In this paper we consider a viscoelastic three dimensional body (of Maxwell–Boltzmann type) controlled on (part of) the boundary. We assume that the material is isotropic and homogeneous. If the body is elastic (. no dissipation due to past memory), controllability has been studied by several authors. We prove that the viscoelastic body inherits the controllability properties of the corresponding purely elastic system. The proof relays on cosine operator methods combined with moment theory.
Accepté le :
DOI : 10.1051/cocv/2016068
Mots clés : Controllability, systems with persistent memory, viscoelasticity
@article{COCV_2017__23_4_1649_0,
author = {Pandolfi, L.},
title = {Controllability of isotropic viscoelastic bodies of {Maxwell{\textendash}Boltzmann} type},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1649--1666},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016068},
zbl = {1398.93048},
mrnumber = {3716936},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016068/}
}
TY - JOUR AU - Pandolfi, L. TI - Controllability of isotropic viscoelastic bodies of Maxwell–Boltzmann type JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1649 EP - 1666 VL - 23 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016068/ DO - 10.1051/cocv/2016068 LA - en ID - COCV_2017__23_4_1649_0 ER -
%0 Journal Article %A Pandolfi, L. %T Controllability of isotropic viscoelastic bodies of Maxwell–Boltzmann type %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1649-1666 %V 23 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016068/ %R 10.1051/cocv/2016068 %G en %F COCV_2017__23_4_1649_0
Pandolfi, L. Controllability of isotropic viscoelastic bodies of Maxwell–Boltzmann type. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1649-1666. doi : 10.1051/cocv/2016068. http://www.numdam.org/articles/10.1051/cocv/2016068/
S. Agmon, Lectures on elliptic boundary value problems. D. Van Nostrand Co., Princeton (1965). | MR | Zbl
, and , Controllability of a nonhomogeneous string and ring under time dependent tension. MMNP 5 (2010) 4–31. | MR | Zbl
, Recent progresses in the boundary control method. Inv. Probl. 23 (2007) R1–R67. | DOI | MR | Zbl
and , The dynamical Lamé system: Regularity of solutions, boundary controllability and boundary data continuation. ESAIM: COCV 8 (2002) 143–167. | Numdam | MR | Zbl
and , Universal inequalities for eigenvalues of a system of elliptic equations. Proc. R. Soc. Edinb. Sec. A 139 (2009) 273–285. | DOI | MR | Zbl
, and , Boundary value problems for systems of elastostatics in Lipschitz domains. Duke Math. J. 57 (1988) 795–818. | DOI | MR | Zbl
, and , An inverse source problem for the Lamé system with variable coefficients. Appl. Anal. 84 (2005) 357–375. | DOI | MR | Zbl
and , Identifying a spatial body force in linear elastodynamics via traction measurements. SIAM J. Control Optim. 36 (1998) 1190–1206. | DOI | MR | Zbl
and , Erratum for “Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions”. Math. Res. Lett. 17 (2010) 793–794. | DOI | MR | Zbl
I. Lasiecka and R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems. Vol. 74 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2000). | MR | Zbl
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1. Vol. 8 of Recherches en Mathématiques Appliquées. Masson, Paris (1988). | MR | Zbl
, and , Boundary controllability and observability of a viscoelastic string. SIAM J. Control Optim. 50 (2012) 820–844. | DOI | MR | Zbl
H. Kolsky, Stress waves in solids. Dover publ., New York (1963). | Zbl
, The controllability of the Gurtin-Pipkin equation: a cosine operator approach. Appl. Math. Optim. 52 (2005) 143–165 (a correction in Appl. Math. Optim. 64 (2011) 467–468). | DOI | MR | Zbl
, Riesz systems and controllability of heat equations with memory. Int. Equ. Oper. Theory 64 (2009) 429–453. | DOI | MR | Zbl
, Riesz systems and moment method in the study of heat equations with memory in one space dimension. Discrete Contin. Dyn. Syst. Ser. B. 14 (2010) 1487–1510. | MR | Zbl
, Sharp control time for viscoelastic bodies. J. Int. Equ. Appl. 27 (2015) 103–136. | MR | Zbl
L. Pandolfi, Distributed systems with persistent memory. Control and moment problems. Springer Briefs in Electrical and Computer Engineering. Control, Automation and Robotics. Springer, Cham (2014). | MR | Zbl
L. Pandolfi, Cosine operator and controllability of the wave equation with memory revisited. Preprint (2016). | arXiv
L. Pandolfi, Controllability of a viscoelastic plate using one boundary control in displacement or bending. Preprint (2016). | arXiv | MR
, Propriétés asymptotique des fonctions fondamentales du problème des vibrations dans un corps élastique. Ark. Mat. Astron. Fysik 26 (1939) 19. | JFM | MR
J.J Telega and W.R. Bielski, Exact controllability of anisotropic elastic bodies, in Modelling and optimization of distributed parameter systems, Warsaw, 1995. Chapman & Hall, New York (1996) 254–262. | MR | Zbl
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