On wellposedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients
Authors:
BaoZhu Guo and ZhiChao Shao
Journal:
Quart. Appl. Math. 65 (2007), 705736
MSC (2000):
Primary 35L35, 93C20, 93D15, 93B05, 93B07
Published electronically:
October 5, 2007
MathSciNet review:
2370357
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Abstract: A system of transmission of EulerBernoulli 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 wellposed in the sense of D. Salamon. This establishes the equivalence between the exact controllability of an openloop system and the exponential stability of a closedloop 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 finitedimensional ones. Finally, the exact controllability of an openloop system is developed under a uniqueness assumption by establishing the observability inequality for the dual system.
 1.
Kais
Ammari, Dirichlet boundary stabilization of the wave equation,
Asymptot. Anal. 30 (2002), no. 2, 117–130. MR 1919338
(2003f:93072)
 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
(2002f:93104), http://dx.doi.org/10.1051/cocv:2001114
 3.
Mohammed
Aassila, Exact boundary controllability of the plate equation,
Differential Integral Equations 13 (2000), no. 1012,
1413–1428. MR 1787074
(2002e:93012)
 4.
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
(2003d:93045), http://dx.doi.org/10.1023/A:1016330420910
 5.
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
(2006f:93089)
 6.
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
(2008c:74012)
 7.
Shu
Gen Chai, Boundary feedback stabilization of Naghdi’s
model, Acta Math. Sin. (Engl. Ser.) 21 (2005),
no. 1, 169–184. MR 2128833
(2005k:93171), http://dx.doi.org/10.1007/s1011400404081
 8.
Ruth
F. Curtain, The SalamonWeiss class of wellposed
infinitedimensional 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
(99a:93054), http://dx.doi.org/10.1093/imamci/14.2.207
 9.
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
(2002k:93022), http://dx.doi.org/10.1007/s49800180394
 10.
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
(91d:93027)
 11.
BaoZhu
Guo and YueHu
Luo, Controllability and stability of a secondorder hyperbolic
system with collocated sensor/actuator, Systems Control Lett.
46 (2002), no. 1, 45–65. MR 2011071
(2004i:93015), http://dx.doi.org/10.1016/S01676911(01)002018
 12.
BaoZhu
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
(2006j:93058), http://dx.doi.org/10.1137/040610702
 13.
BaoZhu
Guo and ZhiChao
Shao, Regularity of a Schrödinger equation with Dirichlet
control and colocated observation, Systems Control Lett.
54 (2005), no. 11, 1135–1142. MR 2170295
(2006d:35208), http://dx.doi.org/10.1016/j.sysconle.2005.04.008
 14.
BaoZhu
Guo and ZhiChao
Shao, Regularity of an EulerBernoulli equation with Neumann
control and collocated observation, J. Dyn. Control Syst.
12 (2006), no. 3, 405–418. MR 2233027
(2007b:93076), http://dx.doi.org/10.1007/s104500060006x
 15.
B. Z. Guo and Z. X. Zhang, On the wellposedness and regularity of wave equations with variable coefficients and partial boundary Dirichlet control and colocated observation, ESAIM Control Optim. Calc. Var., to appear.
 16.
B. Z. Guo and Z. X. Zhang, Wellposedness and regularity for an EulerBernoulli plate with variable coefficients and boundary control and observation, Mathematics of Control, Signals, and Systems, to appear.
 17.
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
(92f:93019)
 18.
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 (electronic). MR 1487483
(98k:35114), http://dx.doi.org/10.1051/cocv:1997112
 19.
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 (2005i:35164), http://dx.doi.org/10.1155/S1085337503305032
 20.
J.L.
Lions and E.
Magenes, Nonhomogeneous boundary value problems and applications.
Vol. I, SpringerVerlag, New York, 1972. Translated from the French by
P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181. MR 0350177
(50 #2670)
 21.
Weijiu
Liu and Graham
H. Williams, Exact controllability for problems of transmission of
the plate equation with lowerorder terms, Quart. Appl. Math.
58 (2000), no. 1, 37–68. MR 1738557
(2001d:93015)
 22.
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
(99b:35112), http://dx.doi.org/10.1017/S0004972700031683
 23.
Higidio
Portillo Oquendo, Nonlinear boundary stabilization for a
transmission problem in elasticity, Nonlinear Anal.
52 (2003), no. 4, 1331–1345. MR 1941260
(2003i:93069), http://dx.doi.org/10.1016/S0362546X(02)001694
 24.
