Abstract
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.
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References
Adams JL, Hartley TT (2008) Finite time controllability of fractional order systems. J Comput Nonlinear Dyn 3:021402-1–021402-5
Anantharaman N, Léautaud M (2014) Sharp polynomial decay rates for the damped wave equation on the torus. Anal PDE 7(1):159–214
Antil H, Otárola E (2014) A FEM for an optimal control problem of fractional powers of elliptic operators. arXiv:1406.7460v2
Antil H, Otárola E, Salgado AJ (2015) A fractional space-time optimal control problem: analysis and discretisation. arXiv:1504.00063v1
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
Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge
Berberan-Santos MN (2005) Properties of the Mittag-Leffler relaxation function. J Math Chem 38(4):629–635
Bettayeb M, Djennoune S (2008) New results on the controllability and observability of fractional dynamical systems. J Vib Control 14(9–10):1531–1541
Biccari U (2014) Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator (preprint)
Bouchaud J-P, Georges A (1990) Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep 195(4–5):127–293
Burq N, Hitrik M (2007) Energy decay for damped wave equations on partially rectangular domains. Math Res Lett 14(1):35–47
Caffarelli L, Chan C, Vasseur A (2011) Regularity theory for parabolic nonlinear integral operators. J Am Math Soc 24(3):849–869
Caffarelli L, Vasseur A (2010) Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann Math. (2) 171(3):1903–1930
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):425–461
Caffarelli L, Silvestre L (2007) An extension problem related to the fractional Laplacian. Commun Partial Differ Equ 32(7–9):1245–1260
Constantin P, Kiselev A, Ryzhik L, Zlatoš A (2008) Diffusion and mixing in fluid flow. Ann Math. (2) 168(2):643–674
Fridman E (2014) Introduction to time-delay systems. Analysis and control. Systems & control: foundations & applications. Birkhäuser/Springer, Cham
Kalman RE (1960) Contributions to the theory of optimal control. Bol Soc Mat Mex (2) 5:102–119
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
Kiselev A, Nazarov F, Volberg A (2007) Global well-posedness for the critical \(2\)D dissipative quasi-geostrophic equation. Invent Math 167(3):445–453
Lebeau G, Robbiano L (1995) Contrôle exact de l’équation de la chaleur. Commun Partial Differ Equ 20(1–2):335–356
Lebeau G, Robbiano L (1997) Stabilisation de l’quation des ondes par le bord. Duke Math J 86(3):465–491
Lebeau G, Zuazua E (1998) Null-controllability of a system of linear thermoelasticity. Arch Ration Mech Anal 141(4):297–329
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):33–43
Lü Q (2010) Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta Math Sin 26(12):2377–2386
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
Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77
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):161–208
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
Micu S, Zuazua E (2006) On the controllability of a fractional order parabolic equation. SIAM J Control Optim 44(6):1950–1972
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractionaldifferential equations. A Wiley-Interscience Publication. Wiley, New York
Miller L (2006) On the controllability of anomalous diffusions generated by the fractional Laplacian. Math Control Signals Syst 18(3):260–271
Nochetto RH, Otarola E, Salgado AJ (2014) A PDE approach to space-time fractional parabolic problems. arXiv:1404.0068v3
Phung K-D (2007) Polynomial decay rate for the dissipative wave equation. J Differ Equ 240(1):92–124
Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Theory and applications. Gordon and Breach Science Publishers, Yverdon
Yi S, Nelson PW, Ulsoy AG (2010) Time-delay systems. Analysis and control using the Lambert W function. World Scientific Publishing Co. Pte. Ltd., Hackensack
Acknowledgments
This work is supported by the Advanced Grant FP7-246775 NUMERIWAVES of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306-C02-00 and SEV-2013-0323 Grants of the MINECO and a Humboldt Award at the University of Erlangen-Nuremberg, and the NSF of China under Grant 11471231, and the Fundamental Research Funds for the Central Universities in China under Grant 2015SCU04A02. This work was initiated while the authors were visiting the CIMI-Toulouse in the context of the activities of the Excellence Chair on ”PDE, Control and Numerics”. The authors acknowledge the CIMI for the hospitality and support and D. Matignon for fruitful discussions.
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Lü, Q., Zuazua, E. On the lack of controllability of fractional in time ODE and PDE. Math. Control Signals Syst. 28, 10 (2016). https://doi.org/10.1007/s00498-016-0162-9
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DOI: https://doi.org/10.1007/s00498-016-0162-9