Received: 23 April 2019 Revised: 29 July 2019 Accepted: 26 October 2019

DOI: 10.1002/asjc.2286

R E G U L A R PA P E R

Dynamic and static feedback control for second orderinfinite-dimensional systems

Jing Wei1 Bao-Zhu Guo2,3

1School of Mathematical Sciences, ShanxiUniversity, Taiyuan, 030006, Shanxi,P.R. China2Department of Mathematics and Physics,North China Electric Power University,Beijing 102206, China3Key Laboratory of System and Control,Academy of Mathematics and SystemsScience, Beijing, Academia Sinica, China

CorrespondenceJing Wei, School of MathematicalSciences, Shanxi University, Taiyuan,Shanxi, 030006, P.R. China.Email: wuhenshuijing@163.com

Funding informationProject of Department of Education ofGuangdong Province, Grant/AwardNumber: 2017KZDXM087; NationalNatural Science Foundation of China,Grant/Award Number: 61873153,11671240

Abstract

This paper considers dynamic feedback stabilization for abstract second-ordersystems, where the dynamic feedback controller is designed as another abstractsecond order infinite/finite-dimensional system. This makes the closed-loop sys-tem PDE-PDE or PDE-ODE coupled. The stability of the closed-loop system isfound to have three different cases. We first consider the dynamic feedback con-trol in a general Hilbert space which is usually different from the control space.It is shown that the stability of the closed-loop systems under the dynamic andstatic feedbacks are usually not equivalent. However, if the dynamic control lawis a copy of the original system, we deduce, under some conditions, that the cou-pled system is exponentially stable if and only if the static feedback closed-loopsystem is exponentially stable. When the dynamic feedback is designed in thecontrol space, the closed-loop system is asymptotically stable if and only if staticfeedback closed-loop system is asymptotically stable.

KEYWORDSdynamic feedback, stability, static feedback

1 INTRODUCTION

The notion of dynamic feedback controller or dynamicfeedback compensator is widely used in modern controltheory (see e.g. [1,2]). It is well known that the robustcontrollers for output regulation exist only by dynamicsfeedback controls, which can be obtained by the inter-nal model principle ([3], Lemma 1.21, p. 19]). From thecontrol point of view, an observer based feedback con-trol is a special dynamic feedback control, which has beenused in active disturbance rejection control theory [4], andadaptive control theory [5]. Moreover, an observer basedcontrol leads naturally to a coupled closed-system. On theother hand, many papers were devoted to the study ofthe coupled second order infinite-dimensional systems inboth control field and material sciences. Some examplescan be found in [6] for coupled beam-wave system, [7]

for coupled hyperbolic systems, and [8] for coupled platesystem.

It has been known that the asymptotic stability of afinite-dimensional system under the static feedback con-troller and the dynamic feedback controller is equiva-lent [9]. The same question could be asked for linearinfinite-dimensional systems and in paper [9], it wasshown that the equivalence of the asymptotic stability stillholds when the dynamic feedback control is chosen inthe form of the first-order system. It was proved in [9]that in all cases the closed-loop system under a first-orderdynamic output feedback is asymptotically stable if andonly if the Hautus test is satisfied. In this paper, we try todevelop this equivalence for the dynamic feedback controlwhich is chosen in the form of the second-order system.We show that if the controlled system satisfies the Haustuscondition, the second order dynamic output feedback in

© 2020 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd.

wileyonlinelibrary.com/journal/asjcAsian J Control. 2021;23:1431–1440. 1431

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the control space and the static output feedback are equiva-lent to achieve the asymptotic stability, which is consistentwith the conclusion of the first order dynamic feedbackdeveloped in [9]. However, if we design a second-orderfeedback compensator in a general Hilbert space, theresults become more complicated because the Haustuscondition is no longer sufficient for the stability of thedynamic feedback closed-loop systems. And in addition,we achieve exponential stability of the closed-loop sys-tem under the dynamic feedback, which is stronger thanasymptotic stability of [9].

To the best of our knowledge, there are few workson the study of the dynamic feedback control forinfinite-dimensional systems. The present paper is there-fore significant to the design of dynamic controller forinfinite-dimensional systems.

The paper is organized as follows. In Section 2, we designa second order dynamic feedback control in a Hilbert spacewhich is different from the control space. We establish thewell-posedness, asymptotic stability and the controllabil-ity of the closed-loop system. When the dynamic feedbackhas a special form, we obtain, in Section 3, the equiva-lence of the exponential stability under dynamic feedbackand static feedback. In Section 4, by designing a dynamicfeedback controller in the control space, we obtain similarconclusions for asymptotic stability. Some simulations arepresented in Section 5 to illustrate the theoretical results,following up concluding remarks in Section 6.

2 DYNAMIC OUTPUT FEEDBACKIN DIFFERENT SPACES

Consider the following second order infinite-dimensionalsystem in a complex Hilbert space and control space U:{

x(t) +x(t) + u(t) = 0,𝑦(t) = ∗ .x(t), (1)

where u(t) is the control and y(t) is the measurement. Here-after, the norm and the dual pairing on a Hilbert spaceare denoted respectively by symbols || · || and ⟨·, ·⟩. For theoperators and , we make assumptions as follows:

(H1) is a linear unbounded, densely defined,self-adjoint and strictly positive operator in . LetD(1∕2) be the domain of 1∕2. satisfies thefollowing Gelfand's triple compact inclusions:

D(1∕2) → → D(1∕2)′. (2)

(H2) The control operator ∈ (U,D(1∕2)′). Theobservation operator ∗ ∈ (D(1∕2),U) satisfies

⟨∗z,u⟩U = ⟨z,u⟩D(1∕2),D(1∕2)′ , ∀ z ∈ D(1∕2), u ∈ U.(3)

(H3) An extension ∈ (D(1∕2),D(1∕2)′) of isdefined by

⟨x, z⟩D(1∕2)′,D(1∕2) = ⟨1∕2x,1∕2z⟩ , ∀ x, z ∈ D(1∕2)(4)

and hence −1 ∈ (U,D(1∕2)).

