# Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control

Authors: Xue-Lian Jin; Yan Li; Fu Zheng
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions.


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## Body

## 1. Introduction

This paper is devoted to analyzing 1-d heat-wave coupled system with a dynamical boundary feedback (see [1–3]). More precisely, we study the following coupled system:(1)ytt,x-yxxt,x=0,x∈-1,0,t>0,zttt,x-zxxt,x=0,x∈0,1,t>0,yt,-1=0,zxt,1=-ηtηtt-ztt,1=-βηtyt,0=ztt,0,yxt,0=zxt,0y0,x=y0x,η0=η0,z0,x=z0x,zt0,x=z1x,where β is a positive constant, η denotes the dynamical control, and -βη(t) is the negative proportional feedback. This system consists of a wave equation, arising on the interval (0,1) with state (z,zt) and a heat equation that holds on the interval (-1,0) with state y. The wave and heat components are coupled through an interface, the point x=0, with transmission conditions imposing the continuity of (y,zt) and (yx,zx).The wave component in (1) is similar to that of [2] and the coupled heat-wave model without dynamical control comes from [1]. However, the coupled heat-wave model described by (1) has a slight difference with that of [1]. The zero Dirichlet boundary at the end point of the wave component was considered in [1]. But the Neumann boundary control is imposed at the same position in (1). This means that a dynamic boundary controller is applied at the free end of the system.In recent years, the hyperbolic-parabolic coupled models have been studied extensively due to their applications in analyzing fluid-structure interactions, which are crucial in many scientific and engineering areas, such as air flow along the aircraft, deformation of heart valves, and the process of mixing [1, 4–6].Some researchers studied the decay rates for the heat-wave coupled systems with various boundary conditions. In 2003, Zhang and Zuazua discussed the polynomial decay and controllability of the systems similar to (1) by the theory of operator semigroup and Riesz basis method in [7, 8]. In 2007, they analyzed the long-time behavior of a coupled heat-wave system in which a wave and a heat equation evolve in two bounded domains of Rn, with natural transmission conditions at a common interface in [9]. In 2009, Dalsen considered the question of stabilization of a fluid-structure model that describes the interaction between a 3D incompressible fluid and a 2D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid through a frequency domain approach in [10].In 2013, using the same approach, Avalos and Triggiani gave the rational decay rate for a PDE heat-structure interaction in [11]. In 2016, by the frequency domain approach together with multipliers, Han and Zuazua discussed the large time decay rate of a transmission problem coupled heat and wave equations on a planar network in [12]. In 2018, Peralta and Kunisch studied a coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid in [13]. The system incorporates delay in the interaction on the interface between the fluid and the solid. They discussed the stability property of the interaction model under suitable assumptions between the competing strength of the delay and the feedback control. For more results on the heat-wave coupled systems, one can also refer to [14, 15] and references therein.The stabilities or stabilizations were obtained through static boundary conditions or feedbacks in the literatures above. However, the dynamical boundary controls were widely used in the beam equations or the wave equations [16–19]. Morgül proposed a dynamic boundary controller applied at the free end of the 1-d linear wave equation in [17]. The stabilities of the system were analyzed in light of the transfer function. For example, the transfer function of the wave equation controlled by the dynamic boundary controller in [2] is g(s)=1/s+β; then the stabilization is asymptotic but not exponential since the transfer function g(s) is strictly positive real. Furthermore, under some conditions, the exponential stability of the wave equation with similar dynamic boundary controller was obtained in [18]. In fact, the dynamical control forms a part of indirect mechanism proposed by Russell (see [19–21] and the references therein). More recently, Wehbe studied the exact controllability of a wave equation with dynamical boundary control in [3]. In 2015, Feng and Guo systematically investigated stability equivalence between dynamic output feedback and static output feedback for a class of second-order systems in [22].The main contributions of this note are as follows. When the heat equation is not present, the system (with boundary feedback) was discussed in [2], and the polynomial (but not exponential) stability was shown. In the same time, when the heat equation was present (without dynamical boundary feedback), the system was analyzed in [1] and the asymptotical (but not exponential) stability was shown again. We combine these two results and show that the similar results hold in this note. But this extension is not trivial and should give strict proof. Moreover, we give the asymptotic expansions of the spectrum of the system operator. In light of the spectral analysis, we show that the system is not exponential stabilization. Though we believe that it is not hard to obtain the rational decay rate of the system (1) by the frequency domain method (see [23–25]) or Riesz basis method (see [2, 7, 8, 12, 26]), we do not explicitly elaborate because the information reflected by the spectrum is enough in some cases (see [2, 7, 8, 14, 15, 27, 28]).The outline of this paper is as follows. In Section2, we transform (1) into an abstract Cauchy problem on the energy space. The well-posedness and asymptotic stability of system (1) are given through the theory of operator semigroup and the spectral configuration of the system operator. In Section 3, the spectral expansions, which imply nonexponential decay rate of energy of system (1), of system operator are given. In the last section, we make a concise concluding remark.


