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March  2021, 11(1): 189-209. doi: 10.3934/mcrf.2020033

## Fractional optimal control problems on a star graph: Optimality system and numerical solution

 1 Department of Mathematics, Indian Institute of Technology Delhi, 110016, Delhi, India 2 Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik Ⅱ, Cauerstr. 11, 91058 Erlangen, Germany

* Corresponding author: Mani Mehra

Received  December 2019 Revised  May 2020 Published  March 2021 Early access  June 2020

In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the $L2$ scheme and the Grünwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.

Citation: Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control and Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
##### References:
 [1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4. [2] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6. [3] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003. [4] O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 14 (2008), 1291-1299.  doi: 10.1177/1077546307087451. [5] R. Almeida and D. F. M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with {C}aputo derivatives, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1490-1500.  doi: 10.1016/j.cnsns.2010.07.016. [6] H. W. Berhe, S. Qureshi and A. A. Shaikh, Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis, Chaos, Solitons & Fractals, 131 (2020), 109536, 13 pp. doi: 10.1016/j.chaos.2019.109536. [7] T. Blaszczyk and M. Ciesielski, Fractional oscillator equation–transformation into integral equation and numerical solution, Applied Mathematics and Computation, 257 (2015), 428-435.  doi: 10.1016/j.amc.2014.12.122. [8] G. W. Bohannan, Analog fractional order controller in temperature and motor control applications, Journal of Vibration and Control, 14 (2008), 1487-1498.  doi: 10.1177/1077546307087435. [9] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [10] A. Debbouche, J. J. Nieto and D. F. M. Torres, Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations, Journal of Optimization Theory and Applications, 174 (2017), 7-31.  doi: 10.1007/s10957-015-0743-7. [11] T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, Journal of Optimization Theory and Applications, 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0. [12] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [13] A. A. Kilbas and H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006. [14] D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004. [15] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling and controllability of networks of thin beams, Lect. Notes Control Inf. Sci., 180 (1992), 467-480.  doi: 10.1007/BFb0113314. [16] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Control of planar networks of Timoshenko beams, SIAM J. Control Optim., 31 (1993), 780-811.  doi: 10.1137/0331035. [17] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503. [18] J. E. Lagnese and G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. [19] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, On the analysis and control of hyperbolic systems associated with vibrating networks, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 77-104.  doi: 10.1017/S0308210500029206. [20] G. Leugering, On the semi-discretization of optimal control problems for networks of elastic strings:global optimality systems and domain decomposition, J. Comput. Appl. Math., 120 (2000), 133-157.  doi: 10.1016/S0377-0427(00)00307-1. [21] G. Leugering, Domain decomposition of an optimal control problem for semi-linear elliptic equations on metric graphs with application to gas networks, Applied Mathematics, 8 (2017), 1074-1099.  doi: 10.4236/am.2017.88082. [22] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Taylor and Francis group, 2015. doi: 10.1201/b18503. [23] A. A. Lotfi and S. A. Yousefi, A numerical technique for solving a class of fractional variational problems, Journal of Computational and Applied Mathematics, 237 (2013), 633-643.  doi: 10.1016/j.cam.2012.08.005. [24] G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234. [25] R. L. Magin and M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, Journal of Vibration and Control, 14 (2008), 1431-1442. [26] F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [27] V. Mehandiratta, M. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, Journal of Mathematical Analysis and Applications, 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011. [28] G. Mophou, G. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, (2020), 1–29. doi: 10.1080/02331934.2020.1730371. [29] G. Mophou, Optimal control for fractional diffusion equations with incomplete data, Journal of Optimization Theory and Applications, 174 (2017), 176-196.  doi: 10.1007/s10957-015-0817-6. [30] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1. [31] K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, J. Comput. Appl. Math., 380 (2020), 112963. doi: 10.1016/j.cam.2020.112963. [32] Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), Journal of Mathematical Sciences, 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa. [33] S. Qureshi and A. Atangana, Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Physica A: Statistical Mechanics and its Applications, 526 (2019), 121127, 19 pp. doi: 10.1016/j.physa.2019.121127. [34] S. Qureshi and P. Kumar, Using Shehu integral transform to solve fractional order Caputo type initial value problems, Journal of Applied Mathematics and Computational Mechanics, 18 (2019), 75-83.  doi: 10.17512/jamcm.2019.2.07. [35] S. Qureshi and A. Yusuf, Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator, Chaos, Solitons & Fractals, 126 (2019), 32-40.  doi: 10.1016/j.chaos.2019.05.037. [36] S. Qureshi, A. Yusuf, A. A. Shaikh and M. Inc, Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data, Physica A: Statistical Mechanics and its Applications, 534 (2019), 122149, 22 pp. doi: 10.1016/j.physa.2019.122149. [37] K. Sayevand and M. Rostami, Fractional optimal control problems: Optimality conditions and numerical solution, IMA Journal of Mathematical Control and Information, 35 (2016), 123-148.  doi: 10.1093/imamci/dnw041. [38] H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Physical Review B, 12 (1975), 2455. doi: 10.1103/PhysRevB.12.2455. [39] A. Shukla, M. Mehra and G. Leugering, A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph, Mathematical Methods in the Applied Sciences, (2019). doi: 10.1002/mma.5907. [40] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd, 2014. doi: 10.1142/9069.

