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1 Explicit formulae for J pectral factor for well-poed linear ytem Ruth F. Curtain Amol J. Saane Department of Mathematic Department of Mathematic Univerity of Groningen Univerity of Twente P.O. Box 800 P.O. Box AV Groningen 7500 AE Enchede The Netherland The Netherland R.F.Curtain@math.rug.nl A.J.Saane@math.utwente.nl Abtract The tandard way to obtain explicit formula for pectral factorization problem for rational tranfer function i to ue a minimal realization and then obtain formulae in term of the generator A B C and D. For well-poed linear ytem with unbounded generator thee formulae will not alway be well-defined. Intead we ugget another approach for the cla of well-poed linear ytem for which zero i in the reolvent et of A. Such a ytem i related to a reciprocal ytem having bounded generating operator depending on B C D and the invere of A. There are nice connection between well-poed linear ytem and their reciprocal ytem which allow u to tranlate a factorization problem for the well-poed linear ytem into one for it reciprocal ytem the latter having bounded generating operator. We illutrate thi general approach by giving explicit olution to the ub-optimal Nehari problem. Introduction The tandard way to obtain explicit formula for pectral factorization problem for rational tranfer function i to ue a minimal realization G) = D + CI A) B and then obtain formulae in term of the generator A B C and D ee Ball and Ran ). Thee formula typically depend on the controllability and obervability Gramian L B L C or on olution of variou Lyapunov equation. Such an approach ha been extended to certain clae of infinite-dimenional linear ytem ee Curtain and Ran 5 Curtain and Zwart 6 Kaahoek et al. 9) but the limiting factor i the difficulty in manipulating with the unbounded operator B and C. For example in Saane and Curtain and 0 where explicit olution to the ub-optimal Hankel norm approximation problem for exponentially table mooth Pritchard-Salamon ytem and exponentially table analytic ytem were obtained via the olution to the appropriate J pectral factorization problem uing the moothing propertie of thee clae. However it wa not poible to extend thi technique to more general well-poed linear ytem. In general it i not clear that the candidate pectral factor i even well-poed ee Staffan 3). While an obviou approach to get around the problem with unbounded operator would be to obtain factorization via the

2 dicretized verion obtained by the uual Cayley tranform thi lead to horrible formulae. Intead we ugget tranlating the problem to the analogou one for reciprocal ytem which we now define. To motivate our definition we recall the definition of the tranfer function of a table well-poed linear ytem from Staffan 2: G) Gβ) = β )CβI A) I A) B for all β C + 0..) If 0 ρa) then we can ubtitute β = 0 to obtain the identity ) G) = G0) CA A A B.2) ) = G..3) We note that G ) i the tranfer function of the linear ytem with bounded generating operator A A B CA G0). We define thi linear ytem to be the reciprocal ytem of the well-poed linear ytem with generating operator A B C with A bounded. In addition to the relationhip.3) between their tranfer function the reciprocal ytem ha the ame controllability and obervability Gramian. Lemma.. Let A B C be generating operator of a well-poed linear ytem with tranfer function G. Suppoe that 0 ρa) and G i the tranfer function of it reciprocal ytem with generating operator A A B CA G0). Then the following hold:. C i an infinite-time admiible obervation operator for A iff CA i an infinitetime admiible obervation operator for A. If either C or CA i infinite-time admiible then the obervability Gramian are identical. 2. B i an infinite-time admiible control operator for A iff A B i an infinite-time admiible obervation operator for A. If either C or A B i infinite-time admiible then the obervability Gramian are identical. 3. G H LU Y )) iff G H LU Y )). Proof. From Hanen and Wei 8 ee alo Grabowki 7) we know that C i an infinite-time admiible obervation operator iff the Lyapunov equation Az L C z 2 + L C z Az 2 = Cz Cz 2.4) for all z and z 2 in DA) ha a nonnegative definite olution L C = L C 0. The equation.4) i clearly equivalent to the Lyapunov equation x L C A x 2 + LC A x x 2 = CA x CA x 2.5) for all x and x 2 in X which etablihe the equivalence. Moreover the obervability Gramian are the mallet poitive olution and o the Gramian are identical. 2

3 2. Thi i dual to part above. 3. Thi follow from.3). The idea i then to tranlate a factorization problem for the well-poed linear ytem with tranfer function G into one for the ytem with tranfer function G the latter having bounded generating operator. We illutrate thi general approach by giving explicit olution to the ub-optimal Nehari problem for the cla of well-poed linear ytem with B C finite rank A bounded which are input output and input-output table. Thi cla include exponentially table well-poed linear ytem with finite rank B C. The aumption that A i not eential and can be removed. A mentioned in Curtain 2 for any iω in the reolvent et of A it i alway poible to define a reciprocal ytem baed on A ω = A iωi. For then the new reciprocal ytem with generating operator A ω A ω B CA ω Giω) and tranfer function G ω atifie G ω and an analogou verion of Lemma. hold. ) = G + iω) 2 The Nehari problem We olve thi problem for the well-poed linear ytem on a Hilbert pace X with generating operator A B C) under the following aumption: A0. The input and output pace are finite-dimenional that i U = C m and Y = C p. A. 0 ρa). A2. B i an infinite-time admiible control operator for {T t)} t 0. A3. C i an infinite-time admiible obervation operator for {T t)} t 0. A4. G ) H C p m ). The ub-optimal Nehari problem i the following: If σ > H G the Hankel norm of the ytem then find all K ) H C p m ) uch that Gi ) + Ki ) σ. K i then called a olution of the ub-optimal Nehari problem. 3

