Time optimal control problem of a class of fractional distributed systems

International Journal of Dynamical Systems and Differential Equations

Volumn 3, Issue 3, 2011, Pages 363-382

  Wang, Jinrong ^a   Zhou, Yong ^b  

^a GUIZHOU UNIVERSITY   (China)

^b XIANGTAN UNIVERSITY   (China)

Author keywords

Existence; Fractional distributed systems; Mild solutions; Time optimal control

Indexed keywords

DIFFERENTIATION (CALCULUS);

CAPUTO FRACTIONAL DERIVATIVES; DISTRIBUTED SYSTEMS; EXISTENCE; EXISTENCE AND UNIQUENESS; MILD SOLUTION; SEMILINEAR EVOLUTION EQUATION; TIME OPTIMAL CONTROL; TIME-OPTIMAL CONTROL PROBLEMS;

OPTIMAL CONTROL SYSTEMS;

EID: 80051740339     PISSN: 17523583     EISSN: 17523591     Source Type: Journal    
DOI: 10.1504/IJDSDE.2011.041881     Document Type: Article

References (39)

1. Araya, D. and Lizama, C. (2008) 'Almost automorphic mild solutions to fractional differential equations', Nonlinear Anal., Vol. 69, pp.3692-3705.
2. Balachandran, K. and Park, J.Y. (2009) ' Nonlocal Cauchy problem for abstract fractional semilinear evolution equations', Nonlinear Anal., Vol. 71, pp.4471-4475.
3. Barbu, V. (1984) 'The time-optimal control problem for parabolic variational inequalities', Appl. Math. Optim., Vol. 11, pp.1-22. (Pubitemid 14585267)
4. Benchohra, M., Henderson, J., Ntouyas, S.K. and Ouahab, A. (2008a) 'Existence results for fractional order functional differential equations with infinite delay', J. Math. Anal. Appl., Vol. 338, pp.1340-1350.
5. Benchohra, M., Henderson, J., Ntouyas, S.K. and Ouahab, A. (2008b) 'Existence results for fractional functional differential inclusions with infinite delay and application to control theory', Fract. Calc. Appl. Anal., Vol. 11, pp.35-56.
6. Chang, Y.K., Kavitha, V. and Mallika Arjunan, M. (2009) 'Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order', Nonlinear Anal., Vol. 71, pp.5551-5559.
7. Cuevas, C. and Lizama, C. (2008) 'Almost automorphic mild solutions to a class of fractional differential equations', Applied Mathematics Letters, Vol. 21, pp.1315-1319.
8. Diethelm, K. and Freed, A.D. (1999) 'On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity', in Keil, F., Mackens, W., Voss, H. and Werther, J. (Eds.): Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer-Verlag, Heidelberg, pp.217-224.
9. El-Borai, M.M. (2004a) 'Semigroup and some nonlinear fractional differential equations', Applied Math. Comp., Vol. 149, pp.823-831.
10. El-Borai, M.M. (2004b) 'The fundamental solutions for fractional evolution equations of parabolic type', J. Appl. Math. Stoch. Anal., Vol. 3, pp.197-211.
11. El-Borai, M.M. (2005) 'On some fractional evolution equations with nonlocal conditions', Int. J. Pure Applied Math., Vol. 24, pp.405-413.
12. Fattorini, H.O. (1964) 'Time-optimal control of solutions of operational differential equations', J. SIAM Control, Vol. 2, pp.54-59.
13. Fattorini, H.O. (1974) 'The time-optimal control problem in Banach spaces', Applied Math. Optim., Vol. 1, pp.163-188.
14. Gaul, L., Klein, P. and Kempfle, S. (1991) 'Damping description involving fractional operators', Mech. Syst. Signal Process., Vol. 5, pp.81-88.
15. Glockle, W.G. and Nonnenmacher, T.F. (1995) 'A fractional calculus approach of self-similar protein dynamics', Biophys. J., Vol. 68, pp.46-53.
16. Henderson, J. and Ouahab, A. (2009) 'Fractional functional differential inclusions with finite delay', Nonlinear Anal., Vol. 70, pp.2091-2105.
