Controllability for the heat equation with memory: a recent approach
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.
 : 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.
 S. Avdonin and L. Pandolfi, Simultaneous temperature and flux controllability for heat equations with memory, Quart. Appl. Math. 71 (2013), 339–368. MR
 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
 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
 J. Baumeister, Boundary control of an integro-differential equation, J. Math. Anal. Appl. 93 (1983), 550–570. MR
 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
 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
 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
 J.-L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, Vol. 1, Masson, Paris 1988. MR
 V. Komornik and P. Loreti, Fourier series in control theory, Springer Monographs in Mathematics, Springer-Verlag, New York 2005. MR
 J. U. Kim, Control of a second-order integro-differential equation, SIAM J. Control Optim. 31 (1993), 101–110. MR
 L. Pandolfi, Boundary controllability and source reconstruction in a viscoelastic string under external traction, J. Math. Anal. Appl. 407 (2013), 464–479. MR
 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)
 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
 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
 L. Pandolfi, Traction, deformation and velocity of deformation in a viscoelastic string, Evol. Equ. Control Theory 2 (2013), 471–493. MR
 L. Pandolfi, Sharp control time for viscoelastic bodies, J. Integral Equations Appl. 27 (2015), 103–136. MR
 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
 L. Pandolfi, Controllability of isotropic viscoelastic bodies of Maxwell-Boltzmann type, ESAIM Control Optim. Calc. Var., DOI: 10.1051/cocv/2016068, to appear.
 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
 M. Tucsnak and G. Weiss, Observation and control for operator semigroups, Birkhäuser, Basel 2009. MR