On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients
Authors:
Bao-Zhu Guo and Zhi-Chao Shao
Journal:
Quart. Appl. Math. 65 (2007), 705-736
MSC (2000):
Primary 35L35, 93C20, 93D15, 93B05, 93B07
DOI:
https://doi.org/10.1090/S0033-569X-07-01069-9
Published electronically:
October 5, 2007
MathSciNet review:
2370357
Full-text PDF Free Access
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Abstract: A system of transmission of Euler-Bernoulli plate equation with variable coefficients under Neumann control and collocated observation is studied. Using the multiplier method on a Riemannian manifold, it is shown that the system is well-posed in the sense of D. Salamon. This establishes the equivalence between the exact controllability of an open-loop system and the exponential stability of a closed-loop system under the proportional output feedback. The regularity of the system in the sense of G. Weiss is also proved, and the feedthrough operator is found to be zero. These properties make this PDE system parallel in many ways to the finite-dimensional ones. Finally, the exact controllability of an open-loop system is developed under a uniqueness assumption by establishing the observability inequality for the dual system.
References
- Kais Ammari, Dirichlet boundary stabilization of the wave equation, Asymptot. Anal. 30 (2002), no. 2, 117–130. MR 1919338
- 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. MR 1836048, DOI https://doi.org/10.1051/cocv%3A2001114
- Mohammed Aassila, Exact boundary controllability of the plate equation, Differential Integral Equations 13 (2000), no. 10-12, 1413–1428. MR 1787074
- 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, DOI https://doi.org/10.1023/A%3A1016330420910
- Shu Gen Chai and Kang Sheng Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A 26 (2005), no. 5, 605–612 (Chinese, with English and Chinese summaries); English transl., Chinese J. Contemp. Math. 26 (2005), no. 4, 337–346 (2006). MR 2186628
- Shugen Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control, Indian J. Pure Appl. Math. 36 (2005), no. 5, 227–249. MR 2179402
- Shu Gen Chai, Boundary feedback stabilization of Naghdi’s model, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 1, 169–184. MR 2128833, DOI https://doi.org/10.1007/s10114-004-0408-1
- 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, DOI https://doi.org/10.1093/imamci/14.2.207
- 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, DOI https://doi.org/10.1007/s498-001-8039-4
- 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
- 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, DOI https://doi.org/10.1016/S0167-6911%2801%2900201-8
- 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, DOI https://doi.org/10.1137/040610702
- 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, DOI https://doi.org/10.1016/j.sysconle.2005.04.008
- 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, DOI https://doi.org/10.1007/s10450-006-0006-x
- B. Z. Guo and Z. X. Zhang, On the well-posedness and regularity of wave equations with variable coefficients and partial boundary Dirichlet control and colocated observation, ESAIM Control Optim. Calc. Var., to appear.
- B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Mathematics of Control, Signals, and Systems, to appear.
- John E. Lagnese, Recent progress in exact boundary controllability and uniform stabilizability of thin beams and plates, Distributed parameter control systems (Minneapolis, MN, 1989) Lecture Notes in Pure and Appl. Math., vol. 128, Dekker, New York, 1991, pp. 61–111. MR 1108855
- John E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems, ESAIM Control Optim. Calc. Var. 2 (1997), 343–357. MR 1487483, DOI https://doi.org/10.1051/cocv%3A1997112
- I. Lasiecka and R. Triggiani, $L_2(\Sigma )$-regularity of the boundary to boundary operator $B^\ast L$ for hyperbolic and Petrowski PDEs, Abstr. Appl. Anal. 19 (2003), 1061–1139. MR 2041290, DOI https://doi.org/10.1155/S1085337503305032
- 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
- Weijiu Liu and Graham H. Williams, Exact controllability for problems of transmission of the plate equation with lower-order terms, Quart. Appl. Math. 58 (2000), no. 1, 37–68. MR 1738557, DOI https://doi.org/10.1090/qam/1738557
- Weijiu Liu and Graham Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc. 57 (1998), no. 2, 305–327. MR 1617324, DOI https://doi.org/10.1017/S0004972700031683
- Higidio Portillo Oquendo, Nonlinear boundary stabilization for a transmission problem in elasticity, Nonlinear Anal. 52 (2003), no. 4, 1331–1345. MR 1941260, DOI https://doi.org/10.1016/S0362-546X%2802%2900169-4