4 Now ) G) + K) = G + K) = CA I A ) A B + K) + G0) = CA I A ) A B + K where K ) := G0) + K ). Clearly G H C p m ) iff G H C p m ) and K ) H C p m ) iff K ) H C p m ) and in the L -norm G + K = G + K. Thi mean that the Hankel norm of G i equal to that of G. So intead of olving the uboptimal Nehari problem for G we olve the uboptimal Nehari problem for the reciprocal ytem with the bounded generating operator A A B CA. Thi ytem atifie all the condition in Curtain and Ootveen 4: it tranfer function i in H C p m ) and the operator A B and CA are infinite-time admiible with the ame Gramian L B and L C a the original ytem. So we tranlate the reult to thi ytem. Let N σ = ) I L σ 2 B L C LX). Then all olution K to the ub-optimal Nehari problem CA i I A ) A B + K i ) σ are given by where K ) = R )R 2 ) R ) R 2 ) Q ) H C p m ) atifie Q and Λ ) = + CA L B 0 σi m σ 2 σ A B) = Λ ) Q ) I m ) N σ I + A ) CA ) L C A B. Appealing to Theorem. in Wei and Wei 4 we ee that Λ ) H 2 C p+m) p+m) ) 0 σi m but not all component need be in H. Λ ) i invertible for every C + 0 and Λ ) I p H 2 H C p ) and Λ ) i invertible for every C + 0. Furthermore Λ ) atifie the following J pectral factorization problem Ip G iω) I p 0 G iω) I m 0 σ 2 I m 0 I m 4 = Λiω) 0 I m Λiω)

5 for ω R and ince G ) = G ) we obtain a J pectral factorization over H2 for G: I p 0 Ip Giω) Giω) I m 0 σ 2 = Λ I m 0 I m iω ) Λ 0 I m iω ) for ω R. The final olution to our original uboptimal Nehari problem i K ) = G0) + K ) = G0) + R ) R2 where ) R ) R 2 = Λ ) Q0 ) I m and Q 0 ) H C p m ) atifie Q 0. From Curtain and Ootveen 4 we have the following formula. Λ = ) 0 I σ m + σ 2 CA L B σ A B) I A ) ) N σ ) CA ) σ L CA B for all C + 0. While it i tempting to try to write thi in term of it reciprocal we know that thi will not in general be well-defined ee Staffan 3). So we leave the explicit olution a it tand. A imilar approach to the optimal Hankel norm problem for well-poed linear ytem ha been taken in Curtain and Saane 3. For other application of reciprocal ytem ee Curtain 2. Reference J.A. Ball and A.C.M. Ran. Optimal Hankel norm model reduction and Wiener-Hopf factorization I: The canonical cae. SIAM Journal on Control and Optimization 252): R.F. Curtain. Reciprocal of Regular linear ytem: a urvey. Proceeding MTNS R.F. Curtain and A.J. Saane. Hankel norm approximation for well-poed linear ytem. manucript 4 R.F. Curtain and J.C. Ootveen. The Nehari problem for nonexponentially table ytem. Integral Equation and Operator Theory 3: R.F. Curtain and A. Ran. Explicit formula for Hankel norm approximation of infinitedimenional ytem. Integral Equation and Operator Theory 3:

6 6 R.F. Curtain and H.J. Zwart. The Nehari problem for the Pritchard-Salamon cla of infinite-dimenional linear ytem: a direct approach. Integral Equation and Operator Theory 8: P. Grabowki. On the pectral-lyapunov approach to parametric optimization of ditributed parameter ytem. IMA Journal of Mathematical Control and Information 7: S. Hanen and G. Wei. New reult on the operator Carleon meaure criterion. IMA Journal of Mathematical Control and Information 4: M.A. Kaahoek C.V.M van der Mee and A.C.M. Ran. Wiener-Hopf factorization of tranfer function of extended Pritchard-Salamon realization. Math. Nachrichten 96: A.J. Saane and R.F. Curtain. Sub-optimal Hankel norm approximation for the analytic cla of infinite-dimenional ytem. Integral Equation and Operator Theory to appear. A.J. Saane and R.F.Curtain. Optimal Hankel norm approximation for the Pritchard- Salamon cla of infinite-dimenional ytem. Integral Equation and Operator Theory 39: O.J. Staffan. Well-Poed Linear Sytem I: General Theory. To be publihed draft available at taffan/publ.htm. 3 O.J. Staffan. Admiible factorization of Hankel operator induced by well-poed linear ytem. Sytem and Control Letter 37: M. Wei and G. Wei. Optimal control of table weakly regular linear ytem. Mathematic of Control Signal and Sytem 0: