On the controllability of anomalous diffusions generated by the fractional Laplacian

Mathematics of Control, Signals, and Systems

Volumn 18, Issue 3, 2006, Pages 260-271

  Miller, Luc ^a,b  

^a UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^b CENTRE DE MATHÉMATIQUES LAURENT SCHWARTZ   (France)

Author keywords

Control cost; Fractional calculus; Interior controllability; Parabolic equation; Spectral observability

Indexed keywords

BOUNDARY VALUE PROBLEMS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; OBSERVABILITY; SET THEORY;

CONTROL COST; FRACTIONAL CALCULUS; INTERIOR CONTROLLABILITY; PARABOLIC EQUATION; SPECTRAL OBSERVABILITY;

CONTROLLABILITY;

EID: 33748366552     PISSN: 09324194     EISSN: 1435568X     Source Type: Journal    
DOI: 10.1007/s00498-006-0003-3     Document Type: Article

References (25)

1. Applebaum D (2004) Lévy processes - from probability to finance and quantum groups. Notices Amer Math Soc 51(11):1336-1347
2. Avdonin SA, Ivanov SA (1995) Families of exponentials, the method of moments in controllability problems for distributed parameter systems. Cambridge University Press, Cambridge
3. 1 [0, 1]. J London Math Soc (2) 54(1):102-110
4. Dolecki S, Russell DL (1977) A general theory of observation and control. SIAM J Control Optim 15(2):185-220
5. Fattorini HO (1966) Control in finite time of differential equations in Banach space. Comm Pure Appl Math 19:17-34
6. Fattorini HO, Russell DL (1971) Exact controllability theorems for linear parabolic equations in one space dimension. Arch Ration Mech Anal 43:272-292
7. Fernández-Cara E, Zuazua E (2000) The cost of approximate controllability for heat equations: The linear case. Adv Differential Equations 5(4-6):465-514
8. Gorenflo R, Mainardi F (2003) Fractional diffusion processes: probability distributions and continuous time random walk. Processes with long range correlations. In: Rangarajan G, Ding M (eds). lecture notes in physics, vol 621 pp 148-166
9. Hanyga A (2001) Multidimensional solutions of space-fractional diffusion equations. R Soc Lond Proc Ser A Math Phys Eng Sci 457(2016):2993-3005
10. Jacob B, Partington JR (2006) On controllability of diagonal systems with one-dimensional input space. Systems Control Lett 55(4):321-328
11. Jacob B, Zwart H (2001) Exact observability of diagonal systems with a finite-dimensional output operator. Systems Control Lett 43(2):101-109
12. Jerison D, Lebeau G (1996) Nodal sets of sums of eigenfunctions. Harmonic analysis and partial differential equations (Chicago, IL, 1996). University of Chicago Press, Chicago, pp 223-239
13. Lebeau G, Robbiano L (1995) Contrôle exact de l'équation de la chaleur. Comm Partial Differential Equations 20(1-2):335-356
14. Lebeau G, Zuazua E (1998) Null-controllability of a system of linear thermoelasticity. Arch Ration Mech Anal 141(4):297-329
15. 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): R161-R208
16. Micu S, Zuazua E (2006) On the controllability of a fractional order parabolic equation. SIAM J Control Optim. 44(6):1950-1972
17. Miller L (2004) Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time. J Differential Equations 204(1):202-226
18. Miller L (2005) Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds. Math Res Lett 12(1):37-47
19. Rebarber R, Weiss G (2000) Necessary conditions for exact controllability with a finite-dimensional input space. Systems Control Lett 40(3):217-227
20. Redheffer RM (1977) Completeness of sets of complex exponentials. Adv Math 24(1):1-62
21. Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge studies in advanced mathematics, vol 68. Cambridge University Press, Cambridge
22. Seidman TI (1998) How violent are fast controls?. Math Control Signals Systems 1(1):89-95
23. Sokolov I, Klafter J, Blumen A (2002) Fractional kinetics. Physics Today 55:48-54
24. Song R, Vondraček Z (2003) Potential theory of subordinate killed Brownian motion in a domain. Probab Theory Related Fields 125(4):578-592
25. Weiss G (1989) Admissible observation operators for linear semigroups. Israel J Math 65(1):17-43