(H4) ∈ (U,D(1∕2)′) or −1 ∈ (U,D(1∕2)) iscompact.

We first propose the following static proportional feed-back:

u(t) = k∗ .x(t), (5)where Re(k) > 0. Then, the closed-loop of system (1) underfeedback (5) is

x + x + k∗ .x = 0 in D(1∕2)′. (6)

We consider system (6) in the state space s = D(1∕2)× equipped with the usual inner product

⟨(𝑓1, g1)⊤, (𝑓2, g2)⊤⟩s

= ⟨1∕2𝑓1,1∕2𝑓2⟩+ ⟨g1, g2⟩ , ∀ (𝑓i, gi)⊤ ∈ s, i = 1, 2.

(7)

Denote the operator s ∶ D(s)(⊂ s) → s as

⎧⎪⎨⎪⎩s(𝑓, g)⊤ = (g,−𝑓 − k∗g)⊤, ∀ ( 𝑓, g)⊤ ∈ D(s),D(s) = {( 𝑓, g)⊤ ∈ D(1∕2) × D(1∕2)|

−𝑓 − k∗g ∈ }.(8)

Then, system (6) can be written as an evolution equationin s: .

X(t) = sX(t), X(0) = X0, (9)where X(t) = (x(t), .x(t))⊤.

Lemma 2.1. Suppose the assumptions (H1)–(H4). Letthe operator s be given by (8). Then, s generates anasymptotically stable C0-semigroup on s:

||estx||s → 0 as t → ∞, ∀ x ∈ s, (10)

if and only if the following Hautus test

Ker(𝜆 −) ∩ Ker(∗) = {0}, ∀ 𝜆 > 0 (11)

holds.

Proof. See Theorem 2.2 of [9].

Now, we design a dynamic output feedback control forsystem (1) as

u(t) = ∗ .z(t), (12)where z(t) is governed by the following second orderequation in a complex Hilbert space :

z(t) + z(t) +∗ .z(t) −∗ .x(t) = 0. (13)

Similarly as and, the operators and are supposedto satisfy the following properties:

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(H5) is a linear densely defined, self-adjoint andstrictly positive operator in and satisfiesGelfand's triple compact inclusions in :D(1∕2) → → D(1∕2)′.

(H6) ∈ (U,D(1∕2)′) and ∗ ∈ (D(1∕2),U) satisfy

⟨∗z,u⟩U = ⟨z,u⟩D(1∕2),D(1∕2)′ , ∀ z ∈ D(1∕2), u ∈ U.(14)

(H7) An extension ∈ (D(1∕2),D(1∕2)′) of isdefined by⟨x, z⟩D(1∕2)′,D(1∕2) = ⟨1∕2x,1∕2z⟩ , ∀ x, z ∈ D(1∕2).

(15)(H8) ∈ (U,D(1∕2)′) or −1 ∈ (U,D(1∕2)) is

compact.

The closed-loop of system (1) under feedback (12) is{x(t) + x(t) + ∗ .z(t) = 0, in D(1∕2)′,z(t) + z(t) +∗ .z(t) −∗ .x(t) = 0, in D(1∕2)′.

(16)We consider system (16) in the state space d =

D(1∕2) × × D(1∕2) × equipped with the usual innerproduct ⟨(𝑓1, g1, 𝜙1, 𝜓1)⊤, (𝑓2, g2, 𝜙2, 𝜓2)⊤⟩d

= ⟨1∕2𝑓1,1∕2𝑓2⟩ + ⟨g1, g2⟩+ ⟨1∕2𝜙1,1∕2𝜙2⟩ + ⟨𝜓1, 𝜓2⟩ ,

(17)

for (𝑓i, gi, 𝜙i, 𝜓i)⊤ ∈ d, i = 1, 2. If we introduce the newvariable X(t) = (x(t), .x(t), z(t), .z(t))⊤, then, system (16) canbe written as a first order system in d{ .

X(t) = dX(t),X(0) = X0,

(18)

where the operator d ∶ D(d)(⊂ d) → d is defined by

⎧⎪⎪⎨⎪⎪⎩

d( 𝑓, g, 𝜙, 𝜓)⊤= (g,−𝑓 − ∗𝜓,𝜓,−𝜙 −∗𝜓 +∗g)⊤,

D(d) = {(𝑓, g, 𝜙, 𝜓) ∈ [D(1∕2)]2

× [D(1∕2)]2| − 𝑓 − ∗𝜓 ∈ ,− 𝜙 −∗𝜓 +∗g ∈ },

∀ (𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d).(19)

Theorem 2.1. Suppose the assumptions (H1)–(H3),(H5)–(H7). Let d be given by (19). Then, d generatesa C0-semigroup of contractions edt on d.

