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September  2017, 7(3): 449-464. doi: 10.3934/mcrf.2017016

Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative

Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

* Corresponding author: Rafał Kamocki

Received  February 2016 Revised  November 2016 Published  July 2017

Fund Project: The project was financed with funds of National Science Centre, granted on the basis of decision DEC-2011/01/B/ST7/03426.

In the paper, a nonlinear control system containing the Riemann-Liouville derivative of order $α∈(0, 1)$ with a nonlinear integral performance index is studied. We discuss the existence of optimal solutions to such problem under some convexity assumption. Our study relies on the implicit function theorem for multivalued mappings.

Citation: 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
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]

T. Akbarian and M. Keyanpour, A new approach to the numerical solution of fractional order optimal control problems, Applications and Applied Mathematics, 8 (2013), 523-534. 

[3]

R. AlmeidaS. Pooseh and D. F. M. Torres, Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75 (2012), 1009-1025.  doi: 10.1016/j.na.2011.02.028.

[4]

R. Almeida and D. F. M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22 (2009), 1816-1820.  doi: 10.1016/j.aml.2009.07.002.

[5]

R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behaviour, Journal of Rheology, 30 (1986), 133-155. 

[6]

S. Banach, Teoria Operacyj I, Warsaw, 1931 (in Polish).

[7]

L. Bourdin and D. Idczak, Fractional fundamental lemma and fractional integration by parts formula -Applications to critical points of Bolza functionals and to linear boundary value problems, Adv. Differential Equations, 20 (2015), 213-232. 

[8]

L. BourdinT. Odzijewicz and D. F. Torres, Existence of minimizers for fractional variational problems containing Caputo derivatives, Advances in Dynamical Systems and Applications, 8 (2013), 3-12. 

[9]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007), 033504, 34pp. doi: 10.1063/1.2483292.

[10]

A. Dzielinski, G. Sarwas and D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models, Advances in Difference Equations, 2011 (2011), 15pp.

[11]

A. DzielinskiD. Sierociuk and G. Sarwas, Some applications of fractional order calculus, Bulletin of the Polish Academy of Sciences: Technical Sciences, 58 (2010), 583-592.  doi: 10.2478/v10175-010-0059-6.

[12]

G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334 (2007), 834-846.  doi: 10.1016/j.jmaa.2007.01.013.

[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]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033.  doi: 10.1016/j.amc.2010.01.100.

[15]

F. Ghomanjani, A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations, Journal of the Egyptian Mathematical Society, 24 (2016), 638-643.  doi: 10.1016/j.joems.2015.12.003.

[16]

R. Herrmann, Fractional Calculus. An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2014. doi: 10.1142/8934.

[17]

R. Hilfer, Applications of Fractional Calculus in Physics, 463 pages, World Sci. Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[18]

D. Idczak and M. Majewski, Fractional fundamental lemma of order $α∈ (n-\frac{1}{2}, n)$ with $n∈\mathbb{N}$, $n≥q 2$, Dynamic Systems and Applications, 21 (2012), 251-268. 

[19]

D. Idczak and S. Walczak, A fractional imbedding theorem, Fract.Calc.Appl.Anal., 15 (2012), 418-425.  doi: 10.2478/s13540-012-0030-3.

[20]

A. D. Ioffe and V. M. Tikchomirov, Theory of Extremal Problems, North-Holland, 1979.

[21]

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

[22]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences, 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[23]

R. Kamocki, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 235 (2014), 94-104.  doi: 10.1016/j.amc.2014.02.086.

[24]

M. M. Khader and A. S. Hendy, An efficient numerical scheme for solving fractional optimal control problems, International Journal of Nonlinear Science, 14 (2012), 287-296. 

[25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 523 pages, Elsevier, Amsterdam, 2006.

[26]

M. Kisielewicz, Differential Inclusions and Optimal Control, 240 pages, PWN with Kluwer Academic Puplishers, Warsaw-Dordrecht, 1991.

[27]

M. Klimek, On Solutions of Linear Fractional Differential Equations of A Variational Type, The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009.

[28]

T. A. M. Langlands, Solution of a modified fractional diffusion equation, Physica a Statistical Mechanics and its Applications, 367 (2006), 136-144.  doi: 10.1016/j.physa.2005.12.012.

[29]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Applied Mathematics Letters, 25 (2012), 1941-1946.  doi: 10.1016/j.aml.2012.03.006.

[30]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, 300 pages, Imperial College Press, London, 2012. doi: 10.1142/p871.

[31]

A. B. Malinowska and D. F. M. Torres, Multiobjective fractional variational calculus in terms of a combined Caputo derivative, Applied Mathematics and Computation, 218 (2012), 5099-5111.  doi: 10.1016/j.amc.2011.10.075.

[32]

M. W. Michalski, Derivatives of noninteger order and their applications, Dissertations Mathematicae, (Rozprawy Mat. ), 328 (1993), 47pp. doi: 94k: 26011.

[33]

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.

[34]

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

[35]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592.  doi: 10.1103/PhysRevE.55.3581.

[36]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives -Theory and Applications, 976 pages, Gordon and Breach: Amsterdam, 1993.

[37]

Ch. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics International Journal of Differential Equation (2010), Article ID 461048, 16 pages. doi: 2011c: 49051.

