February  2018, 11(1): 59-76. doi: 10.3934/dcdss.2018004

A necessary condition of Pontryagin type for fuzzy fractional optimal control problems

1. 

Department of Applied Mathematics, School of Mathematics and Computer Science, Damghan University, Damghan, Iran

2. 

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

* Corresponding author: Delfim F. M. Torres

Received  June 2016 Revised  November 2016 Published  January 2018

Fund Project: This work is part of second author's PhD project. It was partially supported by Damghan University, Iran; and CIDMA-FCT, Portugal, under project UID/MAT/04106/2013.

We prove necessary optimality conditions of Pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. The new results are illustrated by computing the extremals of three fuzzy optimal control systems, which improve recent results of Najariyan and Farahi.

Citation: 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
References:
[1]

T. AllahviranlooA. Armand and Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J. Intell. Fuzzy Systems, 26 (2014), 1481-1490. 

[2]

R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations Imp. Coll. Press, London, 2015. doi: 10.1142/p991.

[3]

S. Arshad and V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Anal., 74 (2011), 3685-3693.  doi: 10.1016/j.na.2011.02.048.

[4]

D. Baleanu and P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak J. Phys., 56 (2006), 1087-1092.  doi: 10.1007/s10582-006-0406-x.

[5]

B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.  doi: 10.1016/j.fss.2012.10.003.

[6]

J. J. Buckley and T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), 241-248.  doi: 10.1016/S0165-0114(98)00323-6.

[7]

R. A. El-Nabulsi and D. F. M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, β), Math. Methods Appl. Sci., 30 (2007), 1931-1939.  doi: 10.1002/mma.879.

[8]

O. S. Fard and M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math., 271 (2014), 71-82.  doi: 10.1016/j.cam.2014.03.019.

[9]

O. S. FardD. F. M. Torres and M. R. Zadeh, A Hukuhara approach to the study of hybrid fuzzy systems on time scales, Appl. Anal. Discrete Math., 10 (2016), 152-167.  doi: 10.2298/AADM160311004F.

[10]

O. S. Fard and M. S. Zadeh, Note on ''Necessary optimality conditions for fuzzy variational problems'', J. Adv. Res. Dyn. Control Syst., 4 (2012), 1-9. 

[11]

B. Farhadinia, Necessary optimality conditions for fuzzy variational problems, Inform. Sci., 181 (2011), 1348-1357.  doi: 10.1016/j.ins.2010.11.027.

[12]

B. Farhadinia, Pontryagin's minimum principle for fuzzy optimal control problems, Iran. J. Fuzzy Syst., 11 (2014), 27-43. 

[13]

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.

[14]

Y. Gao and Y.-J. Liu, Adaptive fuzzy optimal control using direct heuristic dynamic programming for chaotic discrete-time system, J. Vib. Control, 22 (2016), 595-603.  doi: 10.1177/1077546314534286.

[15]

R. Goetschel, Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43.  doi: 10.1016/0165-0114(86)90026-6.

[16]

J. Y. Halpern, Reasoning about Uncertainty MIT Press, Cambridge, MA, 2003.

[17]

R. Hilfer, Applications of Fractional Calculus in Physics World Sci. Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[18]

N. V. Hoa, Fuzzy fractional functional differential equations under Caputo gH-differentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1134-1157.  doi: 10.1016/j.cnsns.2014.08.006.

[19]

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

[20]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC, Boca Raton, FL, 2015.

[21]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations Imp. Coll. Press, London, 2012. doi: 10.1142/p871.

[22]

A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations Springer Briefs in Applied Sciences and Technology, Springer, Cham, 2015. doi: 10.1007/978-3-319-14756-7.

[23]

M. Mazandarani and A. V. Kamyad, Modified fractional Euler method for solving fuzzy fractional initial value problem, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 12-21.  doi: 10.1016/j.cnsns.2012.06.008.

[24]

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

[25]

S. I. Muslih and D. Baleanu, Formulation of Hamiltonian equations for fractional variational problems, Czechoslovak J. Phys., 55 (2005), 633-642.  doi: 10.1007/s10582-005-0067-1.

[26]

S. I. MuslihD. Baleanu and E. Rabei, Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 73 (2006), 436-438.  doi: 10.1088/0031-8949/73/5/003.

[27]

M. Najariyan and M. H. Farahi, Optimal control of fuzzy linear controlled system with fuzzy initial conditions, Iran. J. Fuzzy Syst., 10 (2013), 21-35. 

[28]

M. Najariyan and M. H. Farahi, A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients, J. Comput. Appl. Math., 259 (2014), part B, 682-694.  doi: 10.1016/j.cam.2013.04.029.

[29]

M. Najariyan and M. H. Farahi, A new approach for solving a class of fuzzy optimal control systems under generalized Hukuhara differentiability, J. Franklin Inst., 352 (2015), 1836-1849.  doi: 10.1016/j.jfranklin.2015.01.006.

