

L2((f0,oo):lRm) lL2((*0,cc):IR

m) inf j

~ ~~— . • 1 II (A B C\ A,B,C with exponentially stable evolution > > \\Ht0' '

B. Let (A,B,C) be a node with antistable evolution and

/

CO

XA(t,T)B{T)B'{T)X*A{t,T)dT

Q(t)= f XA(r,t)C*(T)C(r)XA(T,t}dT .

If we have uniform controllability and uniform observability, then

P(t)>Kl,'Q(t)>-Kl,K>Q;''P,Q

are bounded on lit solutions for the Liapunov equations

P' = A(i)P + PA*(t) -B{t)B*(t)

Q' + QA(t) + AHt)Q - C*(t)C(t) = 0 .

340

We deduce that A - BB*P~X = P'P-1 + P ( - A * ) P - 1 hence A — BB*P~l is obtained via a Liapunov transformation from (—A*) which defines an exponentially stable evolution. It follows that

XA-BB-P-^T) = P(t)XA(Ttt)P-i(r)

and in the same way

A - Q-*C*C = Q-iQ' - Q-M*Q

XA-Q->c.c(t,T) = Q-*(t)X*A(T,t)Q(r).

We deduce that

XA(T,t) = QWA-Q->C-c(t,T)Q-l(T)

= P-\t)XA_BB.p.1{t,T)P{T)

XA-BB-P-^T) = P(t)Q(t)XA_Q-lc.c(ttT)[P(T)Q(T)]-i

and P, Q perform the same type of stabilization for A.

Consider now the Hankel operator

roo

CH\*'B-C)u)(t) = - / C(t)XA(t,s)B(s)u(s)ds

and look for the adjoint

A direct calculation shows that

(H\*

341

We have next

[{n{A,B,C)Yn(A,B,C)u]{s)

= B*(s)XA(tQ,s)Q(t0) f" XA(t0,s)B(s)u(s)ds . Jto

We are now in position to prove

THEOREM (Glover, [8]).

The nonzero eigenvalues of the operator (WJ ' ' J 7i\0 ' ' are the

eigenvalues of P(t0)Q(t0) (called the singular values of(A,B,C)).

Proof. Let o2{t0) be a nonzero eigenvalue of \Hto ' ' ) 7i/0 ' ' \vtQ a

corresponding eigenvector; we have

roo

B*(s)XA(t0,s)Q(t0) XA(t0,T)B(a)vto(T)dr =

342

Define

»i.W = ^qj-;B*(s)X'A(t0,s)Q(t0)v(t0)

then

= --^TrC(t)XA(t,t0) / XA(i0,s)B(s)B'(s)XA(t0,s)dsQ(t0)v(t0) a (lo) Jto

= -^^f(t)XA(t,t0)P(t0)Q(t0)v(t0) = -C(t)XA(t,t0)v(t0)

i{n(A,B,C)rniA,B,C)vj{s)

roo

= B'(s)XA(t0,s) / XA(r,t0)C'(r)C(T)XA(r,t0)dTv(t0) J to

= B'(S)XA(t0,s)Q(t0)v(t0) = ) X W ) )

max

343

and we may compute the norm of the Hankel operator associated to (A, B,C) by using the Liapunov equations.

C. We may repeat the same considerations if (A,ByC) corresponds to an exponentially stable evolution. Define

^ ( 0 = / XA(t,s)B(s)B'(s)XA(t,S)ds J—oo

/

oo

X*A(syt)C*(s)C(s)XA(s,t)ds

P'(t) = A(t)P(t) + P(t)A*(t) + B(t)B*{t)

Qf(t) + Q(t)A(t)^A*(t)Q(t) + C^t)C(t) = 0.

We define the Hankel operator

^r{A,B,c)uyt) = Jt0^cmA{t!S)B{s)u{s)ds

for t > t0,

(r jV'o )t . ) (0 = 0, t

344

A direct calculation gives

= B*(s)XA(t0,s)Q(t0) I'" XA(t0,T)B(r)u(T)dr . J — OO

The Theorem of Glover states that the nonzero eigenvalues of

( T{to'B'C)) TitolB'C) a r e t h e eigenvalues of P(t0)Q(t0).

The proof is quite similar to the one above.