- David L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, Differential games and control theory (Proc. NSF—CBMS Regional Res. Conf., Univ. Rhode Island, Kingston, R.I., 1973) Dekker, New York, 1974, pp. 291–319. Lecture Notes in Pure Appl. Math., Vol. 10. MR 0467472
- David L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), no. 4, 639–739. MR 508380, DOI https://doi.org/10.1137/1020095
- Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI https://doi.org/10.1007/BF01762360
- 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, DOI https://doi.org/10.1007/s004980200012
- Michael E. Taylor, Partial differential equations. I, Applied Mathematical Sciences, vol. 115, Springer-Verlag, New York, 1996. Basic theory. MR 1395148
- George Weiss, Transfer functions of regular linear systems. I. Characterizations of regularity, Trans. Amer. Math. Soc. 342 (1994), no. 2, 827–854. MR 1179402, DOI https://doi.org/10.1090/S0002-9947-1994-1179402-6
- George Weiss and Richard Rebarber, Optimizability and estimatability for infinite-dimensional linear systems, SIAM J. Control Optim. 39 (2000), no. 4, 1204–1232. MR 1814273, DOI https://doi.org/10.1137/S036301299833519X
- George Weiss, Olof J. Staffans, and Marius Tucsnak, Well-posed linear systems—a survey with emphasis on conservative systems, Int. J. Appl. Math. Comput. Sci. 11 (2001), no. 1, 7–33. Mathematical theory of networks and systems (Perpignan, 2000). MR 1835146
- Hong Xi Wu, The Bochner technique in differential geometry. I, Adv. in Math. (Beijing) 10 (1981), no. 1, 57–76 (Chinese). MR 691910
- H. Wu, C. L. Shen and Y. L. Yu, An Introduction to Riemannian Geometry, Beijing University Press, Beijing, 1989 (in Chinese).
- 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, DOI https://doi.org/10.1090/conm/268/04320
- 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. MR 1710233, DOI https://doi.org/10.1137/S0363012997331482
References
- K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptotic Analysis, 30 (2002), 117-130. MR 1919338 (2003f:93072)
- K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361-386. MR 1836048 (2002f:93104)
- M. Aassila, Exact boundary controllability of the plate equation, Differential Integral Equations, 13 (2000), 1413-1428. MR 1787074 (2002e:93012)
- C. I. Byrnes, D. S. Gilliam, V. I. Shubov and G. Weiss, Regular linear systems governed by a boundary controlled heat equation, Journal of Dynamical and Control Systems, 8 (2002), 341-370. MR 1914447 (2003d:93045)
- S. G. Chai and K. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser. A, 26(5) (2005), 605-612 (in Chinese). Translation in Chinese J. Contemp. Math., 26 (2005), no. 4, 337-346. MR 2186628 (2006f:93089)
- S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control, Indian J. Pure Appl. Math., 36 (2005), 227-249. MR 2179402
- S. G. Chai, Boundary feedback stabilization of Naghdi’s model, Acta Math. Sin. (Engl. Ser.), 21(1) (2005), 169-184. MR 2128833 (2005k:93171)
- R. F. Curtain, The Salamon-Weiss class of well-posed infinite dimensional linear systems: A survey, IMA J. of Math. Control and Inform., 14 (1997), 207-223. MR 1470034 (99a:93054)
- R. F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems, Math. Control Signals Systems, 14 (2001), 299-337. MR 1868533 (2002k:93022)
- R. F. Curtain and G. Weiss, Well-posedness of triples of operators (in the sense of linear systems theory), in Control and Estimation of Distributed Parameter Systems (F. Kappel, K. Kunisch and W. Schappacher, Eds.), Vol. 91, Birkhäuser, Basel, 1989, 41-59. MR 1033051 (91d:93027)
- B. Z. Guo and Y. H. Luo, Controllability and stability of a second order hyperbolic system with colocated sensor/actuator, Systems and Control Letters, 46 (2002), 45-65. MR 2011071 (2004i:93015)
- B. Z. Guo and X. Zhang, The regularity of the wave equation with partial Dirichlet control and colocated observation, SIAM J. Control Optim., 44 (2005), 1598-1613. MR 2193497 (2006j:93058)
- B. Z. Guo and Z. C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems and Control Letters, 54 (2005), 1135-1142. MR 2170295 (2006d:35208)
- B. Z. Guo and Z. C. Shao, Regularity of an Euler-Bernoulli plate equation with Neumann control and colocated observation, J. Dyn. Control Syst., 12 (2006), no. 3, 405-418. MR 2233027 (2007b:93076)
- B. Z. Guo and Z. X. Zhang, On the well-posedness and regularity of wave equations with variable coefficients and partial boundary Dirichlet control and colocated observation, ESAIM Control Optim. Calc. Var., to appear.
