In this study we apply the Adomian decomposition method (ADM) to approximate the solution of fractional optimal control problems (FOCPs) where the dynamic of system is a linear control system with constant coefficient and the cost functional is defined in a quadratic form. First we stated the necessary optimality conditions in a form of fractional two point boundary value problem (TPBVP), then the ADM is used to solve the resulting fractional differential equations (FDEs). Some examples are provided to demonstrate the validity and applicability of the proposed method.
Alam, K. N., Uddin, K. N., Asmat, A., and Muhammad, J. (2012). Approximate analytical solutions of fractional reaction-diffusion equations. J King Saud Unive Sci., 24, 111-118.
Arikoglu, A., and Ozkol, I. (2007). Solution of a fractional differential equations by using differential transform method. Chaos Solit Fractals, 34, 1473-1481.
Alipour, M., Rostamy, D. and Baleanu, D. (2013). Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. Journal of Vibration and Control, 19, 2523-2540.
Alizadeh, A., and Effati, S. (2016). An iterative approach for solving fractional optimal control problems. Journal of Vibration and Control, 8, 1-19.
Adomian, G. (1989). Nonlinear Stochastic Systems Theory and Applications to Physics. Kluwer Academic Publishers Kluwer.
Adomian, G. (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135, 501-544.
Adomian, G. (1991). Solving frontier problems modelled by nonlinear partial differential equations. Computers and Mathematics with Applications, 22, 91-94.
Ahmed, E., and Elgazzar, A. S. (2007). On fractional order differential equations model for nonlocal epidemics. Physica A, 379, 607-614.
Agrawal, O. P. (2004). A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics, 38, 323-337.
Agrawal, O. P. (2008). A Formulation and Numerical Scheme for Fractional Optimal Control Problems. Journal of Vibration and Control, 14, 1291-1299.
Abbaoui, K., and Cherruault, Y. (1994). Convergence of Adomian Method Applied to Nonlinear Equations. Mathematical and Computer Modelling, 20, 69-73.
Akbarian, T., and Keyanpour, M. (2013). A New Approach to the Numerical Solution of Fractional Order Optimal Control Problems. Applications and Applied Mathematics, 8, 523-534.
Bhrawy, A. H., Doha, E. H., Machado, J. A. T. and Ezz-Eldien, S. S. (2015). An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian Journal of Control, 18, 1-14.
Bohannan, G. W. (2008). Analog fractional order controller in temperature and motor control applications. Journal of Vibration and Control, 14, 1487-1498.
Changpin, Li., and Yihong, W. (2009). Numerical algorithm based on Adomian decomposition for fractional differential equations. Comput Math Appl, 57, 1672-1681.
Cole, K. S. (1993). Electric conductance of biological systems. in: Proc. Cold Spring Harbor Symp. Quant. Biol, Cold Spring Harbor, New York, 107-116.
Duan, J., Jianye, An., and Mingyu, Xu. (2007). Solution of system of fractional differential equations by Adomian decomposition method. Appl Mathematics-A J Chin Univ, 22, 7-12.
Duan, J., Temuer, C., Randolph, R., and Lei, L. (2013). The Adomian decomposition method with convergence acceleration techniques for nonlinear fractional differential equations. Comput Math Appl, 66, 728-736.
Das, S. (2009). Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl, 57, 483-437.
Duan, J.S., Rach, R., Baleanu, D., and Wazwaz A. M. (2012). A review of the Adomian decomposition method and its applications to fractional differential equations. Commun. Fractional Calculus, 3, 73-99.
Erturk, V. S., and Momani, S. (2008). Solving systems of fractional differential equations using differential transform method. J Comput Appl Math, 215, 142-151.
Hosseini, M. M., and Nasabzadeh, H. (2006). On the convergence of Adomian decomposition method. Applied Mathematics and Computation, 182, 536-543.
Hilfer, R. (2000). Applications of fractional calculus in physics. Singapore, Word Scientific Company.
Inc, M. (2008). The approximate and exact solutions of the space- and time-fractional Burger’s equations with initial conditions by variational iteration method. J Math Anal Appl, 345, 476-484.
Jesus, I. S., Machado, J. A. T., and Cunha J. B. (2008). Fractional electrical impedances in botanical elements. Journal of Vibration and Control, 14, 1389-1402.
Jesus, I. S., Machado, J. A. T., and Cunha J. B. (2006). Fractional order electrical impedance of fruits and vegetables. in: Proceedings of the 25th IASTED International Conference MODELLING, IDENTIFICATION, AND CONTROL, February 6-8, Lanzarote, Canary Islands, Spain.
Jiang, Y., and Ma, J. (2011). Higher order finite element methods for time fractional partial differential equations. J Comput Appl Math, 235, 3285-3290.
Kilbas, A. A. A., Srivastava, H. M., and Trujillo J. J. (2006). Theory and applications of fractional differential equations. (Vol. 204). Elsevier Science Limited.
Liu, J., and Hou, G. (2011). Numerical solutions of the space and time fractional coupled Burgers equation by generalized differential transform method. Appl Math Comput, 217, 7001-7008.
Momani, S., and Odibat, Z. (2007). Numerical approach to differential equations of fractional orders. J Comput Appl Math, 207, 96-110.
Meerschaert, M., and Tadjeran, C. (2006). Finite difference approximations for two sided space fractional partial differential equations. Appl Numer Math, 56, 80-90.
Nemati, A., and Yousefi, S.A. (2016). A Numerical Method for Solving Fractional Optimal Control Problems Using Ritz Method. Journal of Computational and Nonlinear Dynamics, 11, 051015-1-051015-7.
Nigmatullin, R. R., and Nelson, S.O. (2006). Recognition of the fractional kinetics in complex systems: Dielectric properties of fresh fruits and vegetables form 0.01 to1.8 GHz. Signal Processing, 86, 2744-2759.
Odibat, Z., and Momani, S. (2008). Numerical methods for nonlinear partial differential equations of fractional order. Appl Math Model, 32, 28-39.
Odibat, Z., Momani, S., and Erturk, V. S. (2008). Generalized differential transform method: application to differential equations of fractional order. Appl Math Comput, 197,467-477.
Pandey, R. K., Singh, O. P., and Baranwal, V. K. (2011). An analytic algorithm for the space-time fractional advectionedispersion equation. Comput Phys Commun, 182, 1134-44.
Petrovic, L. M., Spasic, D. T., and Atanackovic, T. M. (2005). On a mathematical model of a human root dentin. Dental Materials, 21, 125-128.
Ray, S. S., and Bera, R. K. (2005). Analytical solution of BagleyeTorvik equation by Adomian decomposition method. Appl Math Comput, 168, 398-410.
Saadatmandi, A., and Dehghan, M. (2010). A new operational matrix for solving fractional order differential equations. Comput Math Appl, 59, 1326-36.
Surez, J. I., Vinagre, B. M., and Chen, Y. (2008). A fractional adaptation scheme for lateral control of an AGV. Journal of Vibration and Control, 14, 1499-1511.
Samko, S. G., Kilbas, A. A., and Marichev O. I. (1993). Fractional integrals and derivatives. Theory and Applications, Gordon and Breach, Yverdon.
Wazwaz, A. M., and El-Sayed, S. M. (2001). A new modification of the Adomian decomposition method for linear and nonlinear operators. Applied Mathematics and Computation, 122, 393-405.
Zamani, M., Karimi-Ghartemani, M., and Sadati, N. (2007). FOPID controller design for robust performance using particle swarm optimization. Fractional Calculus and Applied Analysis, 10, 169-187.