David
L. Russell, Exact boundary value controllability theorems for wave
and heat processes in starcomplemented 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
(57 #7329)
 25.
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
(80c:93032), http://dx.doi.org/10.1137/1020095
 26.
Jacques
Simon, Compact sets in the space
𝐿^{𝑝}(0,𝑇;𝐵), Ann. Mat. Pura Appl. (4)
146 (1987), 65–96. MR 916688
(89c:46055), http://dx.doi.org/10.1007/BF01762360
 27.
Olof
J. Staffans, Passive and conservative continuoustime impedance and
scattering systems. I. Wellposed systems, Math. Control Signals
Systems 15 (2002), no. 4, 291–315. MR 1942089
(2003i:93024), http://dx.doi.org/10.1007/s004980200012
 28.
Michael
E. Taylor, Partial differential equations. I, Applied
Mathematical Sciences, vol. 115, SpringerVerlag, New York, 1996.
Basic theory. MR
1395148 (98b:35002b)
 29.
George
Weiss, Transfer functions of regular linear
systems. I. Characterizations of regularity, Trans. Amer. Math. Soc. 342 (1994), no. 2, 827–854. MR 1179402
(94f:93074), http://dx.doi.org/10.1090/S00029947199411794026
 30.
George
Weiss and Richard
Rebarber, Optimizability and estimatability for
infinitedimensional linear systems, SIAM J. Control Optim.
39 (2000), no. 4, 1204–1232 (electronic). MR 1814273
(2001m:93021), http://dx.doi.org/10.1137/S036301299833519X
 31.
George
Weiss, Olof
J. Staffans, and Marius
Tucsnak, Wellposed 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
(2002f:93068)
 32.
Hong
Xi Wu, The Bochner technique in differential geometry. I, Adv.
in Math. (Beijing) 10 (1981), no. 1, 57–76
(Chinese). MR
691910 (84m:53054)
 33.
H. Wu, C. L. Shen and Y. L. Yu, An Introduction to Riemannian Geometry, Beijing University Press, Beijing, 1989 (in Chinese).
 34.
PengFei
Yao, Observability inequalities for the EulerBernoulli plate with
variable coefficients, equations (Boulder, CO, 1999) Contemp.
Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000,
pp. 383–406. MR 1804802
(2001m:93025), http://dx.doi.org/10.1090/conm/268/04320
 35.
PengFei
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
(2000m:93027), http://dx.doi.org/10.1137/S0363012997331482
 1.
 K. Ammari, Dirichlet boundary stabilization of the wave equation, Asymptotic Analysis, 30 (2002), 117130. MR 1919338 (2003f:93072)
 2.
 K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var., 6 (2001), 361386. MR 1836048 (2002f:93104)
 3.
 M. Aassila, Exact boundary controllability of the plate equation, Differential Integral Equations, 13 (2000), 14131428. MR 1787074 (2002e:93012)
 4.
 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), 341370. MR 1914447 (2003d:93045)
 5.
 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), 605612 (in Chinese). Translation in Chinese J. Contemp. Math., 26 (2005), no. 4, 337346. MR 2186628 (2006f:93089)
 6.
 S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control, Indian J. Pure Appl. Math., 36 (2005), 227249. MR 2179402
 7.
 S. G. Chai, Boundary feedback stabilization of Naghdi's model, Acta Math. Sin. (Engl. Ser.), 21(1) (2005), 169184. MR 2128833 (2005k:93171)
 8.
 R. F. Curtain, The SalamonWeiss class of wellposed infinite dimensional linear systems: A survey, IMA J. of Math. Control and Inform., 14 (1997), 207223. MR 1470034 (99a:93054)
 9.
 R. F. Curtain, Linear operator inequalities for strongly stable weakly regular linear systems, Math. Control Signals Systems, 14 (2001), 299337. MR 1868533 (2002k:93022)
 10.
 R. F. Curtain and G. Weiss, Wellposedness 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, 4159. MR 1033051 (91d:93027)
 11.
 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), 4565. MR 2011071 (2004i:93015)
 12.
 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), 15981613. MR 2193497 (2006j:93058)
 13.
 B. Z. Guo and Z. C. Shao, Regularity of a Schrödinger equation with Dirichlet control and colocated observation, Systems and Control Letters, 54 (2005), 11351142. MR 2170295 (2006d:35208)
 14.
