Abstract
In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.
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References
Ammari K, Nicaise S (2010) Stabilization of a transmission wave/plate equation. J Differ Equ 249: 707–727
Avalos G, Lasiecka I, Triggiani R (2009) Beyond lack of compactness and lack of stability of a coupled parabolic–hyperbolic fluid–structure system. In: Optimal control of coupled systems of partial differential equations. International Series of Numerical Mathematics, vol 158. Birkhäuser, Basel, pp 1–33
Bardos C, Lebeau G, Rauch J (1992) Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J Control Optim 30:1024–1065
Belinskiy B, Lasiecka I (2007) Gevrey’s and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions. J Math Anal Appl 332(1):137–154
Bekiaris-Liberis N, Krstic M (2010) Delay-adaptive feedback for linear feedforward systems. Syst Control Lett 59:277–283
Blum H, Rannacher R (1980) On the boundary value problem of the biharmonic operator on domains with angular corners. Math Methods Appl Sci 2:556–581
Chen S, Liu K, Liu Z (1998) Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping. SIAM J Appl Math 59:651–668
Curtain RF, Zwart H (1995) An introduction to infinitedimensional linear systems theory. Springer, New York
Dauge M (1988) Elliptic boundary vlaue problems on corner domains-smoothness and asymptotics of solutions. Lecture Notes in Mathematics, vol 1341. Springer, Berlin
de Teresa L, Zuazua E (2000) Null controllability of linear and semilinear heat equations in thin domains. Asymptot Anal 24(3–4):295–317
Dunford N, Schwartz JT (1971) Linear operators, Part III. John Wiley & Sons, Inc., New York/ London/Sydney
Gohberg IC, Krein MG (1969) Introduction to the theory of linear nonself adjoint operators. Translation of Mathematics Monographs, vol 18. AMS, Providence
Grisvard P (1985) Elliptic problems in nonsmooth domains. Pitman, London
Guo BZ, Wang JM, Yung SP (2005) On the \(C_0\)-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam. Syst control Lett 54(6):557–574
Guo BZ, Yang KY (2010) Output feedback stabilization of a one-dimensional Schrödinger equation by boundary observation with time delay. IEEE Trans Automat Control 55:1226–1232
Guo BZ, Xu CZ, Hammouri H (2012) Output feedback stabilization of a one-dimensional wave equation with an arbitrary time delay in boundary observation. ESAIM Control Optim Calc Var 18:22–35
Huang FL (1985) Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann Differ Equ 1(1):43–56
Kim JU (1992) On the energy decay of a linear thermoelastic bar and plate. SIAM J Math Anal 23(4):889–899
Komornik V (1994) Exact controllability and stabilization: the multiplier method, research in applied mathematics, vol 36. John Wiley, Masson, Paris/Chichester
Krstic M (2009) Control of an unstable reaction-diffusion PDE with long input delay. Syst Control Lett 58(10–11):773–782
Krstic M (2009) Delay compensation for nonlinear, adaptive, and PDE aystems, systems & control: foundations & applications. Birkhäuser Boston, Inc., Boston
Krstic M (2009) Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst Control Lett. 58:372–377
Krstic M (2009) Copmensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Trans Automat Control 54:1362–1368
Langer RE (1931) On the zeros of exponential sum and integrals. Bull Am Math Soc 37:213–239
Lasiecka I (2002) Mathematical control theory of coupled PDEs. Society for industrial and applied mathematics (SIAM), Philadelphia
Lebeau G (1992) Contrôle de l’équation de Schrödinger. J Math Pures Appl (9), 71(3):267–291
Lebeau G, Zuazua E (1999) Decay rates for the three-dimensional linear system of thermoelasticity. Arch Ration Mech Anal 148(3):179–231
Levin B Ya (1996) Lectures on entire functions, translations of mathematical monographs, vol 150. American Mathematical Society, Providence
Liu K, Liu Z (1998) Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping. SIAM J Control Optim 36:1086–1098
Locker J (2000) Spectral theory of non-self-adjoint two-point differential operators, mathematical surveys and monographs, vol 73. American Mathematical Society, Providence
Luo ZH, Guo BZ (1997) Shear force feedback c ontrol of a single link flexible robot with revolute joint. IEEE Trans Autom Control 42(1):53–65
Luo ZH, Guo BZ, Morgül O (1999) Stability and stabilization of infinite dimensional systems with applications. Springer, London
Naimark MA (1967) Linear differential operators, vol I. Frederick Ungar Publishing Company, New York
Nicaise S (1992) About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate eqaution I: regularity of the solutions. Ann Sc Norm Super, Pisa CI. Sci. (4), 19(3):327–361
Nicaise S (1994) Polyginal interface problems for the biharmonic operators. Math Methods Appl Sci 17:21–39
Nicaise S, Valein J (2010) Stabilization of second order evolution equations with unbounded feedback with delay. ESAIM Control Optim Calc Var 16:420–456
Opmeer MR (2008) Nuclearity of Hankel operators for ultradifferentiable control systems. Syst Control Lett 57(11):913–918
Pr\(\ddot{u}\)ss J (1984) On the spectrum of \(C_0-\)semigroups. Trans Am Math Soc 284:847–857
Pazy A (1983) Semigroups of linear operators and applications to partial differential equations. Springer, New York
Rao B (1998) Stabilization of elastic plates with dynamical boundary control. SIAM J Control Optim 36:148–163
Shkalikov AA (1986) Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J Sov Math 33:1311–1342
Taylor S (1989) Gevrey regularity of solutions of evolution equations and boundary controllability, Gevrey Semigroups (Chapter 5), PhD thesis, School of Mathematics, University of Minnesota
Wang JM, Guo BZ (2008) Analyticity and dynamic behavior of a damped three-layer sandwich beam. J Optim Theory Appl 137(3):675–689
Wang JM, Guo BZ, Chentouf B (2006) Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach. ESAIM Control Optim Calc Var 12:12–34
Wang JM, Krstic M. Stability of an interconnected system of Euler–Bernoulli beam and heat equation with boundary coupling (submitted)
Wang JM, Ren B, Krstic M (2012) Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans Automat Control 57:179–185
Wang SL, Yao CZ (2000) Numerical solution to the optimal feedback gain of a beam equation under boundary linear feedback control. Acta Anal Funct. Appl 2(2):123–132
Xu GG, Yung SP, Li LK (2006) Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim Calc Var 12:770–785
Young RM (2001) An introduction to nonharmonic fourier series. Academic Press, Inc., London
Zhang X, Zuazua E (2004) Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system. J Differ Equ 204(2):380–438
Zhang X, Zuazua E (2006) Asymptotic behavior of a hyperbolic–parabolic coupled system arising in fluid–structure interaction. Int Ser Numer Math 154:445–455
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The authors would like to thank the referees and associate editor for their very helpful suggestions and comments.
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This work was carried out with the support of the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa.
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Zhang, Q., Wang, JM. & Guo, BZ. Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation. Math. Control Signals Syst. 26, 77–118 (2014). https://doi.org/10.1007/s00498-013-0107-5
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DOI: https://doi.org/10.1007/s00498-013-0107-5