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doi: 10.3934/jimo.2021182
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Optimality conditions of singular controls for systems with Caputo fractional derivatives

1. 

Baku State University, Department of Mechanics and Mathematics, Baku, Azerbaijan

2. 

Department of Mathematics, Istanbul Technical University, Istanbul, Turkey, Azerbaijan National Academy of Sciences, Institute of Control Systems, Baku, Azerbaijan

*Corresponding author: Elimhan N. Mahmudov

Received  September 2020 Revised  July 2021 Early access November 2021

In this paper, we consider an optimal control problem in which a dynamical system is controlled by a nonlinear Caputo fractional state equation. The problem is investigated in the case when the Pontryagin maximum principle degenerates, that is, it is satisfied trivially. Then the second order optimality conditions are derived for the considered problem.

Citation: Shakir Sh. Yusubov, Elimhan N. Mahmudov. Optimality conditions of singular controls for systems with Caputo fractional derivatives. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021182
References:
[1]

O. P. AgrawalO. 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.

[2]

N. U. Ahmed and C. D. Charalambous, Filtering for linear systems driven by fractional Brownian motion, SIAM J. Control Optim., 41 (2002), 313-330.  doi: 10.1137/S0363012900368715.

[3]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.

[4]

M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constaints, ESAIM Control Optim. Calc. Var., 26 (2020), 1-38.  doi: 10.1051/cocv/2019021.

[5]

A. Carpinteri, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, 378 (1997), (291–348). doi: 10.1007/978-3-7091-2664-6_7.

[6]

K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Matematics, Vol.2004, Spinger-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[7]

R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.

[8]

Z. GongC. LiuK. L. TeoS. Wang and Y. Wu, Numerical solution of free final time fractional optimal control problems, Appl. Math. Comput., 405 (2021), 1-15.  doi: 10.1016/j.amc.2021.126270.

[9]

M. I. Gomoyunov, On representation formulas for solutions of linear differential equations with Caputo fractional derivatives, Fract. Calc. Appl. Anal., 23 (2020), 1141-1160.  doi: 10.1515/fca-2020-0058.

[10]

T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Aappl., 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[12]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[13]

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

[14]

W. LiS. Wang and V. Rehbock, A 2nd-order one-point numerical integration scheme for fractional ordinary differential equation, Numer. Algebra Control Optim., 7 (2017), 273-287.  doi: 10.3934/naco.2017018.

[15]

W. LiS. Wang and V. Rehbock, Numerical solution of fractional optimal control, J. Optim. Theory Appl., 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y.

[16]

P. Louhan and S. K. Suneja, On fractional vector optimization over cones with support functions, J. Ind. Manag. Optim., 13 (2017), 549-572.  doi: 10.3934/jimo.2016031.

[17]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Inc., Amsterdam, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.

[18]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: COCV. doi: 10.1051/cocv/2019018.

[19]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Ind. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.

[20]

E. N. Mahmudov, Approximation and optimization of higher order discrete and differential inclusions, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.

[21]

B. S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moskow, 1988.

[22]

P. MuL. Wang and C. Liu, A control parameterization method to solve the fractional-order optimal control problem, J. Optim. Theory Appl., 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7.

[23] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Aapplications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishenko, The Mathematical Theory of Optimal Processes, 4$^th$ edition, Nauka, Moskow, 1983,392pp.

[25]

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.

[26]

E. RentsenJ. Zhou and K. L. Teo, A global optimization approach to fractional optimal control, J. Ind. Manag. Optim., 12 (2016), 73-82.  doi: 10.3934/jimo.2016.12.73.

[27]

L. I. Rozonoer, The maximum principle by L. S. Pontryagin in the theory of optimal systems, Ⅰ, Ⅱ, Ⅲ, Automatics and Remote Control, 1959 (1959), 10-12. 

[28]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[29] V. E. Tarasov, Fractional Dynamics; Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Hedelberg, Higher Education Press, Beijing, 2010.  doi: 10.1007/978-3-642-14003-7.
[30]

N. H. TuanD. O'Regan and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory, 9 (2020), 773-793.  doi: 10.3934/eect.2020033.

[31]

S. Westerlund, Dead matter has memory!, Physical Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.

[32]

Z. WuY. Zou and N. Huang, A new class of global fractional-order projective dynamical system with an application, J. Ind. Manag. Optim., 16 (2020), 37-53.  doi: 10.3934/jimo.2018139.

[33]

X. YangS. Y. Wang and X. T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl., 274 (2002), 279-295.  doi: 10.1016/S0022-247X(02)00299-8.

[34]

X. YangX. Q. Yang and K. L. Teo, Duality and saddle-point type optimality for generalized nonlinear fractional programming, J. Math. Anal. Appl., 289 (2004), 100-109.  doi: 10.1016/j.jmaa.2003.08.029.

[35]

X. Yang and S. H. Hou, On minimax fractional optimality and duality with generalized convexity, J. Global Optim., 31 (2005), 235-252.  doi: 10.1007/s10898-004-5698-4.

[36]

C. YuK. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Process., 95 (2014), 32-42. 

[37]

S. S. Yusubov, Necessary optimality conditions for systems with impulsive actions, Comput. Math and Math. Phys., 45 (2005), 222-226. 

[38]

S. S. Yusubov, Necessary optimality conditions for singular controls, Comput. Math. Math. Phys., 47 (2007), 1446-1451.  doi: 10.1134/S0965542507090060.

[39]

S. S. Yusubov, Boundary value problems for hyperbolic equations with a Caputo fractional derivative, Advanced Mathematical Models and Applications, 5 (2020), 192-204. 

