Luciano Pandofi[1]
Controllability for the heat equation with memory: a recent approach
Pages: 259-277
Received: 12 November 2015
Accepted: 29 February 2016
Mathematics Subject Classification (2010): 45K05, 93B03, 93B05, 93C22.
Keywords: Controllability, systems with persistent memory, thermodynamics of materials with memory, viscoelasticity.
Author address:
[1] : Politecnico di Torino,
Dipartimento di Scienze Matematiche "G. L. Lagrange",
Corso Duca degli Abruzzi 24,
Torino, 10129, Italy
Abstract: We present some recent ideas and new results in the study of controllability of a distributed system with persistent memory, which is encountered in several applications, most noticeably thermodynamics of systems with memory and viscoelasticity.
References
[1]
S. Avdonin and L. Pandolfi,
Simultaneous temperature and flux controllability for heat equations with memory,
Quart. Appl. Math. 71 (2013), 339–368.
MR
[2]
S. Avdonin and L. Pandolfi,
Temperature and heat flux dependence/independence for heat equations with memory,
in "Time Delay Systems - Methods, Applications and New Trend", R. Sipahi, T. Vyhlidal, S.-I. Niculescu and P. Pepe, eds.,
Lecture Notes in Control and Inform. Sci., 423, Springer, Berlin 2012, 87–101.
MR
[3]
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary,
SIAM J. Control Optim. 30 (1992), 1024–1065.
MR
[4]
J. Baumeister,
Boundary control of an integro-differential equation,
J. Math. Anal. Appl. 93 (1983), 550–570.
MR
[5]
X. Fu, J. Yong and X. Zhang,
Controllability and observability of a heat equation with hyperbolic memory kernel,
J. Differential Equations 247 (2009), 2395–2439.
MR
[6]
A. Hassel and T. Tao,
Erratum for "Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions",
Math. Res. Lett. 17 (2010), 793–794.
MR
[7]
G. Leugering,
On boundary controllability of viscoelastic systems,
in "Control of partial differential equations" (Santiago de Compostela, 1987),
Lecture Notes in Control and Inform. Sci., 114, Springer, Berlin 1989, 190–201.
MR
[8]
J.-L. Lions,
Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, Vol. 1,
Masson, Paris 1988.
MR
[9]
V. Komornik and P. Loreti,
Fourier series in control theory,
Springer Monographs in Mathematics, Springer-Verlag, New York 2005.
MR
[10]
J. U. Kim,
Control of a second-order integro-differential equation,
SIAM J. Control Optim. 31 (1993), 101–110.
MR
[11]
L. Pandolfi,
Boundary controllability and source reconstruction in a viscoelastic string under external traction,
J. Math. Anal. Appl. 407 (2013), 464–479.
MR
[12]
L. Pandolfi,
The controllability of the Gurtin-Pipkin equation: a cosine operator approach,
Appl. Math. Optim. 52 (2005), 143–165, (Errratum in: Appl. Math. Optim. 64 (2011), 467–468).
MR
(MR)
[13]
L. Pandolfi,
Riesz systems, spectral controllability and a source identification problem for heat equations with memory,
Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 745–759.
MR
[14]
L. Pandolfi,
On-line input identification and application to Active Noise Cancellation,
Annual Reviews in Control 34 (2010), 245–261.
DOI: 10.1016/j.arcontrol.2010.07.001
[15]
L. Pandolfi,
Traction, deformation and velocity of deformation in a viscoelastic string,
Evol. Equ. Control Theory 2 (2013), 471–493.
MR
[16]
L. Pandolfi,
Sharp control time for viscoelastic bodies,
J. Integral Equations Appl. 27 (2015), 103–136.
MR
[17]
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
[18]
L. Pandolfi,
Controllability of isotropic viscoelastic bodies of Maxwell-Boltzmann type,
ESAIM Control Optim. Calc. Var.,
DOI: 10.1051/cocv/2016068,
to appear.
[19]
M. Renardy,
Mathematical analysis of viscoelastic fluids,
Handbook of differential equations: evolutionary equations, Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam 2008, 229–265.
MR
[20]
M. Tucsnak and G. Weiss,
Observation and control for operator semigroups,
Birkhäuser, Basel 2009.
MR