doi: 10.3934/mcrf.2022015
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Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph

1. 

Department of Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11 (03.322), 91058 Erlangen, Germany

2. 

Laboratoire L.A.M.I.A., Département de Mathématiques et Informatique, Université des Antilles, Campus Fouillole, 97159 Pointe-à-Pitre, (FWI), Guadeloupe, Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347 Ouagadougou 06, Burkina Faso

3. 

Department of Mathematics, University of Buea, Camerron, African Institute for Mathematical Sciences (AIMS), P.O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon

4. 

Department of Mathematical Sciences, Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, 4400 University Dr, Fairfax, VA 22030, USA

* Corresponding author: Mahamadi Warma

Received  May 2021 Revised  February 2022 Early access March 2022

Fund Project: The third author is supported by the Deutscher Akademischer Austausch Dienst/German Academic Exchange Service (DAAD). The fourth author is partially supported by the AFOSR under Award NO: FA9550-18-1-0242 and by the US Army Research Office (ARO) under Award NO: W911NF-20-1-0115

In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.

Citation: Günter Leugering, Gisèle Mophou, Maryse Moutamal, Mahamadi Warma. Optimal control problems of parabolic fractional Sturm-Liouville equations in a star graph. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022015
References:
[1]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.

[3]

Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, Int. J. Comput. Math., 87 (2010), 2837-2845.  doi: 10.1080/00207160802562549.

[4]

F. Ali Mehmeti, Nonlineares in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.

[5]

E. AlvarezC. G. GalV. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.

[6]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and optimization. Second edition. MOS-SIAM Series on Optimization, 17. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[7]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001.

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

[9]

U. Brauer and G. Leugering, On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings, Recent Advances in Control of PDEs. Control Cybernet, 28 (1999), 421-447. 

[10]

Z.-Q. ChenM. M. Meerschaert and E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., 393 (2012), 479-488.  doi: 10.1016/j.jmaa.2012.04.032.

[11]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623.  doi: 10.1137/S0363012903421844.

[12]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], 50. doi: 10.1007/3-540-37726-3.

[13]

L. de Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223. 

[14]

R. DorvilleG. Mophou and V. S. Valmorin, Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1472-1481.  doi: 10.1016/j.camwa.2011.03.025.

[15]

C. G. Gal and M. Warma, Fractional-in-time Semilinear Parabolic Equations and Applications, Mathématiques & Applications (Berlin) [Mathematics & Applications], 84. Springer, Cham, [2020], ⓒ2020. doi: 10.1007/978-3-030-45043-4.

[16]

N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45 (2006), 765-771. 

[17]

D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Spaces Appl., 2013 (2013), Art. ID 128043, 15 pp. doi: 10.1155/2013/128043.

[18]

H. Khosravian-ArabM. Dehghan and M. R. Eslahchi, Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications, J. Comput. Phys., 299 (2015), 526-560.  doi: 10.1016/j.jcp.2015.06.030.

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[20]

M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.  doi: 10.1016/j.camwa.2012.12.011.

[21]

M. KlimekA. B. Malinowska and T. Odzijewicz, Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain, Fract. Calc. Appl. Anal., 19 (2016), 516-550.  doi: 10.1515/fca-2016-0027.

[22]

M. KlimekT. Odzijewicz and A. B. Malinowska, Variational methods for the fractional Sturm-Liouville problem, J. Math. Anal. Appl., 416 (2014), 402-426.  doi: 10.1016/j.jmaa.2014.02.009.

[23]

J. E. Lagnese and G. Leugering, Domain Decomposition Methods in Optimal Control of Partial Differential Equations, International Series of Numerical Mathematics, 148. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7885-2.

[24]

G. Leugering, Reverberation analysis and control of networks of elastic strings, Control of Partial Differential Equations and Applications (Laredo, 1994), 193–206, Lecture Notes in Pure and Appl. Math., 174, Dekker, New York, 1996.

[25]

G. Leugering and G. Mophou, Instantaneous optimal control of friction dominated flow in a gas-network, Shape Optimization, Homogenization and Optimal Control, 75–88, Internat. Ser. Numer. Math., 169, Birkhäuser/Springer, Cham, 2018.

[26]

J.-L. Lions, Ëquations Différentielles Opérationnelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Band 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[27]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Avant propos de P. Lelong Dunod, Paris; Gauthier-Villars, Paris, 1968.

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972.

