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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 361904, 5 pages
Observability Estimate for the Fractional Order Parabolic Equations on Measurable Sets
1College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2School of Computational and Applied Mathematics, University of the Witwatersrand (Wits), Johannesburg 2050, South Africa
3TCSE, Faculty of Engineering and Built Environment, University of the Witwatersrand (Wits), Johannesburg 2050, South Africa
Received 10 January 2014; Accepted 3 March 2014; Published 27 March 2014
Academic Editor: Sheng-Jie Li
Copyright © 2014 Guojie Zheng and M. Montaz Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp. R161–R208, 2004.
- K. Sato, Levy Processes and infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, Mass, USA, 1999.
- J. Apraiz, L. Escauriaza, G. Wang, and C. Zhang, “Observability inequalities and measurable sets,” Journal of the European Mathematical Society. In press.
- A. V. Fursikov and O. Yu. Imanuvilov, “Controllability of Evolution Equations,” in Lecture Notes Series, vol. 34, Seoul National University, Seoul, South Korea, 1996.
- G. Wang, “-null controllability for the heat equation and its consequences for the time optimal control problem,” SIAM Journal on Control and Optimization, vol. 47, no. 4, pp. 1701–1720, 2008.
- G. Wang and L. Wang, “The Carleman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations,” Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 429–461, 2003.
- C. Zhang, “An observability estimate for the heat equation from a product of two measurable sets,” Journal of Mathematical Analysis and Applications, vol. 396, no. 1, pp. 7–12, 2012.
- J. Apraiz and L. Escauriaza, “Null-control and measurable sets,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 19, no. 1, pp. 239–254, 2013.
- S. Micu and E. Zuazua, “On the controllability of a fractional order parabolic equation,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 1950–1972, 2006.
- L. Miller, “On the controllability of anomalous diffusions generated by the fractional Laplacian,” Mathematics of Control, Signals, and Systems, vol. 18, no. 3, pp. 260–271, 2006.
- J. L. Lions, “Exact controllability, stabilization and perturbations for distributed systems,” SIAM Review, vol. 30, no. 1, pp. 1–68, 1988.
- A. Pazy, Semigroups of Linear OperaTors and Applications To Partial Di Erential Equations, Springer, New York, NY, USA, 1983.
- K. D. Phung and G. Wang, “An observability estimate for the parabolic equations from a measurable set in time and its applications,” Journal of the European Mathematical Society, vol. 15, pp. 681–703, 2013.