The Necessary and Sufficient Optimality Conditions for a System of FOCPs with Caputo–Katugampola Derivatives
Keywords:Calculus of variations, Caputo–Katugampola fractional derivative, Hamiltonian system, Necessary and sufficient optimality conditions, Optimal control
The necessary optimality conditions with Lagrange multipliers are studied and derived for a new class that includes the system of Caputo–Katugampola fractional derivatives to the optimal control problems with considering the end time free. The formula for the integral by parts has been proven for the left Caputo–Katugampola fractional derivative that contributes to the finding and deriving the necessary optimality conditions. Also, three special cases are obtained, including the study of the necessary optimality conditions when both the final time and the final state are fixed. According to convexity assumptions prove that necessary optimality conditions are sufficient optimality conditions.
Received 8/6/2022, Revised 19/7/2022, Accepted 21/7/2022, Published Online First 20/2/2023
Hoang MT, Nagy AM. Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes. Chaos Solitons Fractals. 2019; 123: 24-34.
Mohammadi F, Moradi L, Baleanu D, Jajarmi A. A hybrid functions numerical scheme for fractional optimal control problems: application to nonanalytic dynamic systems. J Vib Control. 2018; 24(21): 5030-5043.
Kahouli O, Jmal A, Naifar O, Nagy AM, Ben Makhlouf AB. New Result for the Analysis of Katugampola Fractional-Order Systems—Application to Identification Problems. Mathematics. 2022 May 25; 10(11): 1-17.
Lin Z, Wang H. Modeling and application of fractional-order economic growth model with time delay. Fractal Fract. 2021; 5(3): 1-18.
Heydari MH, Razzaghi M. A new class of orthonormal basis functions: application for fractional optimal control problems. Int J Syst Sci. 2022; 53(2): 240-252.
Luo D, Wang JR, Fečkan M. Applying Fractional Calculus to Analyze Economic Growth Modelling. JAMSI. 2018; 14(1): 25-36.
Heydari MH, Razzaghi M. Piecewise Chebyshev cardinal functions: Application for constrained fractional optimal control problems. Chaos Solitons Fractals. 2021; 150: 1-11.
Agrawal OP. A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 2004; 38(1): 323-337.
Alizadeh A, Effati S. Modified Adomian decomposition method for solving fractional optimal control problems. Trans Inst Meas Control. 2018; 40(6): 2054-2061.
Hasan SQ, Abbas Holel MA. Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems. J Control Sci Eng. 2018; 1-12.
Chiranjeevi T, Biswas RK. Closed-form solution of optimal control problem of a fractional order system. J King Saud Univ Sci. 2019; 31(4): 1042-1047.
Yildiz TA, Jajarmi A, Yildiz B, Baleanu D. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discrete Contin Dyn Syst - S. 2020; 13(3): 407–428.
Gong Z, Liu C, Teo KL, Wang S, Wu Y. Numerical solution of free final time fractional optimal control problems. Appl Math Comput. 2021; 405: 1-15.
Azar AT, Serrano FE, Kamal NA. Optimal fractional order control for nonlinear systems represented by the Euler-Lagrange formulation. Int J Model Identif Control. 2021; 37(1): 1-9.
Salman NK, Mustafa MM. Numerical solution of fractional Volterra-Fredholm integro-differential equation using Lagrange polynomials. Baghdad Sci J. 2020; 17: 1234-1240.
AL-Safi MG. An efficient numerical method for solving Volterra-Fredholm integro-differential equations of fractional order by using shifted Jacobi-spectral collocation method. Baghdad Sci J. 2018; 15(3): 344-351.
Katugampola UN. New approach to a generalized fractional integral. Appl Math Comput. 2011; 218(3): 860-865.
Katugampola UN. A new approach to generalized fractional derivatives. Bull Math Anal Appl. 2014; 6(4): 1-15.
Almutairi O, Kılıçman A. New generalized Hermite-Hadamard inequality and related integral inequalities involving Katugampola type fractional integrals. Symmetry. 2020; 12(4): 1-14.
Dineshkumar C, Udhayakumar R, Vijayakumar V, Nisar KS, Shukla A. A note concerning to approximate controllability of Atangana-Baleanu fractional neutral stochastic systems with infinite delay. Chaos Solitons Fractals. 2022; 157: 1-17.
Almeida R, Malinowska AB, Odzijewicz T. Fractional Differential Equations with Dependence on the Caputo–Katugampola Derivative. J Comput Nonlinear Dynam. November 2016; 11(6): 1-11.
Rao SS. Engineering optimization: Theory and Practice. 5th edition. USA: John Wiley & Sons; 2019: 781-782.
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