Sufficient conditions for the finite and infinite-time admissibility of an observation operator are given. It is shown that the estimates of Weiss are close to being sufficient. If the semigroup is surjective, then the estimate is sufficient

Zwart, H.J.

Zwart, Heiko J.

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Sufficient conditions for admissibilityHans Zwart aaUniversity of TwenteDepartment of Applied MathematicsP.O. Box 217NL-7500 AE EnschedeThe Netherlandsh.j.zwart@math.utwente.nlAbstractSufficient conditions for the finite and infinite-time admissibility of an observationoperator are given. It is shown that the estimates of Weiss are close to being suffi-cient. If the semigroup is surjective, then the estimate is sufficient.Key words: Infinite-dimensional system, admissible observation operator,infinite-time admissible observation operator.1 IntroductionConsider the systemż(t) = Az(t), y(t) = Cz(t), z(0) = z0, (1)where A is the infinitesimal generator of a C0-semigroup T (t) on the Hilbertspace Z, and C is a bounded operator from the domain of A, D(A), to a secondoutput space Y . An important question is whether the system as defined in(1) is well-posed. Of course since A generates a C0-semigroup on Z, we knowthat the state equation has the unique solution z(t) = T (t)z0. However, sinceC is not a bounded operator on Z, it is not clear whether the output equationis well-posed. Pointwise in time the output equation only make sense if C isbounded operator on Z. However, one could relax this to the question if theoutput trajectory is locally square integrable. If this holds, then C is calledan admissible observation operator, see Weiss [6].Preprint submitted to Elsevier ScienceIn [7] it was conjectured 1 that C is an admissible output operator if and onlyif there exists an M such‖C(sI − A)−1‖2 ≤ M/Re(s) (2)for s is some right-half plane. Already in [7] one can find that admissibilityalways implies (2). However, recently Zwart, Jacob and Staffans [9], Jacob andZwart [4], and Jacob, Partington and Pott [3] showed that the converse doesnot hold. Hence the estimate (2) is not sufficient, and so it is natural to askif there are sufficient conditions. Using the Hille-Yosida estimate, Grabowskiand Callier [2] derived a necessary and sufficient condition. However, since thiscondition involves all the powers of (sI − A)−1, it is not so simple to test. Sothere is a need for simple sufficient conditions. In this paper we present someof these conditions. For instance, we show that if there exists a γ > 1 suchthat Re(s)γ‖C(sI −A)−1‖2 is bounded on some right half plane, then C is anadmissible observation operator.Apart from the well-posedness of the equation (1), one would like to knowwhen the output is square integrable over the whole time axis. This is knownas infinite-time admissibility, see Grabowski [1]. In [8] it was conjectured thata scalar observation operator is infinite-time admissible if and only if‖C(sI − A)−1‖2 ≤ M/Re(s) (3)for all s real part positive. For general output operators, Zwart, Jacob andStaffans [9] showed that condition (3) is not sufficient. For one-dimensionaloutput operators one may a similar result in Jacob and Zwart [4]. In Section4 we present a simple sufficient condition when an observation operator isinfinite-time admissible. Furthermore, we