## 2. Well-Posedness and Asymptotic Stability

In this section, we shall transform (1) into an abstract Cauchy problem on a suitable state space. For this purpose, let Hm(a,b) be the Sobolev spaces of positive integer order m on the open interval (a,b). The energy space of (1) is the Hilbert space (2)H≔L2-1,0×H10,1×L20,1×Cwith the norm (3)y,z,w,ηH2=∫-10yx2dx+∫01zxx2dx+∫01wx2dx+η2,∀y,z,w,η∈H.Introduce the operatorA in H as follows:(4)DA=yzwη∈Hy∈H2-1,0,z∈H20,1,w∈H10,1,y-1=0,y0=w0,yx0=zx0,zx1=-η,(5)Au=yxxwzxxw1-βη,∀u=yzwη∈DA.Settingu=(y,z,w,η)⊤∈H, then we formally transform system (1) into an evolution equation on H(6)u′t=Aut,u0=u0∈H.Now we give one of the main results of this section.Theorem 1.
The operator(A,D(A)) generates a contracted C0-semigroup (T(t))t≥0 on H.Proof.
It is easy to see that the operatorA is densely defined and closed. It follows from a straightforward computation that(7)ReAu,uH=-∫-10yxx2dx-βη2≤0,∀u∈DA.This means that the operator A is dissipative.
Moreover, for anyu=(y,z,w,η)⊥∈D(A) and u~=(y~,z~,w~,η~)⊥∈H, we have (8)Au,u~H=∫-10yxxxy~xdx+∫01wxxz~xdx+∫01zxxxw~xdx+w1-βηη~=∫-10yxy~xxxdx-∫01wxz~xxdx-∫01zxxw~xxdx+w1-βηη~+yxxy~x-10-yxy~xx-10+wxz~xx01+zxxw~x01=∫-10yxy~xxxdx-∫01wxz~xxdx-∫01zxxw~xxdx-ηw~1+βη~+yx0y~0-w~0-y0y~x0+z~x0+w1z~x1+η~-yx-1y~-1It follows from the above identities that the adjoint A∗ of the operator A is given as (9)DA∗=yzwη∈Hy∈H2-1,0,z∈H20,1,w∈H10,1,y-1=0,y0=w0,yx0=-zx0,zx1=-η,A∗u=yxx-w-zxx-w1-βη,∀u=yzwη∈DA∗.Similarly, we also have (10)ReA∗u,uH=-∫-10yxx2dx-βη2≤0,∀u∈DA∗.This shows that the operator A∗ is dissipative as well. It follows form Pro.3.1.11 of [29] that the operator A is further m-dissipative. Thus the Lumer-Phillips theorem (Theorem 3.8.4. of [29]) implies that the statement of the theorem holds.To show the asymptotic stability of system (1), we should analyze the spectral configuration of the operator A.Lemma 2.
LetA and H be defined as above; then we have 0∈ρ(A). Furthermore, A-1 is compact on H. Therefore, the spectrum of A consists only of the isolated eigenvalues of finite multiplicity, i.e., σ(A)=σp(A).Proof.
For any fixedu~∈H, we consider the solvability of the resolvent equation Au=u~, which is equivalent to(11)yxxx=y~x,x∈-1,0;wx=z~x,x∈0,1;zxxx=w~x,x∈0,1;w1-βη=η~;with the known conditions(12)y-1=0,zx1=-η,yx0=zx0,y0=w0.By (11) and (12), a straightforward computation yields(13)yx=∫-1x∫0sy~rdrds-x+1∫01w~rdr-z~1-η~βx+1,x∈-1,0;zx=∫0x∫1sw~rdrds-z~1-η~βx,x∈0,1;wx=z~xx∈0,1;η=z~1-η~β.From the expressions of y(x), z(x), and w(x) in (13), we see that u=(y,z,w,η)∈D(A) and Au=u~. This means that there exists a nonzero element u∈D(A) such that Au=u~. Moreover, it is easy to see that ker(A)=0. Therefore, by the inverse operator theorem, 0∈ρ(A). According to the Sobolev embedding theorem, we have that D(A) is a compact subspace of H, which implies that A-1 is compact on H. Hence, A is a discrete operator in H. Therefore, the spectrum of A consists only of isolated eigenvalues of finite multiplicity (see [27, 28]). The proof of the lemma is complete.Furthermore, we have the following spectral distribution result.Lemma 3.
There is no eigenvalue ofA on the imaginary axis.Proof.
We use a contradiction argument and assume that the conclusion of the lemma is false. That is to say there exist0≠b∈R and 0≠u=(y,z,w,η)∈D(A) such that Au=ibu. It is easy to see that (14)ReAu,u=Reibu,u=0,which together with (7) implies that yx(x)=0 and η=0. On the other hand, Au=ibu is equivalent to(15)yxxx=ibyx,wx=ibzx,zxxx=-bzxwith the known conditions(16)y-1=yx0=zx0=zx1=w1=z1=0,w0=y0.Obviously, we have (17)zx=ceibx+de-ibxand (18)ceib+de-ib=0,ceib-de-ib=0.This means that c=d=0 and z(x)=0. By the same argument, we have y(x)=0 and w(x)=0. Thus we show that u=0. This contradicts u≠0. Therefore, the statement of the lemma holds.Theorem1 and Lemmas 2 and 3 imply that system (1) is asymptotically stable. Certainly, one can also obtain the same result by using LaSalle’s invariance principle (see [17] or [30]).


