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Polynomial energy decay of a wave–Schrödinger transmission system
Boundary Value Problems volume 2018, Article number: 60 (2018)
Abstract
We study in this paper a wave–Schrödinger transmission system for its stability. By analyzing carefully Green’s functions for the infinitesimal generator of the semigroup associated with the system under consideration, we obtain a useful resolvent estimate on this generator which can be applied to derive the decaying property. Our study is inspired by L. Lu & J.-M. Wang [Appl. Math. Lett., 54:7–14, 2016] whose energy decay result is improved upon in our paper. Our method, different from the one used in the previous reference, can be adapted to study stability problems for other 1-D transmission systems.
1 Introduction
Thanks to its wide applicability, the Schrödinger equation
where \(\varDelta=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x^{2}_{j}}\) is the Laplacian on \(\mathbb{R}^{n}\), has been receiving extensive attention from the mathematical control community; see [2–8] and the references cited therein. Specifically, the systems described by the Schrödinger equation have received extensive studies for their stability in the past three decades. Among the vast references in this direction, Lagnese [9] proved a stability result via “connecting” it to the stability property of the plate equation \(\partial_{t}^{2}u+\varDelta^{2}u+\text{l.o.t}=0\) (while the study of the stability and stabilization of the plate equation has a relatively long history). Machtyngier and Zuazua [4] studied the boundary and internal stabilization problem via the multiplier method (the main idea has originated from stability studies for wave equations). In [7, 10], some collocated boundary stabilization problems were investigated. Zuazua [2] provided a nice survey on the recent studies on the control properties for the Schrödinger equation.
This paper is devoted to the study of the stabilization of the Schrödinger equation via a damped wave equation through a common end point. More precisely, we are concerned in this paper with the system
where \(\mathrm{i}=\sqrt{-1}\) is the imaginary unit, and \(k\in\mathbb {R}\setminus\{0\}\) and \(b\in(0,\infty)\) are fixed arbitrarily. System (1.1) was recently studied by Lu and Wang [1] with the intension to understand better the transmission of dissipation effect from a damped wave equation to a damping-free Schrödinger equation where the energy can be exchanged by (1.1)1.
The natural phase space for system (1.1) is
Let us define an unbounded linear operator A in H by
We can prove as in [1] that A is the infinitesimal generator of a strongly continuous semigroup \(\{e^{tA}\}\) on H. Therefore, (1.1) admits for every triple \((u^{0},u^{1},v^{0})\in H\) a unique solution \((u,v)\in\mathbb{S}^{0}\); if further \((u^{0},u^{1},v^{0})\in\mathcal{D}(A)\), then \((u,v)\in\mathbb{S}^{1}\). Here \(\mathbb{S}^{0}\) and \(\mathbb{S}^{1}\) are defined by
We associate with system (1.1) the following energy functional:
As indicated before, the study of this paper is directly inspired by [1]. And therefore, it is worth recalling the main results in [1] as follows.
Theorem A
(see [1])
Let A be defined as in (1.3), E as in (1.6), and H as in (1.2).
-
A is the infinitesimal generator of a strongly continuous semigroup \(\{e^{tA} \}_{t\in[0,\infty)}\) of contractions on H. In particular, we have: For every triple \((u^{0},v^{0},v^{1})\in H\), the boundary value problem (1.1) admits a unique solution \((u,v)\in\mathbb{S}^{0}\) such that \(u(\cdot,0)=u^{0}\), \(v(\cdot,0)=v^{0}\) and \(\partial_{t}v(\cdot,0)=v^{1}\); if, in addition, \((u^{0},v^{0},v^{1})\in\mathcal{D}(A)\) (see (1.3)), then \((u,v)\in\mathbb{S}^{1}\).
