August  2022, 15(8): 1883-1918. doi: 10.3934/dcdss.2022012

A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs

1. 

The Center for Mathematics and Artificial Intelligence, Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

2. 

Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

Received  October 2021 Published  August 2022 Early access  January 2022

Fund Project: The first author is partially supported by NSF grants DMS-2110263, DMS-1913004 and the Air Force Office of Scientific Research under Award NO: FA9550-19-1-0036. The last author is partially supported by the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242 and the Army Research Office (ARO) under Award NO: W911NF-20-1-0115

We consider optimal control of fractional in time (subdiffusive, i.e., for $ 0<\gamma <1 $) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we $\mathsf{first\;show}$ the existence and regularity of solutions to the forward and the associated $\mathsf{backward\;(adjoint)}$ problems. In the second part, we prove existence of optimal $\mathsf{controls }$ and characterize the associated $\mathsf{first\;order}$ optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.

Citation: Harbir Antil, Ciprian G. Gal, Mahamadi Warma. A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 1883-1918. doi: 10.3934/dcdss.2022012
References:
[1]

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.

[2]

O. P. Agrawal, A general finite element formulation for fractional variational problems, J. Math. Anal. Appl., 337 (2008), 1-12.  doi: 10.1016/j.jmaa.2007.03.105.

[3]

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.

[4]

H. Antil, H. C. Elman, A. Onwunta and D. Verma, Novel deep neural networks for solving bayesian statistical inverse problems, arXiv preprint arXiv: 2102.03974, 2021.

[5]

H. AntilR. KhatriR. Löhner and D. Verma, Fractional deep neural network via constrained optimization, Machine Learning: Science and Technology, 2 (2020), 015003.  doi: 10.1088/2632-2153/aba8e7.

[6]

H. AntilR. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 35 pp. 

[7] H. AntilD. P. KouriM.-D. Lacasse and D. Ridzal, Frontiers in PDE-Constrained Optimization, Springer, New York, 2018. 
[8]

H. Antil, D. P. Kouri and D. Ridzal, ALESQP: An augmented Lagrangian equality-constrained sqp method for optimization with general constraints, Submitted to SIOPT, 2021.

[9]

H. AntilR. Nochetto and P. Venegas, Controlling the Kelvin force: Basic strategies and applications to magnetic drug targeting, Optim. Eng., 19 (2018), 559-589.  doi: 10.1007/s11081-018-9392-7.

[10]

H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: Analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.

[11]

H. Antil, D. Verma and M. Warma, External optimal control of fractional parabolic PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 20, 33 pp. doi: 10.1051/cocv/2020005.

[12]

H. Antil and M. Warma, Optimal control of fractional semilinear PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 5, 30pp.

[13]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, 2$^{nd}$ edition, Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[14]

E. BarkaiR. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138.  doi: 10.1103/PhysRevE.61.132.

[15]

E. G. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Technische Universiteit Eindhoven, Eindhoven, 2001. Eindhoven University of Technology, Eindhoven, 2001.

[16]

P. Blanchard and E. Bruning, Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics, volume 69 of Progress in Mathematical Physics., 2$^nd$ edition, Birkhäuser, 2015. doi: 10.1007/978-3-319-14045-2.

[17]

A. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics 659, Springer, Berlin, 1978.

[18]

P. M. d. Carvalho Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de S ão Paulo, 2013.

[19]

E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J. Control Optim., 31 (1993), 993-1006.  doi: 10.1137/0331044.

[20]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637.

[21]

J. Diestel and J. J. Uhl, Jr, Vector Measures, American Mathematical Society, Providence, R. I., 1977.

[22]

K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[23]

S. Elagan, On the invalidity of semigroup property for the mittag-leffler function with two parameters, J. Egyptian Math. Soc., 24 (2016), 200-203.  doi: 10.1016/j.joems.2015.05.003.

[24]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the cahn-hilliard equation, Math. Comp., 58 (1992), 603-630.  doi: 10.1090/S0025-5718-1992-1122067-1.

[25]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[26]

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

[27]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Cnstraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.

[28]

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718614.

[29]

B. JinB. Li and Z. Zhou, Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint, IMA J. Numer. Anal., 40 (2020), 377-404.  doi: 10.1093/imanum/dry064.

[30]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys., 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[31]

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

[32]

Y. Shin, J. Darbon and G. E. Karniadakis, A caputo fractional derivative-based algorithm for optimization, arXiv preprint arXiv: 2104.02259, 2021.

[33]

T. TangH. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757-A3778.  doi: 10.1137/18M1203560.

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Theory, methods and applications. Translated from the 2005 German original by Jürgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[35]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum, 73 (2006), 10-30.  doi: 10.1007/s00233-006-0617-2.

[36]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.

show all references

References:
[1]

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.

[2]

O. P. Agrawal, A general finite element formulation for fractional variational problems, J. Math. Anal. Appl., 337 (2008), 1-12.  doi: 10.1016/j.jmaa.2007.03.105.