Proof. By a simple calculation, for any ( 𝑓, g, 𝜙, 𝜓)⊤ ∈D(d),⟨d( 𝑓, g, 𝜙, 𝜓)⊤, ( 𝑓, g, 𝜙, 𝜓)⊤⟩d

= ⟨(g,−𝑓 − ∗𝜓,𝜓,−𝜙 −∗𝜓

+∗g)⊤, ( 𝑓, g, 𝜙, 𝜓)⊤⟩d

= ⟨1∕2g,1∕2𝑓⟩ − ⟨1∕2𝑓,1∕2g⟩− ⟨∗𝜓,∗g⟩U

+ ⟨1∕2𝜓,1∕2𝜙⟩ − ⟨1∕2𝜙,1∕2𝜓⟩− ||∗𝜓||2U + ⟨∗g,∗𝜓⟩U ,

(20)

from which, it yields

Re⟨d( 𝑓, g, 𝜙, 𝜓)⊤, (𝑓, g, 𝜙, 𝜓)⊤⟩d = −||∗𝜓||2U ≤ 0.(21)

Since is an isometric mapping from D(1∕2) toD(1∕2)′ and is an isometric mapping from D(1∕2)to D(1∕2)′, for any (𝑓1, g1, 𝜙1, 𝜓1)⊤ ∈ d, it has

−1d (𝑓1, g1, 𝜙1, 𝜓1)⊤ = (−−1(g1 + ∗𝜙1), 𝑓1,

− −1(𝜓1 +∗𝜙1 −∗𝑓1), 𝜙1)⊤.(22)

Hence,−1d exists and is bounded ond. Since D(d)

is densely defined in d by [[10], Proposition 3.1.6,p. 71], we complete the proof by the Lumer-PhillipsTheorem [[11], Theorem 4.3, p. 14].

Theorem 2.2. Suppose the assumptions (H1)–(H8).Then, d given by (19) generates an asymptotically sta-ble C0-semigroup on d:

||edtx||d → 0 as t → ∞, ∀ x ∈ d (23)

if and only if for any 𝜆 ∈ R, 𝜆 ≠ 0, the followingequations

⎧⎪⎨⎪⎩(𝜆2 −)𝑓 = 0,∗𝜙 = 0,(𝜆2 − )𝜙 + i𝜆∗𝑓 = 0,

(24)

has null solution only.

Proof. “Sufficiency”: Recall the assumptions (H4) and(H8) to see that d is compact on d. The proofwill be accomplished if we can show that there is noeigenvalue of d located on the imaginary axis ([12],Theorem 3.26, p. 130). Suppose that

d( 𝑓, g, 𝜙, 𝜓)⊤ = i𝜔(𝑓, g, 𝜙, 𝜓)⊤, ∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),(25)

where 𝜔 ∈ R, 𝜔 ≠ 0. It follows that

⎧⎪⎪⎨⎪⎪⎩

g = i𝜔𝑓, in D(1∕2),𝜓 = i𝜔𝜙, in D(1∕2),−𝑓 − i𝜔∗𝜙 = −𝜔2𝑓, in ,−𝜙 − i𝜔∗𝜙 + i𝜔∗𝑓 = −𝜔2𝜙, in .

(26)Since d is dissipative in (21), we obtain

∗𝜙 = 0 in U (27)

which comes from0 = Re⟨i𝜔( 𝑓, g, 𝜙, 𝜓)⊤, ( 𝑓, g, 𝜙, 𝜓)⊤⟩d

= Re⟨d( 𝑓, g, 𝜙, 𝜓)⊤, ( 𝑓, g, 𝜙, 𝜓)⊤⟩d

= −||∗𝜓||2U .(28)

Plug (27) back into equation (26) to conclude that𝑓 = 𝜔2𝑓 and (𝜔2−)𝜙+i𝜔∗𝑓 = 0. Thus, (𝑓, 𝜙) ∈

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Ker(𝜔2 − ) × Ker(∗) is a solution of (24). This factimplies immediately that f = 0 and 𝜙 = 0. Therefore,there is no eigenvalue of d on the imaginary.

“Necessity”: Suppose that (f, 𝜙) satisfies (24) for any𝜆 ∈ R, 𝜆 ≠ 0. Since

d

⎛⎜⎜⎜⎝𝑓

i𝜆𝑓𝜙

i𝜆𝜙

⎞⎟⎟⎟⎠=

⎛⎜⎜⎜⎜⎝

i𝜆𝑓−𝑓 − i𝜆∗𝜙

i𝜆𝜙−𝜙 − i𝜆∗𝜙 + i𝜆∗𝑓

⎞⎟⎟⎟⎟⎠= i𝜆

⎛⎜⎜⎜⎝𝑓

i𝜆𝑓𝜙

i𝜆𝜙

⎞⎟⎟⎟⎠,

(29)i𝜆 is an eigenvalue of d. Since edt is asymptoti-cally stable, it must have f = 𝜙 = 0. This ends theproof.

Remark 2.1. It is worth mentioning that the con-dition in Theorem 2.2 is stronger than Hautus test(11). This means that there is no equivalence betweenasymptotic stability under static feedback (5) andsecond-order dynamic feedback (12) in general sit-uation. However, by choosing a special dynamicfeedback in next section, we can establish not onlythe equivalence of the asymptotic stability, but alsothe equivalence of the exponential stability. This isthe key feature of our second-order dynamic feedbackcontroller.

Theorem 2.3. Suppose the assumptions (H1)–(H8).Then, the following assertions are equivalent:

(i). System (1) is approximately observable;

(ii). Ker(𝜆 −) ∩ Ker(∗) = {0}, ∀ 𝜆 > 0; (30)

(iii). System (16) is approximately observable with theoutput 𝑦z(t) = 1∕2z(t).

Proof. (i) ⇔ (ii) is the consequence of [[9], Theorem2.3].(ii) ⇔ (iii). If the output 𝑦z(t) = 1∕2z(t) ≡ 0 for all

t ≥ 0, then, z(t) ≡ 0 for all t ≥ 0. System (16) withthe output 𝑦z(t) = 1∕2z(t) is approximately observableif and only if the following system{ .

X(t) = AX(t),𝑦(t) = B∗X(t),

(31)

is approximately observable, where

X(t) = (x(t), .x(t))⊤, A =(

0 I− 0

), B

∗ = (0,∗)⊤.