 B. Z. Guo and Z. C. Shao, Regularity of an EulerBernoulli plate equation with Neumann control and colocated observation, J. Dyn. Control Syst., 12 (2006), no. 3, 405418. MR 2233027 (2007b:93076)
 15.
 B. Z. Guo and Z. X. Zhang, On the wellposedness and regularity of wave equations with variable coefficients and partial boundary Dirichlet control and colocated observation, ESAIM Control Optim. Calc. Var., to appear.
 16.
 B. Z. Guo and Z. X. Zhang, Wellposedness and regularity for an EulerBernoulli plate with variable coefficients and boundary control and observation, Mathematics of Control, Signals, and Systems, to appear.
 17.
 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, 61111. MR 1108855 (92f:93019)
 18.
 John E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems, ESAIM Control Optim. Calc. Var., 2 (1991), 343357. MR 1487483 (98k:35114)
 19.
 I. Lasiecka and R. Triggiani, regularity of the boundary to boundary operator for hyperbolic and Petrowski PDEs, Abstr. Appl. Anal., No. 19, 2003, 10611139. MR 2041290 (2005i:35164)
 20.
 J. L. Lions and E. Magenes, NonHomogeneous Boundary Value Problems and Applications, Vol. I, SpringerVerlag, Berlin, 1972. MR 0350177 (50:2670)
 21.
 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), 3768. MR 1738557 (2001d:93015)
 22.
 W. Liu and G. H. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austra. Math. Soc., 57 (1998), 305327. MR 1617324 (99b:35112)
 23.
 H. P. Oquendo, Nonlinear boundary stabilization for a transmission problem in elasticity, Nonlinear Analysis, 52 (2003), 13311345. MR 1941260 (2003i:93069)
 24.
 D. L. Russell, Exact boundary value controllability theorems for wave and heat processes in starcomplemented regions, in Differential Games and Control Theory (Roxin, Liu and Sternberg, Eds.), Lecture Notes in Pure Appl. Math., 10, Marcel Dekker, New York, 1974, 291319. MR 0467472 (57:7329)
 25.
 D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), no. 4, 639739. MR 508380 (80c:93032)
 26.
 J. Simon, Compact sets in the space , Ann. Mat. Pura. Appl., 146(4) (1987), 6596. MR 916688 (89c:46055)
 27.
 O. J. Staffans, Passive and conservative continuoustime impedance and scattering systems. Part I: wellposed systems, Math. Control, Signals, and Systems, 15 (2002), 291315. MR 1942089 (2003i:93024)
 28.
 M. E. Taylor, Partial Differential Equations I: Basic Theory, SpringerVerlag, New York, 1996. MR 1395148 (98b:35002b)
 29.
 G. Weiss, Transfer functions of regular linear systems I: characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827854. MR 1179402 (94f:93074)
 30.
 G. Weiss and R. Rebarber, Optimizability and estimatability for infinitedimensional linear systems, SIAM J. Control Optim., 39 (2000), 12041232. MR 1814273 (2001m:93021)
 31.
 G. Weiss, O. J. Staffans and M. Tucsnak, Wellposed linear systemsa survey with emphasis on conservative systems, Int. J. Appl. Math. Comput. Sci., 11 (2001), 733. MR 1835146 (2002f:93068)
 32.
 H. Wu, Bochner's skills in differential geometry (Part I), Advances in Mathematics, Vol. 10, No. 1 (1981), 5776 (Chinese). MR 691910 (84m:53054)
 33.
 H. Wu, C. L. Shen and Y. L. Yu, An Introduction to Riemannian Geometry, Beijing University Press, Beijing, 1989 (in Chinese).
 34.
 P. F. Yao, Observability inequalities for the EulerBernoulli plate with variable coefficients, Contemporary Mathematics, Vol. 268, Amer. Math. Soc., Providence, RI, 2000, 383406. MR 1804802 (2001m:93025)
 35.
 P. F. Yao, On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Contr. and Optim., 37 (1999) 15681599. MR 1710233 (2000m:93027)
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Additional Information
BaoZhu 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
ZhiChao Shao
Affiliation:
School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Email:
zcshao@cam.wits.ac.za
DOI:
http://dx.doi.org/10.1090/S0033569X07010699
PII:
S 0033569X(07)010699
Keywords:
EulerBernoulli plate equation,
wellposedness 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. ZhiChao 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.