[30]

E. R. Pinch, Optimal Control and the Calculus of Variations Oxford Science Publications, Oxford Univ. Press, New York, 1993.

[31]

I. Podlubny, Fractional Differential Equations Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999.

[32]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical Theory of Optimal Processes Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York, 1962.

[33]

S. PoosehR. Almeida and D. F. M. Torres, Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10 (2014), 363-381.  doi: 10.3934/jimo.2014.10.363.

[34]

S. SalahshourT. Allahviranloo and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372-1381.  doi: 10.1016/j.cnsns.2011.07.005.

[35]

S. Salahshour, T. Allahviranloo, S. Abbasbandy and D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty Adv. Difference Equ. 2012 (2012), 12 pp. doi: 10.1186/1687-1847-2012-112.

[36]

J. SoolakiO. S. Fard and A. H. Borzabadi, Generalized Euler-Lagrange equations for fuzzy variational problems, SeMA Journal, 73 (2016), 131-148.  doi: 10.1007/s40324-015-0060-y.

[37]

J. SoolakiO. S. Fard and A. H. Borzabadi, Generalized Euler-Lagrange equations for fuzzy fractional variational calculus, Math. Commun., 21 (2016), 199-218. 

[38]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387-403.  doi: 10.1016/B978-1-4832-1450-4.50045-6.

[39]

V. E. Tarasov, Fractional variations for dynamical systems: Hamilton and Lagrange approaches, J. Phys. A, 39 (2006), 8409-8425.  doi: 10.1088/0305-4470/39/26/009.

[40]

G. S. Taverna and D. F. M. Torres, Generalized fractional operators for nonstandard Lagrangians, Math. Meth. Appl. Sci., 38 (2015), 1808-1812.  doi: 10.1002/mma.3188.

[41]

J. XuZ. Liao and J. J. Nieto, A class of linear differential dynamical systems with fuzzy matrices, J. Math. Anal. Appl., 368 (2010), 54-68.  doi: 10.1016/j.jmaa.2009.12.053.

[42]

D. Yang and K.-Y. Cai, Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems, Expert Systems with Applications, 37 (2010), 6963-6967.  doi: 10.1016/j.eswa.2010.03.024.

show all references

References:
[1]

T. AllahviranlooA. Armand and Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J. Intell. Fuzzy Systems, 26 (2014), 1481-1490. 

[2]

R. Almeida, S. Pooseh and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations Imp. Coll. Press, London, 2015. doi: 10.1142/p991.

[3]

S. Arshad and V. Lupulescu, On the fractional differential equations with uncertainty, Nonlinear Anal., 74 (2011), 3685-3693.  doi: 10.1016/j.na.2011.02.048.

[4]

D. Baleanu and P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak J. Phys., 56 (2006), 1087-1092.  doi: 10.1007/s10582-006-0406-x.

[5]

B. Bede and L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.  doi: 10.1016/j.fss.2012.10.003.

[6]

J. J. Buckley and T. Feuring, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems, 105 (1999), 241-248.  doi: 10.1016/S0165-0114(98)00323-6.

[7]

R. A. El-Nabulsi and D. F. M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, β), Math. Methods Appl. Sci., 30 (2007), 1931-1939.  doi: 10.1002/mma.879.

[8]

O. S. Fard and M. Salehi, A survey on fuzzy fractional variational problems, J. Comput. Appl. Math., 271 (2014), 71-82.  doi: 10.1016/j.cam.2014.03.019.

[9]

O. S. FardD. F. M. Torres and M. R. Zadeh, A Hukuhara approach to the study of hybrid fuzzy systems on time scales, Appl. Anal. Discrete Math., 10 (2016), 152-167.  doi: 10.2298/AADM160311004F.

[10]

O. S. Fard and M. S. Zadeh, Note on ''Necessary optimality conditions for fuzzy variational problems'', J. Adv. Res. Dyn. Control Syst., 4 (2012), 1-9. 

[11]

B. Farhadinia, Necessary optimality conditions for fuzzy variational problems, Inform. Sci., 181 (2011), 1348-1357.  doi: 10.1016/j.ins.2010.11.027.

[12]

B. Farhadinia, Pontryagin's minimum principle for fuzzy optimal control problems, Iran. J. Fuzzy Syst., 11 (2014), 27-43. 

[13]

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.

[14]

Y. Gao and Y.-J. Liu, Adaptive fuzzy optimal control using direct heuristic dynamic programming for chaotic discrete-time system, J. Vib. Control, 22 (2016), 595-603.  doi: 10.1177/1077546314534286.

[15]

R. Goetschel, Jr. and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43.  doi: 10.1016/0165-0114(86)90026-6.

[16]

J. Y. Halpern, Reasoning about Uncertainty MIT Press, Cambridge, MA, 2003.

[17]

R. Hilfer, Applications of Fractional Calculus in Physics World Sci. Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[18]

N. V. Hoa, Fuzzy fractional functional differential equations under Caputo gH-differentiability, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1134-1157.  doi: 10.1016/j.cnsns.2014.08.006.