According to [[9], Theorem 2.3], Σo(B∗,A) is approxi-mately observable if and only if (30) holds. This provesthe theorem.

Example 1. Consider the following wave equationwith boundary control

⎧⎪⎨⎪⎩wtt(x, t) = wxx(x, t), x ∈ (0, 1), t ∈ (0,∞),w(0, t) = 0, t ∈ [0,∞),wx(1, t) = u(t), t ∈ [0,∞),𝑦(t) = −wt(1, t), t ∈ [0,∞),

(32)

where u(t) is the control and y(t) is the measured out-put. Let = L2(0, 1) and denote ∶ D() → as

{𝑓 = −𝑓 ′′, ∀ 𝑓 ∈ D(),D() = {𝑓 ∈ H2(0, 1)|𝑓 ′(1) = 𝑓 (0) = 0}.

(33)

The control space is U = C and the control operatoris = −𝛿(x − 1), where 𝛿(·) is the Dirac distribu-tion. Clearly, ∗𝑓 = −𝑓 (1),∀ 𝑓 ∈ H1(0, 1). We shallpresent two different stabilizing controllers to illus-trate our results. First, we design the following directproportional control law:

u(t) = −kwt(1, t), ∀ t ≥ 0, (34)

where k > 0. Then, the closed-loop of system (32)under the feedback (34) is

⎧⎪⎨⎪⎩wtt(x, t) = wxx(x, t), x ∈ (0, 1), t ∈ (0,∞),w(0, t) = 0, t ∈ [0,∞),wx(1, t) = −kwt(1, t), t ∈ [0,∞),

(35)

which is well known to be exponentially stable for allk > 0.

Next, we design a dynamic feedback controller sat-isfying Theorem 2.2. Define the operator ∶ D() → = L2(0, 1) by{ 𝑓 = −𝑓 ′′, ∀ 𝑓 ∈ D(),

D() = {𝑓 ∈ H2(0, 1)|𝑓 ′(0) = q𝑓 (0), 𝑓 ′(1) = 0},(36)

and = = −𝛿(x − 1), ∗𝑓 = −𝑓 (1),∀ 𝑓 ∈H1(0, 1). It is seen that such defined ,,, satisfythe assumptions (H1)–(H8) and ≠ . The cor-responding stabilizing controller is then designed as

u(t) = −zt(1, t), ∀ t ≥ 0, (37)

where z(x, t) is governed by

⎧⎪⎨⎪⎩ztt(x, t) = zxx(x, t), x ∈ (0, 1), t ∈ (0,∞),zx(0, t) = qz(0, t), t ∈ [0,∞),zx(1, t) = −zt(1, t) + wt(1, t), t ∈ [0,∞).

(38)

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Thus, the closed-loop system under feedback(37)–(38) reads:

⎧⎪⎪⎪⎨⎪⎪⎪⎩

wtt(x, t) = wxx(x, t), x ∈ (0, 1), t ∈ (0,∞),w(0, t) = 0, t ∈ [0,∞),wx(1, t) = −zt(1, t), t ∈ [0,∞),ztt(x, t) = zxx(x, t), x ∈ (0, 1), t ∈ (0,∞),zx(0, t) = qz(0, t), t ∈ [0,∞),zx(1, t) = −zt(1, t) + wt(1, t), t ∈ [0,∞).

(39)We consider system (39) in the state space d =

H1(0, 1)×L2(0, 1)×H1(0, 1)×L2(0, 1). Define the systemoperator d ∶ D(d) → d for (39) as follows

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d( 𝑓, g, 𝜙, 𝜓)⊤

= (g, 𝑓 ′′, 𝜓, 𝜙′′)⊤, ∀ (𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),D(d) = {( 𝑓, g, 𝜙, 𝜓)⊤ ∈

[H2(0, 1) × H1(0, 1)

]2|𝑓 (0) = 0, 𝑓 ′(1) = −𝜓(1), 𝜙′(0) = q𝜙(0),𝜙′(1) = −𝜓(1) + g(1)}.

(40)We now consider the eigenvalue problem of d, i.e.

d( 𝑓, g, 𝜙, 𝜓)⊤ = 𝜆(𝑓, g, 𝜙, 𝜓)⊤, ∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),(41)

which is equivalent to

⎧⎪⎨⎪⎩𝑓 ′′(x) = 𝜆2𝑓 (x), 𝜙′′(x) = 𝜆2𝜙(x), x ∈ [0, 1],𝑓 (0) = 0, 𝑓 ′(1) = −𝜆𝜙(1),𝜙′(0) = q𝜙(0), 𝜙′(1) = −𝜆𝜙(1) + 𝜆𝑓 (1).

(42)

Let

𝑓 (x) = c1e𝜆x + c2e−𝜆x and 𝜙(x) = c3e𝜆x + c4e−𝜆x, (43)

where c1, c2, c3, c4 are constants. Substituting (43) intothe boundary conditions of (42) yields

⎧⎪⎪⎨⎪⎪⎩

c1 + c2 = 0,(𝜆 − q)c3 − (𝜆 + q)c4 = 0,(e𝜆 + e−𝜆)c1 + e𝜆c3 + e−𝜆c4 = 0,(e−𝜆 − e𝜆)c1 + 2e𝜆c3 = 0.

(44)

Hence, (44) has nonzero solution if and only if thecharacteristic determinant det(Δ(𝜆)) = 0, where

Δ(𝜆) =

⎡⎢⎢⎢⎢⎣

1 1 0 00 0 (𝜆 − q) −(𝜆 + q)

(e𝜆 + e−𝜆) 0 e𝜆 e−𝜆

(e−𝜆 − e𝜆) 0 2e𝜆 0

⎤⎥⎥⎥⎥⎦, (45)

from which we find the characteristic equation as

(q − 𝜆)e−2𝜆 + 3(q + 𝜆)e2𝜆 + 2𝜆 = 0, ∀ 𝜆 ∈ 𝜎(d). (46)

Proposition 2.1. Let d be given by (40). Then, theC0-semigroup edt generated by d is asymptoticallystable but not exponentially stable on d.