-
The spectrum \(\sigma(A)\) of A consists merely of eigenvalues of A, and is distributed as follows:
$$ \left . \textstyle\begin{array}{l} \lambda_{1j}= - \frac{b}{2} \pm\frac{\sqrt{b^{2}-4j^{2}\pi^{2}}}{2} +\mathcal{O}(j^{-1}), \\ \lambda_{2j}= - \vert j-\frac{1}{2} \vert ^{2} \pi^{2}\mathrm{i}+\mathcal{O}(j^{-1} ),\quad\operatorname{\Re}\mathfrak{e} \lambda_{2j}< 0 \end{array}\displaystyle \right \},\quad\textit{as }j\nearrow\infty. $$(1.7) -
\(E(t)\searrow0\) as \(t\nearrow\infty\).
Note especially that Lu and Wang [1] proved that \(E(t)\) decreases to 0 as \(t\rightarrow+\infty\). But due to the fact that \(\lim_{j\rightarrow\infty}\operatorname{\Re} \mathfrak{e} \lambda_{2j} =0\), \(E(t)\) cannot decay uniformly (see the last section of the paper for a brief proof of this statement). Recently, the non-uniform decay properties have been investigated extensively in the literature for PDEs; see [11, 12]. Our main result gives a more accurate decay rate for the energy \(E(t)\).
Theorem 1.1
Let E, defined as in (1.6), be the energy associated with system (1.1). There exists \(M\in(0,\infty)\) such that, for every solution \((u,v)\in\mathbb{S}^{1}\) with \(u(\cdot,0)=u^{0}\), \(v(\cdot,0)=v^{0}\), and \(\partial_{t}v(\cdot,0)=v^{1}\),
By [13, Theorem 2.4], this theorem follows immediately from the following theorem.
Theorem 1.2
Let A be defined by (1.3). There exists \(C\in(0,\infty)\) such thatFootnote 1
Throughout this paper, C is a generic constant which can assume a different value at each occurrence.
The rest of the paper is organized as follows. With the aid of the idea of Green’s functions, we provide in Sect. 2 an explicit formulae for the resolvent \(R(\mathrm{i}\gamma;A)\). The main results of this paper are proved in Sect. 3. Some concluding remarks are included in Sect. 4.
2 Green’s functions and the resolvent \(R(\mathrm{i}\gamma;A)\)
We would like to calculate in this section the resolvent \(R(\mathrm{i}\gamma;A)\) with \(\gamma\in\mathbb{R}\) by using the idea of Green’s functions. Let \((\phi,\psi,\eta)\in H\). Consider the equation \((\lambda\mathrm{id}_{H}-A)(f,g,h)= (\phi,\psi,\eta)\) with \(\mathrm{id}_{H}\) denoting the identity operator on H, or equivalently, the boundary value problem (BVP)
Denote by \(F^{j}\) and \(G^{j}\), \(j=1,2,3\), the Green’s functions for BVP (2.1). By using the idea of Green’s functions, every solution \((f,g,h)\) to BVP (2.1) can be expressed by
The Green’s functions for BVP (2.1) should assume the form
where \(\mathfrak{h}\) is the Heaviside function, namely
and the coefficients \(\sigma_{jk}\), \(\varsigma_{jk}\) (\(j=1,2,3\), \(k=1,2\)), \(\breve{\sigma}_{11}\), \(\breve{\sigma}_{12}\), \(\breve{\varsigma}_{jk}\) (\(j=2,3\), \(k=1,2\)) are yet to be determined later (see (2.4), (2.5), (2.6), and (2.7)). The Green’s functions should also satisfy
This, together with the notion of Green’s functions, implies
and
By Cramer’s rule, we can deduce from (2.4) that
We deduce \(\sigma_{11}\) from (2.5) by Cramer’s rule that
where Δ is given by
Similarly, we can deduce from (2.5) that \(\sigma_{12}\), \(\varsigma_{11}\), \(\varsigma_{12}\) can be expressed as follows:
and
We can deduce from (2.6) that \(\sigma_{21}\), \(\sigma_{22}\), \(\varsigma_{21}\), \(\varsigma_{22}\) can be expressed as follows:
We can deduce from (2.7) that \(\sigma_{31}\), \(\sigma_{32}\), \(\varsigma_{31}\), \(\varsigma_{32}\) can be expressed as follows:
Let us remind that Δ in the above formulas is given explicitly by (2.13).