[29]

G. Lumer, Connecting of local operators and evolution equations on networks, Potential Theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), 219–234, Lecture Notes in Math., 787, Springer, Berlin, 1980.

[30]

V. MehandirattaM. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, J. Math. Anal. Appl., 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011.

[31]

V. MehandirattaM. Mehra and G. Leugering, Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: a study of fractional calculus on metric graph, Netw. Heterog. Media, 16 (2021), 155-185.  doi: 10.3934/nhm.2021003.

[32]

V. MehandirattaM. Mehra and G. Leugering, Fractional optimal control problems on a star graph: Optimality system and numerical solution, Math. Control Relat. Fields, 11 (2021), 189-209.  doi: 10.3934/mcrf.2020033.

[33]

V. MehandirattaM. Mehra and G. Leugering, Optimal control problems driven by time-fractional diffusion equations on metric graphs: optimality system and finite difference approximation, SIAM J. Control Optim., 59 (2021), 4216-4242.  doi: 10.1137/20M1340332.

[34]

G. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl., 61 (2011), 68-78.  doi: 10.1016/j.camwa.2010.10.030.

[35]

G. MophouG. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, 70 (2021), 659-687.  doi: 10.1080/02331934.2020.1730371.

[36]

J. NakagawaK. Sakamoto and M. Yamamoto, Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration, J. Math-for-Ind., 2A (2010), 99-108. 

[37]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[38]

M. RiveroJ. Trujillo and M. Velasco, A fractional approach to the Sturm-Liouville problem, Open Physics, 11 (2013), 1246-1254.  doi: 10.2478/s11534-013-0216-2.

[39]

S. Samko, A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and applications. Edited and with a foreword by S. M. Nikol'skii. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993.

[40]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.  doi: 10.1137/0330015.

[41]

A. P. S. Selvadurai, Partial Differential Equations in Mechanics. 1, Fundamentals, Laplace's equation, diffusion equation, wave equation. Springer-Verlag, Berlin, 2000.

[42]

M. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.  doi: 10.1016/j.cam.2006.04.018.

[43]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[44]

M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252 (2013), 495-517.  doi: 10.1016/j.jcp.2013.06.031.

[45]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/121.

show all references

References:
[1]

O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337.  doi: 10.1007/s11071-004-3764-6.

[2]

O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.  doi: 10.1088/1751-8113/40/24/003.

[3]

Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, Int. J. Comput. Math., 87 (2010), 2837-2845.  doi: 10.1080/00207160802562549.

[4]

F. Ali Mehmeti, Nonlineares in Networks, Mathematical Research, 80. Akademie-Verlag, Berlin, 1994.

[5]

E. AlvarezC. G. GalV. Keyantuo and M. Warma, Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Anal., 181 (2019), 24-61.  doi: 10.1016/j.na.2018.10.016.

[6]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and optimization. Second edition. MOS-SIAM Series on Optimization, 17. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[7]

E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Dissertation, Technische Universiteit Eindhoven, Eindhoven, 2001.

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.

[9]

U. Brauer and G. Leugering, On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings, Recent Advances in Control of PDEs. Control Cybernet, 28 (1999), 421-447. 

[10]

Z.-Q. ChenM. M. Meerschaert and E. Nane, Space-time fractional diffusion on bounded domains, J. Math. Anal. Appl., 393 (2012), 479-488.  doi: 10.1016/j.jmaa.2012.04.032.

[11]

R. Dáger, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control Optim., 43 (2004), 590-623.  doi: 10.1137/S0363012903421844.

[12]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], 50. doi: 10.1007/3-540-37726-3.

[13]

L. de Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rend. Sem. Mat. Univ. Padova, 34 (1964), 205-223. 

[14]

R. DorvilleG. Mophou and V. S. Valmorin, Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation, Comput. Math. Appl., 62 (2011), 1472-1481.  doi: 10.1016/j.camwa.2011.03.025.

[15]

C. G. Gal and M. Warma, Fractional-in-time Semilinear Parabolic Equations and Applications, Mathématiques & Applications (Berlin) [Mathematics & Applications], 84. Springer, Cham, [2020], ⓒ2020. doi: 10.1007/978-3-030-45043-4.

[16]

N. Heymans and I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45 (2006), 765-771. 

[17]

D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Spaces Appl., 2013 (2013), Art. ID 128043, 15 pp. doi: 10.1155/2013/128043.