Proof. If (f, 𝜙) satisfies (24), then,

⎧⎪⎪⎨⎪⎪⎩

𝑓 ′′(x) = −𝜆2𝑓 (x), 𝜙′′(x) = −𝜆2𝜙(x), x ∈ [0, 1],𝑓 (0) = 0, 𝑓 ′(1) = 0,𝜙′(0) = q𝜙(0), 𝜙(1) = 0,𝜙′(1) = i𝜆𝑓 (1).

(47)The general solution to system (47) is{

𝑓 (x) = c1 sin 𝜆x + c2 cos 𝜆x, x ∈ [0, 1],𝜙(x) = c3 sin 𝜆x + c4 cos 𝜆x, x ∈ [0, 1], (48)

where the coefficients cj(j = 1, 2, 3, 4) satisfy

⎧⎪⎨⎪⎩c2 = 0, 𝜆c3 − qc4 = 0,c3 sin 𝜆 + c4 cos 𝜆 = 0,c3 cos 𝜆 − c4 sin 𝜆 − ic1 sin 𝜆 = 0,c1 cos 𝜆 = 0.

(49)

If cos 𝜆 ≠ 0, one has c1 = 0 and c3 sin 𝜆 + c4 cos 𝜆 =0, c3 cos 𝜆 − c4 sin 𝜆 = 0. The last two equations yieldc3 = c4 = 0. If cos 𝜆 = 0 and hence sin 𝜆 ≠ 0, one hasc3 sin 𝜆 = 0, 𝜆c3 − qc4 = 0 and c4 sin 𝜆 + ic1 sin 𝜆 = 0.Thus, c3 = c4 = c1 = 0. Therefore, c1 = c2 = c3 = c4 =0 and hence edt is asymptotically stable.

On the other hand, by a simple calculation for (46),we obtain

e2𝜆 =−𝜆 ±

√−3q2 + 4𝜆2

3(q + 𝜆)= −1 ± 2

3+

q𝜆+( 1

𝜆2

), |𝜆| → ∞.

(50)Since the roots of e2𝜆1 = −1 are 𝜆1 = (n − 1

2)𝜋i. By

Rouché's theorem, the roots of (50) have the followingasymptotic expansion:

𝜆1n =(

n − 12

)𝜋i + (n−1), n = 1, 2, … . (51)

Thus, Re(𝜆1n) → 0 as n → ∞ and as a result, edt isnot exponentially stable.

It follows from Theorem 2.3 that system (39) isapproximately observable, that is,

∫t

0

(∫

1

0z2

x(x, 𝜏)dx + qz2(0, t))

d𝜏 = 0, ∀ t ≥ 0

⇒ (w(·, 0),wt(·, 0), z(·, 0), zt(·, 0)) = 0.(52)

Remark 2.2. As indicated in Example 2.1, the expo-nential stability between the static feedback and thedynamic feedback has no equivalence in general case ≠ .

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3 EQUIVALENCE OFEXPONENTIAL STABILITY: = AND = In this section, we discuss the relationship between thestatic feedback closed-loop system and the dynamic feed-back closed-loop system under the condition = and = . In this way, system (16) is reduced to{

x(t) + x(t) + ∗ .z(t) = 0, in D(1∕2)′,z(t) + z(t) + ∗ .z(t) − ∗ .x(t) = 0, in D(1∕2)′.

(53)We consider system (53) in the state space d =

D(1∕2) × × D(1∕2) × . Define the operator d ∶D(d)(⊂ d) → d by

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d(𝑓, g, 𝜙, 𝜓)⊤

= (g,−𝑓 − ∗𝜓,𝜓,−𝜙 − ∗𝜓 + ∗g)⊤,∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ d,

D(d)= {( 𝑓, g, 𝜙, 𝜓)⊤ ∈ [D(1∕2)]4| − 𝑓 − ∗𝜓 ∈ ,

−𝜙 − ∗𝜓 + ∗g ∈ }.(54)

Theorem 3.1. Suppose the assumptions (H1)–(H4).If is admissible, and the transfer function isbounded, that is, supRe(s)=𝛼||s∗(s2 + )−1||U <

∞, for some 𝛼 > 0, then, the following assertions areequivalent:

(i) The s defined by (9) generates an exponentiallystable C0-semigroup est on s;

(ii) Thed defined by (54) generates an exponentiallystable C0-semigroup edt on d;

(iii) Σo(∗,) is exactly observable on [0, 𝜏] for some𝜏 > 0.

Proof. (i) ⇔ (iii). The case of k = 1 has been provedin [[13], Theorems 2,3]. It is not difficult to provethat such a conclusion can be extended to Re(k) > 0analogously.(i) ⇒ (ii). Define the following new coordinates

(𝜀

𝛿

)=

⎛⎜⎜⎝1 − 1+

√3i

2

1 − 1−√

3i2

⎞⎟⎟⎠(

xz

)=∶ P

(xz

), (55)

where P =⎛⎜⎜⎝

1 − 1+√

3i2

1 − 1−√

3i2

⎞⎟⎟⎠ is an invertible operator. Then,

(𝜀, 𝛿) satisfies

⎧⎪⎨⎪⎩�� + 𝜀 + 1+

√3i

2∗ .

𝜀 = 0, in D(1∕2)′,

𝛿 + 𝛿 + 1−√

3i2

∗ .𝛿 = 0, in D(1∕2)′.