3 Proof of the main results
We seek to obtain in this section the lower bound for \(\vert \varDelta \vert \) (see (2.13) for the definition of Δ). As mentioned in Sect. 2, we need merely consider the situation \(\lambda\in\mathrm{i}\mathbb{R}\). For the sake of clarity, we distinguish λ into two cases.
Case 1
(\(\lambda\in\mathbb{C}\setminus\{0\}\) and \(\lambda= \vert \lambda \vert \mathrm{i}\))
In this case, \(\sqrt{\lambda(\lambda+b)}=\mathfrak{p}( \vert \lambda \vert )+ \mathrm{i}\mathfrak{q}( \vert \lambda \vert )\), where
Obviously, we have
Mainly using the triangle inequality, we can deduce from (2.13) that
But
where the “⩾” in the second line follows if and only if \(\vert \lambda \vert \geqslant\frac{b}{\sqrt{48}}\), and \(\mathfrak{p}(\cdot)\) is given by (3.1). And similarly, we have
and
Therefore,
Case 2
(\(\lambda\in\mathbb{C}\setminus\{0\}\) and \(\lambda=- \vert \lambda \vert \mathrm{i}\))
In this case, \(\sqrt{\lambda(\lambda+b)} =\mathfrak{p}( \vert \lambda \vert )- \mathrm{i}\mathfrak{q}( \vert \lambda \vert )\), where \(\mathfrak{p}\) and \(\mathfrak{q}\) are given by (3.1). Substitute this into (2.13) to obtain
Here α and β are given explicitly as
and satisfy
where the “=” in the first line follows from a series of elementary calculations and rearrangements, the “⩾” in the third line follows from (3.2) and
By (3.10), we deduce from (3.8) that
Having the above analysis results at our disposal, we are now in a position to prove the main results.
Proof of Theorem 1.2
It is equivalent to proving
where \((f,g,h)\) and \((\phi,\psi,\eta)\) are related by (2.2).
Let us consider first the term \(\int_{0}^{1}G^{2}(x,\xi)\psi(\xi)\,d\xi\). Combine (2.2), (2.3), (2.19), and (2.20), to obtain
The derivative \(\widehat{ \psi}'\) of ψ̂ reads
Since \(g\in H^{1}(0,1)\) satisfies \(g(1)=0\) in the trace sense, it suffices to estimate \(\Vert g' \Vert _{L^{2}(0,1)}\) instead of \(\Vert g \Vert _{H^{1}(0,1)}\). Therefore, we only need to analyze \(\| \widehat{ \psi}'\|_{L^{2}(0,1)}\).
By a density argument, we can prove
By Young’s inequality (see [14, Theorem 2.24, p. 33]), we have
where the “⩽” in the second line follows from
in which we used (3.2) when we establish the last “<”.
Mainly using Hölder’s inequality, we have
By some routine calculations, we have
(3.14), together with (3.15), (3.16), (3.17), and (3.18), implies
whenever \(\lambda\in\mathrm{i}\mathbb{R}\) satisfies \(\vert \lambda \vert \geqslant\max ( \frac{16e^{2b}k^{4}}{b^{2}},\frac{1}{k^{4}},b, \frac{288(b+2)^{2}}{k^{4}b^{2}},\frac{6(2+b)}{b} )\).
Applying the approach used in deducing (3.19) from (3.14) via the “steps” (3.15), (3.16), (3.17), and (3.18), we can prove
which, together with (2.2)3, implies
We can also prove
where the constant \(C>0\) is independent of \((\phi,\psi,\eta)\) and λ.
Now it remains to analyze the term \(\int_{0}^{1}F^{1}(x,\xi)\phi(\xi)\,d\xi\). But
To provide in detail a way to analyze \(\int_{0}^{1}F^{1}(x,\xi)\phi(\xi)\,d\xi\), we continue as follows:
Employing the same idea, we analyze the rest of (3.23) term-by-term, and then collect all the information together to obtain
This, together with (3.22), implies
where the constant \(C>0\) is independent of \((\phi,\psi,\eta)\) and λ.
