Chapter 4

Hoo-control with state-feedback

In this chapter we formulate and solve the ?too-control problem with state­feedback for the Pritchard-Salamon class. In particular, we show that under certain conditions there exists a state-feedback that solves the sub-optimal ?too-control problem if and only if there exists a solution to a certain operator Riccati equation. The main result is given in Section 4.1 (see Theorem 4.4) and proved in Section 4.2.

The result is a generalization of the finite-dimensional results of Doyle et al. [26] and Tadmor [87] (see also the summary of results in Section 1.1) and the proof follows roughly the lines of the proof of Tadmor [87]. Pritchard­Salamon systems are infinite-dimensional systems with unbounded input and output operators and the consequences of these facts for the proof shall be stressed. As in the previous chapter on the LQ-problem, we shall deal with the unboundedness of input and output operators by considering initial conditions

in the 'larger space' V, so that in fact only the output operators are unbounded (with respect to V).

In order to alleviate the notational complexity we shall first suppose that a certain simplifying assumption is satisfied. However, this assumption is not necesary, as will be shown in Section 4.3 (see Theorem 4.20) .

The result in this chapter is the first step towards solving the ?too -control problem with measurement-feedback in Chapter 5.

4.1 Problem formulation and main result

Let W,?t and V be real separable Hilbert spaces, satisfying W L-t ?t L-t V, let S(·) be a Co-semigroup on V which restricts to Co-semigroups on Wand 1l and suppose that D(AV) L-t W. Let BJ E £(W, V), B2 E £(U, V), C1 E £(W, Z), Dll E £(W, Z) and D12 E £(U, Z), where U, Wand Z are also real separable Hilbert spaces. We identify the dual of ?t with itself and the duals of Wand V are determined by the pivot space formulation of Section 2.5 (see

B. Keulen (ed.), H∞-Control for Distributed Parameter Systems: A State-Space Approach© Birkhäuser Boston 1993

102 1too -controJ with state-feedback

formula (2.69)). Furthermore, the duals of U, Wand Z are assumed to be identified with themselves.

Assuming that BI and B2 are admissible input operators and that CI is an admissible output operator, we define the smooth Pritchard-Salamon system EG = E(S(·), (BI B2), CI , (Dll D12 )) of the form

{ x(t) = S(t)xo + J; S(t - s)(Btw(s) + B2u(s))ds

EG: z(t) = Clx(t) + DllW(t) + D12u(t),

(4.1)

where Xo E V, t ~ 0 and (w(·), u(·)) E L~C(O, 00; W x U). We call x(t) E V the state of the system, u(t) E U is the control input, w(t) E W is the disturbance input and z(t) E Z is the to-be-controlled output.

The purpose of this chapter is to find a state-feedback that exponentially stabilizes the system, such that the corresponding closed-loop map from w(·) to z(·) satisfies a certain norm bound, or, to be more precise: Suppose that F E £(W, U) is an admissible output operator for this system, such that the Co-semigroup S B2F(') is exponentially stable on V. If we apply the admissible state-feedback u(·) = Fx(·) to system (4.1) we obtain the Pritchard-Salamon system E(SB2F('), BI, (CI + D I2 F), Dn) (d. Lemma 2.14):

{ x(t) = SB2F(t)XO + J~ SB2F(t - s)Btw(s)ds

EGB2F :

z(t) = (Cl + D12F)x(t) + Dnw(t). ( 4.2)

It follows from Lemma 3.2 that the linear map G B2F (corresponding to Xo = 0) defined by

(GB2FW)(t) = z(t) = (CI+D12F) it SB2F(t-s)Bl w(s)ds+Dllw(t)(4.3)

satisfies GB2F E £(L2(0, 00; W), L2(0, 00; Z)). It follows from Lemma 3.4 that

IIGB2F II = IIGB2F(-)lloo, where GB2F is the closed-loop transfer function given by

Definition 4.1 A ,-admissible state-feedback for EG given by (4.1) is an admissible output operator F E £(W, U) such that the Co-semigroup SB2FC) is exponentially stable on V and the corresponding linear map G B2F defined by (4.3) satisfies

IIGB2F II < ,.

4.1 Problem formulation and main result 103

Remark 4.2 We recall that if a Co-semigroup is exponentially stable on V, it need not be exponentially stable on Wand vice versa (see [15]). For the moment we only ask for stability on V, which is sufficient to render the closed-loop map given by (4.3) bounded. If, in addition, we know that there exist some admissible input and output operators F and G such that SGF(-) is exponentially stable on both Wand V, then any admissible perturbation of S(·) is exponentially stable on W if and only if it is exponentially stable on V (see Theorem 2.20 item (iv)).

Finally, we note that the approach here is similar to the one that we had for the LQ-problem in Chapter 3. In particular, since we consider initial conditions in the 'larger space' V and stability on V, we shall only have to worry about unboundedness of the output operators.

Below we shall relate the existence of a ,-admissible state-feedback to the

solvability of an operator Riccati equation, but first we have to make two regularity assumptions and a simplifying assumption:

there is an (' > 0 such that for all (w, x, u) E IR x D(Av) x U with

D~2Dl2 is coercive, (4.5)

Du = O. (4.6)

Remark 4.3 Assumptions (4.4) and (4.5) are the generalizations of the weak finite-dimens­ional regularity assumptions in [37]. In finite-dimensional terms, (4.4) means that the system E(A, B2, Cl , D12 ) has no invariant zeros on the imaginary axis (whence the term invariant zeros condition). The Hoo-control problem with assumptions (4.4) and (4.5) is usually called regular. We note that (4.4),(4.5) and (4.6) imply that the LQ-problem with stability is solvable, where the cost

is given by Ilz(')IIL(o,oo;z) (see Chapter 3). It follows from Appendix C.l that if D~2Cl = 0 is satisfied and D~2D12 is

coercive, then (4.4) is implied by the detect ability assumption that there exists an admissible output operator G E £( Z, V) such that SGC, (.) is exponentially stable on V.

The assumption that Dll = 0 is merely made to reduce the complexity of the formulas. In Section 4.3 we shall show that without this assumption we can still relate the existence of a ,-admissible state-feedback to the solvability of an operator Riccati equation. Hence, we have a perfect generalization of

104 'Hoo-control with state-feedback

the weakest a priori regularity assumptions that have been used in the finite­

dimensional case. Finally, in Section 4.3 we show that the regularity assumptions (4.4)-(4.5)

can be removed as well, however at the expense of introducing an extra, reg­ularizing parameter into the problem.

Next, we present the main result of this chapter. It is a perfect generalization

of the finite-dimensional state-feedback result in [37J.