[19]

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

[20]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC, Boca Raton, FL, 2015.

[21]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations Imp. Coll. Press, London, 2012. doi: 10.1142/p871.

[22]

A. B. Malinowska, T. Odzijewicz and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations Springer Briefs in Applied Sciences and Technology, Springer, Cham, 2015. doi: 10.1007/978-3-319-14756-7.

[23]

M. Mazandarani and A. V. Kamyad, Modified fractional Euler method for solving fuzzy fractional initial value problem, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 12-21.  doi: 10.1016/j.cnsns.2012.06.008.

[24]

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

[25]

S. I. Muslih and D. Baleanu, Formulation of Hamiltonian equations for fractional variational problems, Czechoslovak J. Phys., 55 (2005), 633-642.  doi: 10.1007/s10582-005-0067-1.

[26]

S. I. MuslihD. Baleanu and E. Rabei, Hamiltonian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 73 (2006), 436-438.  doi: 10.1088/0031-8949/73/5/003.

[27]

M. Najariyan and M. H. Farahi, Optimal control of fuzzy linear controlled system with fuzzy initial conditions, Iran. J. Fuzzy Syst., 10 (2013), 21-35. 

[28]

M. Najariyan and M. H. Farahi, A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients, J. Comput. Appl. Math., 259 (2014), part B, 682-694.  doi: 10.1016/j.cam.2013.04.029.

[29]

M. Najariyan and M. H. Farahi, A new approach for solving a class of fuzzy optimal control systems under generalized Hukuhara differentiability, J. Franklin Inst., 352 (2015), 1836-1849.  doi: 10.1016/j.jfranklin.2015.01.006.

[30]

E. R. Pinch, Optimal Control and the Calculus of Variations Oxford Science Publications, Oxford Univ. Press, New York, 1993.

[31]

I. Podlubny, Fractional Differential Equations Mathematics in Science and Engineering, 198, Academic Press, San Diego, CA, 1999.

[32]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical Theory of Optimal Processes Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt, Interscience Publishers John Wiley & Sons, Inc. New York, 1962.

[33]

S. PoosehR. Almeida and D. F. M. Torres, Fractional order optimal control problems with free terminal time, J. Ind. Manag. Optim., 10 (2014), 363-381.  doi: 10.3934/jimo.2014.10.363.

[34]

S. SalahshourT. Allahviranloo and S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1372-1381.  doi: 10.1016/j.cnsns.2011.07.005.

[35]

S. Salahshour, T. Allahviranloo, S. Abbasbandy and D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty Adv. Difference Equ. 2012 (2012), 12 pp. doi: 10.1186/1687-1847-2012-112.

[36]

J. SoolakiO. S. Fard and A. H. Borzabadi, Generalized Euler-Lagrange equations for fuzzy variational problems, SeMA Journal, 73 (2016), 131-148.  doi: 10.1007/s40324-015-0060-y.

[37]

J. SoolakiO. S. Fard and A. H. Borzabadi, Generalized Euler-Lagrange equations for fuzzy fractional variational calculus, Math. Commun., 21 (2016), 199-218. 

[38]

T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387-403.  doi: 10.1016/B978-1-4832-1450-4.50045-6.

[39]

V. E. Tarasov, Fractional variations for dynamical systems: Hamilton and Lagrange approaches, J. Phys. A, 39 (2006), 8409-8425.  doi: 10.1088/0305-4470/39/26/009.

[40]

G. S. Taverna and D. F. M. Torres, Generalized fractional operators for nonstandard Lagrangians, Math. Meth. Appl. Sci., 38 (2015), 1808-1812.  doi: 10.1002/mma.3188.

[41]

J. XuZ. Liao and J. J. Nieto, A class of linear differential dynamical systems with fuzzy matrices, J. Math. Anal. Appl., 368 (2010), 54-68.  doi: 10.1016/j.jmaa.2009.12.053.

[42]

D. Yang and K.-Y. Cai, Finite-time quantized guaranteed cost fuzzy control for continuous-time nonlinear systems, Expert Systems with Applications, 37 (2010), 6963-6967.  doi: 10.1016/j.eswa.2010.03.024.

Figure 1.  The fuzzy extremals for the fuzzy optimal control problem (12) of Example 1 under $[(1)-gH]_{\beta}^{C}$-differentiability of $\tilde{x}$
Figure 2.  The fuzzy extremals for the fuzzy optimal control problem (12) of Example 1 under $[(2)-gH]$-differentiability of $\tilde{x}$
Figure 3.  The extremals for the crisp optimal control problem (19) of Example 2
Figure 4.  The fuzzy extremals for the fuzzy optimal control problem (18) of Example 2
Figure 5.  The extremals for the crisp optimal control problem (22) of Example 3
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