Well-posedness and regularity for non-uniform Schrödinger and Euler-Bernoulli equations with boundary control and observation

Authors:
Bao-Zhu Guo and Zhi-Chao Shao

Journal:
Quart. Appl. Math. **70** (2012), 111-132

MSC (2000):
Primary 35L35, 93C20, 93D15

Published electronically:
August 26, 2011

MathSciNet review:
2920619

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The open-loop systems of a Schrödinger equation and an Euler-Bernoulli equation with variable coefficients and boundary controls and collocated observations are considered. It is shown, with the help of a multiplier method on a Riemannian manifold, that both systems are well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The feed-through operators are found to be zero. The results imply particularly that the exact controlability of each open-loop system is equivalent to the exponential stability of the associated closed-loop system under the output proportional feedback.

**1.**Kais Ammari,*Dirichlet boundary stabilization of the wave equation*, Asymptot. Anal.**30**(2002), no. 2, 117–130. MR**1919338****2.**Kais Ammari and Marius Tucsnak,*Stabilization of second order evolution equations by a class of unbounded feedbacks*, ESAIM Control Optim. Calc. Var.**6**(2001), 361–386 (electronic). MR**1836048**, 10.1051/cocv:2001114**3.**Shugen Chai and Bao-Zhu Guo,*Feedthrough operator for linear elasticity system with boundary control and observation*, SIAM J. Control Optim.**48**(2010), no. 6, 3708–3734. MR**2606833**, 10.1137/080729335**4.**Shugen Chai and Bao-Zhu Guo,*Well-posedness and regularity of Naghdi’s shell equation under boundary control and observation*, J. Differential Equations**249**(2010), no. 12, 3174–3214. MR**2737426**, 10.1016/j.jde.2010.09.012**5.**C. I. Byrnes, D. S. Gilliam, V. I. Shubov, and G. Weiss,*Regular linear systems governed by a boundary controlled heat equation*, J. Dynam. Control Systems**8**(2002), no. 3, 341–370. MR**1914447**, 10.1023/A:1016330420910**6.**Ruth F. Curtain,*The Salamon-Weiss class of well-posed infinite-dimensional linear systems: a survey*, IMA J. Math. Control Inform.**14**(1997), no. 2, 207–223. Distributed parameter systems: analysis, synthesis and applications, Part 2. MR**1470034**, 10.1093/imamci/14.2.207**7.**Ruth F. Curtain,*Linear operator inequalities for strongly stable weakly regular linear systems*, Math. Control Signals Systems**14**(2001), no. 4, 299–337. MR**1868533**, 10.1007/s498-001-8039-4**8.**Ruth F. Curtain and George Weiss,*Well posedness of triples of operators (in the sense of linear systems theory)*, Control and estimation of distributed parameter systems (Vorau, 1988), Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 41–59. MR**1033051****9.**P. Grisvard,*Caractérisation de quelques espaces d’interpolation*, Arch. Rational Mech. Anal.**25**(1967), 40–63 (French). MR**0213864****10.**Bao-Zhu Guo and Yue-Hu Luo,*Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator*, Systems Control Lett.**46**(2002), no. 1, 45–65. MR**2011071**, 10.1016/S0167-6911(01)00201-8**11.**Bao-Zhu Guo and Zhi-Chao Shao,*Regularity of a Schrödinger equation with Dirichlet control and colocated observation*, Systems Control Lett.**54**(2005), no. 11, 1135–1142. MR**2170295**, 10.1016/j.sysconle.2005.04.008**12.**Bao-Zhu Guo and Zhi-Chao Shao,*Regularity of an Euler-Bernoulli equation with Neumann control and collocated observation*, J. Dyn. Control Syst.**12**(2006), no. 3, 405–418. MR**2233027**, 10.1007/s10450-006-0006-x**13.**Bao-Zhu Guo and Zhi-Chao Shao,*On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients*, Quart. Appl. Math.**65**(2007), no. 4, 705–736. MR**2370357**, 10.1090/S0033-569X-07-01069-9**14.**Bao-Zhu Guo and Xu Zhang,*The regularity of the wave equation with partial Dirichlet control and colocated observation*, SIAM J. Control Optim.**44**(2005), no. 5, 1598–1613. MR**2193497**, 10.1137/040610702**15.**Bao-Zhu Guo and Zhi-Xiong Zhang,*On the well-posedness and regularity of the wave equation with variable coefficients*, ESAIM Control Optim. Calc. Var.**13**(2007), no. 4, 776–792. MR**2351403**, 10.1051/cocv:2007040**16.**Bao-Zhu Guo and Zhi-Xiong Zhang,*Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation*, Math. Control Signals Systems**19**(2007), no. 4, 337–360. MR**2354054**, 10.1007/s00498-007-0017-5**17.**Bao-Zhu Guo and Zhi-Xiong Zhang,*Well-posedness of systems of linear elasticity with Dirichlet boundary control and observation*, SIAM J. Control Optim.**48**(2009), no. 4, 2139–2167. MR**2520323**, 10.1137/070705593**18.