[18]

H. Khosravian-ArabM. Dehghan and M. R. Eslahchi, Fractional Sturm-Liouville boundary value problems in unbounded domains: theory and applications, J. Comput. Phys., 299 (2015), 526-560.  doi: 10.1016/j.jcp.2015.06.030.

[19]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[20]

M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.  doi: 10.1016/j.camwa.2012.12.011.

[21]

M. KlimekA. B. Malinowska and T. Odzijewicz, Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain, Fract. Calc. Appl. Anal., 19 (2016), 516-550.  doi: 10.1515/fca-2016-0027.

[22]

M. KlimekT. Odzijewicz and A. B. Malinowska, Variational methods for the fractional Sturm-Liouville problem, J. Math. Anal. Appl., 416 (2014), 402-426.  doi: 10.1016/j.jmaa.2014.02.009.

[23]

J. E. Lagnese and G. Leugering, Domain Decomposition Methods in Optimal Control of Partial Differential Equations, International Series of Numerical Mathematics, 148. Birkhäuser Verlag, Basel, 2004. doi: 10.1007/978-3-0348-7885-2.

[24]

G. Leugering, Reverberation analysis and control of networks of elastic strings, Control of Partial Differential Equations and Applications (Laredo, 1994), 193–206, Lecture Notes in Pure and Appl. Math., 174, Dekker, New York, 1996.

[25]

G. Leugering and G. Mophou, Instantaneous optimal control of friction dominated flow in a gas-network, Shape Optimization, Homogenization and Optimal Control, 75–88, Internat. Ser. Numer. Math., 169, Birkhäuser/Springer, Cham, 2018.

[26]

J.-L. Lions, Ëquations Différentielles Opérationnelles et Problémes aux Limites, Die Grundlehren der mathematischen Wissenschaften, Band 111 Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.

[27]

J.-L. Lions, Contrôle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Avant propos de P. Lelong Dunod, Paris; Gauthier-Villars, Paris, 1968.

[28]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972.

[29]

G. Lumer, Connecting of local operators and evolution equations on networks, Potential Theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), 219–234, Lecture Notes in Math., 787, Springer, Berlin, 1980.

[30]

V. MehandirattaM. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, J. Math. Anal. Appl., 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011.

[31]

V. MehandirattaM. Mehra and G. Leugering, Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge: a study of fractional calculus on metric graph, Netw. Heterog. Media, 16 (2021), 155-185.  doi: 10.3934/nhm.2021003.

[32]

V. MehandirattaM. Mehra and G. Leugering, Fractional optimal control problems on a star graph: Optimality system and numerical solution, Math. Control Relat. Fields, 11 (2021), 189-209.  doi: 10.3934/mcrf.2020033.

[33]

V. MehandirattaM. Mehra and G. Leugering, Optimal control problems driven by time-fractional diffusion equations on metric graphs: optimality system and finite difference approximation, SIAM J. Control Optim., 59 (2021), 4216-4242.  doi: 10.1137/20M1340332.

[34]

G. Mophou, Optimal control of fractional diffusion equation, Comput. Math. Appl., 61 (2011), 68-78.  doi: 10.1016/j.camwa.2010.10.030.

[35]

G. MophouG. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, 70 (2021), 659-687.  doi: 10.1080/02331934.2020.1730371.

[36]

J. NakagawaK. Sakamoto and M. Yamamoto, Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration, J. Math-for-Ind., 2A (2010), 99-108. 

[37]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

[38]

M. RiveroJ. Trujillo and M. Velasco, A fractional approach to the Sturm-Liouville problem, Open Physics, 11 (2013), 1246-1254.  doi: 10.2478/s11534-013-0216-2.

[39]

S. Samko, A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and applications. Edited and with a foreword by S. M. Nikol'skii. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993.

[40]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.  doi: 10.1137/0330015.

[41]

A. P. S. Selvadurai, Partial Differential Equations in Mechanics. 1, Fundamentals, Laplace's equation, diffusion equation, wave equation. Springer-Verlag, Berlin, 2000.

[42]

M. Steinbach, On PDE solution in transient optimization of gas networks, J. Comput. Appl. Math., 203 (2007), 345-361.  doi: 10.1016/j.cam.2006.04.018.

[43]

J. von Below, Sturm-Liouville eigenvalue problems on networks, Math. Methods Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[44]

M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252 (2013), 495-517.  doi: 10.1016/j.jcp.2013.06.031.

[45]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/121.

Figure 1.  A sketch of a star graph with n edges
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