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On the Regional Controllability of the Sub-Diffusion Process with Caputo Fractional Derivative

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Abstract

This paper is devoted to the investigation of regional controllability of the fractional order sub-diffusion process. We first derive the equivalent integral equations of the abstract sub-diffusion systems with Caputo and Riemann-Liouville fractional derivatives by utilizing the Laplace transform. The new definitions of regional controllability of the system studied are introduced by extending the existence contributions. Then we analyze the regional controllability of the fractional order sub-diffusion system with minimum energy control in two different cases: B ∈ L (Rm, L2(Ω)) and B ∉ L (Rm, L2(Ω)). T he adjoint system of fractional order sub-diffusion system is also presented at the same time. Two applications are worked out in the end to verify the effectiveness of our results.

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References

  1. J. Cao, Y.Q. Chen, C. Li, Multi-UAV-based optimal crop-dusting of anomalously diffusing infestation of crops. In: 2015 American Control Conference, Palmer House Hilton, July 1-3, 2015, Chicago, IL-USA; See also: arXiv:1411.2880.

  2. K. Cao, Y.Q. Chen, D. Stuart, D. Yue, Cyber-physical modeling and control of crowd of pedestrians: a review and new framework. IEEE/CAA J. of Automatica Sinica 2, No 3 (2015), 334–344.

    Article  MathSciNet  Google Scholar 

  3. A. Cartea, D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general L´evy jump distribution functions. Phys. Review E 76, No 4 (2007), 041105.

  4. Y. Chitour, E. Tr´elat, Controllability of partial differential equations. In: Advanced Topics in Control Systems Theory, Springer (2006), 171-198.

  5. A. Debbouche, J.J. Nieto, Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls. Appl. Math. and Computation 245 (2014), 74–85.

    Article  MathSciNet  Google Scholar 

  6. S. Dolecki, D.L. Russell, A general theory of observation and control. SIAM J. on Control and Optimization 15, No 2 (1977), 185–220.

    Article  MathSciNet  Google Scholar 

  7. A. El Jai, A.J. Pritchard, Sensors and Controls in the Analysis of Distributed Systems. Halsted Press (1988).

    MATH  Google Scholar 

  8. A. El Jai, M. Simon, E. Zerrik, A. Pritchard, Regional controllability of distributed parameter systems. Internat. J. of Control 62, No 6 (1995), 1351–1365.

    Article  MathSciNet  Google Scholar 

  9. F. Ge, Y.Q. Chen, C. Kou, Cyber-physical systems as general distributed parameter systems: three types of fractional order models and emerging research opportunities. IEEE/CAA J. of Automatica Sinica 2, No 4 (2015), 353–357.

    Article  MathSciNet  Google Scholar 

  10. F. Ge, Y.Q. Chen, C. Kou, Regional controllability of anomalous diffusion generated by the time fractional diffusion equations, ASME IDETC/CIE 2015, Boston, Aug. 2-5, 2015, DETC2015-46697.

    Google Scholar 

  11. F. Ge, Y.Q. Chen, C. Kou, Regional gradient controllability of sub-diffusion processes. J. Math. Anal. Appl. 440, No 2 (2016), 865–884.

    Article  MathSciNet  Google Scholar 

  12. F. Ge, Y.Q. Chen, C. Kou, Regional boundary controllability of time fractional diffusion processes. IMA J. of Math. Control and Information (2016), 1–18.

    Google Scholar 

  13. F. Ge, H. Zhou, C. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique. Appl. Math. and Computation 275 (2016), 107–120.

    Article  MathSciNet  Google Scholar 

  14. R. Gorenflo, F. Mainardi, Fractional diffusion processes: Probability distributions and continuous time random walk. Springer Lecture Notes in Physics 621 (2003), 148–166.

    Article  Google Scholar 

  15. G. Gradenigo, A. Sarracino, D. Villamaina, A. Vulpiani, Einstein relation in systems with anomalous diffusion. Acta Phys. Polonica B 44, No 5 (2013), 899–912.

    Article  MathSciNet  Google Scholar 

  16. S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis: Volume II: Applications. Springer (2013).

    MATH  Google Scholar 

  17. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier (2006).

    MATH  Google Scholar 

  18. M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type. Publ. Office of Czestochowa Univ. of Techn. (2009).

    Google Scholar 

  19. J. E. Lagnese, The Hilbert uniqueness method: a retrospective. In: Optimal Control of Partial Differential Equations, Springer (1991), 158–181.