show that (3) the L2-norm of CT (·)over a time interval of length t grows at most as log(t).In the proof of our theorems the following spaces and results are used fre-quently.Definition 1.1 Let Z and Y be Hilbert spaces, and let L(Z, Y ) denote thebounded linear operators from Z to Y . Furthermore, we defineL2((0,∞); Z) := {h : [0,∞) → Z |∞∫0‖h(t)‖2dt < ∞};1 In [7] Weiss studies admissible input operators. Since they are dual notions, see [6],any result/conjecture for admissible input operator has its equivalent counterpartfor admissible observation operators2H2(Z) := {f : C+0 → Z | f is analytic and supr>0∞∫−∞‖f(r + jω)‖2dω < ∞};H∞(L(Z, Y )) := {G : C+0 → L(Y, Z) | f is analytic and supRe(s)>0‖F (s)‖ < ∞}Here C+0 denotes the open right half plane of the complex plane.It is well-known that the Laplace transform of a f ∈ L2(0,∞); Z) lies in H2(Z)and‖f‖L2(0,∞);Z) =1√2π‖f̂‖H2(Z), (4)where f̂ denotes the Laplace transform of f . Furthermore, one has that forany G ∈ H∞(L(Z, Y )) and any f ∈ H2(Z) the function Gf ∈ H2(Y ) and‖Gf‖H2(Y ) ≤ ‖G‖H∞(L(Z,Y ))‖f‖H2(Z). (5)Hence G defines a bounded linear operator from H2(Z) to H2(Y ).2 General resultsWe begin with the formal definition of an admissible observation operator.Definition 2.1 Let T (t) be a C0-semigroup on the Hilbert space Z, and let Cbe a bounded linear operator from the domain of A, D(A) to a second Hilbertspace Y . Then C is an admissible observation operator if there exist t1 > 0and M > 0 such that for all z0 ∈ D(A)t1∫0‖CT (t)z0‖2Y dt ≤ M‖z0‖2.Thus if C is an admissible observation operator, then the mapping z0 7→CT (·)z0 can be extended to a bounded mapping from Z to L2((0, t1); Y ).It is easy to see that if C is an admissible observation operator for T (t), thenit is also an admissible observation operator for eωtT (t). Hence if we want toinvestigate admissible observation operators, then we may assume that thesemigroup is exponentially stable.3Now we can formulate our first result. Note that this result was first obtainedby Weiss [7]. However, our proof is much shorter.Theorem 2.2 Let T (t) be an exponential stable, and surjective C0-semigroupwith generator A on the Hilbert space Z, and let C be a bounded linear operatorfrom D(A) to Y . If C(sI −A)−1 is bounded on the right half-plane, then C isan admissible observation operator.Proof Let x0 be an element of Z. Since T (t) is exponentially stable, we havethat ((s+1)I−A)−1x0 ∈ H2(Z). Using now that C(sI−A)−1 ∈ H∞(L(Z, Y )),we obtain thatC(sI − A)−1((s + 1)I − A)−1x0 ∈ H2(Z).By the resolvent identity, we see that this expression equalsC(sI − A)−1x0 − C((s + 1)I − A)−1x0.Thus by taking the inverse Laplace transform, we have thatCT (t)x0 − e−tCT (t)x0 = (1 − e−t)CT (t)x0 ∈ L2((0,∞); Y ). (6)Since 1 − e−t is bounded away from zero on [1,∞), we get thatCT (t)T (1)x0 = CT (t + 1)x0 ∈ L2((0,∞); Y ).Now using the fact that T (1) is surjective we have that for every z0 ∈ Z,(z0 = T (1)x0), CT (t)z0 ∈ L2((0,∞); Y ). And hence we have admissibility.3 Sufficient conditions for admissibilityIn Weiss [7] it was conjectured that C is an admissible observation operatorif and only if‖C(sI − A)−1‖ ≤ K√Re(s), (7)for all complex s in some right half-plane. In Zwart, Jacob and Staffans [9] it isshown that this does not hold in general. Here we show that the estimate (7)almost implies admissibility. For the proof of this result we need the followingsimple observation.4Lemma 3.1 Let A be a closed linear operator on Z, and let C ∈ L(D(A), Y ).Then the following are equivalent.