Bilinear optimal control of a reaction-diffusion equation: overcoming boundedness constraints on controls

Document Type : Research Article

Authors

• R. El Mezegueldy
• Y. Ouakrim
• M. Ouzahra

Laboratory MMPA, Department of Mathematics and Informatics, ENS, University of Sidi Mohamed Ben Abdellah, Fez, Morocco.

10.22067/ijnao.2025.90287.1538

Abstract

We study an optimal control problem for bilinear reaction-diffusion equa-tions. The novelty of our approach lies in considering controls from the space of essentially bounded functions without imposing a priori bounds on the admissible set. We introduce an auxiliary problem with controls in Lp spaces $(p > 1 + N /2, where N $ is the spatial dimension) and demon-strate that both formulations are equivalent in terms of optimal solutions. Under suitable assumptions on the system parameters, we establish the ex-istence of optimal controls and derive first-order optimality conditions. Our theoretical findings are supported by numerical simulations that validate the practical effectiveness of the proposed approach. This work provides a new framework for handling bilinear control problems without artificial constraints, offering potential applications in population dynamics and eco-logical management.

Keywords

• Bilinear optimal control
• Reaction-diffusion equations
• Essen-tially bounded controls
• Optimality conditions

Main Subjects

• Control and Optimization

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References

[1] Allegretto, W. and Nistri, P. Existence and optimal control for periodic parabolic equations with nonlocal terms, IMA J. Math. Control Inf., 16(1) (1999), 43–58.
[2] Banerjee, M., Kuznetsov, M., Udovenko, O. and Volpert, V. Nonlocal reaction–diffusion equations in biomedical applications, Acta Biotheor., 70(2) (2022), 12.

[3] Biccari, U. and Hernández-Santamaría, V. Null controllability of linear and semilinear nonlocal heat equations with an additive integral kernel, SIAM J. Control Optim., 57(4) (2019), 2924–2938.
[4] Bradley, M.E. and Lenhart, S. Bilinear optimal control of a Kirchhoff plate, Syst. Control Lett., 22(1) (1994), 27–38.
[5] Breiten, T., Kunisch, K., and Pfeiffer, L. Infinite-horizon bilinear opti-mal control problems: Sensitivity analysis and polynomial feedback laws, SIAM J. Control Optim., 56(5) (2018), 3184–3214.
[6] Brezis, H. Analyse fonctionnelle, Théorie et applications (1983).
[7] Cannarsa, P., Floridia, G. and Khapalov, A.Y. Multiplicative control-lability for semilinear reaction–diffusion equations with finitely many changes of sign, J. Math. Pures Appl., 108(4) (2017), 425–458.
[8] Casas, E. and Kunisch, K. Infinite horizon optimal control problems for a class of semilinear parabolic equations, SIAM J. Control Optim., 60(4) (2022), 2070–2094.
[9] Casas, E. and Kunisch, K. Optimal control of semilinear parabolic equa-tions with non-smooth pointwise-integral control constraints in time-space, Appl. Math. Optim., 85(1) (2022), 12.
[10] Colbrook, M.J. Computing spectral measures and spectral types, Com-mun. Math. Phys., 384 (2021), 433–501.
[11] Corrêa Vianna Filho, A.L. and Guillén-González, F. A Review on the Analysis and Optimal Control of Chemotaxis-Consumption Models, SeMA J. (2024), 1–34.
[12] Daubechies, I., Defrise, M. and De Mol, C. An iterative thresholding al-gorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., 57(11) (2004), 1413–1457.
[13] El Boukhari, N. and Zerrik, E.H. Optimal control of distributed semilin-ear systems with essentially bounded measurable control functions, Int. J. Dyn. Control, 11(6) (2023), 2809–2819.

[14] El Kabouss, A. Fractional Optimal Control Problem of Parabolic Bilinear Systems with Bounded Controls, Vibration Control of Structures (2023).
[15] Evans, L.C. Partial differential equations, American Mathematical So-ciety, 19 (2022).
[16] Fleig, A. and Guglielmi, R. Optimal control of the Fokker–Planck equa-tion with space-dependent controls, J. Optim. Theory Appl., 174 (2017), 408–427.
[17] Furter, J. and Grinfeld, M. Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65–80.
[18] Glowinski, R., Song, Y., Yuan, X., and Yue, H. Bilinear optimal control of an advection-reaction-diffusion system, SIAM Rev., 64(2) (2022), 392–421.
[19] Guillén-González, F. and Corrêa Vianna Filho, A.L. An optimal control problem subject to strong solutions of chemotaxis-consumption models, SIAM J. Control Optim., 61(5) (2023), 3156–3182.
[20] Hillen, T., Enderling, H. and Hahnfeldt, P. The tumor growth paradox and immune system-mediated selection for cancer stem cells, Bull. Math. Biol., 75 (2013), 161–184.
[21] Ladyzhenskaya, O.A. Linear and quasilinear elliptic equations, trans-lated by Scripta Technica (1968).
[22] Li, X. and Yong, J. Optimal control theory for infinite dimensional sys-tems, Springer Science & Business Media (2012).
[23] Liang, M. Bilinear optimal control for a wave equation, Math. Models Methods Appl. Sci., 9(01) (1999), 45–68.
[24] Lions, J.L. Optimal control of systems governed by partial differential equations, Springer, Berlin, 170 (1971).
[25] Martini, A., Müller, D. and Golo, S.N. Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type, J. Eur. Math. Soc., 25(3) (2022), 785–843.

[26] Mophou, G., Kenne, C. and Warma, M. Bilinear optimal control for a fractional diffusive equation, J. Optim., (2024), 1–27.
[27] Ouzahra, M. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control, Discrete Contin. Dyn. Syst. Ser. B, 27(2) (2022), 1075–1090.
[28] Pal, S. and Melnik, R. Nonlocal Models in Biology and Life Sci-ences: Sources, Developments, and Applications, arXiv preprint arXiv:2401.14651 (2024).
[29] Sikora, A. On the L2 → L∞ norms of spectral multipliers of ”quasi-homogeneous” operators on homogeneous groups, Trans. Amer. Math. Soc., 351(9) (1999), 3743–3755.
[30] Sikora, A., Yan, L. and Yao, X. Spectral multipliers, Bochner–Riesz means and uniform Sobolev inequalities for elliptic operators, Int. Math. Res. Not., 2018(10) (2018), 3070–3121.
[31] Saporiti, R., Sinigaglia, C., Manzoni, A., and Braghin, F. An optimal control strategy to design passive thermal cloaks of arbitrary shape, Proc. R. Soc. A, 480(2294) (2024), 20230646.
[32] Sogge, C.D. Eigenfunction and Bochner Riesz estimates on manifolds with boundary, arXiv preprint math/0202032 (2002).
[33] Tröltzsch, F. Optimal control of partial differential equations: theory,methods and applications, American Mathematical Society, 112 (2010).
[34] Yahyaoui, S. and Ouzahra, M. Quadratic optimal control and feedback stabilization of bilinear systems, Optim. Control Appl. Methods, 42(4) (2021), 878–890.
[35] Yahyaoui, S. and Ouzahra, M. Optimal control problem for unbounded bilinear systems and applications, J. Control Decis. (2023), 1–13.
[36] Zerrik, E.H. and Boukhari, N.E. Constrained optimal control for a class of semilinear infinite dimensional systems, J. Dyn. Control Syst., 24 (2018), 65–81.