Combining (3.20), (3.21), and (3.24), we know that (3.12) is proved, so is Theorem 1.2. □
4 Concluding comments and an open question
By analyzing carefully Green’s functions for boundary value problems associated with ordinary differential equations (i.e., (2.1)), we prove that the infinitesimal generator of the semigroup associated with system (1.1) satisfies the resolvent estimate (1.9), thereby proving that the energy of system (1.1) decays polynomially.
Having a very simple underlying idea, our method is based on Green’s functions and relies on heavy calculations. Our method can be modified to treat other transmission systems of 1-D partial differential equations where one of the equations is damped in the whole interval. However, according to the deductions based on our idea, it seems very hard to find the optimal decay rate of the energy of system (1.1). Therefore, one of our next concerns is to understand better the following question.
Open question
Could the decay rate \((t+1)^{-1}\) given in estimate (1.8) be improved?
As indicated before, the above question seems difficult to solve with merely the method used in this paper. To close this section, we prove by a contradiction argument that the energy E (defined in (1.6)) can NOT decay exponentially. Assume to the contrary that \(E(t)\) decays exponentially, or equivalently, there exists a pair \((M_{0},\gamma_{0})\in(0,\infty)^{2}\) such that, for every \(w\in H\),
where H is given by (1.2), and A by (1.3).
Write, for every \(\lambda_{0}\in\mathbb{C}\) with \(\gamma<\operatorname{\Re} \mathfrak{e} \lambda<0\),
By (4.1), \(R_{\lambda}\) is well defined and belongs to \(\mathscr{L}(H)\). Moreover, by (4.2), \(R_{\lambda}\) satisfies
Therefore, λ belongs to \(\rho(A)\), the resolvent set of A, and moreover, \(R_{\lambda}=R(\lambda;A)\), the resolvent of A.
Thus, we proved just now that λ belongs to \(\rho(A)\) whenever \(\lambda\in\mathbb{C}\) satisfies \(\gamma_{0}<\operatorname{\Re}\mathfrak{e} \lambda<0\). This contradicts (1.7)2. The proof is complete.
Notes
Throughout this paper, \(\langle{\cdot}\rangle=\sqrt{1+|{\cdot}|^{2}}\).
References
Lu, L., Wang, J.-M.: Transmission problem of Schrödinger and wave equation with viscous damping. Appl. Math. Lett. 54, 7–14 (2016)
Zuazua, E.: Remarks on the controllability of the Schrödinger equation. In: Quantum Control: Mathematical and Numerical Challenges. CRM Proc. Lecture Notes, vol. 33, pp. 193–211. Am. Math. Soc., Providence (2003)
Machtyngier, E.: Exact controllability for the Schrödinger equation. SIAM J. Control Optim. 32, 24–34 (1994)
Machtyngier, E., Zuazua, E.: Stabilization of the Schrödinger equation. Port. Math. 51, 243–256 (1994)
Phung, K.-D.: Observability and control of Schrödinger equation. SIAM J. Control Optim. 40, 211–230 (2001)
Lebeau, G.: Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992)
Guo, B.Z., Shao, Z.C.: Regularity of a Schrödinger equation with Dirichlet control and collocated observation. Syst. Control Lett. 54, 1135–1142 (2005)
Dautray, R., Lions, J.L.: Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, vol. 1. Masson, Paris (1984)
Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM Studies in Appl. Math., vol. 10. SIAM, Philadelphia (1989)
Krstic, M., Guo, B.Z., Smyshlyaev, A.: Boundary controllers and observers for the linearized Schrödinger equation. SIAM J. Control Optim. 49, 1479–1497 (2011)
Liu, Z., Rao, B.: Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56, 630–644 (2005)
Burq, N.: Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180(1), 1–29 (1998)
Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347(2), 455–478 (2010)
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam (2003)
Acknowledgements
The author is supported by NSFC (#11701050 and #11571244), by JG Program (#2017JG13) of Chengdu Normal University, and by SCJYT Program (#18ZB0098) of Sichuan Province, China.
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Wang, C. Polynomial energy decay of a wave–Schrödinger transmission system. Bound Value Probl 2018, 60 (2018). https://doi.org/10.1186/s13661-018-0978-y
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DOI: https://doi.org/10.1186/s13661-018-0978-y