**V. Komornik,*Exact controllability and stabilization*, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR**1359765****19.**I. Lasiecka,*Exponential decay rates for the solutions of Euler-Bernoulli equations with boundary dissipation occurring in the moments only*, J. Differential Equations**95**(1992), no. 1, 169–182. MR**1142282**, 10.1016/0022-0396(92)90048-R**20.**J. Lagnese and J.-L. Lions,*Modelling analysis and control of thin plates*, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 6, Masson, Paris, 1988. MR**953313****21.**J.-L. Lions and E. Magenes,*Non-homogeneous boundary value problems and applications. Vol. I*, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR**0350177****22.**I. Lasiecka and R. Triggiani,*𝐿₂(Σ)-regularity of the boundary to boundary operator 𝐵*𝐿 for hyperbolic and Petrowski PDEs*, Abstr. Appl. Anal.**19**(2003), 1061–1139. MR**2041290**, 10.1155/S1085337503305032**23.**I. Lasiecka and R. Triggiani,*The operator 𝐵*𝐿 for the wave equation with Dirichlet control*, Abstr. Appl. Anal.**7**(2004), 625–634. MR**2084941**, 10.1155/S1085337504404011**24.**S. E. Rebiai,*Uniform energy decay of Schrödinger equations with variable coefficients*, IMA J. Math. Control Inform.**20**(2003), no. 3, 335–345. MR**2009070**, 10.1093/imamci/20.3.335**25.**Dietmar Salamon,*Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach*, Trans. Amer. Math. Soc.**300**(1987), no. 2, 383–431. MR**876460**, 10.1090/S0002-9947-1987-0876460-7**26.**Dietmar Salamon,*Realization theory in Hilbert space*, Math. Systems Theory**21**(1989), no. 3, 147–164. MR**977021**, 10.1007/BF02088011**27.**Olof J. Staffans,*Passive and conservative continuous-time impedance and scattering systems. I. Well-posed systems*, Math. Control Signals Systems**15**(2002), no. 4, 291–315. MR**1942089**, 10.1007/s004980200012**28.**Michael E. Taylor,*Partial differential equations. I*, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR**1395148****29.**George Weiss,*Admissible observation operators for linear semigroups*, Israel J. Math.**65**(1989), no. 1, 17–43. MR**994732**, 10.1007/BF02788172**30.**George Weiss,*Admissibility of unbounded control operators*, SIAM J. Control Optim.**27**(1989), no. 3, 527–545. MR**993285**, 10.1137/0327028**31.**George Weiss,*The representation of regular linear systems on Hilbert spaces*, Control and estimation of distributed parameter systems (Vorau, 1988), Internat. Ser. Numer. Math., vol. 91, Birkhäuser, Basel, 1989, pp. 401–416. MR**1033074**, 10.1016/0039-6028(89)90131-3**32.**George Weiss,*Transfer functions of regular linear systems. I. Characterizations of regularity*, Trans. Amer. Math. Soc.**342**(1994), no. 2, 827–854. MR**1179402**, 10.1090/S0002-9947-1994-1179402-6**33.**Peng-Fei Yao,*On the observability inequalities for exact controllability of wave equations with variable coefficients*, SIAM J. Control Optim.**37**(1999), no. 5, 1568–1599 (electronic). MR**1710233**, 10.1137/S0363012997331482**34.**Peng-Fei Yao,*Observability inequalities for the Euler-Bernoulli plate with variable coefficients*, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999) Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000, pp. 383–406. MR**1804802**, 10.1090/conm/268/04320

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Additional Information

**Bao-Zhu Guo**

Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People’s Republic of China, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, People’s Republic of China, and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

Email:
bzguo@iss.ac.cn

**Zhi-Chao Shao**

Affiliation:
School of Information Technology and Management and China Service Industry Research Center, University of International Business and Economics, Beijing 100029, People’s Republic of China

Email:
zcshao@amss.ac.cn

DOI:
https://doi.org/10.1090/S0033-569X-2011-01243-0

Keywords:
Schrödinger equation,
Euler-Bernoulli equation,
well-posed and regular system,
variable coefficients,
boundary control and observation.

Received by editor(s):
August 9, 2010

Published electronically:
August 26, 2011

Additional Notes:
This work was carried out with the support of the National Natural Science Foundation of China, the National Research Foundation of South Africa, and the National Basic Research Program of China (973 Program: 2011CB808002).

Zhi-Chao Shao also acknowledges the support of the Scientific Research Foundation for the Returned Overseas Scholar by Education Ministry of China and the program for Innovative Research Team in UIBE

Article copyright:
© Copyright 2011
Brown University