Research Article | Open Access
Xue-Lian Jin, Yan Li, Fu Zheng, "Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control", Mathematical Problems in Engineering, vol. 2019, Article ID 5716729, 9 pages, 2019. https://doi.org/10.1155/2019/5716729
Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1- 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.
This paper is devoted to analyzing 1- heat-wave coupled system with a dynamical boundary feedback (see [1–3]). More precisely, we study the following coupled system:where is a positive constant, denotes the dynamical control, and - is the negative proportional feedback. This system consists of a wave equation, arising on the interval with state and a heat equation that holds on the interval with state . The wave and heat components are coupled through an interface, the point , with transmission conditions imposing the continuity of and .
The wave component in (1) is similar to that of  and the coupled heat-wave model without dynamical control comes from . However, the coupled heat-wave model described by (1) has a slight difference with that of . The zero Dirichlet boundary at the end point of the wave component was considered in . 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 , with natural transmission conditions at a common interface in . 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 .
In 2013, using the same approach, Avalos and Triggiani gave the rational decay rate for a PDE heat-structure interaction in . 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 . 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 . 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 . 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  is ; then the stabilization is asymptotic but not exponential since the transfer function is strictly positive real. Furthermore, under some conditions, the exponential stability of the wave equation with similar dynamic boundary controller was obtained in . 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 . 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 .
The main contributions of this note are as follows. When the heat equation is not present, the system (with boundary feedback) was discussed in , 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  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 Section 2, 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 be the Sobolev spaces of positive integer order on the open interval . The energy space of (1) is the Hilbert space with the norm
Introduce the operator in as follows:
Setting , then we formally transform system (1) into an evolution equation on
Now we give one of the main results of this section.
Theorem 1. The operator generates a contracted -semigroup on .
Proof. It is easy to see that the operator is densely defined and closed. It follows from a straightforward computation thatThis means that the operator is dissipative.
Moreover, for any and , we have It follows from the above identities that the adjoint of the operator is given as Similarly, we also have This shows that the operator is dissipative as well. It follows form Pro.3.1.11 of  that the operator is further m-dissipative. Thus the Lumer-Phillips theorem (Theorem 3.8.4. of ) 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 .
Lemma 2. Let and be defined as above; then we have . Furthermore, is compact on . Therefore, the spectrum of A consists only of the isolated eigenvalues of finite multiplicity, i.e., .
Proof. For any fixed , we consider the solvability of the resolvent equation , which is equivalent towith the known conditionsBy (11) and (12), a straightforward computation yieldsFrom the expressions of , , and in (13), we see that and . This means that there exists a nonzero element such that . Moreover, it is easy to see that . Therefore, by the inverse operator theorem, . According to the Sobolev embedding theorem, we have that is a compact subspace of , which implies that is compact on . Hence, is a discrete operator in . Therefore, the spectrum of 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 of 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 exist and such that . It is easy to see that which together with (7) implies that and . On the other hand, is equivalent towith the known conditionsObviously, we have and This means that and . By the same argument, we have and . Thus we show that . This contradicts . Therefore, the statement of the lemma holds.
3. Spectral Expansion of the System Operator
Firstly, we reduce the eigenvalue problem of to the problem of finding nonzero root of a transcendental function. For convenience, we introduce the following functions:
Lemma 4. The set consisting of the eigenvalues of the operator is given by whereMoreover, for any , the corresponding eigenvector isin which
Proof. Let be an eigenvalue of and the corresponding eigenvector; then it is easy to check thatBy the first two equations in (24), we conclude that there exist four constants , , , and such thatUsing , we have andUsing other known conditions in (24) and (26)-(27), we getSystem (28) is seemed as the equations on the unknown parameters , , and . It is easy to see that is nonzero vector if and only if the coefficient determinant of (28) is zero, i.e.,However, it is easy to see that is a root of if and only if it is that of , which is defined in (21). Thus the first statement of the lemma holds. In the moreover part, we observe that and cannot be zero simultaneously. Without loss of generality, we may choose . By (28) and some simple calculations, we obtain and . 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" roots of 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 is sufficiently large, the leading term of isIt is easy to see that has two classes of nonzero roots, i.e., the roots of “” and “”, respectively.
Obviously, “” has the rootsNote that are typically the eigenvalues of the classical heat equation with mixed Neumann-Dirichlet boundary conditions (see Ex.2.6.10 in  ). In the meanwhile, the roots of “” are just the eigenvalues of the wave equation with dynamical boundary control (see pp.360 of ). According to lemma 3.3 of  we know the roots of “” have the asymptotic expansions:If we letthen we have the following asymptotic expansions of the roots of .
Lemma 5. There exist and such that has two sequences of roots, and , which satisfyandwhere denotes the closed disk in centered at and with radius .
Proof. The first case is when “”. In this case, we shall show that has a sequence of roots satisfying (34). To this end, we setObviously, is a root of iff it is a root of since the roots of satisfy (32). For any fixed , let ; then we have Now, let us define a function For any , then it is easy to see that By the similar method of Proof of Lem.2.2 in , we havewhich implies that It follows from the last inequality that there exists a sufficiently large such that Therefore, by means of Brouwer fixed point theorem, we conclude that there exists a such that . This implies that is a root of and (34) holds.
The second case is when “". In this case, we shall show that has a sequence of roots satisfying (35). To this end, we setObviously, is a root of iff it is a root of since the roots of satisfy (31). For any fixed , let ; in this case, we have Define a function For any , by the definition of the function in (19), then we havein which , when and are used. Moreover, by proposition 2.1 and its proof of , we knowNow, combining (42) and (43), we get It follows from the last inequality the there exists a sufficiently large such that Thus, by means of Brouwer fixed point theorem, we conclude that there exists a such that . This implies that is a root of and (35) holds.
Furthermore, we are able to obtain more precisely asymptotic expansions on and .
Lemma 6. The following asymptotic expansions hold:
Proof. From (36), we know that satisfiesBy proposition 2.2 in , we haveMoreover, it is easy to see thatThus, by (48)-(50), we have Taking logarithm in the above identity and noting that when , we conclude that (46) holds for any sufficiently large .
Similarly, from (40) and (19), we know that satisfiesBy (43), we getIt is easy to see from proposition 2.2 in  thatThus, by (52)-(54), we have Taking logarithm in the above identity, and noting that when , we conclude that (47) holds for any sufficiently large .
Proof. By (46) and (31), we get the asymptotic expansions on :However, by (33) and some simple computations, we knowThus, inserting into (47), we obtain the asymptotic estimates on :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.
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.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work is supported by the National Natural Science Foundation of China under Grants 11871117 and by the Natural Science Foundation of Liao Ning province under Grants 201602190, respectively.
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