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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

Vaibhav Mehandiratta 1, , Mani Mehra 1,, and  Günter Leugering 2,

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  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.

Keywords: Fractional optimal control problem, Caputo fractional derivative, Right Riemann-Liouville derivative, Optimality system, $ L2 $ scheme, Grünwald-Letnikov formula.
Mathematics Subject Classification: Primary:34A08, 49J15, 26A33;Secondary:49K15, 93C15.
Citation: Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar

[13]

A. A. Kilbas and H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.  Google Scholar

[14]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Taylor and Francis group, 2015. doi: 10.1201/b18503.  Google Scholar

[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.  Google Scholar

[24]

G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234.   Google Scholar

[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.   Google Scholar

[26]

F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd, 2014. doi: 10.1142/9069.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.  Google Scholar

[13]

A. A. Kilbas and H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.  Google Scholar

[14]

D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Taylor and Francis group, 2015. doi: 10.1201/b18503.  Google Scholar

[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.  Google Scholar

[24]

G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234.   Google Scholar

[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.   Google Scholar

[26]

F. Mainardi and P. Paradisi, Fractional diffusive waves, Journal of Computational Acoustics, 9 (2001), 1417-1436.  doi: 10.1142/S0218396X01000826.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Springer, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd, 2014. doi: 10.1142/9069.  Google Scholar

Figure 1.  A sketch of the star graph with $ k $ edges along with boundary control
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Figure 2.  Convergence of $ y_i(x) $, $ i=1,2,3 $ for the optimality system $ (50) $ for $ \alpha=3/2 $
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Figure 3.  State variables $ y_i(x) $, $ i=1,2,3 $, for different fractional order $ \alpha $ for the optimality system $ (50) $ with $ N=64 $
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Figure 4.  Convergence of $ y_i(x) $, $ i=1,2,3 $ for the optimality system $ (54) $ for $ \alpha=3/2 $
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Table 1.  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
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$ 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
Table 2.  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
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$ \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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