[38]

S. Westerlund and L. Ekstam, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1 (1994), 826-839.  doi: 10.1109/94.326654.

[39]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

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]

T. Akbarian and M. Keyanpour, A new approach to the numerical solution of fractional order optimal control problems, Applications and Applied Mathematics, 8 (2013), 523-534. 

[3]

R. AlmeidaS. Pooseh and D. F. M. Torres, Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75 (2012), 1009-1025.  doi: 10.1016/j.na.2011.02.028.

[4]

R. Almeida and D. F. M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22 (2009), 1816-1820.  doi: 10.1016/j.aml.2009.07.002.

[5]

R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behaviour, Journal of Rheology, 30 (1986), 133-155. 

[6]

S. Banach, Teoria Operacyj I, Warsaw, 1931 (in Polish).

[7]

L. Bourdin and D. Idczak, Fractional fundamental lemma and fractional integration by parts formula -Applications to critical points of Bolza functionals and to linear boundary value problems, Adv. Differential Equations, 20 (2015), 213-232. 

[8]

L. BourdinT. Odzijewicz and D. F. Torres, Existence of minimizers for fractional variational problems containing Caputo derivatives, Advances in Dynamical Systems and Applications, 8 (2013), 3-12. 

[9]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys. 48 (2007), 033504, 34pp. doi: 10.1063/1.2483292.

[10]

A. Dzielinski, G. Sarwas and D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models, Advances in Difference Equations, 2011 (2011), 15pp.

[11]

A. DzielinskiD. Sierociuk and G. Sarwas, Some applications of fractional order calculus, Bulletin of the Polish Academy of Sciences: Technical Sciences, 58 (2010), 583-592.  doi: 10.2478/v10175-010-0059-6.

[12]

G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 334 (2007), 834-846.  doi: 10.1016/j.jmaa.2007.01.013.

[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]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033.  doi: 10.1016/j.amc.2010.01.100.

[15]

F. Ghomanjani, A numerical technique for solving fractional optimal control problems and fractional Riccati differential equations, Journal of the Egyptian Mathematical Society, 24 (2016), 638-643.  doi: 10.1016/j.joems.2015.12.003.

[16]

R. Herrmann, Fractional Calculus. An Introduction for Physicists, Second edition. World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2014. doi: 10.1142/8934.

[17]

R. Hilfer, Applications of Fractional Calculus in Physics, 463 pages, World Sci. Publishing, River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[18]

D. Idczak and M. Majewski, Fractional fundamental lemma of order $α∈ (n-\frac{1}{2}, n)$ with $n∈\mathbb{N}$, $n≥q 2$, Dynamic Systems and Applications, 21 (2012), 251-268. 

[19]

D. Idczak and S. Walczak, A fractional imbedding theorem, Fract.Calc.Appl.Anal., 15 (2012), 418-425.  doi: 10.2478/s13540-012-0030-3.

[20]

A. D. Ioffe and V. M. Tikchomirov, Theory of Extremal Problems, North-Holland, 1979.

[21]

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

[22]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences, 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[23]

R. Kamocki, On the existence of optimal solutions to fractional optimal control problems, Appl. Math. Comput., 235 (2014), 94-104.  doi: 10.1016/j.amc.2014.02.086.

[24]

M. M. Khader and A. S. Hendy, An efficient numerical scheme for solving fractional optimal control problems, International Journal of Nonlinear Science, 14 (2012), 287-296. 

[25]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 523 pages, Elsevier, Amsterdam, 2006.

[26]

M. Kisielewicz, Differential Inclusions and Optimal Control, 240 pages, PWN with Kluwer Academic Puplishers, Warsaw-Dordrecht, 1991.

[27]

M. Klimek, On Solutions of Linear Fractional Differential Equations of A Variational Type, The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009.

[28]

T. A. M. Langlands, Solution of a modified fractional diffusion equation, Physica a Statistical Mechanics and its Applications, 367 (2006), 136-144.  doi: 10.1016/j.physa.2005.12.012.

[29]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Applied Mathematics Letters, 25 (2012), 1941-1946.  doi: 10.1016/j.aml.2012.03.006.

[30]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, 300 pages, Imperial College Press, London, 2012. doi: 10.1142/p871.

[31]

A. B. Malinowska and D. F. M. Torres, Multiobjective fractional variational calculus in terms of a combined Caputo derivative, Applied Mathematics and Computation, 218 (2012), 5099-5111.  doi: 10.1016/j.amc.2011.10.075.

[32]

M. W. Michalski, Derivatives of noninteger order and their applications, Dissertations Mathematicae, (Rozprawy Mat. ), 328 (1993), 47pp. doi: 94k: 26011.

[33]

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.

[34]

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

[35]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592.  doi: 10.1103/PhysRevE.55.3581.

[36]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives -Theory and Applications, 976 pages, Gordon and Breach: Amsterdam, 1993.

[37]

Ch. Tricaud and Y. Chen, Time-optimal control of systems with fractional dynamics International Journal of Differential Equation (2010), Article ID 461048, 16 pages. doi: 2011c: 49051.

[38]

S. Westerlund and L. Ekstam, Capacitor theory, IEEE Transactions on Dielectrics and Electrical Insulation, 1 (1994), 826-839.  doi: 10.1109/94.326654.

[39]

H. YeJ. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.

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