    Chapter  Google Scholar 

  20. J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer (1971).

    Book  Google Scholar 

  21. J. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Review 30, No 1 (1988), 1–68.

    Article  MathSciNet  Google Scholar 

  22. F. Mainardi, P. Paradisi, R. Gorenflo, Probability distributions generated by fractional diffusion equations (2007). arXiv:0704.0320.

    MATH  Google Scholar 

  23. B. Mandelbrot, The Fractal Geometry of Nature. Freeman & Co. (1982).

    MATH  Google Scholar 

  24. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports 339, No 1 (2000), 1–77.

    Article  MathSciNet  Google Scholar 

  25. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer (2012).

    MATH  Google Scholar 

  26. I. Podlubny, Fractional Differential Equations. Academic Press (1999).

    MATH  Google Scholar 

  27. I. Podlubny, Y.Q. Chen, Adjoint fractional differential expressions and operators. In: ASME 2007 Intern. Design Engineering Techn. Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers (2007), 1385–1390.

    Google Scholar 

  28. A. Pritchard, A. Wirth, Unbounded control and observation systems and their duality. SIAM J. on Control and Optimization 16, No 4 (1978), 535–545.

    Article  MathSciNet  Google Scholar 

  29. M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations. Springer (2006).

    MATH  Google Scholar 

  30. K. Schlacher, M. Sch¨oberl, Modelling, analysis and control of distributed parameter systems. Mathematical and Computer Modelling of Dynamical Systems 17, No 1 (2011), 1–2.

    Article  MathSciNet  Google Scholar 

  31. W.M. Spears, D.F. Spears, Physicomimetics: Physics-Based Swarm Intelligence. Springer (2012).

    Book  Google Scholar 

  32. P.J. Torvik, R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials. J. of Appl. Mechanics 51, No 2 (1984), 294–298.

    Article  Google Scholar 

  33. V. Uchaikin, R. Sibatov, Fractional Kinetics in Solids. World Sci. (2013).

    Book  Google Scholar 

  34. M. Valadier, Young measures, weak and strong convergence and the visintin-balder theorem. Set-Valued Anal. 2, No 1-2 (1994), 357–367.

    Article  MathSciNet  Google Scholar 

  35. M. Weiss, M. Elsner, F. Kartberg, T. Nilsson, Anomalous sub-diffusion is a measure for cytoplasmic crowding in living cells. Biophysical J. 87, No 5 (2004), 3518–3524.

    Article  Google Scholar 

  36. Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations. Computers and Math. with Applications 59, No 3 (2010), 1063–1077.

    Article  MathSciNet  Google Scholar 

  37. C. Zeng, Y.Q. Chen, Q. Yang, Almost sure and moment stability properties of fractional order Black-Scholes model. Fract. Calc. Appl. Anal. 16, No 2 (2013), 317–331; DOI: 10.2478/s13540-013-0020-0; http://www.degruyter.com/view/j/fca.2013.16.issue-2/

    Article  MathSciNet  Google Scholar 

  38. C. Zeng, Y.Q. Chen, Optimal random search, fractional dynamics and fractional calculus. Fract. Calc. Appl. Anal. 17, No 2 (2014), 321–332; DOI: 10.2478/s13540-014-0171-7; http://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.

    Article  MathSciNet  Google Scholar 

  39. E. Zerrik, A. Boutoulout, A. E. Jai, Actuators and regional boundary controllability of parabolic systems. International J. of Systems Science 31, No 1 (2000), 73–82.

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Computer Science, China University of Geosciences, Wuhan, 430074, China

    Fudong Ge

  2. University of California, Merced, CA 95343, United States of America

    YangQuan Chen

  3. Department of Applied Mathematics, Donghua University, Shanghai, 201620, China

    Chunhai Kou

  4. Institute of Control and Informatization of Production Processes BERG Faculty, Technical University of Kosice B. Nemcovej 3, 04200, Kosice, Slovakia

    Igor Podlubny

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  1. Fudong Ge
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  2. YangQuan Chen
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  3. Chunhai Kou
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  4. Igor Podlubny
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Correspondence to Fudong Ge.

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Ge, F., Chen, Y., Kou, C. et al. On the Regional Controllability of the Sub-Diffusion Process with Caputo Fractional Derivative. FCAA 19, 1262–1281 (2016). https://doi.org/10.1515/fca-2016-0065

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