(56)

Note that (𝜀, 𝛿) is decoupled and hence (𝜀, 𝛿) is expo-nentially stable due to (i). Since (x, z)⊤ = P−1(𝜀, 𝛿)⊤, the(x, z)-system is exponentially stable as well.(ii) ⇒ (iii). By (55), the 𝜀-subsystem is exponentially

stable. By “(i)⇒ (iii)", Σo(∗,) is exactly observableon [0, 𝜏]. This ends the proof.

Example 2. We consider Example 1 in amulti-dimensional case. Let Ω be an open boundeddomain of Rn(n ≥ 1) with a smooth C2-boundary𝜕Ω = Γ0 ∪ Γ1, where Γ0 and Γ1 are open and disjoint,with meas(Γ0) > 0 and 𝜈 being the unit outer normalto 𝜕Ω. We now study the following multi-dimensionalwave system

⎧⎪⎪⎨⎪⎪⎩

wtt(x, t) − Δw(x, t) = 0, in Ω × (0,∞),w(x, t) = 0, on Γ0 × [0,∞),𝜕w𝜕𝜈(x, t) = u(x, t), on Γ1 × [0,∞),

𝑦(x, t) = −wt(x, t), on Γ1 × [0,∞),

(57)

where u(x, t) is the control, y(x, t) is the measurement(output). Let us first select the general proportionaloutput control as

u(x, t) = −kwt(x, t) on Γ1 × [0,∞), (58)

where k > 0 is a constant. Then, the closed-loop takesthe form:

⎧⎪⎨⎪⎩wtt(x, t) − Δw(x, t) = 0, in Ω × (0,∞),w(x, t) = 0, on Γ0 × [0,∞),𝜕w𝜕𝜈(x, t) = −kwt(x, t), on Γ1 × [0,∞).

(59)

By applying the multiplier method ([14], Theorem8.6), we know that system (59) is exponentially stablein the state space s = H1(Ω) × L2(Ω) if Ω satisfies thegeometric condition

(x − x0) · 𝜈 ≤ 0 on Γ0, (x − x0) · 𝜈 ≥ 0 on Γ1, (60)

where x0 is a fixed point in Rn.Now, choose = L2(Ω) and U = L2(Γ1). The

corresponding operator is given by

⎧⎪⎨⎪⎩𝑓 = −Δ𝑓, ∀ 𝑓 ∈ D(),

D() ={𝑓 ∈ H2(Ω) × H1(Ω)|||𝑓 |Γ0 = 0, 𝜕𝑓

𝜕𝜈

|||Γ1= 0

}.

(61)The control operator is defined by

u = −Υu, ∀ u ∈ U, (62)

where Υ ∈ (L2(Γ1),H3∕2(Ω)) is the Neumann map([9], p. 1942). Then, ∗ ∈ (D(1∕2),L2(Γ1)) satis-fies ∗𝑓 = −𝑓 |Γ1 , and ,,, with = , = satisfy the assumptions of Theorem 3.1. Under the

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WEI AND GUO

feedback

u(x, t) = −zt(x, t) on Γ1 × [0,∞), (63)

where⎧⎪⎨⎪⎩ztt(x, t) − Δz(x, t) = 0, in Ω × (0,∞),z(x, t) = 0, on Γ0 × [0,∞),𝜕z𝜕𝜈(x, t) = −zt(x, t) + wt(x, t), on Γ1 × [0,∞),

(64)

the corresponding closed-loop system becomes

⎧⎪⎪⎪⎨⎪⎪⎪⎩

wtt(x, t) − Δw(x, t) = 0, in Ω × (0,∞),ztt(x, t) − Δz(x, t) = 0, in Ω × (0,∞),w(x, t) = z(x, t) = 0, on Γ0 × [0,∞),𝜕w𝜕𝜈(x, t) = −zt(x, t), on Γ1 × [0,∞),

𝜕z𝜕𝜈(x, t) = −zt(x, t) + wt(x, t), on Γ1 × [0,∞),

(65)

with the state space d = [H1(Ω) × L2(Ω)]2. By theequivalence between the exponential stability of theclosed-loop system under the dynamic feedback (63)and the exponential stability of the closed-loop systemunder the static feedback (58), we have the followingProposition 3.1.

Proposition 3.1. For any initial value(w(·, 0),wt(·, 0), z(·, 0), zt(·, 0)) ∈ d, if Ω satisfies thegeometric condition (60), then, the solution of system(65) is exponentially stable:

||∇w(·, t)||2L2(Ω) + ||wt(·, t)||2L2(Ω)

+ ||∇z(·, t)||2L2(Ω) + ||zt(·, t)||2L2(Ω)

≤ Ce−𝜔t[||∇w(·, 0)||2L2(Ω) + ||wt(·, 0)||2L2(Ω)

+ ||∇z(·, 0)||2L2(Ω) + ||zt(·, 0)||2L2(Ω)

],

(66)

where C and 𝜔 are positive constants.

4 DYNAMIC OUTPUT FEEDBACKIN CONTROL SPACE

In this section, we discuss dynamic output feedback con-trol in the control space U. Inspired by the controllerdesign (12)–(13), a dynamic output feedback control isproposed for system (1) as

u(t) = .z(t), (67)

where z(t) is given by the following second order equationin the control space U:

z(t) + z(t) + .z(t) − ∗ .x(t) = 0 in U, (68)

with satisfying assumption (H9).(H9). The operator ∈ (U) is compact, strictly positive

on U.

Under the feedback control (67), closed-loop of system(1) is{

x(t) + x(t) + .z(t) = 0, in D(1∕2)′,z(t) + z(t) + .z(t) − ∗ .x(t) = 0, in U.