Theorem 4.4 Let EG be the smooth Pritchard-Salamon system given by (4.1) and suppose that assumptions (4.4),(4.5) and (4.6) hold. The following are equivalent state­ments:

(i) there exists a ,-admissible state-feedback for EG,

(ii) there exists an admissible output operator FE £(W, U) such that SB2 F (-)

is exponentially stable on V and there exists a 8 > 0 such that for all w(·) E L2(0, 00; W) there exists a u(·) E L2(0, 00; U) such that x(·) de­fined by

x(t):= it S(t - s)(B1w(s) + B2u(s))ds (4.7)

satisfies x(·) E L2(0, 00; V) and zw,u(·) defined by

zw,u(·) := C1x(·) + D12u( ·)

satisfies

IIZw,uOIlL(o,oo;z) ::; (l- 82 )llw(·)IIL(o,oo;w),

(iii) there exists aPE £(V, V') with P = pi ~ 0 satisfying

< Px, (Av - B2(D~2D12rl D~2Cdy ><v',v> +

< (Av - B2(D~2Dd-1 D~2Cdx, Py ><v,v'> +

< p(,-2 BIB~ - B2(D~2DI2rl B~)Px, y ><v',v> +

< (I - D12(D~2D12rl D~2)CIX,

(J - DI2(D~2DI2rl D~2)ClY >z= 0

for all x,y E D(AV), such that A~ given by D(A~) = D(AV) and

( 4.8)

( 4.9)

A~ = AV - B2(D~2Dd-1 D~2CI + (,-2 BIB~ - B2(D~2DI2rl B~)P

is the infinitesimal generator of a Co-semigroup S p(.) which is exponen­tially stable on V.

4.2 Proof of the state-feedback result

In this case, a ,-admissible state-feedback is given by u(·)

F E £(W, U) is the admissible output operator given by

105

Fx(·), where

(4.10)

Moreover, if there exists an operator P satisfying item (iii), then it is unique.

Remark 4.5 In the proof of this result, which is given in Section 4.2, we shall partly use the approach of [87J for the finite-dimensional case. There, Tadmor relates the H(Xl-control problem with state-feedback to the solvability of a certain sup-inf-problem (see (4.15)), which can be solved using some arguments from game theory. We also refer to [70J and [86J where several parts of the proof in [87J have been worked out in detail. It is because of this approach that we are

able to prove the equivalence with (ii) (this was also realized in [86]). The nice thing about this fact is that it can be used in the proof of the measurement­

feedback result in Chapter 5: if there exists a dynamic measurement-feedback that solves the problem, then there also exists a static state-feedback that solves the problem and a solution to the state-feedback Riccati equation.

Hence, it also shows that if there exists a ,-admissible dynamic state-feedback, then there also exists a ,-admissible static state-feedback. It follows from Sec­tion 5.4 that this assertion remains true without any a priori assumptions.

Remark 4.6 We could also have formulated the Riccati equation in terms of the operator Po := c-1p E £(V), where c = i'/i iVd (ivtl is the isometry from V to V' (d. (2.69)). It is easy to see that B; = B~c and B; = B~c. Supposing that D;2[C1 Dd = [0 I] (this simplifies the formulas considerably), (4.9) can be reformulated as

+ < C1x, Cly >z= 0 for all x, y E D(A). (4.11 )

We have chosen the formulation of (4.9) because it is more convenient in the derivation of the measurement-feedback result. In the proof of Theorem 4.4, we shall in fact derive the existence of a Po as above and then define P by

P:= cPo·

106 1too -control with state-feedback

4.2 Proof of the state-feedback result

It is easy to see that item (i) implies item (ii). Therefore, we shall only prove the implications (ii) ::} (iii) and (iii) ::} (i). We shall first prove these implications for the special case that

(4.12)

Then we remove this extra assumption by introducing an admissible prelimi­nary feedback, using the results of Lemma 2.14. It is easy to see that (4.12) implies that (4.5) holds and that the to-be-controlled output satisfies

< z,z >z=< C1X,C1X >z + < u,u >u, ( 4.13)

for all x E Wand u E U. So for the moment we have that (4.4) holds and that (4.12) is satisfied.

a) Proof of (ii) ::} (iii)

In this part we assume that (ii) is satisfied. Consider the system (4.1), where

the initial condition is given by Xo = ~ for some ~ E V and w(·) is some element in L2(0, OOj W). Furthermore, let u E L2(0, OOj U) be any input such that the corresponding state from (4.1) satisfies x(·) E L 2(0,oojV) (the existence of such u(·) is guaranteed by the assumption that (ii) is satisfied). Note that the to-be-controlled output z(·) E L2(0, OOj Z) (see Lemma 3.2) and so the following cost functional is well-defined:

J(~, w(·), u(·)) := Ilz(')IIL(o,oo;z) -lllw(')IIL(o,oo;w)' ( 4.14)

Because of (4.12) and (4.13), Ilz(')IIL(o,oo;z) is given by

IIz(')IIL(o,oo;z) = 100 « C1x(r),C1x(r) >z + < u(r),u(r) >u)dr.

Using the assumption that (ii) holds, we will solve the following sup-inf­problem:

sup inf J(e, w(·), u(·)), w(')EL2(O,00;W) u(')E L2(O,00;U)

( 4.15)

where u(·) E L2(0, OOj U) must be such that x(·) E L2(0, OOj V). The solution of this problem will lead us to the solution of (4.9) with the required properties.

First we consider the following optimal control problem.

Given (e, w(·)) E V X L2(0, OOj W), find the infimum of

P: {IIz(-)IIL(o,oo;z) I (x(-),z(·)) satisfy (4.1) with Xo = e

and u E L2(0, OOj U) is such that x(·) E L2(0, OOj V)}.

4.2 Proof of the state-feedback result 107

To solve this problem we need Lemma 3.10. We know that there exists an

admissible output operator F E £(W, U) such that S B2F(') is exponentially stable on V. Remark 2.19 then implies that the pair (AV, B2 ) is exponentially stabilizable. Furthermore, we conclude from assumption (404) combined with assumption (4.12) that condition (iii) in Theorem 3.10 is satisfied with

Hence, using Remark 3.14, formula (3.66), this theorem implies the existence of some L E £(V) with L = L' :::; 0 such that

(4.16)

for all x, y E D(AV) and Ar = AV + B2B'7,L is the generator of a Co-semigroup Sd·), which is exponentially stable on V. It follows that Ar is the infinites­imal generator of the exponentially stable semi group SI(·) on V and the fol­

lowing is well-defined for all w(-) E L2(0, 00; W).

( 4.17)

We conclude from Appendix Bo4 that r(·) E L2(0 , 00; V), r(·) is strongly con­

tinuous (w.r.t. the topology of V) and r(t) ~ ° as t --t 00.

Next, given e E V and w(·) E L 2(0 , 00; W) , we define

( 4.18)

(this is (4.1) with Xo = e and u(·) = B'7,(Lx(·) + r(·))). It follows from

Appendix B.1 that xC) E L2(0, 00; V) is strongly continuous and x(t) ~ ° as t --t 00.

Now we define

1](') := Lx(·) + r(·), (4.19)

(so 1](') E L2(0, 00; V), 1](') is strongly continuous and 1](t) ~ 0 as t --t 00). In the following lemma we prove an important property of 1/(') for smooth

disturbances and smooth initial conditions (1] satisfies a 'Hamilton-Jacobi e­quation' corresponding to the optimal control problem P, as explained in [87] for the finite-dimensional case).