17. Hernández, E., O'Regan, D. and Balachandran, K. (2010) 'On recent developments in the theory of abstract differential equations with fractional derivatives', Nonlinear Anal., Vol. 73, pp.3462-3471.
18. Hilfer, R. (2000) Applications of Fractional Calculus in Physics, World Scientific, Singapore.
19. Hu, L., Ren, Y. and Sakthivel, R. (2009) 'Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays', Semigroup Forum, Vol. 79, pp.507-514.
20. Jaradat, O.K., Al-Omari, A. and Momani, S. (2008) 'Existence of the mild solution for fractional semilinear initial value problems', Nonlinear Anal., Vol. 69, pp.3153-3159.
21. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) 'Theory and applications of fractional differential equations', North-Holland Mathematics Studies, Vol. 204, Elsevier Science B.V., Amsterdam.
22. Lakshmikantham, V., Leela, S. and Devi, J.V. (2009) Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge.
23. LaSalle, J.P. (1960) The Time Optimal Control Problem, Contributions to the Theory of Nonlinear Oscillations, Princeton University Press, Princeton, Vol. 5, pp.1-24.
24. Li, X. and Yong, J. (1995) Optimal Control Theory for Infinite Dimensional Systems, Birkhauser Boston.
25. Li, X. and Yong, J. (1995) Optimal Control Theory for Infinite Dimensional Systems, Birkhauser Boston. Liu, H. and Chang, J.C. (2009) 'Existence for a class of partial differential equations with nonlocal conditions', Nonlinear Anal., Vol. 70, pp.3076-3083.
26. Mainardi, F. (1997) 'Fractional calculus, Some basic problems in continuum and statistical mechanics', in Carpinteri, A. and Mainardi, F. (Eds.): Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, pp.291-348.
27. Metzler, F., Schick, W., Kilian, H.G. and Nonnenmache, T.F. (1995) 'Relaxation in filled polymers: a fractional calculus approach', J. Chem. Phys., Vol. 103, pp.7180-7186.
28. Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York.
29. Mophou, G.M. and N'Guérékata, G.M. (2009) 'Existence of mild solution for some fractional differential equations with nonlocal conditions', Semigroup Forum, Vol. 79, pp.315-322.
30. Mophou, G.M. and N'Guérékata, G.M. (2010) 'Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay', Applied Math. Comp., Vol. 216, pp.61-69.
31. N'Guerekata, G.M. (2009) 'A Cauchy problem for some fractional abstract differential equation with nonlocal conditions', Nonlinear Anal., Vol. 70, pp.1873-1876.
32. Ozdemir, N., Karadeniz, D. and ?Iskender, B.B. (2009) 'Fractional optimal control problem of a distributed system in cylindrical coordinates', Physics Letters A, Vol. 373, pp.221-226.
33. Podlubny, I. (1999) Fractional Differential Equations, Academic Press, San Diego.
34. Wang, J., Xiang, X. and Wei, W. (2009) 'The constructive approach on existence of time optimal controls of system governed by nonlinear equations on Banach spaces', E. J. Qualitative Theory of Diff. Equ., Vol. 45, pp.1-10.
35. Wang, J. and Zhou, Y. (2011) 'A class of fractional evolution equations and optimal controls', Nonlinear Anal., Vol. 12, pp.262-272.
36. Yong, J. (1991) 'Time optimal controls for semilinear distributed parameter systems-existence theory and necessary conditions', J. Kodai Math., Vol. 14, pp.239-253.
37. Zeidler, E. (1990) Nonlinear Functional Analysis and Its Application II/A, Springer-Verlag, New York.
38. Zhou, Y. and Jiao, F. (2010a) 'Nonlocal Cauchy problem for fractional evolution equations', Nonlinear Anal., Vol. 11, pp.4465-4475.
39. Zhou, Y. and Jiao, F. (2010b) 'Existence of mild solutions for fractional neutral evolution equations', Comp. Math. Appl., Vol. 59, pp.1063-1077.