show all references

##### References:
 [1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379.  doi: 10.1016/S0022-247X(02)00180-4. [2] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6. [3] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, Journal of Physics A: Mathematical and Theoretical, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003. [4] O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 14 (2008), 1291-1299.  doi: 10.1177/1077546307087451. [5] R. Almeida and D. F. M. Torres, Necessary and sufficient conditions for the fractional calculus of variations with {C}aputo derivatives, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 1490-1500.  doi: 10.1016/j.cnsns.2010.07.016. [6] H. W. Berhe, S. Qureshi and A. A. Shaikh, Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis, Chaos, Solitons & Fractals, 131 (2020), 109536, 13 pp. doi: 10.1016/j.chaos.2019.109536. [7] T. Blaszczyk and M. Ciesielski, Fractional oscillator equation–transformation into integral equation and numerical solution, Applied Mathematics and Computation, 257 (2015), 428-435.  doi: 10.1016/j.amc.2014.12.122. [8] G. W. Bohannan, Analog fractional order controller in temperature and motor control applications, Journal of Vibration and Control, 14 (2008), 1487-1498.  doi: 10.1177/1077546307087435. [9] R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [10] A. Debbouche, J. J. Nieto and D. F. M. Torres, Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations, Journal of Optimization Theory and Applications, 174 (2017), 7-31.  doi: 10.1007/s10957-015-0743-7. [11] T. L. Guo, The necessary conditions of fractional optimal control in the sense of Caputo, Journal of Optimization Theory and Applications, 156 (2013), 115-126.  doi: 10.1007/s10957-012-0233-0. [12] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747. [13] A. A. Kilbas and H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006. [14] D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004. [15] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling and controllability of networks of thin beams, Lect. Notes Control Inf. Sci., 180 (1992), 467-480.  doi: 10.1007/BFb0113314. [16] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Control of planar networks of Timoshenko beams, SIAM J. Control Optim., 31 (1993), 780-811.  doi: 10.1137/0331035. [17] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling of dynamic networks of thin thermoelastic beams, Math. Methods Appl. Sci., 16 (1993), 327-358.  doi: 10.1002/mma.1670160503. [18] J. E. Lagnese and G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8. [19] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, On the analysis and control of hyperbolic systems associated with vibrating networks, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 77-104.  doi: 10.1017/S0308210500029206. [20] G. Leugering, On the semi-discretization of optimal control problems for networks of elastic strings:global optimality systems and domain decomposition, J. Comput. Appl. Math., 120 (2000), 133-157.  doi: 10.1016/S0377-0427(00)00307-1. [21] G. Leugering, Domain decomposition of an optimal control problem for semi-linear elliptic equations on metric graphs with application to gas networks, Applied Mathematics, 8 (2017), 1074-1099.  doi: 10.4236/am.2017.88082. [22] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Taylor and Francis group, 2015. doi: 10.1201/b18503. [23] A. A. Lotfi and S. A. Yousefi, A numerical technique for solving a class of fractional variational problems, Journal of Computational and Applied Mathematics, 237 (2013), 633-643.  