108 Hoo -control with state-feedback

Lemma 4.7 Suppose that w(·) E L2( -00, OJ W) is continuously differentiable such that w(·) E Ll ( -oo,Oj W) and let e E D(AV). Then we have r(t) E D(Ar), i(t) E D(At), r(·) and i(·) are both continuously differentiable (w.r.t. the topology of V) and there holds

d _ v < e, dt1/(t) >v=< Cl e,CIX(t) >z - < A e,1/(t) >v· ( 4.20)

Proof First of all, if w( ·) E L 2 ( -00 ,OJ W) is such that w(·) E L} (-00, OJ W), then

r(t) E D((At)*) for all t ~ 0, r(·) is continuously differentiable w.r.t. the topology of V and

r = -(A~)*r - LBl w. (4.21 )

The proof of these facts follows from Appendix B.5 with X = V, T(t) = Si(t) and

Now since r(·) and w(·) are both continuously differentiable we can infer from Appendix 8.3 that i(·) given by (4.18) satisfies i(t) E D(AD for all t ~ 0, that x(·) is continuously differentiable w.r.t. the topology of V and that

( 4.22)

Since L E C(V), it follows that 1/0 given by 1/0 = Li(·) + r(-) is continuously differentiable w.r.t . the topology of V. Thus, we can calculate

d) . < x, dt1/(t >v=< x,Li(t) >v + < x,r(t) >v=

=< Lx, (Av + B2B;L)i(t) + B2B;r(') + BIW(t) >v

- < (Av +B2B;L)x,r(t) >v - < x,LBIW(t) >v

using (4.21) and (4.22)

where in the last step we have used the Riccati equation for L given by (4.16) .

• Using Lemma 4.7 we can prove the following result, which is a type of

maximum principle solution of the optimal control problem P (1/0 in (4.20) plays the role of the 'adjoint variable') .

4.2 Proof of the state-feedback result 109

Lemma 4.8 The optimal control for problem P is given by

( 4.23)

and the infimum is in fact a minimum.

Proof For arbitrary ~ E V and w( ·) E L2(0,oojW) we define r(·), x(·), 7](') and

u(·) as in (4.17), (4.18), (4.19) and (4.23) . Suppose that we have a u( ·) E L2(0,oojU) such that x(·) in problem P satisfies x(·) E L2(0 ,OO jV). We wish to prove that

< 7](0),~ >v= - LX> « C1x(r),C1x(r) >z

+ < 7](r),B2u(r) >v + < 7](r),B1w(r) >v)dr. (4.24 )

In the finite-dimensional case one differentiates < 7]( t), x( t) >v with respect

to t and then (after some manipulations using Lemma 4.7, formula (4.20))

integrating from 0 to 00 gives the result . However, (4.20) only holds for smooth w(·) and ~ E D(AV) . Moreover, for x(-) to be continuously differentiable we

would need that u(·) be smooth. We overcome this difficulty by introducing

sequences ~n E D(AV), un(-) E CCD(O,oojU) and wn(·) E CCD(O,oojW) v

such that ~n -t x, Ilun(-)-u(·)IIL2(O,oo;U) -t 0 and Ilwn(-)-w(')II L2(O,OO;W) -t 0 as n -t 00. Using these sequences we derive a formula which provides the

result via a limiting argument (note that these sequences exist, since D(AV) is dense in V and CC D(O, OOj Y) is dense in L2(0, OOj Y), where Y is any separable Hilbert space):

To this end, we define for t E [0, TJ, T > 0

xn(t) := S(t)en + l S(t - s) (BIWn(S) + B2un(s)) ds.

Since en E D(AV), un(-) E CCD(O,oojU) and wn(-) E CCD(O,oojW) it follows from Appendix B.3 that xn (-) is continuously differentiable and

Xn = AVxn + B1wn + B2unj xn(O) = en,xn(t) E D(AV).

Furthermore, Appendix B.6 tells us that IIxn(-) - x(· )IIL2(O,T;V) -t 0 and v xn(T) -t x(T) as n -t 00.

Similarly, the sequences en(-) and wn(-) induce sequences rn(-), xn(-) and 7]n(-) via (4.17), (4.18) and (4.19) which are also continuously differentiable w.r.t. the topology of V and which converge to the corresponding r(-) , x(·) and 7](')

110 1ioo -control with state-feedback

in the LTnorm and pointwise on [0, T] (see Appendices B.l,B.3,B.6 and B.7). Furthermore, we have

d _ v < x, dt7Jn(t) >v=< C1X,C1Xn(t) >z - < A X,7Jn(t) >v,

for all x E D(AV) (cf. Lemma 4.7). Using these results, we obtain

d dt < 7Jn(t) , xn(t) >v=< C1Xn(t),C1xn(t) >z-

< 7Jn(t), AV xn( t) >v + < 7Jn(t), AV xn(t) + BI wn(t) + B2un(t) >v=

Integrating this from 0 to T we obtain

(4.25)

We claim that the limit as n -t 00 of this last expression can be obtained by just removing the subscript n. Here the only difficulty is the 'unbounded term' JoT « C1Xn(r),C1Xn(r) >z dr, but using Appendix B.6 (and, in particular, formula (B.20)), we conclude that the limit of (4.25) as n -t 00 is given by

< 7J(T),x(T) >v - < 7J(O),x(O) >v= IT (< C1x(r),C1x(r) >z +

< 7J(r),B2u(r) >v + < 7J(r), Blw(r) >v)dr.

In order to obtain (4.24), we let T tend to infinity in the above expression,

using the facts that x(T) ~ 0, 7J(T) ~ 0 as T -t 00 (see Appendices B.l,B.4), 7J( ·),B2u(·),B2w(·) E L2(0, 00; V) and C1X('),C1x(') E L2(0,00;Z) (see Lem­ma 3.2).

Recall that we want to minimize

IIz(')IIL(o,oo;z) = 100 « C1x(r),C1x(r) >z + < u(r),u(r) >u)dr.

Now using (4.24), we find that

IIz(')lIiz(o,oo;z) + 2 < 7J(O),t >=

100 « C1x(r) - C1x(r) ,C1x(r) - C1x(r) >z

+ < u(r) - u(r),u(r) - u(r) >u - < C1x(r),C1x(r) >z-

4.2 Proof of the state-feedback result 111

< u(r),u(r) >u -2 < 1/(r), Blw(r) >v)dr

and this expression is minimized for uO = uO = B;1/0· • Before we continue with the proof of the necessity part, we introduce the

operators :F : V x L2(0, 00; W) --+ L2(0, 00; V) x L2(0, 00; U) x L2(0, 00; V) and

9 : V x L2 (0, 00; W) --+ L2 (0, 00; Z) defined by

F(~,w(·)):= (x( ' ),u('),1/0)

g(~, w(·)) := z(·) = C1x(·) + D12U( ' )'

(4.26)

(4.27)

F and 9 are well-defined, bounded linear operators: indeed, since SL(·) is exponentially stable on V, the map from w(-) to rO in (4.17) is linear and

bounded from L2(0, 00; W) to L2(0 ,00; V) and therefore the map from (~,w(·)) to x(·) in (4.18) is linear and bounded from V x L2(0 , 00; W) to L2(0, 00; V). Now since L E C(V), 1/0 = Lx(·) + r(·) and u(·) = B;1/(')' it follows that :F is linear and bounded. The linearity of 9 is now obvious and the bounded ness follows from the above with Lemma 3.2. Next, we consider the problem of finding the so-called worst disturbance, i.e., we set u(·) = u( .) in (4.15) and we consider solving the optimization problem

sup J(~,w( ' )'u(·)) = W(·)eL2(O,OO;W)

sup IIz(')IIL(o,oo;z) - ,,? IIw(,) IIL(o,oo;w) = W(·)eL 2(O,OO;W)

sup IIg(~, w('))IIL(o,oo;z) - ,,2I1w(')lIi2(o,oo;W)' (4.28) w(·)eL2 (O,oo;W)

In the following lemma we need the assumption that (ii) is satisfied. Defining

(4.29)

we obtain the following result:

Lemma 4.9 Let C be defined by (4.29). Then C(O, w(·)) 2: ° and C(O, w( . ))~ defines a norm on L2(0, 00; W), which is equivalent to the usual L2-norm on L2(0, 00; W).

