Fractional order optimal control problems with free terminal time

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	A. C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, Inc., Singapore, 1992.
[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.
[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.
[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.
[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.
[12]	D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall Inc., Englewood Cliffs, NJ, 1970.
[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.
[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.
[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.
[16]	A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[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.
[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.
[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.
[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.
[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.
[7]	A. C. Chiang, Elements of Dynamic Optimization, McGraw-Hill, Inc., Singapore, 1992.
[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.
[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.
[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.
[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.
[12]	D. E. Kirk, Optimal Control Theory: An Introduction, Prentice-Hall Inc., Englewood Cliffs, NJ, 1970.
[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.
[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.
[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.
[16]	A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, London, 2012.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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