Abstract
In this paper we study controllability of a linear equation with persistent memory when the control belongs to \( H^k_0(0,T;L^2(\Omega )) \). In the case the memory is zero, our equation is reduced to the wave equation and a result due to Everdoza and Zuazua informally states that smoother targets can be reached by using smoother controls. In this paper we prove that this result can be partially extended to systems with memory, but that the memory is an obstruction to a complete extensions.
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References
Agmon, S.: Lectures on Elliptic Boundary Value Problems. D. Van Nostrand Co., Princeton (1965)
Avdonin, S.A., Ivanov, S.A.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, New York (1995)
Bardos, C., Lebeau, G., Rauch, J.: sharp sufficient conditions for observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 305, 1024–1065 (1992)
Ervedoza, S., Zuazua, E.: A systematic method for building smooth controls from smooth data. Discrete Contin. Dyn. Syst. Ser. B 14, 1375–1401 (2010)
Ervedoza, S., Zuazua, E.: Numerical Approximation of Exact Controls for Waves. Springer, New York (2013)
Gohberg, I.C., Krejn, M.G.: Opèrateurs Linèaires non Auto-adjoints dans un Espace Hilbertien. Dunod, Paris (1971)
Lasiecka, I., Triggiani, R.: A cosine operator approach to \(L_2(0, T;L_2(\Gamma ))\)-boundary input hyperbolic equations. Appl. Math. Optim. 7, 35–93 (1981)
Lions J.-L.: Contrôlabilitè exacte, perturbations et stabilization de systémes distribuès. Vol. 1, Recherches en Mathé matiques Appliqué e 9, Masson, Paris (1988)
Komornik, V.: Exact Controllability and Stabilization. The Multiplier Method. RAM: Research in Applied Mathematics. Wiley, Chichester (1994)
Pandolfi, L.: The controllability of the Gurtin–Pipkin equation: a cosine operator approach. Appl. Math. Optim. 52, 143–165 (2005). (a correction. Appl. Math. Optim. 64, 467–468 (2011))
Pandolfi, L.: Riesz systems and moment method in the study of heat equations with memory in one space dimension. Discrete Contin. Dyn. Syst. Ser. B. 14, 1487–1510 (2010)
Pandolfi, L.: Sharp control time for viscoelastic bodies. J. Integral Equ. Appl. 27, 103–136 (2015)
Pandolfi, L.: Distributed Systems with Persistent Memory. Control and Moment Problems. Springer Briefs in Electrical and Computer Engineering. Control, Automation and Robotics. Springer, Cham (2014)
Pandolfi, L.: Controllability for the heat equation with memory: a recent approach. Riv. Matem. Univ. Parma 7, 259–277 (2016)
Pandolfi, L.: Controllability of isotropic viscoelastic bodies of Maxwell–Boltzmann type. ESAIM Control. Calc. Var. 23, 1649–1666 (2017)
Pandolfi, L., Triulzi, D.: Regularity of the steering control for systems with persistent memory. Appl. Math. Lett. 51, 34–40 (2016)
Russel, D.A.: Unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52, 189–211 (1973)
Soriano, J.A.: Exact controllability of the wave equation. Rev. Mat. Univ. Complut. Madrid 8, 459–493 (1995)
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Supported by GNAMPA-INDAM in the framework of the “Groupement de Recherche en Contrôle des EDP entre la France et l’Italie (CONEDP-CNRS)”.
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Pandolfi, L. Controllability and Lack of Controllability with Smooth Controls in Viscoelasticity via Moment Methods. Integr. Equ. Oper. Theory 90, 33 (2018). https://doi.org/10.1007/s00020-018-2462-6
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DOI: https://doi.org/10.1007/s00020-018-2462-6