- B. Z. Guo and Z. X. Zhang, Well-posedness and regularity for an Euler-Bernoulli plate with variable coefficients and boundary control and observation, Mathematics of Control, Signals, and Systems, to appear.
- John E. Lagnese, Recent progress in exact boundary controllability and uniform stabilizability of thin beams and plates, in Distributed Parameter Control Systems, Lecture Notes in Pure and Appl. Math., 128, Dekker, New York, 1991, 61-111. MR 1108855 (92f:93019)
- John E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems, ESAIM Control Optim. Calc. Var., 2 (1991), 343-357. MR 1487483 (98k:35114)
- I. Lasiecka and R. Triggiani, $L^2 (\Sigma )$-regularity of the boundary to boundary operator $B^* L$ for hyperbolic and Petrowski PDEs, Abstr. Appl. Anal., No. 19, 2003, 1061-1139. MR 2041290 (2005i:35164)
- J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin, 1972. MR 0350177 (50:2670)
- W. Liu and G. H. Williams, Exact controllability for problems of transmission of the plate equation with lower order terms, Quart. Appl. Math., 58 (2000), 37-68. MR 1738557 (2001d:93015)
- W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austra. Math. Soc., 57 (1998), 305-327. MR 1617324 (99b:35112)
- H. P. Oquendo, Nonlinear boundary stabilization for a transmission problem in elasticity, Nonlinear Analysis, 52 (2003), 1331-1345. MR 1941260 (2003i:93069)
- D. L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, in Differential Games and Control Theory (Roxin, Liu and Sternberg, Eds.), Lecture Notes in Pure Appl. Math., 10, Marcel Dekker, New York, 1974, 291-319. MR 0467472 (57:7329)
- D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), no. 4, 639-739. MR 508380 (80c:93032)
- J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura. Appl., 146(4) (1987), 65-96. MR 916688 (89c:46055)
- O. J. Staffans, Passive and conservative continuous-time impedance and scattering systems. Part I: well-posed systems, Math. Control, Signals, and Systems, 15 (2002), 291-315. MR 1942089 (2003i:93024)
- M. E. Taylor, Partial Differential Equations I: Basic Theory, Springer-Verlag, New York, 1996. MR 1395148 (98b:35002b)
- G. Weiss, Transfer functions of regular linear systems I: characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854. MR 1179402 (94f:93074)
- G. Weiss and R. Rebarber, Optimizability and estimatability for infinite-dimensional linear systems, SIAM J. Control Optim., 39 (2000), 1204-1232. MR 1814273 (2001m:93021)
- G. Weiss, O. J. Staffans and M. Tucsnak, Well-posed linear systems-a survey with emphasis on conservative systems, Int. J. Appl. Math. Comput. Sci., 11 (2001), 7-33. MR 1835146 (2002f:93068)
- H. Wu, Bochner’s skills in differential geometry (Part I), Advances in Mathematics, Vol. 10, No. 1 (1981), 57-76 (Chinese). MR 691910 (84m:53054)
- H. Wu, C. L. Shen and Y. L. Yu, An Introduction to Riemannian Geometry, Beijing University Press, Beijing, 1989 (in Chinese).
- P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, Contemporary Mathematics, Vol. 268, Amer. Math. Soc., Providence, RI, 2000, 383-406. MR 1804802 (2001m:93025)
- P. F. Yao, On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Contr. and Optim., 37 (1999) 1568-1599. MR 1710233 (2000m:93027)
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Additional Information
Bao-Zhu Guo
Affiliation:
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, P.R. 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 Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Email:
zcshao@cam.wits.ac.za
Keywords:
Euler-Bernoulli plate equation,
well-posedness and regularity,
boundary control and observation,
exact controllability,
exact observability,
multiplier method on Riemannian manifold.
Received by editor(s):
June 15, 2006
Published electronically:
October 5, 2007
Additional Notes:
This work was carried out with the support of the National Natural Science Foundation of China and the National Research Foundation of South Africa. Zhi-Chao Shao acknowledges the support of the Postdoctoral Fellowship of the Claude Leon Foundation of South Africa.
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.