Control of the Fisher-Stefan System

Abstract

This paper addresses the exact controllability of trajectories in the one-dimensional Fisher-Stefan problem—a reaction-diffusion equation that models the spatial propagation of biological, chemical, or physical populations within a free-end domain, governed by Stefan’s law. We establish the local exact controllability of the trajectories by reformulating the problem as the local null controllability of a nonlinear system with distributed controls. Our approach leverages the Lyusternik-Graves theorem to achieve local inversion, leading to the desired controllability result. Finally, we illustrate our theoretical findings through several numerical experiments based on the physics-informed neural networks (PINNs) approach.

Keywords

Reaction-diffusion system / Fisher-Stefan model / Control to trajectories / Free boundary problems for PDEs / Physics-informed neural networks (PINNs) / 35K57 / 93B05 / 35R35 / 93C20 / 35K58 / 47H99 / 65M99

References

[1] Bárcena-Petisco JA, Fernández-Cara E, Souza DA. Exact controllability to the trajectories of the one-phase Stefan problem. J. Differ. Equ., 2023, 376: 126-153.

[2] Behzadan A, Holst M. Multiplication in Sobolev spaces, revisited. Ark. Mat., 2021, 59(2): 275-306.

[3] Brosa PF, Please CP, Van Gorder RA. Extended Stefan problem for solidification of binary alloys in a finite planar domain. SIAM J. Appl. Math., 2019, 79(3): 876-913.

[4] Brosa PF, Please CP, Van Gorder RA. Extended Stefan problem for the solidification of binary alloys in a sphere. Eur. J. Appl. Math., 2021, 32(2): 242-279.

[5] Canosa J. On a nonlinear diffusion equation describing population growth. IBM J. Res. Dev., 1973, 17(4): 307-313.

[6] Crank J. Free and Moving Boundary Problems, 1984. Oxford, Oxford University Press

[7] Dalwadi MP, Waters SL, Byrne HM, Hewitt IJ. A mathematical framework for developing freezing protocols in the cryopreservation of cells. SIAM J. Appl. Math., 2020, 80(2): 657-689.

[8] Demarque R, Fernández-Cara E. Local null controllability of one-phase Stefan problems in 2D star-shaped domains. J. Evol. Equ., 2018, 18(1): 245-261.

[9] Du Y, Guo Z. Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary, II. J. Differ. Equ., 2011, 250(12): 4336-4366.

[10] Du, Y., Guo, Z.: The Stefan problem for the Fisher-KPP equation. J. Differ. Equ. 253(3), 996–1035 (2012)

[11] Du, Y., Lin, Z.: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42(1), 377–405 (2010)

[12] Et-Tahri, F., Chorfi, S.-E., Maniar, L., Boutaayamou, I.: Null controllability of a volume-surface reaction-diffusion equation with dynamic boundary conditions. J. Math. Anal. Appl. 542(2), 128793 (2025)

[13] Fadai, N.T., Simpson, M.J.: New travelling wave solutions of the Porous-Fisher model with a moving boundary. J. Phys. A: Math. Theor. 53(9), 095601 (2020)

[14] Fernández-Cara E, de Sousa IT. Local null controllability of a free-boundary problem for the semilinear 1D heat equation. Bull. Braz. Math. Soc. New Ser., 2017, 48(2): 303-315.

[15] Fernández-Cara E, Hernández F, Límaco J. Local null controllability of a 1D Stefan problem. Bull. Braz. Math. Soc. New Ser., 2019, 50: 745-769.

[16] Fernández-Cara, E., Límaco, J., de Menezes, S.B.: On the controllability of a free-boundary problem for the 1D heat equation. Syst. Control Lett. 87, 29–35 (2016)

[17] Fernández-Cara E, Límaco J, Thamsten Y, Menezes D. Local null controllability of a quasi-linear system and related numerical experiments. ESAIM Control Optim. Calc. Var., 2023, 29: 27.