Proof

According to our assumption that (ii) is satisfied, there exists a {j > Osuch that for all w(·) E L2(0, 00; W) there exists a u(·) E L2(0, 00; U) such that x(·) given by

x(t) = it S(t - S)(BIW(S) + B2U(s))ds

112 Hoo-control with state-feedback

satisfies x(·) E L2(0, 00; V) and zw,u(-) = C}x(·) + D12u(·) satisfies

(4.30)

Note that g(O, w(·)) denotes the output that corresponds to the optimal con­

trol in problem P with ~ = 0, and zw,u(-) just denotes the output that corresponds to some stabilizing control in (4.1) with ~ = 0. Thus for all w(·) E L2(0,00;W) there holds

(4.31)

Hence we conclude from (4.30) and (4.31) that

'lllw(')IIL(o,oo;w) -llg(O, w('))IIL(o,oo;w) 2: 821Iw(')lli2(O,OO ;W) (4.32)

for all w(·) E L2(0, 00; W). It follows from the definition of C in (4.29) and

(4.32) that C(O,w(·)) 2: ° and that

'Yllw(')112 2: C(O,w(.))t 2: 8I1w(')112 for all w(·) E L2 (0, 00; W). (4.33)

Finally, we note that C(O, w(·)) is a quadratic form in w(·) and so (4.33) implies that C(O, w(.))t defines a norm on L2(0, 00; W) and that it is equivalent to

the L2-norm on L2(0, 00; W). •

Finding the worst disturbance is now equivalent to finding the infimum of

C(~, w(·)) for w(·) E L2 (0, 00; W). In doing so, we deviate from the approach in [87] by using the following lemma from Yakubovich (see e.g. [101] or [61]):

Lemma 4.10 Consider a Hilbert space V with a quadratic form

J(v) =< J(v,v > with J( E C(V); J( = J(*. (4.34)

Let Mo be a closed subspace of V and M the translation of Mo by some element m in V (i.e. M=Mo+m). Suppose that the following condition holds:

. f < J(vo,vo > In > O.

voEMo < Vo, Vo > ( 4.35)

Then there exists a unique element u E M such that

J(u) = inf J(v), vEM

where u is of the form u = m* + m with m* E Mo and m* = Tm for some T E C(V).

The following resuft shows that the infimum of C (~, w( . )) is in fact a minimum.

4.2 Proof of the state-feedback result 113

Lemma 4.11 Given ~ E V there exists a unique w* (.) E L2 (0,00; W) such that

inf C(~,w(·)) = C(~,w*(-)) w(-jEL2(O,OO;W) ( 4.36)

and there exists an H E .c(V, L2(0, 00; W)) such that

w*(·) = H~. ( 4.37)

Proof We apply Lemma 4.10 with V := V x L2(0, 00; W), Mo := {(a, w(·)) I w(·) E

L2(0,00;W)} c V, m := (~,O), M := {(~,w(·)) I we) E L2(0,00;W)} = (~,O) + Mo c V and (using v = (~,w(·)))

J(v) = J(e,w(')) :=C(~,w(·)) =

,2/1w(')/IL(o,oo;W) - /Iy(~,W(·))I/i2(O,OO;Z) =

,2 < w(.),w(.) >L2(O,OO;W) - < Y(~,w(·)),Y(~,w(·)) >L2(O,oo;Z)=

,2 < w(.),w(·) >L2(O,oo;W) - < Y*Y(~,w(·)),(~,w(-)) >v=

< J{(e,w(')), (e,w(·)) >v=< J{v,v >v,

where we have defined J{ v = J{(e, w(·)) := (0, ,2w(·)) - y*y(e, w(·)) (so that J{ = J{* E .c(V)). Condition (4.35) is satisfied because there exists a I) > ° such that for Vo = (O,wC)) E Mo there holds

< Kvo,vo >v= J(O,w(·)) =C(O,w(,)) ~ 1)2/1w(')/IL(o,oo;w)

(using Lemma 4.9)

= 1)2 < Vo, Vo >v .

Hence, Lemma 4.10 implies that there exists a unique element of M of the form m* + m = (0, w*(·)) + (e, 0) = (e, w*(·)) such that

inf C(e, w(·)) = inf J(v) = J(m* + m) = C(e, w*(·)). W(')EL2(O,OO;W) vEM

Moreover, there exists aTE .c(V) such that

m* = (O,w*(·)) = Tm = T(~,O)

and this readily implies the lemma. •

114 1too -control with state-feed back

As mentioned before, the minimizing w' (.) from Lemma 4.11 is called the worst disturbance because it maximizes J(e,w(·),u(·)) in (4.28). In the fol­lowing lemma we find a characterization of it in terms of 7](.). In the sequel we shall use the definitions

(x*(.),u*(.),7]*(.)):= F(e,w*(·)) ( 4.38)

and

z*(·) := 9(e, w*(·)), (4.39)

for a given e E V.

Lemma 4.12 Let e E V be given and let w*(·) denote the unique worst disturbance from Lemma 4.11. For arbitrary w(·) E L2(0, 00; W) we define 7](.) as in (4.19) (see also (4.26)). Considering 7](.) as a function ofw(·), it follows that w*(·) is the unique element of L2(0, 00; W) satisfying

( 4.40)

Proof First of all we shall prove that w· (.) satisfies (4.40). Note that according to (4.38) this is equivalent to proving that w*(·) = _,-2 B;7]*(·). Denoting wOO:= -,-2B;1/*(·), we shall therefore show that woO = w*(·).

