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Fractional order optimal control problems with free terminal time
1. | CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal, Portugal, Portugal |
References:
[1] |
O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems,, Nonlinear Dynam., 38 (2004), 323.
doi: 10.1007/s11071-004-3764-6. |
[2] |
O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives,, J. Phys. A, 40 (2007), 6287.
doi: 10.1088/1751-8113/40/24/003. |
[3] |
O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems,, J. Vib. Control, 14 (2008), 1291.
doi: 10.1177/1077546307087451. |
[4] |
O. P. Agrawal, O. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables,, J. Vib. Control, 16 (2010), 1967.
doi: 10.1177/1077546309353361. |
[5] |
T. M. Atanackovic and B. Stankovic, On a numerical scheme for solving differential equations of fractional order,, Mech. Res. Comm., 35 (2008), 429.
doi: 10.1016/j.mechrescom.2008.05.003. |
[6] |
S. N. Avvakumov and Yu. N. Kiselev, Boundary value problem for ordinary differential equations with applications to optimal control,, in Spectral and Evolution Problems, (1999), 147.
|
[7] |
A. C. Chiang, Elements of Dynamic Optimization,, McGraw-Hill, (1992). Google Scholar |
[8] |
G. S. F. Frederico and D. F. M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether's theorem,, Int. Math. Forum, 3 (2008), 479.
|
[9] |
G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory,, Nonlinear Dynam., 53 (2008), 215.
doi: 10.1007/s11071-007-9309-z. |
[10] |
Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems,, Struct. Multidiscip. Optim., 38 (2009), 571.
doi: 10.1007/s00158-008-0307-7. |
[11] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).
|
[12] |
D. E. Kirk, Optimal Control Theory: An Introduction,, Prentice-Hall Inc., (1970). Google Scholar |
[13] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pac. J. Optim., 7 (2011), 63.
|
[14] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria,, Automatica J. IFAC, 48 (2012), 2116.
doi: 10.1016/j.automatica.2012.06.055. |
[15] |
S. Liu, Q. Hu and Y. Xu, Optimal inventory control with fixed ordering cost for selling by Internet auctions,, J. Ind. Manag. Optim., 8 (2012), 19.
doi: 10.3934/jimo.2012.8.19. |
[16] |
A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations,, Imperial College Press, (2012).
|
[17] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,, A Wiley-Interscience Publication, (1993).
|
[18] |
D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative,, Carpathian J. Math., 26 (2010), 210.
|
[19] |
D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems,, Signal Process., 91 (2011), 379.
doi: 10.1016/j.sigpro.2010.07.016. |
[20] |
S. Pooseh, R. Almeida and D. F. M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative,, Numer. Funct. Anal. Optim., 33 (2012), 301.
doi: 10.1080/01630563.2011.647197. |
[21] |
S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives,, Comput. Math. Appl., 64 (2012), 3090.
doi: 10.1016/j.camwa.2012.01.068. |
[22] |
S. Pooseh, R. Almeida and D. F. M. Torres, Numerical approximations of fractional derivatives with applications,, Asian J. Control, 15 (2013), 698.
doi: 10.1002/asjc.617. |
[23] |
C. Tricaud and Y. Chen, An approximate method for numerically solving fractional order optimal control problems of general form,, Comput. Math. Appl., 59 (2010), 1644.
doi: 10.1016/j.camwa.2009.08.006. |
[24] |
C. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics,, Int. J. Differ. Equ., 2010 (2010).
doi: 10.1155/2010/461048. |
show all references
References:
[1] |
O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems,, Nonlinear Dynam., 38 (2004), 323.
doi: 10.1007/s11071-004-3764-6. |
[2] |
O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives,, J. Phys. A, 40 (2007), 6287.
doi: 10.1088/1751-8113/40/24/003. |
[3] |
O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems,, J. Vib. Control, 14 (2008), 1291.
doi: 10.1177/1077546307087451. |
[4] |
O. P. Agrawal, O. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables,, J. Vib. Control, 16 (2010), 1967.
doi: 10.1177/1077546309353361. |
[5] |
T. M. Atanackovic and B. Stankovic, On a numerical scheme for solving differential equations of fractional order,, Mech. Res. Comm., 35 (2008), 429.
doi: 10.1016/j.mechrescom.2008.05.003. |
[6] |
S. N. Avvakumov and Yu. N. Kiselev, Boundary value problem for ordinary differential equations with applications to optimal control,, in Spectral and Evolution Problems, (1999), 147.
|
[7] |
A. C. Chiang, Elements of Dynamic Optimization,, McGraw-Hill, (1992). Google Scholar |
[8] |
G. S. F. Frederico and D. F. M. Torres, Fractional optimal control in the sense of Caputo and the fractional Noether's theorem,, Int. Math. Forum, 3 (2008), 479.
|
[9] |
G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory,, Nonlinear Dynam., 53 (2008), 215.
doi: 10.1007/s11071-007-9309-z. |
[10] |
Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems,, Struct. Multidiscip. Optim., 38 (2009), 571.
doi: 10.1007/s00158-008-0307-7. |
[11] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, North-Holland Mathematics Studies, (2006).
|
[12] |
D. E. Kirk, Optimal Control Theory: An Introduction,, Prentice-Hall Inc., (1970). Google Scholar |
[13] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, A new computational method for a class of free terminal time optimal control problems,, Pac. J. Optim., 7 (2011), 63.
|
[14] |
Q. Lin, R. Loxton, K. L. Teo and Y. H. Wu, Optimal control computation for nonlinear systems with state-dependent stopping criteria,, Automatica J. IFAC, 48 (2012), 2116.
doi: 10.1016/j.automatica.2012.06.055. |
[15] |
S. Liu, Q. Hu and Y. Xu, Optimal inventory control with fixed ordering cost for selling by Internet auctions,, J. Ind. Manag. Optim., 8 (2012), 19.
doi: 10.3934/jimo.2012.8.19. |
[16] |
A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations,, Imperial College Press, (2012).
|
[17] |
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,, A Wiley-Interscience Publication, (1993).
|
[18] |
D. Mozyrska and D. F. M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative,, Carpathian J. Math., 26 (2010), 210.
|
[19] |
D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems,, Signal Process., 91 (2011), 379.
doi: 10.1016/j.sigpro.2010.07.016. |
[20] |
S. Pooseh, R. Almeida and D. F. M. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative,, Numer. Funct. Anal. Optim., 33 (2012), 301.
doi: 10.1080/01630563.2011.647197. |
[21] |
S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives,, Comput. Math. Appl., 64 (2012), 3090.
doi: 10.1016/j.camwa.2012.01.068. |
[22] |
S. Pooseh, R. Almeida and D. F. M. Torres, Numerical approximations of fractional derivatives with applications,, Asian J. Control, 15 (2013), 698.
doi: 10.1002/asjc.617. |
[23] |
C. Tricaud and Y. Chen, An approximate method for numerically solving fractional order optimal control problems of general form,, Comput. Math. Appl., 59 (2010), 1644.
doi: 10.1016/j.camwa.2009.08.006. |
[24] |
C. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics,, Int. J. Differ. Equ., 2010 (2010).
doi: 10.1155/2010/461048. |
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