[18] Fernández-Cara E, Münch A. Strong convergent approximations of null controls for the 1D heat equation. SeMA J., 2013, 61(1): 49-78.

[19] Fernández-Cara E, Nina-Huamán D, Núñez-Chávez MR, Vieira FB. On the theoretical and numerical control of a one-dimensional nonlinear parabolic partial differential equation. J. Optim. Theory Appl., 2017, 175: 652-682.

[20] Fursikov AV, Imanuvilov OY. Controllability of Evolution Equations. Lecture Notes Series, 1996. Seoul, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center. 34

[21] Gaffney EA, Maini PK, McCaig CD, Zhao M, Forrester JV. Modelling corneal epithelial wound closure in the presence of physiological electric fields via a moving boundary formalism. Math. Med. Biol., 1999, 16(4): 369-393.

[22] García-Cervera CJ, Kessler M, Periago F. Control of partial differential equations via physics-informed neural networks. J. Optim. Theory Appl., 2023, 196(2): 391-414.

[23] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, JMLR Workshop and Conference Proceedings, pp. 249–256 (2010)

[24] Grindrod, P.: The Theory and Applications of Reaction-Diffusion Equations: Patterns and Waves. Clarendon Press, Oxford University Press, New York (1991)

[25] Kimpton, L.S., Whiteley, J.P., Waters, S.L., King, J.R., Oliver, J.M.: Multiple travelling-wave solutions in a minimal model for cell motility. Math. Med. Biol. (IMA) 30(3), 241–272 (2013)

[26] Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. In: 3rd International Conference on Learning Representations (2015)

[27] Kolmogorov AN. A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Mosc. Univ. Math. Bull., 1937, 1: 1-25

[28] Lewis, M.A., Petrovskii, S.V., Potts, J.R.: The Mathematics Behind Biological Invasions. Springer International Publishing, Cham (2016)

[29] Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. I. Springer-Verlag, Berlin (2012)

[30] Mitchell SL, Myers TG. Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions. Int. J. Heat Mass Transf., 2010, 53(17/18): 3540-3551.

[31] Mitchell SL, Vynnycky M. Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems. Appl. Math. Comput., 2009, 215(4): 1609-1621.

[32] Murray J.D. Mathematical Biology: I. An Introduction. Springer-Verlag, New York (2002)

[33] Nina HD, Núñez-Chávez MR, Límaco J, Carvalho PP. Local null controllability for the thermistor problem. Nonlinear Anal., 2023, 236. ArticleID: 113330

[34] Peng WQ, Pu JC, Chen Y. PINN deep learning method for the Chen-Lee-Liu equation: rogue wave on the periodic background. Commun. Nonlinear Sci. Numer. Simul., 2022, 105. ArticleID: 106067

[35] Pinkus, A.: Approximation theory of the MLP model in neural networks. Acta Numer. 8, 143–195 (1999)

[36] Raisi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys., 2019, 378: 686-707.

[37] Shuttleworth R, Trucu D. Multiscale modelling of fibres dynamics and cell adhesion within moving boundary cancer invasion. Bull. Math. Biol., 2019, 81: 2176-2219.

[38] Trélat E. Contrôle Optimal: Théorie & Applications, 2005. Paris, Vuibert

[39] Ward JP, King JR. Mathematical modelling of avascular-tumour growth. Math. Med. Biol. (IMA), 1997, 14(1): 39-69.

[40] Zhang Q, Qiu C, Hou J, Yan W. Advanced physics-informed neural networks for numerical approximation of the coupled Schrödinger-KdV equation. Commun. Nonlinear Sci. Numer. Simul., 2024, 138. ArticleID: 108229

Funding

Ministry of Science and Technology(PID2022-141957OA-C22)

Fundación Séneca(21996/PI/22)

Universidad Politécnica de Cartagena

RIGHTS & PERMISSIONS

The Author(s)