Define (XOO, unO, 1/0(.)) := F(e, WO(.)) and zoO := 9(e, WO (.)). Consider the time interval [0, T] for some T > O. We claim that

,2I1w*OIlL(o,T;W) -llz*(·)IIL(o,T;z) -,21IwO(·)IIL(o,T;W)+

II z0(·)IIL(o,T;z)+ < 17*(T),x*(T) >v - < 17*(T),xO(T) >v=

lllw°(-) - w*UIIL(o,T;W) + IlzoU - z*OIIL(o,T;Z) · (4.41 )

In the finite-dimensional case, this result follows from a completion of the squares argument, using the differentiability of x°(-),x*O and 17*(·). Howev­er, here these funcions are not differentiable in general (only for smooth input functions and smooth initial conditions). As before, this difficulty can be cir­

cumvented by introducing sequences en E D(AV) and w~(-) E CCD(O, 00; W),

such that en ~ e and IIw~O - w*(·)II L2(O,OO;W) -+ 0 as n -+ 00. Furthermore, the property of 170 in Lemma 4.7 is needed. A limiting argument (with n

tending to 00) gives formula (4.41). This method was also used in the proof Lemma 4.8 and so we can safely omit the details. In (4.41), we can now take

the limit as T -+ 00 and obtain (using 17*(T) ~ 0, x*(T) ~ 0 and xO(T) ~ 0

4.2 Proof of the state-feedback result 115

as T -t 00, see Appendices B.6,B.7 and the argument in the proof of Lemma

4.8)

This can be reformulated as

from which we conclude that woO = Wo(.) = -,-2B~17"(')'

Next we shall give (a sketch of) the proof of uniqueness (we follow the

finite-dimensional proof in [70, 86]). Suppose that there exists another ele­

ment w(·) E L2(0,oojW) that satisfies (4.40). Defining (xC),u('),i](')) :=

F(Cw(')) and z(·):= 9(e,w(·)) we shall show that w(·) = w"(')' Using some

arguments similar to those in the proof of (4.41), it can be shown that for

arbitrary T > 0

lllw"(,) - w(')IIL(o,T;W) -llz"(·) - z(')IIL(o,T;Z)+

2 < 17"(T) - i](T), x"(T) - x(T) >v=

-,2I1w"(,) - w(')IIL(o,T;W) + Ilz"(·) - zOIlL(o,T;Z)'

Using the fact that 17"(T) ~ 0, x"(T) ~ 0, i](T) ~ 0 and x(T) ~ 0 as T -t 00, it follows that

,21Iw"(.) - w(')IIL(o,oo ;w) -llz"(,) - z(')IIL(o,oo;z) = 0. (4.42)

Since

z"(·) - z(.) = 9(e,w"(·)) - 9(e,w(·)) = 9(O,w"(·) - w(·))

(using the linearity of 9), (4.42) can be reformulated as

lllw"(.) - w(')IIL(o,oo;w) -119(0, w"(·) - w('))IIL(o,oo;z) = O.

Finally, Lemma 4.9 implies that w"(·) = w(·) and so uniqueness is proved. _

116 1ioo -controi with state-feedback

Summary 4.13 Before continuing, we recapitulate the results obtained so far: we have solved the sup-inf optimization problem (4.15)

sup inf J(~,w(·),u(·)), w(')EL2(O,OO;W) u(')EL2(O,OO;U)

for fixed ~ E V and J given by (4.14). The infimization part with respect

to u E L2(0, 00; U) (for fixed w(·) E L2(0, 00; W) and ~ E V) was treated in Lemma 4.8 and we characterized the optimal control u(·) = Bi1]('), where 1](') is given by (4.19) (and of course 1](') depends on w(·) and O. Furthermore, the infimum was in fact a minimum. Then in Lemmas 4.11 and 4.12 we treated the part of finding the worst disturbance in

sup J(~,w(')'uC)) w(')EL2(O,oo;W)

(compare with (4.28) and (4.29)). We have shown that in fact the max­

imum is attained by a unique w*(·). Furthermore, w*(·) = H~ for some H E £(V, L 2(0, 00; W)) and w*(·) is the unique solution to w(·) = _,-2 B;1](')' Finally, we recall the definitions of (4.38) and (4.39): (x*C), u*(·), 1]*(')) = .r(~,w*(·)) and z*(·) = g(~,w*(')) (we shall use these definitions throughout the rest of this section).

The next step is the feedback synthesis part: we show that 1]*(') (the adjoint variable 1](') corresponding to the worst disturbance) satisfies 1]* (.) =

- Pox*(') for some Po E £(V). This implies that the optimal control and the worst disturbance are in fact of a feedback form, the gain depending on Po· Furthermore, we show that this Po is nonnegative definite and that it is a stabilizing solution to the Riccati equation (4.11). Let us define Po: V ~ V by

( 4.43)

First of all, we show that Po is bounded and derive the relationship between

1)*(') and x*('):

Lemma 4.14 Let ~ E V be arbitrary and let L, H, w*(·), 1]*0 and x*O be defined as in Summary 4.13. The operator Po defined by (4.43) satisfies Po E £(V) and there holds

1]*(') = -Pox*(')'

4.2 Proof of the state-feedback result 117

Proof Since L E £(V), HE £(V, L2(0, 00; W)) (Lemma 4.11) and Si,U is exponen­tially stable on V, we conclude that Po E £(V).

Next we note that 7]*0 and x*O are determined by (4.17),(4.18) and (4.19) where w(·) is replaced by w*U = H~. This implies in particular that 7]*(') and x*(·) are both continuous functions (with respect to the topology of V) and there holds

7]*(0) = L~ + L)O SjJr) LBI (HO(T)dT

and so

7]*(0) = -Po~ = -Pox*(O). ( 4.44)

In order to prove 7]*(t) = -Pox*(t) for t > 0 we use an argument from [86]. Having arrived at x*(t), at some time t > 0, we can consider a new sup­inf optimization problem starting at time t with the initial value x*(t) E V, etc. For this problem we also find the optimal control and the worst disturbance as in Lemma 4.11 (use time-invariance). It follows that this new worst disturbance also satisfies equation (4.40) and so the uniqueness result of Lemma 4.12 implies that the new worst disturbance is given by w*( T) for all T E (t,oo). Hence the adjoint variable and the state corresponding to the new worst disturbance are given by 7]*0 and x* (.), respectively. Since the new sup-inf optimization problem starts at time t with initial condition x*(t) it follows from (4.44) that 7]*(t) = -Pox*(t). Since t was arbitrary the result follows. •

Next we show that u*(·) and w*(·) are in feedback form and that the corresponding closed-loop map is exponentially stable on V:

Lemma 4.15 Let ~ E V be arbitrary and suppose that w*(·), 7]*(')' x*(·) and u*(·) are de­fined as in Summary 4.13. There holds u*(·) = -B;Pox*(') and w*(·) =

,-2B;Pox*(.). Furthermore, ( ,~~~o ) E £(V, W x U) is an admissible

output operator for the system (4.1) and the Co-semigroup

Sp(·):= S ( -2B*P, )(.) (Bl B2) '_B;~oo

is exponentially stable on V.

Proof The statements for u*(-) and w*(·) follow immediately from Lemmas 4.8, 4.12

118 Hoo-control with state-feedback

and 4.14. In fact, since x*(') and .,,*(-) are both continuous functions with respect to the topology of V (as explained in the proof of Lemma 4.14), u*(.) and w*(-) are also continous (as elements in L2 they can be chosen as contin­uous functions).

It is easy to see that ( ,~~~o ) is an admissible output operator for (4.1)

because it is bounded on V, and so Sp(·) is a well defined Co-semigroup on both Wand V (see Lemma 2.13). Furthermore, we have

x*(t) = S(t)e + 1t S(t - s) (BIW*(S) + B2U*(S)) ds =

S(t)e + 1t S(t - s) (,-2 BIB; Pox*(s) - B2B;Pox*(s)) ds.

