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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 & Management Optimization, doi: 10.3934/jimo.2021182
##### References:
 [1] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.  Google Scholar [8] Z. Gong, C. Liu, K. L. Teo, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Li, S. 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.  Google Scholar [15] W. Li, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: COCV. doi: 10.1051/cocv/2019018.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] B. S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moskow, 1988.  Google Scholar [22] P. Mu, L. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [25] S. Pooseh, R. 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.  Google Scholar [26] E. Rentsen, J. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [30] N. H. Tuan, D. 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.  Google Scholar [31] S. Westerlund, Dead matter has memory!, Physical Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.  Google Scholar [32] Z. Wu, Y. 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.  Google Scholar [33] X. Yang, S. 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.  Google Scholar [34] X. Yang, X. 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.  Google Scholar [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.  Google Scholar [36] C. Yu, K. 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.   Google Scholar [37] S. S. Yusubov, Necessary optimality conditions for systems with impulsive actions, Comput. Math and Math. Phys., 45 (2005), 222-226.   Google Scholar [38] S. S. Yusubov, Necessary optimality conditions for singular controls, Comput. Math. Math. Phys., 47 (2007), 1446-1451.  doi: 10.1134/S0965542507090060.  Google Scholar [39] S. S. Yusubov, Boundary value problems for hyperbolic equations with a Caputo fractional derivative, Advanced Mathematical Models and Applications, 5 (2020), 192-204.   Google Scholar [40] Y. Zhou, V. 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.  Google Scholar

show all references

##### References:
 [1] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] R. Gabasov and F. M. Kirillova, High-order necessary conditions for optimality, SIAM J. Control, 10 (1972), 127-168.  doi: 10.1137/0310012.  Google Scholar [8] Z. Gong, C. Liu, K. L. Teo, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Li, S. 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.  Google Scholar [15] W. Li, S. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] E. N. Mahmudov, Optimal control of higher order differential inclusions with functional constraints, ESAIM: COCV. doi: 10.1051/cocv/2019018.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] B. S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moskow, 1988.  Google Scholar [22] P. Mu, L. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [25] S. Pooseh, R. 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.  Google Scholar [26] E. Rentsen, J. 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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [30] N. H. Tuan, D. 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.  Google Scholar [31] S. Westerlund, Dead matter has memory!, Physical Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.  Google Scholar [32] Z. Wu, Y. 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.  Google Scholar [33] X. Yang, S. 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.  Google Scholar [34] X. Yang, X. 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.  Google Scholar [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.  Google Scholar [36] C. Yu, K. 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.   Google Scholar [37] S. S. Yusubov, Necessary optimality conditions for systems with impulsive actions, Comput. Math and Math. Phys., 45 (2005), 222-226.   Google Scholar [38] S. S. Yusubov, Necessary optimality conditions for singular controls, Comput. Math. Math. Phys., 47 (2007), 1446-1451.  doi: 10.1134/S0965542507090060.  Google Scholar [39] S. S. Yusubov, Boundary value problems for hyperbolic equations with a Caputo fractional derivative, Advanced Mathematical Models and Applications, 5 (2020), 192-204.   Google Scholar [40] Y. Zhou, V. 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.  Google Scholar
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