## 3. Spectral Expansion of the System OperatorA

Firstly, we reduce the eigenvalue problem ofA to the problem of finding nonzero root of a transcendental function. For convenience, we introduce the following functions:(19)rλ=λ+β+1λ+β-1e2λ,sλ=λ+β-1λ+β+1e-2λ.Lemma 4.
The set consisting of the eigenvalues of the operatorA is given by (20)σpA=λ∈C∣fλ=0where(21)fλ=λ+β+1e2λλe2λ+1+e2λ-1+λ+β-1λe2λ+1-e2λ-1.Moreover, for any λ∈σp(A), the corresponding eigenvector is(22)ay,z,w,η,0≠a∈C,in which(23)y=yx,λ=λ1+sλ1-e2λe-λx-eλx+2,x∈-1,0,z=zx,λ=sλeλx+e-λx,x∈0,1,w=wx,λ=λsλeλx+e-λx,x∈0,1,η=ηλ=-λsλeλ-e-λ.Proof.
Letλ∈C be an eigenvalue of A and y,z,w,η∈DA∖0 the corresponding eigenvector; then it is easy to check that(24)yxxx=λyxx∈-1,0,zxxx=λ2zx,x∈0,1,wx=λzx,x∈0,1,w1-βη=λη,y-1=0,zx1=-η,y0=w0,(25)zx0=yx0.By the first two equations in (24), we conclude that there exist four constants a, b, c, and d such that(26)yx=aeλx+be-λx,x∈-1,0,zx=ceλx+de-λx,x∈0,1.Using y(-1)=0, we have a=-be2λ and(27)yx=be-λx-eλx+2.
Using other known conditions in (24) and (26)-(27), we get(28)λc+λd-1-e2λb=0,λc-λd+1+e2λb=0,λ+β+1eλc-λ+β-1e-λd=0.System (28) is seemed as the equations on the unknown parameters b, c, and d. It is easy to see that (y,z,w,η) is nonzero vector if and only if the coefficient determinant of (28) is zero, i.e.,(29)△λ≔λλe2λ-1λ-λ1+e2λλ+β+1eλ-λ+β-1e-λ0=0.However, it is easy to see that λ∈C is a root of △(λ) if and only if it is that of f(λ), which is defined in (21). Thus the first statement of the lemma holds. In the moreover part, we observe that c and d cannot be zero simultaneously. Without loss of generality, we may choose c=1. By (28) and some simple calculations, we obtain b=λ(1+r(λ))/1-e2λ and d=r(λ). Substituting them into the corresponding equations of (24) and (26), we get (23). The proof of the lemma is complete.We now mainly focus on analyzing the asymptotic behavior of “large" rootsλ of f(λ) since this is essential to both the Riesz bases property of generalized eigenvectors and polynomial stability of system (1) (see [1, 5, 14, 15]). When λ is sufficiently large, the leading term of f(λ) is(30)Lλ=λλ+β+1e2λ+λ+β-11+e2λ.It is easy to see that L(λ) has two classes of nonzero roots, i.e., the roots of “1+e2λ” and “(λ+β+1)e2λ+(λ+β-1)”, respectively.Obviously, “1+e2λ=0” has the roots(31)λl0=-l-122π2,l=1,2,⋯Note that {λl0:l=1,2,⋯} are typically the eigenvalues of the classical heat equation with mixed Neumann-Dirichlet boundary conditions (see Ex.2.6.10 in [29] ). In the meanwhile, the roots of “(λ+β+1)e2λ+(λ+β-1)=0” are just the eigenvalues of the wave equation with dynamical boundary control (see pp.360 of [2]). According to lemma 3.3 of [2] we know the roots of “(λ+β+1)e2λ+(λ+β-1)” have the asymptotic expansions:(32)λn=i12+nπ+O1n,asn→∞.If we let(33)λk1=i12+kπ,k∈Z,then we have the following asymptotic expansions of the roots of f(λ).Lemma 5.