(69)

We consider system (69) in the state space d =D(1∕2) × × U2 equipped with the usual inner product

⟨(𝑓1, g1, 𝜙1, 𝜓1)⊤, (𝑓2, g2, 𝜙2, 𝜓2)⊤⟩d

= ⟨1∕2𝑓1,1∕2𝑓2⟩ + ⟨g1, g2⟩ + ⟨𝜙1, 𝜙2⟩U

+ ⟨𝜓1, 𝜓2⟩U ,∀ (𝑓i, gi, 𝜙i, 𝜓i)⊤ ∈ d, i = 1, 2.(70)

Introduce the new variable X(t) = (x(t), .x(t), z(t), .z(t))⊤.Then, X(t) satisfies { .

X(t) = dX(t),X(0) = X0,

(71)

where the operator d ∶ D(d)(⊂ d) → d is defined by

⎧⎪⎪⎪⎨⎪⎪⎪⎩

d( 𝑓, g, 𝜙, 𝜓)⊤

= (g,−𝑓 − 𝜓,𝜓,−𝜙 − 𝜓 + ∗g)⊤,∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),

D(d)= {(𝑓, g, 𝜙, 𝜓)⊤ ∈ [D(1∕2)]2 × U2| − 𝑓 − 𝜓 ∈ ,

−𝜙 − 𝜓 + ∗g ∈ U}.(72)

Theorem 4.1. Suppose the assumptions (H1)–(H4)and (H9). Then, the following assertions are equivalent:

(i) The d given by (72) generates an asymptoticallystable C0-semigroup edt on d;

(ii) The s given by (8) generates an asymptoticallystable C0-semigroup est on s;

(iii) Ker(𝜆 −) ∩ Ker(∗) = {0}, ∀ 𝜆 > 0; (73)

(iv) System (69) is approximately observable with theoutput yz(t) = z(t).

Proof. (ii) ⇔ (iii) is a consequence of Lemma 2.1.(iii) ⇒ (i). Let Y = ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d). Then,

Re⟨dY ,Y⟩d =Re⟨(g,−𝑓 − 𝜓,𝜓,−𝜙 − 𝜓 + ∗g)⊤,( 𝑓, g, 𝜙, 𝜓)⊤⟩d

=Re[⟨1∕2g,1∕2𝑓⟩ + ⟨−𝑓 − 𝜓, g⟩+ ⟨𝜓, 𝜙⟩U + ⟨−𝜙 − 𝜓 + ∗g, 𝜓⟩U]

= − ⟨𝜓,𝜓⟩U ≤ 0.(74)

For any (𝑓1, g1, 𝜙1, 𝜓1)⊤ ∈ d, solve

d(𝑓, g, 𝜙, 𝜓)⊤ = (𝑓1, g1, 𝜙1, 𝜓1)⊤, (75)

to obtain(𝑓, g, 𝜙, 𝜓)⊤ = −1

d (𝑓1, g1, 𝜙1, 𝜓1)⊤

= (−−1(g1 + 𝜙1), 𝑓1,

−𝜓1 − 𝜙1 + ∗𝑓1, 𝜙1)⊤.(76)

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WEI AND GUO

FIGURE 1 Simulations for system (35) and (65) [Color figure can be viewed at wileyonlinelibrary.com]

Hence, −1d exists and is bounded on d. Note

that D(d) is densely defined in d by [[10], Propo-sition 3.1.6, p. 71]. Thus, d given by (72) gener-ates a C0-semigroup of contractions on d by a theLumer-Phillips Theorem [[11], Theorem 4.3, p. 14].

If condition (73) holds, we show the strong stabilityof d on d. Under the assumption, −1

d is compacton . By [[12], Theorem 3.26, p. 130], we only need toprove that there is no eigenvalue of d located on theimaginary. Suppose that

d( 𝑓, g, 𝜙, 𝜓)⊤ = i𝜔( 𝑓, g, 𝜙, 𝜓)⊤, ∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),(77)

where 𝜔 ∈ R, 𝜔 ≠ 0. Then,

⎧⎪⎪⎨⎪⎪⎩

g = i𝜔𝑓, in D(1∕2),𝜓 = i𝜔𝜙, in U,−𝑓 − i𝜔𝜙 = −𝜔2𝑓, in ,−𝜙 − i𝜔𝜙 + i𝜔∗𝑓 = −𝜔2𝜙, in U.

(78)

Since d is dissipative in (74), we perform the sameas (28) to obtain

⟨𝜙, 𝜙⟩U = 0. (79)

Hence, 𝜙 = 0. Then, (78) is reduced to 𝑓 = 𝜔2𝑓

and ∗𝑓 = 0. By assumption (73), f = 0 and hence(f, g, 𝜙, 𝜓) = 0. Therefore, there is no eigenvalue of dlocated on the imaginary axis.(i) ⇒ (iii). Suppose that 𝑓 ∈ Ker(𝜆 − ) ∩

Ker(∗),∀ 𝜆 > 0. It implies immediately from

d( 𝑓, i√𝜆𝑓, 0, 0)⊤ = (i

√𝜆𝑓,−𝑓, 0, i√𝜆∗𝑓 )⊤

= i√𝜆( 𝑓, i

√𝜆𝑓, 0, 0)⊤

(80)

that i√𝜆 is an eigenvalue of d. Since edt is asymp-

totically stable, there is no eigenvalue of d located onthe imaginary axis, we thus obtain f = 0.(iv) ⇔ (iii). If the output yz(t) = z(t) ≡ 0 for all

t ≥ 0, then, being approximately observable for system(69) with yz(t) = z(t) is equivalent to the approximately

observable of the system Σo(B∗,A) defined by (31).Thus, system (16) is approximately observable with theoutput yz(t) = z(t) from the same arguments presentedin the proof of Theorem 2.3.