[40]

Y. ZhouV. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.

show all references

References:
[1]

O. P. AgrawalO. 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.

[2]

N. U. Ahmed and C. D. Charalambous, Filtering for linear systems driven by fractional Brownian motion, SIAM J. Control Optim., 41 (2002), 313-330.  doi: 10.1137/S0363012900368715.

[3]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.

[4]

M. Bergounioux and L. Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constaints, ESAIM Control Optim. Calc. Var., 26 (2020), 1-38.  doi: 10.1051/cocv/2019021.

[5]

A. Carpinteri, Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, 378 (1997), (291–348). doi: 10.1007/978-3-7091-2664-6_7.

[6]

K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Matematics, Vol.2004, Spinger-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[7]

R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.

[8]

Z. GongC. LiuK. L. TeoS. Wang and Y. Wu, Numerical solution of free final time fractional optimal control problems, Appl. Math. Comput., 405 (2021), 1-15.  doi: 10.1016/j.amc.2021.126270.

[9]

M. I. Gomoyunov, On representation formulas for solutions of linear differential equations with Caputo fractional derivatives, Fract. Calc. Appl. Anal., 23 (2020), 1141-1160.  doi: 10.1515/fca-2020-0058.

[10]

T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, J. Optim. Theory Aappl., 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[12]

R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Methods Appl. Sci., 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.

[13]

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

[14]

W. LiS. Wang and V. Rehbock, A 2nd-order one-point numerical integration scheme for fractional ordinary differential equation, Numer. Algebra Control Optim., 7 (2017), 273-287.  doi: 10.3934/naco.2017018.

[15]

W. LiS. Wang and V. Rehbock, Numerical solution of fractional optimal control, J. Optim. Theory Appl., 180 (2019), 556-573.  doi: 10.1007/s10957-018-1418-y.

[16]

P. Louhan and S. K. Suneja, On fractional vector optimization over cones with support functions, J. Ind. Manag. Optim., 13 (2017), 549-572.  doi: 10.3934/jimo.2016031.

[17]

E. N. Mahmudov, Approximation and Optimization of Discrete and Differential Inclusions, Elsevier, Inc., Amsterdam, 2011. doi: 10.1016/B978-0-12-388428-2.00001-1.

[18]

E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: COCV. doi: 10.1051/cocv/2019018.

[19]

E. N. Mahmudov, Optimization of fourth order Sturm-Liouville type differential inclusions with initial point constraints, J. Ind. Manag. Optim., 16 (2020), 169-187.  doi: 10.3934/jimo.2018145.

[20]

E. N. Mahmudov, Approximation and optimization of higher order discrete and differential inclusions, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 1-26.  doi: 10.1007/s00030-013-0234-1.

[21]

B. S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moskow, 1988.

[22]

P. MuL. Wang and C. Liu, A control parameterization method to solve the fractional-order optimal control problem, J. Optim. Theory Appl., 187 (2020), 234-247.  doi: 10.1007/s10957-017-1163-7.

[23] I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Aapplications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999. 
[24]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishenko, The Mathematical Theory of Optimal Processes, 4$^th$ edition, Nauka, Moskow, 1983,392pp.

[25]

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.

[26]

E. RentsenJ. Zhou and K. L. Teo, A global optimization approach to fractional optimal control, J. Ind. Manag. Optim., 12 (2016), 73-82.  doi: 10.3934/jimo.2016.12.73.

[27]

L. I. Rozonoer, The maximum principle by L. S. Pontryagin in the theory of optimal systems, Ⅰ, Ⅱ, Ⅲ, Automatics and Remote Control, 1959 (1959), 10-12. 

[28]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[29] V. E. Tarasov, Fractional Dynamics; Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Hedelberg, Higher Education Press, Beijing, 2010.  doi: 10.1007/978-3-642-14003-7.
[30]

N. H. TuanD. O'Regan and T. B. Ngoc, Continuity with respect to fractional order of the time fractional diffusion-wave equation, Evol. Equ. Control Theory, 9 (2020), 773-793.  doi: 10.3934/eect.2020033.

[31]

S. Westerlund, Dead matter has memory!, Physical Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.

[32]

Z. WuY. Zou and N. Huang, A new class of global fractional-order projective dynamical system with an application, J. Ind. Manag. Optim., 16 (2020), 37-53.  doi: 10.3934/jimo.2018139.

[33]

X. YangS. Y. Wang and X. T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl., 274 (2002), 279-295.  doi: 10.1016/S0022-247X(02)00299-8.

[34]

X. YangX. Q. Yang and K. L. Teo, Duality and saddle-point type optimality for generalized nonlinear fractional programming, J. Math. Anal. Appl., 289 (2004), 100-109.  doi: 10.1016/j.jmaa.2003.08.029.

[35]

X. Yang and S. H. Hou, On minimax fractional optimality and duality with generalized convexity, J. Global Optim., 31 (2005), 235-252.  doi: 10.1007/s10898-004-5698-4.

[36]

C. YuK. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Process., 95 (2014), 32-42. 

[37]

S. S. Yusubov, Necessary optimality conditions for systems with impulsive actions, Comput. Math and Math. Phys., 45 (2005), 222-226. 

[38]

S. S. Yusubov, Necessary optimality conditions for singular controls, Comput. Math. Math. Phys., 47 (2007), 1446-1451.  doi: 10.1134/S0965542507090060.

[39]

S. S. Yusubov, Boundary value problems for hyperbolic equations with a Caputo fractional derivative, Advanced Mathematical Models and Applications, 5 (2020), 192-204. 

[40]

Y. ZhouV. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Equ. Control Theory, 4 (2015), 507-524.  doi: 10.3934/eect.2015.4.507.

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