There existl1∈N and k1∈N such that f(λ) has two sequences of roots, {λlp}l=l1∞ and {λkh}|k|=k1∞, which satisfy(34)λlp∈Bl-1λl0,l≥l1and(35)λkh∈Bk-1/2λk1,k≥k1.where Br(z) denotes the closed disk in C centered at z and with radius r.Proof.
The first case is when “Reλ→-∞”. In this case, we shall show that f(λ) has a sequence of roots {λlp} satisfying (34). To this end, we set(36)hλ=1+e2λ+rλ-1e2λ-1λrλ+1.Obviously, λ is a root of f(λ) iff it is a root of h(λ) since the roots of (λ+β+1)e2λ+(λ+β-1) satisfy (32). For any fixed l∈N, let μ=λ-(l-1/2)πi; then we have (37)hλ=hμ2+2μl-12πi-l-122π2=1-e2μ-rμ2+2μl-1/2πi-l-1/22π2-11+e2μμ+l-1/2πirμ2+2μl-1/2πi-l-1/22π2+1Now, let us define a function H(μ)=μ+1/2h(μ2+2μ(l-1/2)πi-(l-1/2)2π2). For any μ∈Bl-1(0), then it is easy to see that (38)Hμ≤μ+121-e2μ+122+Ol-1l-1/2πi+Ol-1=μ∫011-e2μsds+122+Ol-1l-1/2πi+Ol-1By the similar method of Proof of Lem.2.2 in [1], we have(39)μ∫011-e2μsds=Ol-2,which implies that |H(μ)|≤1/(l-1/2)π. It follows from the last inequality that there exists a sufficiently large l1∈N such that H(μ)∈Bl-1(0),∀μ∈Bl-1(0),l≥l1. Therefore, by means of Brouwer fixed point theorem, we conclude that there exists a ul0∈Bl-1(0) such that H(ul0)=ul0. This implies that λlp=(μl0)2+2μl0(l-1/2)πi-(l-1/2)2π2 is a root of h(λ) and (34) holds.
The second case is when “Imλ→∞". In this case, we shall show that f(λ) has a sequence of roots {λkh} satisfying (35). To this end, we set(40)gλ=1+rλ+rλ-1e2λ-1λe2λ+1.Obviously, λ is a root of f(λ) iff it is a root of g(λ) since the roots of e2λ+1 satisfy (31). For any fixed k∈Z, let μ=λ-λk1; in this case, we have (41)gλ=gμ+λk1=1+rμ+λk1-rμ+λk1+1e2μ+λk1-1μ+λk1e2μ+λk1+1.Define a function G(μ)=μ+g(μ+λk1). For any μ∈B|k|-1/2(0), by the definition of the function r(λ) in (19), then we have(42)rμ+λk1=-μ+λk1+β+1μ+λk1+β-1e2μ=-1-22k+1π11-2μ-β+1/2k+1πe2μ=-1-Ok-11+Ok-11+Ok-1/2=-1+Ok-1in which 1/1-z=1+z+O(|z|2), ez=1+z+O(|z|2) when |z|<1 and λk1=(k+1/2)πi are used. Moreover, by proposition 2.1 and its proof of [1], we know(43)μ+λk1=λk1+Ok-1e2μ+λk1-1e2μ+λk1+1=sgn⁡k+Ok-2Now, combining (42) and (43), we get (44)Gμ≤μ+Ok-1+2+Ok-1sgn⁡k+Ok-2λk1+Ok-1≤Ok-1/2+2kπIt follows from the last inequality the there exists a sufficiently large k1∈N such that (45)Gμ∈Bk-1/20,∀μ∈Bk-1/20,k≥k1.Thus, by means of Brouwer fixed point theorem, we conclude that there exists a uk1∈B|k|-1/2(0) such that G(uk1)=uk1. This implies that λkh=μk1+(k+1/2)πi is a root of g(λ) and (35) holds.Furthermore, we are able to obtain more precisely asymptotic expansions onλkh and λlp.Lemma 6.
