Stability of an interconnected system of euler−bernoulli beam and heat equation with boundary coupling

Wang, Jun-Min; Krstic, Miroslav

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1029-1052.

We study the stability of an interconnected system of Euler−Bernoulli beam and heat equation with boundary coupling, where the boundary temperature of the heat equation is fed as the boundary moment of the Euler−Bernoulli beam and, in turn, the boundary angular velocity of the Euler−Bernoulli beam is fed into the boundary heat flux of the heat equation. We show that the spectrum of the closed-loop system consists only of two branches: one along the real axis and the another along two parabolas symmetric to the real axis and open to the imaginary axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. With a careful estimate for the resolvent operator, the completeness of the root subspaces of the system is verified. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class $\delta >2$.

Classification : 93D15, 93C20, 35P20
Mots clés : Euler−Bernoulli beam, heat equation, boundary control, stability, spectrum, Gevrey regularity

B. Belinskiy and I. Lasiecka, Gevrey’s and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions. J. Math. Anal. Appl. 332 (2007) 137–154. | DOI | MR | Zbl

N.Dunford and J.T. Schwartz, Linear Operators, Part III. John Wiley & Sons, Inc., New York-London-Sydney (1971). | MR | Zbl

I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. Vol. 18 of Trans. Math. Monogr. AMS Providence, Rhode Island (1969). | MR

B.Z. Guo, J.M. Wang and S.P. Yung, On the $C_0$-semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam. Systems Control Lett. 54 (2005) 557–574. | DOI | MR | Zbl

M. Krstic, Control of an unstable reaction-diffusion PDE with long input delay. Systems Control Lett. 58 (2009) 773–782. | DOI | MR | Zbl

M. Krstic, Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Systems and Control: Foundations and Applications. Birkhäuser Boston, Inc., Boston, MA (2009). | MR | Zbl

R.E. Langer, On the zeros of exponential sum and integrals. Bull. Amer. Math. Soc. 37 (1931) 213–239. | DOI | MR | Zbl

I. Lasiecka, Mathematical control theory of coupled PDEs. Society for Industrial and Applied Mathematics. SIAM, Philadelphia, PA (2002). | MR | Zbl

B.Ya. Levin, Lectures on Entire Functions. Vol. 150 of Trans. Math. Monogr. American Mathematical Society, Providence, Rhode Island (1996). | MR | Zbl

J. Locker, Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators. Vol.73 of Math. Surv. Monogr. American Mathematical Society, Providence, Rhode Island (2000). | MR | Zbl

Z.H. Luo and B.Z. Guo, Shear force feedback control of a single link flexible robot with revolute joint. IEEE Trans. Automat. Control 42 (1997) 53–65. | DOI | MR | Zbl

Z.H. Luo, B.Z. Guo and O. Morgül, Stability and Stabilization of Infinite dimensional Systems with Applications. Springer-Verlag, London (1999). | MR | Zbl

M.A. Naimark, Linear Differential Operators. In vol. I. Frederick Ungar Publishing Company, New York (1967). | MR | Zbl

M.R. Opmeer, Nuclearity of Hankel operators for ultradifferentiable control systems. Systems Control Lett. 57 (2008) 913–918. | DOI | MR | Zbl

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983). | MR | Zbl

A.A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Soviet Math. 33 (1986) 1311–1342. | DOI | Zbl

M.A. Shubov, Generation of Gevrey class semigroup by non-selfadjoint Euler−Bernoulli beam model. Math. Methods Appl. Sci. 29 (2006) 2181–2199. | DOI | MR | Zbl

S. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Gevrey Semigroups. Ph.D. thesis, School of Mathematics, University of Minnesota (1989). | MR

J.M. Wang and B.Z. Guo, Analyticity and dynamic behavior of a damped three-layer sandwich beam. J. Optim. Theory Appl. 137 (2008) 675–689. | DOI | MR | Zbl

J.M. Wang, B.Z. Guo and B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach. ESAIM: COCV 12 (2006) 12–34. | Numdam | MR | Zbl

J.M. Wang, B. Ren and M. Krstic, Stabilization and Gevrey regularity of a Schrödinger equation in boundary feedback with a heat equation. IEEE Trans. Automatic Control 57 (2012) 179–185. | DOI | MR | Zbl

R.M. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, Inc., London (2001). | MR | Zbl

X. Zhang and E. Zuazua, Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system. J. Differ. Equ. 204 (2004) 380–438. | DOI | MR | Zbl

X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction. Internat. Ser. Numer. Math. 154 (2007) 445-455. | DOI | MR | Zbl