Example 3. We still consider the controlled system(32) in Example 1. Let = I. It is obvious that ∈(C) is compact and system (32) satisfies the assump-tions (H1)–(H4). Now, we design a dynamic feedbackin control space U = C as follows:

u(t) = .z(t), ∀ t ≥ 0, (81)

where z(t) satisfies the second order ODE

z(t) + z(t) + .z(t) + wt(1, t) = 0 (82)

in C. Then, the closed-loop system reads:

⎧⎪⎨⎪⎩wtt(x, t) = wxx(x, t), x ∈ (0, 1), t ∈ (0,∞),w(0, t) = 0, t ∈ [0,∞),wx(1, t) =

.z(t), t ∈ [0,∞),z(t) + z(t) + .z(t) + wt(1, t) = 0, t ∈ [0,∞).

(83)We consider the closed-loop system (83) in the state

space d = H1(0, 1) × L2(0, 1) × C2. Define the systemoperator d ∶ D(d) → d for (83) as follows

⎧⎪⎪⎨⎪⎪⎩

d(𝑓, g, 𝜙, 𝜓)⊤ = (g, 𝑓 ′′, 𝜓,−𝜙 − 𝜓 − g(1))⊤,∀ ( 𝑓, g, 𝜙, 𝜓)⊤ ∈ D(d),

D(d) = {( 𝑓, g, 𝜙, 𝜓)⊤ ∈[H2(0, 1) × H1(0, 1) × C2] |

𝑓 (0) = 0, 𝑓 ′(1) = 𝜓}.(84)

Then, the following proposition is a direct resultfrom Theorem 4.1 and [[15], Example 1].

Proposition 4.1. Let d be given by (84). Then, theC0-semigroup edt generated by d is asymptoticallystable but not exponentially stable on d.

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FIGURE 2 Simulations for system (39) [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 3 Simulations for system (83) [Color figure can be viewed at wileyonlinelibrary.com]

From the perspective of approximate observability,we obtain

∫t

0|z(𝜏)|2d𝜏 = 0, ∀ t ≥ 0 ⇒ (w(·, 0),wt(·, 0), z(0),

.z(0)) = 0.

(85)

Remark 4.1. As indicated in Example 3, Theorem 4.1can not achieve the equivalence of the exponential sta-bility of est and edt. Recall Remark 2.2, we therebypay more attention to the equivalence of asymptoticstability in Sections 2 and 4.

5 NUMERICAL SIMULATION

In this section, we present some numerical simulationsfor Examples 1–3 with space dimension n = 1. The finitedifference method is applied in simulation. The time andspace steps are taken as 0.005 and 0.007, respectively. Herethe initial value of w-system in three examples is chosen as

w(x, 0) = cos 2𝜋x − 1, wt(x, 0) = 0. (86)

The initial value of auxiliary system “z” is chosen as{z(x, 0) = cos 2𝜋x − 1, zt(x, 0) = 0,z(0) = 1, .z(0) = 0. (87)

The solution of closed-loop system (35) with static feed-back control is depicted in Figure 1. The solution of threeclosed-loop systems with dynamic feedback control aredepicted in Figure 1–3. It can be seen that system (35) and(65) are indeed exponentially stable and (39) and (83) areasymptotically stable.

6 CONCLUDING REMARKS

This paper addresses the difference between static outputfeedback control and dynamic feedback control for generallinear collocated infinite-dimensional systems. The con-trol u = ∗ .z designed in Section 3 is applicable to establishthe equivalence of exponential stability of est and edt,but it may require some strong geometric assumptions forinstance the condition (60). In Section 2, we can design acontroller u = ∗ .z in a more general Hilbert space than in

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Section 3 without geometric assumptions and the stabilityresult of closed-loop under this feedback works for vari-ous cases. It is seen that the control u = .z designed inSection 4 is simplest to use. Nevertheless, the equivalenceof exponential stability can not be obtained in general bycontroller u = ∗ .z or u = .z.

ACKNOWLEDGEMENTS

This work is supported by the National Natural Sci-ence Foundation of China (No. 61873153, 11671240), andthe Project of Department of Education of GuangdongProvince (No. 2017KZDXM087).

ORCID

Jing Wei https://orcid.org/0000-0003-0953-2583

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AUTHOR BIOGRAPHIES

Jing Wei received the B.Sc.degree in mathematics fromShanxi University, Taiyuan,Shanxi, P.R. China in 2014and is currently pursuing thePh.D. degree at Shanxi Uni-

versity, Taiyuan, Shanxi, P.R. China. Her researchinterests include control and stability analysis of dis-tributed parameter systems and output regulationof infinite-dimensional systems.

Bao-Zhu Guo received thePh.D. degree from the Chi-nese University of Hong Kongin applied mathematics in1991. From 1985 to 1987, hewas a Research Assistant at

the Beijing Institute of Information and Control,China. During the period 1993-2000, he was withthe Beijing Institute of Technology, first as anassociate professor (1993-1998) and subsequentlya professor (1998-2000). Since 2000, he has beenwith the Academy of Mathematics and SystemsScience, the Chinese Academy of Sciences, wherehe is a research professor in mathematical sys-tem theory. From 2004-2019, he had been withSchool of Computer Science and Applied Mathe-matics, the University of the Witwatersrand, SouthAfrica as a chair professor. His research interestsinclude the theory of control and application ofinfinite-dimensional systems.

How to cite this article: Wei J, Guo B-Z.Dynamic and static feedback control for secondorder infinite-dimensional systems. Asian J Control.2021;23:1431–1440. https://doi.org/10.1002/asjc.2286

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