The following asymptotic expansions hold:(46)λlp=λl0+1λl0+Ol-2,l→∞,(47)λkh=λk1-sgn⁡kλk1+Ok-1,k→∞,Proof.
From (36), we know that λlp satisfies(48)e2λlp=-1-rλlp-1e2λlp-1λlprλlp+1.By proposition 2.2 in [1], we have(49)e2λlp=Ol-2,e2λlp=-1+Ol-1.Moreover, it is easy to see that(50)λlp+β+1λlp+β-1=1+Ol-2,rλlp=Ol-2.Thus, by (48)-(50), we have (51)e2λlp-λl0=1+rλlp-1e2λlp-1λlprλlp+1=1+-1+Ol-2-2+Ol-1λl0+Ol-11+Ol-2=1+2λl0+Ol-2.Taking logarithm in the above identity and noting that ln(1+z)=z+O(z2) when z<1, we conclude that (46) holds for any sufficiently large l.
Similarly, from (40) and (19), we know that λkh satisfies(52)e2λkh=λkh+β-1λkh+β+1-1-rλkh-1e2λkh-1λkhe2λkh+1.By (43), we get(53)λkh=λk1+Ok-1e2λkh-1e2λh1+1=sgnk+Ok-2It is easy to see from proposition 2.2 in [1] that(54)λkh+β-1λkh+β+1=1+Ok-1,rλkh=-1+Ok-1/2.Thus, by (52)-(54), we have (55)e2λkh-λk1=λkh+β-1λkh+β+11+rλkh-1e2λkh-1λkhe2λkh+1=1+Ok-11+sgnk+Ok-2-2+Ok-1/2λk1+Ok-1=1-2sgnkλk1+Ok-1.Taking logarithm in the above identity, and noting that ln(1-z)=-z+O(z2) when |z|<1, we conclude that (47) holds for any sufficiently large |k|.Theorem 7.
System (1) or (6) is not exponentially stable.Proof.
By (46) and (31), we get the asymptotic expansions on λlp:(56)λlp=-l-122π2+2+Ol-1.However, by (33) and some simple computations, we know(57)1λk1=11+2kπ1-i,ifk≥0;-11+2kπ1+i,ifk<-1.Thus, inserting 1/λk1 into (47), we obtain the asymptotic estimates on λkh:(58)λkh=-11+2kπ+12+kπi+sgnk1+2kπi+Ok-1.This shows that the spectral bound is zero. The growth bound of the semigroup corresponding to (6) is also zero because the growth bound of the semigroup is greater than or equal to the spectral bound of its generator. However, the exponential stability of the semigroup is equivalent to the fact that its growth bound is negative [29, 30]. Therefore, system (1) or (6) is not exponentially stable.


## 4. Conclusion

In this note, the spectrum and the asymptotic stability of the heat-wave system with dynamical boundary control are investigated through the theory of operator semigroup. The solution to the system is shown to be uniquely existed. The energy of the solution to the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability, which implies that the energy of the system approaches zero as time tends to infinity, of the semigroup is derived in light of the spectral configuration. However, the spectral expansions of the system operator are investigated and the energy of the system does not exponentially decay to zero and is shown in view of the spectral expansions.



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