doi: 10.1016/j.cam.2012.08.005. [24] G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234. [25] R. L. Magin and M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, Journal of Vibration and Control, 14 (2008), 1431-1442. [26] F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826. [27] V. Mehandiratta, M. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, Journal of Mathematical Analysis and Applications, 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011. [28] G. Mophou, G. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, (2020), 1–29. doi: 10.1080/02331934.2020.1730371. [29] G. Mophou, Optimal control for fractional diffusion equations with incomplete data, Journal of Optimization Theory and Applications, 174 (2017), 176-196.  doi: 10.1007/s10957-015-0817-6. [30] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1. [31] K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, J. Comput. Appl. Math., 380 (2020), 112963. doi: 10.1016/j.cam.2020.112963. [32] Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), Journal of Mathematical Sciences, 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa. [33] S. Qureshi and A. Atangana, Mathematical analysis of dengue fever outbreak by novel fractional operators with field data, Physica A: Statistical Mechanics and its Applications, 526 (2019), 121127, 19 pp. doi: 10.1016/j.physa.2019.121127. [34] S. Qureshi and P. Kumar, Using Shehu integral transform to solve fractional order Caputo type initial value problems, Journal of Applied Mathematics and Computational Mechanics, 18 (2019), 75-83.  doi: 10.17512/jamcm.2019.2.07. [35] S. Qureshi and A. Yusuf, Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator, Chaos, Solitons & Fractals, 126 (2019), 32-40.  doi: 10.1016/j.chaos.2019.05.037. [36] S. Qureshi, A. Yusuf, A. A. Shaikh and M. Inc, Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data, Physica A: Statistical Mechanics and its Applications, 534 (2019), 122149, 22 pp. doi: 10.1016/j.physa.2019.122149. [37] K. Sayevand and M. Rostami, Fractional optimal control problems: Optimality conditions and numerical solution, IMA Journal of Mathematical Control and Information, 35 (2016), 123-148.  doi: 10.1093/imamci/dnw041. [38] H. Scher and E. W. Montroll, Anomalous transit-time dispersion in amorphous solids, Physical Review B, 12 (1975), 2455. doi: 10.1103/PhysRevB.12.2455. [39] A. Shukla, M. Mehra and G. Leugering, A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph, Mathematical Methods in the Applied Sciences, (2019). doi: 10.1002/mma.5907. [40] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd, 2014. doi: 10.1142/9069.
A sketch of the star graph with $k$ edges along with boundary control
Convergence of $y_i(x)$, $i=1,2,3$ for the optimality system $(50)$ for $\alpha=3/2$
State variables $y_i(x)$, $i=1,2,3$, for different fractional order $\alpha$ for the optimality system $(50)$ with $N=64$
Convergence of $y_i(x)$, $i=1,2,3$ for the optimality system $(54)$ for $\alpha=3/2$
Control variable $u=(u_1,u_2,u_3)$ for different values of $N$
 $N$ $u_1$ $u_2$ $u_3$ 32 .1867 .1792 .1749 64 .1834 .1762 .1718 128 .1817 .1746 .1702 256 .1808 .1738 .1694 512 .1804 .1734 .1690 1024 .1802 .1732 .1688
 $N$ $u_1$ $u_2$ $u_3$ 32 .1867 .1792 .1749 64 .1834 .1762 .1718 128 .1817 .1746 .1702 256 .1808 .1738 .1694 512 .1804 .1734 .1690 1024 .1802 .1732 .1688
Control variable $u=(u_1,u_2,u_3)$ for different fractional order $\alpha$ with $N=64$
 $\alpha$ $u_1$ $u_2$ $u_3$ 1.2 .2017 .1959 .1910 1.4 .1894 .1824 .1778 1.6 .1775 .1703 .1662 1.8 .1666 .1598 .1563 2 .1572 .1511 .1482
 $\alpha$ $u_1$ $u_2$ $u_3$ 1.2 .2017 .1959 .1910 1.4 .1894 .1824 .1778 1.6 .1775 .1703 .1662 1.8 .1666 .1598 .1563 2 .1572 .1511 .1482
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