[3]

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.

[4]

H. Antil, H. C. Elman, A. Onwunta and D. Verma, Novel deep neural networks for solving bayesian statistical inverse problems, arXiv preprint arXiv: 2102.03974, 2021.

[5]

H. AntilR. KhatriR. Löhner and D. Verma, Fractional deep neural network via constrained optimization, Machine Learning: Science and Technology, 2 (2020), 015003.  doi: 10.1088/2632-2153/aba8e7.

[6]

H. AntilR. Khatri and M. Warma, External optimal control of nonlocal PDEs, Inverse Problems, 35 (2019), 35 pp. 

[7] H. AntilD. P. KouriM.-D. Lacasse and D. Ridzal, Frontiers in PDE-Constrained Optimization, Springer, New York, 2018. 
[8]

H. Antil, D. P. Kouri and D. Ridzal, ALESQP: An augmented Lagrangian equality-constrained sqp method for optimization with general constraints, Submitted to SIOPT, 2021.

[9]

H. AntilR. Nochetto and P. Venegas, Controlling the Kelvin force: Basic strategies and applications to magnetic drug targeting, Optim. Eng., 19 (2018), 559-589.  doi: 10.1007/s11081-018-9392-7.

[10]

H. AntilE. Otárola and A. J. Salgado, A space-time fractional optimal control problem: Analysis and discretization, SIAM J. Control Optim., 54 (2016), 1295-1328.  doi: 10.1137/15M1014991.

[11]

H. Antil, D. Verma and M. Warma, External optimal control of fractional parabolic PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 20, 33 pp. doi: 10.1051/cocv/2020005.

[12]

H. Antil and M. Warma, Optimal control of fractional semilinear PDEs, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 5, 30pp.

[13]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, 2$^{nd}$ edition, Mathematical Optimization Society, Philadelphia, PA, 2014. doi: 10.1137/1.9781611973488.

[14]

E. BarkaiR. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138.  doi: 10.1103/PhysRevE.61.132.

[15]

E. G. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Technische Universiteit Eindhoven, Eindhoven, 2001. Eindhoven University of Technology, Eindhoven, 2001.

[16]

P. Blanchard and E. Bruning, Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics, volume 69 of Progress in Mathematical Physics., 2$^nd$ edition, Birkhäuser, 2015. doi: 10.1007/978-3-319-14045-2.

[17]

A. Bruckner, Differentiation of Real Functions, Lecture Notes in Mathematics 659, Springer, Berlin, 1978.

[18]

P. M. d. Carvalho Neto, Fractional Differential Equations: A Novel Study of Local and Global Solutions in Banach Spaces, PhD thesis, Universidade de S ão Paulo, 2013.

[19]

E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J. Control Optim., 31 (1993), 993-1006.  doi: 10.1137/0331044.

[20]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637.

[21]

J. Diestel and J. J. Uhl, Jr, Vector Measures, American Mathematical Society, Providence, R. I., 1977.

[22]

K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.

[23]

S. Elagan, On the invalidity of semigroup property for the mittag-leffler function with two parameters, J. Egyptian Math. Soc., 24 (2016), 200-203.  doi: 10.1016/j.joems.2015.05.003.

[24]

C. M. Elliott and S. Larsson, Error estimates with smooth and nonsmooth data for a finite element method for the cahn-hilliard equation, Math. Comp., 58 (1992), 603-630.  doi: 10.1090/S0025-5718-1992-1122067-1.

[25]

C. G. Gal and M. Warma, On some degenerate non-local parabolic equation associated with the fractional $p$-Laplacian, Dyn. Partial Differ. Equ., 14 (2017), 47-77.  doi: 10.4310/DPDE.2017.v14.n1.a4.

[26]

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

[27]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Cnstraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.

[28]

K. Ito and K. Kunisch, Lagrange Multiplier Approach to Variational Problems and Applications, Advances in Design and Control, 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718614.

[29]

B. JinB. Li and Z. Zhou, Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint, IMA J. Numer. Anal., 40 (2020), 377-404.  doi: 10.1093/imanum/dry064.

[30]

R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys., 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.

[31]

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

[32]

Y. Shin, J. Darbon and G. E. Karniadakis, A caputo fractional derivative-based algorithm for optimization, arXiv preprint arXiv: 2104.02259, 2021.

[33]

T. TangH. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757-A3778.  doi: 10.1137/18M1203560.

[34]

F. Tröltzsch, Optimal Control of Partial Differential Equations, Theory, methods and applications. Translated from the 2005 German original by Jürgen Sprekels. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[35]

M. Warma, The Robin and Wentzell-Robin Laplacians on Lipschitz domains, Semigroup Forum, 73 (2006), 10-30.  doi: 10.1007/s00233-006-0617-2.

[36]

M. Warma, Approximate controllability from the exterior of space-time fractional diffusive equations, SIAM J. Control Optim., 57 (2019), 2037-2063.  doi: 10.1137/18M117145X.

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