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

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

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.
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.

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.

[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.

[1]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266

[2]

Enkhbat Rentsen, J. Zhou, K. L. Teo. A global optimization approach to fractional optimal control. Journal of Industrial and Management Optimization, 2016, 12 (1) : 73-82. doi: 10.3934/jimo.2016.12.73

[3]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control and Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[4]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013

[5]

Harbir Antil, Ciprian G. Gal, Mahamadi Warma. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1883-1918. doi: 10.3934/dcdss.2022012

[6]

Chongyang Liu, Wenjuan Sun, Xiaopeng Yi. Optimal control of a fractional smoking system. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022071

[7]

Tuğba Akman Yıldız, Amin Jajarmi, Burak Yıldız, Dumitru Baleanu. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 407-428. doi: 10.3934/dcdss.2020023

[8]

Agnieszka B. Malinowska, Tatiana Odzijewicz. Optimal control of the discrete-time fractional-order Cucker-Smale model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 347-357. doi: 10.3934/dcdsb.2018023

[9]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control and Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016

[10]

Z. Foroozandeh, Maria do rosário de Pinho, M. Shamsi. On numerical methods for singular optimal control problems: An application to an AUV problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2219-2235. doi: 10.3934/dcdsb.2019092

[11]

Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004

[12]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

[13]

Iman Malmir. Caputo fractional derivative operational matrices of Legendre and Chebyshev wavelets in fractional delay optimal control. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 395-426. doi: 10.3934/naco.2021013

[14]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[15]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control and Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007

[16]

Jingtao Shi, Juanjuan Xu, Huanshui Zhang. Stochastic recursive optimal control problem with time delay and applications. Mathematical Control and Related Fields, 2015, 5 (4) : 859-888. doi: 10.3934/mcrf.2015.5.859

[17]

Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control and Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83

[18]

Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1305-1320. doi: 10.3934/jimo.2021021

[19]

Tan H. Cao, Boris S. Mordukhovich. Optimal control of a perturbed sweeping process via discrete approximations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3331-3358. doi: 10.3934/dcdsb.2016100

[20]

Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (540)
  • HTML views (0)
  • Cited by (45)

[Back to Top]