Observability estimate for the fractional order parabolic equations on measurable sets. (English) Zbl 1406.93068

Zheng, Guojie; Ali, M. Montaz

Abstr. Appl. Anal. 2014, Article ID 361904, 5 p. (2014).

Summary: We establish an observability estimate for the fractional order parabolic equations evolved in a bounded domain \(\Omega\) of \(\mathbb{R}^n\). The observation region is \(F \times \omega\), where \(\omega\) and \(F\) are measurable subsets of \(\Omega\) and \((0,T)\), respectively, with positive measure. This inequality is equivalent to the null controllable property for a linear controlled fractional order parabolic equation. The building of this estimate is based on the Lebeau-Robbiano strategy and a delicate result in measure theory provided in [K. D. Phung and G. Wang, J. Eur. Math. Soc. (JEMS) 15, No. 2, 681–703 (2013; Zbl 1258.93037)].

MSC:

93B07	Observability
26A33	Fractional derivatives and integrals
35R11	Fractional partial differential equations
93C20	Control/observation systems governed by partial differential equations

Keywords:

observability estimate; fractional order parabolic equations

Citations:

Zbl 1258.93037

References:

[1]	Metzler, R.; Klafter, J., 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, 37, 31, R161-R208 (2004) · Zbl 1075.82018
[2]	Sato, K., Levy Processes and infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics (1999), Cambridge, Mass, USA: Cambridge University Press, Cambridge, Mass, USA · Zbl 0973.60001
[3]	Apraiz, J.; Escauriaza, L.; Wang, G.; Zhang, C., Observability inequalities and measurable sets · Zbl 1302.93040
[4]	Fursikov, A. V.; Imanuvilov, O. Yu., Controllability of Evolution Equations, Lecture Notes Series, 34 (1996), Seoul, South Korea: Seoul National University, Seoul, South Korea · Zbl 0862.49004
[5]	Wang, G., \(L^\infty \)-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM Journal on Control and Optimization, 47, 4, 1701-1720 (2008) · Zbl 1165.93016
[6]	Wang, G.; Wang, L., The Carleman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations, Journal of Optimization Theory and Applications, 118, 2, 429-461 (2003) · Zbl 1041.49024
[7]	Zhang, C., An observability estimate for the heat equation from a product of two measurable sets, Journal of Mathematical Analysis and Applications, 396, 1, 7-12 (2012) · Zbl 1270.35251
[8]	Apraiz, J.; Escauriaza, L., Null-control and measurable sets, ESAIM: Control, Optimisation and Calculus of Variations, 19, 1, 239-254 (2013) · Zbl 1262.35118
[9]	Micu, S.; Zuazua, E., On the controllability of a fractional order parabolic equation, SIAM Journal on Control and Optimization, 44, 6, 1950-1972 (2006) · Zbl 1116.93022
[10]	Miller, L., On the controllability of anomalous diffusions generated by the fractional Laplacian, Mathematics of Control, Signals, and Systems, 18, 3, 260-271 (2006) · Zbl 1105.93015
[11]	Lions, J. L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30, 1, 1-68 (1988) · Zbl 0644.49028
[12]	Pazy, A., Semigroups of Linear OperaTors and Applications To Partial Di Erential Equations (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0516.47023
[13]	Phung, K. D.; Wang, G., An observability estimate for the parabolic equations from a measurable set in time and its applications, Journal of the European Mathematical Society, 15, 681-703 (2013) · Zbl 1258.93037