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April  2014, 10(2): 363-381. doi: 10.3934/jimo.2014.10.363

Fractional order optimal control problems with free terminal time

Shakoor Pooseh 1, , Ricardo Almeida 1, and  Delfim F. M. Torres 1,

1. 

CIDMA — Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal, Portugal, Portugal

Received  December 2012 Revised  July 2013 Published  October 2013

We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal-time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numerical methods to solve the problems are presented, and some computational simulations are discussed in detail.
Keywords: free-time problem, fractional calculus, fractional optimal control, numerical approximations., Optimization and control.
Mathematics Subject Classification: Primary: 26A33, 33F05; Secondary: 49K1.
Citation: Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres. Fractional order optimal control problems with free terminal time. Journal of Industrial & Management Optimization, 2014, 10 (2) : 363-381. doi: 10.3934/jimo.2014.10.363
References:
[1]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6.  Google Scholar

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303. doi: 10.1088/1751-8113/40/24/003.  Google Scholar

[3]

O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vib. Control, 14 (2008), 1291-1299. doi: 10.1177/1077546307087451.  Google Scholar

[4]

O. P. Agrawal, O. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables, J. Vib. Control, 16 (2010), 1967-1976. doi: 10.1177/1077546309353361.  Google Scholar

[5]

T. M. Atanackovic and B. Stankovic, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm., 35 (2008), 429-438. doi: 10.1016/j.mechrescom.2008.05.003.  Google Scholar

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S. N. Avvakumov and Yu. N. Kiselev, Boundary value problem for ordinary differential equations with applications to optimal control, in Spectral and Evolution Problems, Vol. 10 (Sevastopol, 1999), Natl. Taurida Univ. "V. Vernadsky," Simferopol', 2000, 147-155.  Google Scholar

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A. C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, Inc., Singapore, 1992. Google Scholar

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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), 479-493.  Google Scholar

[9]

G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam., 53 (2008), 215-222. doi: 10.1007/s11071-007-9309-z.  Google Scholar

[10]

Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38 (2009), 571-581. doi: 10.1007/s00158-008-0307-7.  Google Scholar

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006.  Google Scholar

[12]

D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall Inc., Englewood Cliffs, NJ, 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), 63-81.  Google Scholar

[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), 2116-2129. doi: 10.1016/j.automatica.2012.06.055.  Google Scholar

[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), 19-40. doi: 10.3934/jimo.2012.8.19.  Google Scholar

[16]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.  Google Scholar

[17]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[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), 210-221.  Google Scholar

[19]

D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Process., 91 (2011), 379-385. doi: 10.1016/j.sigpro.2010.07.016.  Google Scholar

[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), 301-319. doi: 10.1080/01630563.2011.647197.  Google Scholar

[21]

S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64 (2012), 3090-3100. doi: 10.1016/j.camwa.2012.01.068.  Google Scholar

[22]

S. Pooseh, R. Almeida and D. F. M. Torres, Numerical approximations of fractional derivatives with applications, Asian J. Control, 15 (2013), 698-712. doi: 10.1002/asjc.617.  Google Scholar

[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), 1644-1655. doi: 10.1016/j.camwa.2009.08.006.  Google Scholar

[24]

C. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics, Int. J. Differ. Equ., 2010 (2010), Art. ID 461048, 16 pp. doi: 10.1155/2010/461048.  Google Scholar

show all references

References:
[1]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6.  Google Scholar

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303. doi: 10.1088/1751-8113/40/24/003.  Google Scholar

[3]

O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vib. Control, 14 (2008), 1291-1299. doi: 10.1177/1077546307087451.  Google Scholar

[4]

O. P. Agrawal, O. Defterli and D. Baleanu, Fractional optimal control problems with several state and control variables, J. Vib. Control, 16 (2010), 1967-1976. doi: 10.1177/1077546309353361.  Google Scholar

[5]

T. M. Atanackovic and B. Stankovic, On a numerical scheme for solving differential equations of fractional order, Mech. Res. Comm., 35 (2008), 429-438. doi: 10.1016/j.mechrescom.2008.05.003.  Google Scholar

[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, Vol. 10 (Sevastopol, 1999), Natl. Taurida Univ. "V. Vernadsky," Simferopol', 2000, 147-155.  Google Scholar

[7]

A. C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, Inc., Singapore, 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), 479-493.  Google Scholar

[9]

G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam., 53 (2008), 215-222. doi: 10.1007/s11071-007-9309-z.  Google Scholar

[10]

Z. D. Jelicic and N. Petrovacki, Optimality conditions and a solution scheme for fractional optimal control problems, Struct. Multidiscip. Optim., 38 (2009), 571-581. doi: 10.1007/s00158-008-0307-7.  Google Scholar

[11]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006.  Google Scholar

[12]

D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall Inc., Englewood Cliffs, NJ, 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), 63-81.  Google Scholar

[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), 2116-2129. doi: 10.1016/j.automatica.2012.06.055.  Google Scholar

[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), 19-40. doi: 10.3934/jimo.2012.8.19.  Google Scholar

[16]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.  Google Scholar

[17]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.  Google Scholar

[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), 210-221.  Google Scholar

[19]

D. Mozyrska and D. F. M. Torres, Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal Process., 91 (2011), 379-385. doi: 10.1016/j.sigpro.2010.07.016.  Google Scholar

[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), 301-319. doi: 10.1080/01630563.2011.647197.  Google Scholar

[21]

S. Pooseh, R. Almeida and D. F. M. Torres, Approximation of fractional integrals by means of derivatives, Comput. Math. Appl., 64 (2012), 3090-3100. doi: 10.1016/j.camwa.2012.01.068.  Google Scholar

[22]

S. Pooseh, R. Almeida and D. F. M. Torres, Numerical approximations of fractional derivatives with applications, Asian J. Control, 15 (2013), 698-712. doi: 10.1002/asjc.617.  Google Scholar

[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), 1644-1655. doi: 10.1016/j.camwa.2009.08.006.  Google Scholar

[24]

C. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics, Int. J. Differ. Equ., 2010 (2010), Art. ID 461048, 16 pp. doi: 10.1155/2010/461048.  Google Scholar

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