(1) There exists an M > 0 such that for all s ∈ C+0 we have‖C(sI − A)−1‖ ≤ M√Re(s),(2) For all γ > 0 we have that C((· + γ)I − A)−1 ∈ H∞(L(Z, Y )) and‖C((· + γ)I − A)−1‖H∞(L(Z,Y )) ≤M√γ.Furthermore, we need the following facts and notation.(1) There exist (unique) numbers x2 < 1 < x1 such that xie−xi = 1/(2e),i = 1, 2.(2) For x1 and x2 as above, we have thatxe−x > 1/(2e), for all x ∈ (x2, x1).The proof of this follows directly from the fact that xe−x is zero at 0 and atinfinity, and that it is monotonically increasing and decreasing on [0, 1) and(1,∞), respectively.With these facts we prove the following result.Theorem 3.2 Let T (t) be a strongly continuous semigroup on the Hilbertspace Z, with generator A. Assume that C ∈ L(D(A), Y ) satisfies the estimate‖C(sI − A)−1‖ ≤ K√Re(s),for all complex s in some right half-plane. Then the following holds(1) for every r ∈ [1, 2) there exists a Mr ≥ 0 such that for all z0 ∈ Z1∫0‖CT (t)z0‖rdt ≤ Mr‖z0‖r. (8)Thus C is admissible observation operator for Z, Y , T (·) and r for anyr ∈ [1, 2), see section 6 of Weiss [6].(2) There exists a κ > 0 such that for all z0 ∈ Z and all ε ∈ (0, x2) we havethat1∫ε‖CT (t)z0‖2dt ≤ κ| log(ε)|‖z0‖2, (9)5Proof First we see that since the semigroup generated by ωI + A is given byeωtT (t), we can without loss of generality assume that T (t) is exponentiallystable, and‖C(sI − A)−1‖ ≤ K√Re(s),for all complex s ∈ C+0 .For n ∈ N defineλn =(x1x2)n−1, xn =x1λn,so that λn → ∞, xn → 0 (monotone) and λnxn = x1, λnxn+1 = x2. Henceλnxn+1e−λnxn+1 =12e= λnxne−λnxnandλnte−λnt >12e, for all t ∈ (xn+1, xn). (10)Since T (t) is exponentially stable, we have that for all λ > 0 there holds‖((· + λ)I − A)−1z0‖H2 ≤M√2λ· ‖z0‖.Using (7) and Lemma 3.1, it follows that for all λ > 0 we have that‖C((· + λ)I − A)−2z0‖H2(Y ) ≤‖C((· + λ)I − A)−1‖H∞‖((· + λ)I − A)−1z0‖H2(Z)≤ K√λ· M√2λ‖z0‖ ≤K1λ‖z0‖,with K1 > 0 independent of λ. In the time domain this means that, for allλ > 0 and all z0 ∈ D(A)∞∫0‖λte−λtCT (t)z0‖2dt ≤K212π‖z0‖2. (11)Now we shall prove the first assertion.6Part 1. Let r < 2x1∫0‖CT (t)z0‖rdt =∞∑n=1xn∫xn+1‖CT (t)z0‖rdt≤ (2e)r∞∑n=1xn∫xn+1‖λnte−λntCT (t)z0‖rdt,where we have used (10). Now we use the Hölder inequality with q = 2/r (notethat q > 1). With 1/p + 1/q = 1 and using (11),x1∫0‖CT (t)z0‖rdt≤ (2e)r∞∑n=1p√√√√√xn∫xn+11dt · q√√√√√xn∫xn+1‖λnte−λntCT (t)z0‖rqdt≤ (2e)r∞∑n=1p√(xn − xn+1) ·1q√2πq√K21‖z0‖2=∞∑n=1p√(x2x1)n−1(x1 − x2) ·1q√2π(2eK1)r‖z0‖r.Since p√(x2x1)< 1, the sum converges and we obtain that for some M 1r > 0,x1∫0‖CT (t)z0‖rdt ≤ M1r ‖z0‖r, for all z0 ∈ D(A). (12)Since x1 > 1, we see that we have proved the first assertion.Part 2. For N ∈ N we have thatx1∫xN‖CT (t)z0‖2dt =N−1∑n=1xn∫xn+1‖CT (t)z0‖2dt≤ (2e)2N−1∑n=1xn∫xn+1‖λnte−λntCT (t)z0‖2dt,where we have used (10). With (11) we obtainx1∫xN‖CT (t)z0‖2dt≤ (2e)2N−1∑n=1∞∫0‖λnte−λntCT (t)z0‖2dt7≤ (2e)2N−1∑n=1K212π‖z0‖2 =2πe2(N − 1)K21‖z0‖2.By definition of xn we have thatxN = x1(x2x1)N−1.Thuslog(xN ) = log(x1) − (N − 1) log(x1x2) (13)Let ε be a number in the interval (0, x2). Since xn → 0, there exists a N ∈ Nsuch that xN ≤ ε < xN−1. For this N we have that| log(ε)|−11∫ε‖CT (t)z0‖2dt≤ | log(xN−1)|−1x1∫xN‖CT (t)z0‖2dt≤ | log(xN−1)|−12πe2(N − 1)K21‖z0‖2.Now by (13) we see that the assertion of part 2. holds.Hence we have seen that the estimate (7) almost implies the admissibility ofC. In the following theorem we show that a slightly stronger estimate than(7) is sufficient for admissibility.Theorem 3.3 Let A be the infinitesimal generator of the C0-semigroup T (t)on the Hilbert space Z and let C ∈ L(D(A), Y ), where Y is a second Hilbertspace. If there exists a K > 0 such that‖C(sI − A)−1‖ ≤ Klog(Re(s))√Re(s), (14)for all complex s in some right half-plane, then C is an admissible observationoperator.Proof The proof goes along the same lines as the previous proof. Since ad-missibility of C for T (t) is equivalent to admissibility of C for eωtT (t), we canwithout loss of generality assume that T (t) is exponentially stable, and thatfor all γ > 1 there holds‖C((· + γ)I − A)−1‖H∞(L(Z,Y )) ≤Klog(γ)√γ.8Similar as in the proof of Theorem 3.2, we obtain for λ > 1∞∫0‖λe−λtCT (t)z0‖2dt ≤ K2‖z0‖2 log(λ)−2. (15)Now we proceed as in part 2. of the previous proof. For N > 1 we havex1∫xN‖CT (t)z0‖2dt = (2e)2N−1∑n=1xn∫xn+1‖λnte−λntCT (t)z0‖2dt≤ (2e)2N−1∑n=1∞∫0‖λnte−λntCT (t)z0‖2dt≤ (2e)2N−1∑n=1K21 log(λn)−2‖z0‖2.Since log(λn) behaves like n for n large, we see that the above inequalityimplies thatx1∫xN‖CT (t)z0‖2dt ≤ K2‖z0‖2for all N ∈ N and all z0 ∈ D(A). Using now the fact that xN → 0, we concludethat C is admissible.Looking carefully to the above proof, we see that we could conclude admissi-bility if‖C(sI − A)−1‖ ≤ Kg(Re(s))√Re(s),for all s in some right-half plane with g a monotonically increasing functionsatisfying∞∑n=1g(αn)−2 < ∞for some α > 1.94 Sufficient conditions for infinite-time admissibilityIn the previous section we studied the well-posedness of (1) on small timeintervals. In this section, we study the well-posedness on the entire interval[0,∞). By well-posedness we again mean that the output is square integrable.Definition 4.1 Let T (t) be a C0-semigroup on the Hilbert space Z, and let Cbe a bounded linear operator from the domain of A, D(A) to a second Hilbertspace Y . Then C is an infinite-time admissible observation operator if thereexist M > 0 such that for all z0 ∈ D(A)∞∫0‖CT (t)z0‖2Y dt ≤ M‖z0‖2.Thus if C is an infinite-time admissible observation operator, then the mappingz0 7→ CT (·)z0 can be extended to a bounded mapping from Z to L2((0,∞); Y ).It is easy to see that if the C0-semigroup is exponentially stable, then anyadmissible C is infinite-time admissible.Let C be an infinite-time admissible observation operator. Since C(sI −A)−1is the Laplace transform of CT (t)z0 there holds‖C(sI − A)−1z0‖2 = ‖∞∫0e−stCT (t)z0dt‖2≤‖e−s·‖2L2(0,∞)‖CT (t)z0‖2L2(0,∞),Y )≤ 12Re(s)M‖z0‖2, (16)where we have used the Cauchy-Schwarz inequality. Hence we see that if C isinfinite-time admissible, then there exists an K > 0 such that‖C(sI − A)−1‖ ≤ K√Re(s), (17)for all s ∈ C+0 .From Zwart, Jacob and Staffans [9] we have that this condition is not sufficient.However, like estimate (7) it is very close to being sufficient as will becomeclear from Theorems 4.2 and 4.3. The proof of these theorems is very similarto the proof of Theorem 3.2 and 3.3, respectively. Because of this similarity,we present only the details of the first assertion.10Theorem 4.2 Let T (t) be a uniformly