Therefore, x*(-) = Sp(·)e (see Lemma 2.14) and since x*(-) is in L2(0,OO;V) for arbitrary intitial conditions e E V we conclude that 5 p(.) is exponentially stable on V (use Datko's result given in Lemma A.l).

Finally, we show that Po is nonnegative definite and that it satisfies the Riccati equation (4.11).

Lemma 4.16 The operator Po E C(V) defined by (4.43) satisfies

and

+ < CIX,CIy >z= 0 for all X,Y E D(AV). (4.11 )

Proof First we note that the infinitesimal generator of Sp(·) on V is given by D(A~) = D(AV) and A~x = (AV +,-2BIB;Po-B2BiPo)x for all x E D(A~) (cf. Lemma 2.13 and Lemma 4.15). Now let e, Y E D(AV) = D(A~). Then x*(t) = Sp(t)e E D(A~) for all t ;::: 0, x* (.) is strongly continuously differentiable (w.r. t. the topology on V) and, from Lemma 4.14, .,,*(-) = -Pox*(')' Thus we have

< Y, ! .,,*(t) >v=< Y, -PoA~Sp(t)e >v . ( 4.45)

Now .,,*(-) also satisfies (4.19), where x(·) is replaced by x*(') and r(·) is given by (4.17) with w*(·) . Since w*(·) = ,-2B;PoSp(·)e it is strongly continuously

4.2 Proof of the state-feedback result 119

differentiable and w*(·) = "1- 2 B; PoSp(·) A~~ E Ll (0,00; W), so we can appeal to Lemma 4.7 to obtain

< y,~ TJ*(t) >v=< CIY, Clx*(t) >z - < AVy, TJ*(t) >v . (4.46)

Equating (4.45) and (4.46) for t = 0 gives

< y,PoA~~ >v + < AVy,Po~ >v + < Cly,Cle >z= 0 (4.47)

and so

Using (4.47) we will now prove that Po is self-adjoint. For all e,y E D(AV) = D(A~) we have

< y, (Po - p;)A~e >v + < A~y, (Po - p;)e >v=< y, poA~e >v +

This implies that ~ < Sp(t)y, (Po - P;)Sp(t)e >v= 0 and so we see that for all e, y E D(AV), < Sp(t)y, (Po - P;)Sp(t)e >v=< y, (Po - P;)~ >v. But since 5 p( .) is exponentially stable on V and D( A V) is dense in V, this implies that Po = Po. Now (4.11) follows immediately from (4.48). Finally, we have to show that Po is nonnegative definite. Using Po = P; and (4.11) we compute for e E D(AV) = D(A~)

~ < x*(t) , Pox*(t) >v=

< A~x*(t), Pox*(t) >v + < Pox*(t), A~x*(t) >v=

< B;Pox*(t),B;Pox*(t) >u (from (4.11))

= l < w*(t), w*(t) >w - < Clx*(t), Clx*(t) >z - < u*(t), u*(t) >u

(from Lemma 4.15)

120 Hoo-control with state-feedback

= 'l < w*(t),w*(t) >w - < z*(t),z*(t) >z ( 4.49)

(using (4.13)).

Since Sp(') is exponentially stable on V, we may integrate (4.49) over (0,00) to obtain

< (,Po~ >v= Iiz*(')lli2(o,oo;Z)_,21Iw*(')IIi,(o,oo;W) = -C(~,w*(·))(4 .50)

(in the last step we used the definition of C( ~, w(·)) in (4.29) and the fact that z*(·) = g(~,w*(')))' Since w*O is such that

inf C((,w(·)) =C(~,w*(')) W(')EL2(O,OO;W)

(this is the result of Lemma 4.11), it follows that

C(~,w*(·)) ~ C(~,O) = -llg(~,O)IIL(o,oo;z) ~ o. (4.51 )

Finally, combination of (4.50) and (4.51) implies that < ~,Po~ >v~ 0 for all ~ E D(AV) and since D(AV) is dense in V we infer that Po ~ O. •

So far, we have proved that there exists a Po E C(V) with Po = Po ~ 0 satisfying (4.11), with the additional property that the Co-semi group S p(.) is exponentially stable on V. Defining P := cPo, where c = i1t IVI (iv tl is the isometry from V to V', it follows that P E C(V, V') is nonnegative definite, that P satisfies (4.9) and that

Sp(')=S (,-2B;Po)(')=S (,-2B~P)(')' (BI B2) _ B* P.o (BI B2) _ B' P

2 2

is exponentially stable on V (note that assumption (4.12) simplifies the for­mulas in Theorem 4.4 considerably). This concludes part a) (i .e. the proof of the implication (ii) =} (iii)), under assumption (4.12).

b) Proof of (iii) =} (i)

We suppose that (iii) holds and define Po := c-I P (recall that c = i1t IVI (iv tl is the isometry from V to V'). It follows that Po E C(V) is nonnegative definite and that it satisfies the Riccati equation (4.11) such that AV + ,-2 BIB; Po-B2BiPo is the generator of the Co-semi group Sp(·) which is exponentially stable on V (again, we note that (4.12) simplifies the formulas considerably).

We want to find an admissible output operator F E C(W, U) such that the Co-semigroup SB2F (-) is exponentially stable on V and the closed-loop map GB2F defined by (4.3) satisfies IIGB2F II <,. In the following lemma we show that the feedback F = - B~P = - Bi Po satisfies these conditions. We note that because of the definition of Po and the simplifying assumption (4.12), there holds F = -B~P = -(D~2D12tI(B~P + D;2Cd, just as in (4.10).

4.2 Proof of the state-feedback result 121

Lemma 4.17 The state-feedback u(·) = Fx(·) = -B~Px(') = -B;Pox(-) , with P and Po as above, is such that SB2 F(-) is exponentially stable on V and guarantees the closed-loop inequality IIGB2F II < ,.

Proof Note that F = - B;Po E C(V, U) is an admissible output operator because

it is bounded on V. First we show that the semigroup SB2F(') generated by

AV - B2 B;Po is exponentially stable on V. From the Riccati equation (4.11)

we infer that for all X,y E D(AV)

Since AV - B2B;Po = A~ -,-2BlB;Po, and it is given that the semigroup

generated by A~ is exponentially stable on V, we see that the pair (AV -B2B;Po, Bi Po) is exponentially detectable on V. Hence, the fact that SB2F(-) is exponentially stable on V follows from Zabczyk's Lyapunov type result in

Lemma A.3. Next we show that GB2F given by

{ x(t) = J~ SB2F(t - s)Blw(s)ds

(GB2FW)(t) = z(t) = (Cl - D12B;Po)x(t) t ~ 0,

where w(·) E L2(0,00; W), satisfies IIGB2F II < f. First of all, we note that since SB2 FO is exponentially stable on V it

follows that x(·) E L2(0,00;V), x(t) ~ ° as t -t 00 (see Appendix B.1) and

we have GB2F E C(L2 (0, 00; W),L2 (0,00;Z)) (see Lemma 3.2).

Suppose that, in addition, w(·) E L2(0, 00; W) is continuously differen­

tiable and denote woO := w(-) - ,-2 Bi Pox(') E L2(0, 00; W). Using Ap­pendix B.3, we calculate

d dt < x(t), Pox(t) >v=

< (Av - B2B;Po)x(t),Pox(t) >v + < BlW(t),Pox(t) >v +

< Pox(t),(Av - B2B;Po)x(t) >v + < Pox(t), BlW(t) >v=

- < Clx(t), Clx(t) >z - < B;Pox(t), B;Pox(t) >u -

,2 < w(i) _,-2 B; Pox(t), w(t) _,-2 B; Pox(t) >w +

122 'Hoo-control with state-Feedback

'''? < w(t),w(t) >w (using (4.11» = - < z(t),z(t) >z

-"? < wo(t), wo(t) >w +l < w(t), w(t) >w,

using (4.13) and the definition of woO above, Since x(t) ~ 0 as t -+ 00 (see

Appendix 8.2) and x(O) = 0, we may integrate this over (0,00) to obtain

Furthermore, using Lemma 2,14, x(t) can also be written as

x(t) = it Sp(t - S)BIWO(s)ds,

where Sp(') is the semi group generated by A~ = AV - B2B;Po + ,-2 BIB; Po, which is exponentially stable on V. The exponential stability of S p(.) on

V implies the boundedness of the mapping woO E L2(0, 00; W) I-t x(·) E L2(0, 00; V) and therefore also of the mapping wo(·) E L2(0, 00; W) I-t w(·) = wo(-) + ,-2 B; Pox(') E L2(0, 00; W). Hence, there exists a positive constant 8 > 0 independent of wO or woO, such that

Combining this with (4.52) yields

IIz(')IIL(o,oo;Z) = ,21Iw(')IIL(o,oo;w) -lllwo(')IIL(o,oo;w)

:::; (/2 - 82)lIw(')IIL(o,oo;W)' ( 4.53)

As before, (4.53) can be extended to the general case of w(·) E L2(0, 00; W) by the introduction of an approximating sequence of smooth wn (·) which con­verges to w(·). Using Appendix 8.6 (4.53) then holds for wn (·) and we can take the limit for n -+ 00 to obtain

II(GB2FW)(')IIL(o,oo;Z) = Ilz(')IIL(o,oo;z) :::; (,2 - 82)lIw(')IIL(o,oo;w),

for all w(·) E L2(0,00;W). Hence, IIGB2F II <,.

This concludes part b) (the proof of (iii) =} (i», under assumption (4.12).

•

To complete the proof of Theorem 4.4, we show that the implications (ii) =}

(iii) and (iii) =} (i) remain true without the extra assumption (4.12). The idea is to apply a preliminary feedback

( 4.54)

4.2 Proof of the state-feedback result

where v(·) is a new control input and F is given by

F := -(D~2D12rl D~2Cl'

123

( 4.55)

Note that F E £(W, U) is an admissible output operator because Cl E

£(W, Z) is admissible and (D~2Dl2tl D~2 E £(Z, U), so that we can use Lemma 2.14. With the input u(·) = Fx(.) + (D~2D12ttvO, the system (4.1) transforms into the smooth Pritchard-Salamon system Ea = E(S(·), (81 82 ),

C\, (0 D12 )) of the form

{ x( t) S(t)xo + J; S(t - s)(B1w(s) + 82v(s))ds

( 4.56)

z( t) C\x(t) + D12V(t),

where

SO SB2P(') 82 B2(D~2DI2tt 81 Bl C1 (I - D12(D~2D12tl D~2)Cl = Cl + D12F D12 = D12(D~2DI2)-L

Note that we need the assumption that D~2D12 is coercive to guarantee the existence of a bounded inverse and that C1 is an admissible output operator for both SO and S(·). Lemma 2.14 implies that with the input

( 4.57)

in the transformed system (4.56), we obtain the original system (4.1). Fur­thermore, we see that with the state-feedbacks u(·) = Flx(·) in system (4.1) and v(·) = F2x(·) in system (4.56), where Fl E £(W, U) and F2 E £(W, U) are admissible output operators related by

Fl = (D~2DI2rtF2 + For F2 = (D~2D12)t(Fl - F),

we have

SB2F2(') = SB2F,(') and GB2F2 = GB2F,.

It is easy to see that the system Eo given by (4.56) satisfies the assumptions (4.4) and (4.12) (assumption (4.4) is invariant under state-feedback, as is shown in Appendix C).

Now suppose that (ii) is satisfied for Ea. It follows from the above that there exists an admissible output operator F E £(W, U) such that S B2 F(') is exponentially stable on V. Furthermore, using the relation v(·) =

(D~2D12)t(U(') - Fx(.)) it is easy to see that (ii) is also satisfied for Ea.

124 'lice-control with state- feedback

Hence, using the fact that we have proved Theorem 4.4 under the extra as­sumption that (4.12) holds, we conclude that there exists a nonnegative defi­nite P E £(V, V') such that for all x,y E D(AV)

and AV + (-y-2B2B~ - B1BDP is the generator of an exponentially stable semigroup exponentially stable on V (d. Theorem 4.4). It is now clear from the definition of Ea in (4.56) and the perturbation results of Lemma 2.13 that item (iii) of Theorem 4.4 is satisfied.

Now suppose that (iii) is satisfied for Ea. Defining Eo as in 4.56 it follows immediately from the perturbation results in Lemma 2.13 that Eo also satisfies item (iii) of Theorem 4.4. Hence, the feedback v(·) = F2x( ·), with F2 :=

- fj~p, applied to (4.56) is such that S B2F2 (.) is exponentially stable on V and IIGB2F2 11 < ,. It follows from the above that the feedback u(·) = F1x(·) applied to system (4.1), with Fl = -(D~2D12tl(B~P + D~2Cl)X(·), is such that SB2Fl(·) is exponentially stable on V and IIGB2F1 11 < " so that item (i) is satisfied for Ea.

Finally, the uniqueness of P follows immediately from Lemma 2.33 and this concludes the proof of Theorem 4.4.

4.3 Relaxation of the a priori assumptions

In this section we show how we can remove the assumptions (4.4), (4.5) and (4.6). In the first part we show how an 'lice-control problem with state­feedback with feedthrough from disturbance to the to-be-controlled output (i.e. Dn =I 0) can be reduced to an 'lice-control problem without this feed­through. In the second part we show that assumptions (4.4) and (4.5) can be removed by introducing a 'regularizing parameter'. In order to reduce the notational burden we assume in this section that, = 1 (as usual, the general case can be obtained by scaling).

4.3.1 Feedthrough from disturbance to output

Let

4.3 Relaxation of the a priori assumptions 125

given by

Eo {

x(t) ( 4.58)

z(t)

be a smooth Pritchard-Salamon system as in (4.1).

In principle, we could try to derive a result for Ea as we had for the system

Ea with Du = 0 in Theorem 4.4, but there is a problem with condition (ii) of that theorem. Consider the finite-dimensional system

{X = -x

z = w + u.

It is easy to see that for all w(·) E L2(0, 00) there exists a u(·) E L2(0, 00) such that x(·) E L2 (0, 00) and z(·) = 0 (take u(·) = -w( ·)), but there exists no stabilizing state-feedback controller that makes the closed-loop norm equal

to 0 (for Xo = 0 we have u(·) = Fx(·) = 0 so that z(·) = w(·)). However, for Ea we can derive the equivalence of the existence of an ad­

missible state-feedback controller with the solvability of a certain Riccati e­

quation. First of all, we recall that for an admissible state-feedback controller

there holds IICB2F II = IIGB2F lioo < 1 and because of (2.23) and (4.3) this im­

plies that IIDul1 < 1. Apparently, IIDul1 < 1 is a necessary condition for the existence of an admissible state-feedback. Now suppose that IIDnl1 < 1 and define the transformed Pritchard-Salamon system without feed through

given by

Eo {

x(t) S(t)xo + J; S(t - s)(81w(s) + 8 2u(s))ds

z( t) = C'tx(t) + D12u(t),

where

S(-) .- SB1(I-D;IDII)-ID;IC1 (.)

81 .- B1(I - D~lDllt! 82 - B2 + B1(1 - Di1Dnt1Di1D12 ( 4.59) 61 .- (I - DuD~lt!Cl D12 . - (I - Dll Dilt tD12 .

We note that because of Lemma 2.13, Eo is also a smooth Pritchard-Salamon system. The following holds.

126 'Hoo-control with state-feedback

Lemma 4.18 Let F E C(W, U) be an admissible output operator. The state-feedback u(·) = Fx( ·) is admissible for ~G if and only if IIDlll1 < 1 and u(·) = Fx(·) is admissible for ~G'

The proof of this lemma will follow from (the proof of) Lemma 5.17, where we shall transform an 'Hoo-control problem with measurement-feedback with feedthrough from the disturbance to the to-be-controlled output to an equiv­alent problem without this feed through term. We note that similar transfor­mations have been used in [36, 79] for the finite-dimensional case.

In order to derive a result like Theorem 4.4 for ~G, we show that if the reg­ularity assumptions (4.4) and (4.5) are satisfied for ~G, then the corresponding assumptions for ~G are also satisfied:

Lemma 4.19 Suppose that II DIIII < 1 and that the pair (A v, B2 ) is exponentially stabilizable. If ~G satisfies the regularity assumptions (4.4) and (4.5), then

(i) b~2b12 is coercive,

(ii) there exists an E > 0 such that for all (w, x, u) E R X D(AV) X U satisfying iwx = AV x + B2u, there holds

Proof It follows immediately from the definition of b12 that coerciveness of D~2D12 implies that b~2b12 is coercive. Furthermore, it is straightforward to show that for all w E R

(use Lemma 2.13). Since

( 01 BID~I(I - DI~D\lt~ ) E C(V X Z) (I - DllDll ) 2

( ~

and

4.3 Relaxation of the a priori assumptions 127 -----------------------

the rest of the proof follows from Appendix C.l. • The next theorem now follows from Theorem 4.4 and Lemmas 4.18 and

4.19 and requires no further proof.

Theorem 4.20 Suppose that the smooth Pritchard-Salamon system Ee given by (4.58) satis­fies the regularity assumptions (4.4) and (4.5). The following are equivalent statements:

(i) there exists an admissible state-feedback controller u(·) = Fx(·) for Ee ,

(ii) IIDllll < 1 and with the definitions of (4.59) there exists aPE C(V, V') with P = pI 2: 0 satisfying

for all x,y E D(AY) = D(AV), such that A~ given by D(A~) = D(AV) and

is the infinitesimal generator of a Co-semigroup S p(.) which is exponen­tially stable on V.

Furthermore, in this case the state-feedback u(·) = Fx{·), where F E C(W, U) is the admissible output operator given by

is an admissible state-feedback.

128 Hoc-control with state-feedback

4.3.2 How to 'remove' the regularity assumptions

Here we show that the regularity assumptions (4.4) and (4.5) can be removed. Actually we do not really remove the regularity assumptions, because we have to allow for an extra parameter in the problem. Similar tricks have been used in [52, 104] for the finite-dimensional case and in [74, 88] for infinite­dimensional Hoc-control problems.

Consider the smooth Pritchard-Salamon system ~G of the form (4.58) and define the transformed Pritchard-Salamon system

EG, {

x(t) S(t)xo + J~ S(t - s)(Blw(s) + B2u(s))ds ( 4.60)

z,(t)

where

for some ( > O. It is easy to see that ~G. is again a smooth Pritchard­Salamon system. Furthermore, it satisfies the regularity assumptions (4.4) and (4.5) because D~2,DI2' ~ (2Iu and for all x E Wand u E U we have IICI,x + D12,UII~xvxu ~ (21Ixll~· Furthermore, comparison of the original sys­tem (4.58) and the transformed system (4.60) shows that if u(·) E L2(0, OOi U) and w(·) E L2(0,OOiW) are such that x(·) E L2(0,OOiV), then the to-be­controlled output z, satisfies

Ilz,(·)IIL(o.oc;zxvxU) =

(4.61 )

We have the following result.

Theorem 4.21 Let F E C(W, U) be an admissible output operator. If the state-feedback u(·) = Fx(·) is admissible for ~G, then there exists an (I > 0 such that for all o S ( S (I the same state-feedback is admissible for ~G.. Conversely, if for some ( ~ 0, u(·) = Fx(·) is admissible for ~G. then the same state-feedback is admissible for ~G'

4.3 Relaxation of the a priori assumptions 129

Proof The stability of the closed-loop systems does not depend on the operators

corresponding to the to-be-controlled output, so we only have to worry about the norms of the closed-loop maps, GB2F and (G,)B2F (d. (4.3)).

Suppose that u(-) = Fx(·) is an admissible state-feedback for ~G' Then there exists some 8 > 0 such that IIGB2F I12 < 1- 8, so that for zero initial con­

ditions and u(·) = Fx(·) we have Ilz(')IIL(o,oo;z) -::; (1-8)llw(')IIL(o,oo;w)' Fur­thermore, it follows from Lemma 3.2 that there exists an (1 > 0 such that for

zero initial conditions (illx(-)lli2 (o,oo;V) +(illu(')IIL(o,oo;u) -::; ~81Iw(')IIL(o,oo;w)' Now (4.61) implies that for all 0 -::; ( -::; 1'1 and all w(·) E L2 (0, 00; W)

Ilz'(')IIL(o,oo;zxvxU) -::; Ilzq(')lIi2(o,oo ;ZXVXU) -::; (1- ~8)llw(')IIL(o,oo;w) and therefore II(G,)B2FII < 1 for all 0 -::; ( -::; (1'

The second part of the theorem follows from the observation that (4.61)

implies that IIGB2FII -::; II(G,)B2FII· •

Since ~G< satisfies the regularity assumptions, Theorems 4.20 and 4.21 can be used to derive a result of the following form: there exists an admissible state-feedback u(·) = Fx(·) for ~G if and only if there exists a stabilizing solution of a Riccati equation parametrized by a sufficiently small L This

means that in general the Riccati equation is parametrized by two parameters b and f), which is less attractive from an applications point of view. For the finite-dimensional case, other approaches to remove the regularity conditions can be found in Stoorvogel [86J and Scherer [83J . The extension of these ideas to the infinite-dimensional case remains an open problem.