D. If A defines an exponentially stable evolution we have Ylt = I a n ^

(^oA'B-cKyt)=Jt'c(t)XA(t,s)B(s)u(s)ds

hence for nodes with exponentially stable evolution the Toeplitz operator is the input-output operator corresponding to zero initial conditions and future evolution. A direct computation gives in this case

( ^ • B , C ) ) *«/](*) = J™ B*(t)XA(s,t)C*(s)y(s)ds

(?LA'B'C))V^^](0 = £•(

345

with exponentially dichotomic evolution. Define

and perform the node similarity for G*G defined by S: a direct calculation shows that G*G is similar to

Node similarity does not change the operator, hence it does not change the Toeplitz operator. We deduce that we may compute T^*G and we obtain

fT(A,B,C)\ *fT(A,B,C)\ _ T(A,B,Cy(A,B,C)

a result known in the stationary case (see footnote at page 331 in E.A. Jonckheere h Jyh-Ching Juang, [12]).

The same result holds for nodes (A,B,C,D).

E. For a node G and a stabilizing compensator Gc, the coupling is given by uc = y,u = yc; we have y = Gu,yc = Gcuc and after the coupling y = GGcy,yc = GcGyc.

Stabilization of G by Gc will be related to properties of the nodes I—GGC and / — GCG. We may define stability by asking the operator ( / — Ggjp)"

1 to be bounded and such property is satisfied if H&GJI < 1.

We are looking now for perturbations AG of G such that I—(G+AG)GC is still invertible.

Since

[/ - (G + AG)GJ-i = ( / - GGe)-*[I - (AG)GC(/ - GGe)-»]->

we deduce the condition

| | ( A G ) G c ( / - G G c ) - i | | < l .

346

If ||AG|| < u sufficient condition will be

1 £\\'

A compensator Gc stabilizes GUI — GGC and J — GCG are invertible with bounded inverses corresponding to exponentially stable evolutions.

We may express these properties in terms of Hankel operators as H(l-GGC)-* = 0 ? f t ( / - G c G ) - i = Q >

If we assume (A,B) uniformly controllable and (C, A) uniformly observable we may define

M = (A-BF,B,-F,I), M =(A-HC,H,-C,I)

N = (A-BF1B,C,0), N = (A- / /C,J9 ,C,0)

U =XA-nC,H,F,0), U = (A- BF,H,F,0)

V = (A-HCyB,F,I), V = (A-BF,H,CJ)

to define coprime factorizations

G = (N + DM)M~1 , VM + UN = 1

G = M-1(N + MD) „ NU + MV = I.

The node (V + LN)~l(LM - U) = (ML- U)(V + NL)~l defines a family of stabilizing compensators for G, parametrized by the node L with exponentially stable evolution.

For this family of stabilizing compensators

GC{I - GGc)-i = M(L - M-W)M] \{ M*M = l,M*M = I,

then

\\Gn-GGc)-i\\ = \\L-M*V\\.

347

Since L corresponds to an exponentially stable evolution we deduce that

HL-M*U = n;QM'u.

Assume A to be antistable and construct F,H by using the corresponding Liapunov equations, F = B*P~1,H = Q~1C*\ M*U corresponds to x' = (A - BB*P~1)x + Q^C+u, z' = Az - BB*P~1xy y = -B*P~lx + B*P-*z, (x - z)' = A(x -z) + Q^C^u, y = ~B*P^x-zz); M*JJ = (A,Q-lC*,-B*P-x,U). Moreover MM* = M*M = J, M*M = MM* = I.

The corresponding Liapunov equations are

'••?' = AP + PA - Q-iC*CQ~l

Q' + QA + A*Q-P-1BB*P~1

and we deduce that P = Q - 1 , Q = P _ 1 ; we deduce that

1

mm ff(P(t0)Q(t0))

hence

,L-M*Uv 1-

then

II7Y II — "

If there exists X, stable, such that

to

| |G c ( J -GG c ) -» | | = m a x - r * t0 mm a(P(t0)Q(t0))

mmto mm a(P(t0)Q(t0))

348

and stabilization is preserved for

| |AG||(

349

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Aristide HALANAY, Bucharest, Romania.

Lavoro pervenuto in redazione il 16.10.1990,

in forma defmitiva il 17.3.1991.