On the lack of controllability of fractional in time ODE and PDE

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• 作者：Qi Lü ; Enrique Zuazua
• 关键词：Fractional in time ODE ; PDE ; Partial controllability ; Null controllability ; Observability
• 刊名：Mathematics of Control, Signals, and Systems (MCSS)
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：28
• 期：2
• 全文大小：494 KB
• 参考文献：1.Adams JL, Hartley TT (2008) Finite time controllability of fractional order systems. J Comput Nonlinear Dyn 3:021402-1–021402-5CrossRef
  2.Anantharaman N, Léautaud M (2014) Sharp polynomial decay rates for the damped wave equation on the torus. Anal PDE 7(1):159–214MathSciNet CrossRef MATH
  3.Antil H, Otárola E (2014) A FEM for an optimal control problem of fractional powers of elliptic operators. arXiv:​1406.​7460v2
  4.Antil H, Otárola E, Salgado AJ (2015) A fractional space-time optimal control problem: analysis and discretisation. arXiv:​1504.​00063v1
  5.Atanacković TM, Pilipović S, Stanković B, Zorica D (2014) Fractional calculus with applications in mechanics. Wave propagation, impact and variational principles. Mechanical engineering and solid mechanics series. Wiley, Hoboken; ISTE, London
  6.Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, CambridgeCrossRef MATH
  7.Berberan-Santos MN (2005) Properties of the Mittag-Leffler relaxation function. J Math Chem 38(4):629–635MathSciNet CrossRef MATH
  8.Bettayeb M, Djennoune S (2008) New results on the controllability and observability of fractional dynamical systems. J Vib Control 14(9–10):1531–1541MathSciNet CrossRef MATH
  9.Biccari U (2014) Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator (preprint)
  10.Bouchaud J-P, Georges A (1990) Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep 195(4–5):127–293MathSciNet CrossRef
  11.Burq N, Hitrik M (2007) Energy decay for damped wave equations on partially rectangular domains. Math Res Lett 14(1):35–47MathSciNet CrossRef MATH
  12.Caffarelli L, Chan C, Vasseur A (2011) Regularity theory for parabolic nonlinear integral operators. J Am Math Soc 24(3):849–869MathSciNet CrossRef MATH
  13.Caffarelli L, Vasseur A (2010) Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math. (2) 171(3):1903–1930MathSciNet CrossRef MATH
  14.Caffarelli L, Salsa S, Silvestre L (2008) Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent Math 171(2):425–461MathSciNet CrossRef MATH
  15.Caffarelli L, Silvestre L (2007) An extension problem related to the fractional Laplacian. Commun Partial Differ Equ 32(7–9):1245–1260MathSciNet CrossRef MATH
  16.Constantin P, Kiselev A, Ryzhik L, Zlatoš A (2008) Diffusion and mixing in fluid flow. Ann Math. (2) 168(2):643–674MathSciNet CrossRef MATH
  17.Fridman E (2014) Introduction to time-delay systems. Analysis and control. Systems & control: foundations & applications. Birkhäuser/Springer, ChamMATH
  18.Kalman RE (1960) Contributions to the theory of optimal control. Bol Soc Mat Mex (2) 5:102–119MathSciNet MATH
  19.Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland mathematics studies, vol 204. Elsevier Science B.V., Amsterdam
  20.Kiselev A, Nazarov F, Volberg A (2007) Global well-posedness for the critical \(2\) D dissipative quasi-geostrophic equation. Invent Math 167(3):445–453MathSciNet CrossRef MATH
  21.Lebeau G, Robbiano L (1995) Contrôle exact de l’équation de la chaleur. Commun Partial Differ Equ 20(1–2):335–356MathSciNet CrossRef MATH
  22.Lebeau G, Robbiano L (1997) Stabilisation de l’quation des ondes par le bord. Duke Math J 86(3):465–491MathSciNet CrossRef
  23.Lebeau G, Zuazua E (1998) Null-controllability of a system of linear thermoelasticity. Arch Ration Mech Anal 141(4):297–329MathSciNet CrossRef MATH
  24.Li K, Peng J, Gao J (2013) Controllability of nonlocal fractional differential systems of order \(\alpha \in (1,2]\) in Banach spaces. Rep Math Phys 71(1):33–43MathSciNet CrossRef MATH
  25.Lü Q (2010) Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math Sin 26(12):2377–2386MathSciNet CrossRef MATH
  26.Matignon D, d’Andréa-Novel B (1996) Some results on controllability and observability of finite-dimensional fractional differential systems. In: Proceedings of the IAMCS, IEEE Conference on systems, man and cybernetics Lille, France, pp 952-956
  27.Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNet CrossRef MATH
  28.Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A 37(31):161–208MathSciNet CrossRef MATH
  29.Michiels W, Niculescu S-I (2014) Stability, control, and computation for time-delay systems. An eigenvalue-based approach, 2nd edn. Advances in design and control, vol. 27. Society for Industrial and Applied Mathematics (SIAM), Philadelphia
  30.Micu S, Zuazua E (2006) On the controllability of a fractional order parabolic equation. SIAM J Control Optim 44(6):1950–1972MathSciNet CrossRef MATH
  31.Miller KS, Ross B (1993) An introduction to the fractional calculus and fractionaldifferential equations. A Wiley-Interscience Publication. Wiley, New York
  32.Miller L (2006) On the controllability of anomalous diffusions generated by the fractional Laplacian. Math Control Signals Syst 18(3):260–271MathSciNet CrossRef MATH
  33.Nochetto RH, Otarola E, Salgado AJ (2014) A PDE approach to space-time fractional parabolic problems. arXiv:​1404.​0068v3
  34.Phung K-D (2007) Polynomial decay rate for the dissipative wave equation. J Differ Equ 240(1):92–124MathSciNet CrossRef MATH
  35.Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Theory and applications. Gordon and Breach Science Publishers, YverdonMATH
  36.Yi S, Nelson PW, Ulsoy AG (2010) Time-delay systems. Analysis and control using the Lambert W function. World Scientific Publishing Co. Pte. Ltd., HackensackCrossRef MATH
  
• 作者单位：Qi Lü (1)
  Enrique Zuazua (2)
  
  1. School of Mathematics, Sichuan University, Chengdu, 610064, China
  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Communications Engineering and Networks
  Control Engineering
  
• 出版者：Springer London
• ISSN：1435-568X

文摘

This paper is devoted to the analysis of the problem of controllability of fractional (in time) ordinary and partial differential equations (ODE/PDE). The fractional time derivative introduces some memory effects on the system that need to be taken into account when defining the notion of control. In fact, in contrast with the classical ODE and PDE control theory, when driving these systems to rest, one is required not only to control the value of the state at the final time, but also the memory accumulated by the long-tail effects that the fractional derivative introduces. As a consequence, the notion of null controllability to equilibrium needs to take into account both the state and the memory term. The existing literature so far is only concerned with the problem of partial controllability in which the state is controlled, but the behavior of the memory term is ignored. In the present paper, we consider the full controllability problem and show that, due to the memory effects, even at the ODE level, controllability cannot be achieved in finite time. This negative result holds even for finite-dimensional systems in which the control is of full dimension. Consequently, the same negative results hold also for fractional PDE, regardless of their parabolic or hyperbolic nature. This negative result exhibits a completely opposite behavior with respect to the existing literature on classical ODE and PDE control where sharp sufficient conditions for null controllability are well known.