bounded C0-semigroup on the Hilbertspace Z, with generator A. Assume that C ∈ L(D(A), Y ) is an admissibleobservation operator which satisfies the estimate‖C(sI − A)−1‖ ≤ K√Re(s),for all complex s ∈ C+0 . Then there exists a κ > 0 such that for all z0 ∈ D(A)and all Ω ∈ (x1,∞) we have thatΩ∫0‖CT (t)z0‖2dt ≤ κ log(Ω)‖z0‖2. (18)Here x1 is the largest solution of xe−x = 1/(2e), see Section 3.Proof Since T (t) is uniformly bounded, we have that for all λ > 0‖((· + λ)I − A)−1z0‖2H2(Z) = 2π‖e−λ·T (·)z0‖2L2((0,∞);Z)≤ 2π supt≥0‖T (t)‖2‖e−λ·‖2L2(0,∞)‖z0‖2≤ M1λ‖z0‖2. (19)It is easy to see that (17) implies that‖C((· + λ)I − A)−1‖H∞(L(Z,Y )) ≤K√λ(20)for all λ > 0. Combining (19) and (20) we obtain that for all λ > 0‖C((· + λ)I − A)−2z0‖H2(Y ) ≤‖C((· + λ)I − A)−1‖H∞‖((· + λ)I − A)−1z0‖H2(Z)≤ K√λ· M1√2λ‖z0‖ ≤K1λ‖z0‖,with K1 > 0 independent of λ. In the time domain this means that, for allλ > 0 and all z0 ∈ D(A)∞∫0‖λte−λtCT (t)z0‖2dt ≤K212π‖z0‖2. (21)11We define y1 = x2, and y2 = x1 and for n ∈ Nµn =(y1y2)n−1, yn =y1µn,so that µn → 0, yn → ∞ and µnyn = y1, µnyn+1 = y2. Henceµnte−µnt >12e, for all t ∈ (yn, yn+1). (22)Using this we see thatyN∫y1‖CT (t)z0‖2dt =N−1∑n=1yn+1∫yn‖CT (t)z0‖2dt≤ (2e)2N−1∑n=1yn+1∫yn‖µnte−µntCT (t)z0‖2dt.With (21) we obtain thatyN∫y1‖CT (t)z0‖2dt ≤2πe2K21 (N − 1).Since N ∼ log(yN) we have proved the assertion.Theorem 4.3 Let A be the infinitesimal generator of the uniformly boundedC0-semigroup T (t) on the Hilbert space Z and let C ∈ L(D(A), Y ), where Yis a second Hilbert space. If there exists a K > 0 such that for all γ ∈ (0, 1)‖C((γ + ·)I − A)−1‖H∞(L(Z,Y )) ≤Klog(γ)√γ, (23)then C is an infinite-time admissible observation operator.AcknowlegdementsThe author would like to thank Birgit Jacob and George Weiss for the stim-ulating discussions.12References[1] P. Grabowski. Admissibility of observation functionals. Internat. J. of Control,62: 1161–1173, 1995.[2] P. Grabowski and F.M. Callier. Admissible observation operators. Semigroupcriteria of admissibility. Integr. Equat. Oper. Th., 25: 182–198, 1996.[3] B. Jacob, J.R. Partington, and S. Pott. Admissible and weakly admissibleobservation operators for the right shift semigroup. Proc. Edinburgh Math.Soc., 45, pp. 353–362, 2002.[4] B. Jacob and H. Zwart. Counterexamples concerning observation operators forC0-semigroups. SIAM J. Control Optim. 43, pp. 137–153, 2004.[5] J.R. Partington and G. Weiss. Admissible observation operators for the rightshift semigroup. Mathematical Control of Systems and Signals, 13, pp. 179–192,2000.[6] G. Weiss. Admissible observation operators for linear semigroups. Israel Journalof Mathematics, 65:17–43, 1989.[7] G. Weiss. Two conjectures on the admissibility of control operators. In F. Kappel,W. Desch, editors, Estimation and Control of Distributed Parameter Systems,pages 367–378, Basel, 1991. Birkhäuser Verlag.[8] G. Weiss. A powerful generalization of the Carleson measure theorem, InV. Blondel, E. Sontag, M. Vidyasagar and J. Willems, editors, Open Problemsin Mathematical Systems Theory and Control, pages 267–272, 1998, Springer-Verlag.[9] H. Zwart, B. Jacob and O. Staffans. Weak admissibility does not implyadmissibility for analytic semigroups. System & Control Letters, 48, pp. 341–350,2003.13

Sufficient Conditions for Admissibility

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