## Local null-controllability of a nonlocal semilinear heat equation.(English)Zbl 1475.35184

Authors’ abstract: This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small time local null-controllability of a $$2 \times 2$$ reaction-diffusion system, where the second equation is governed by the parabolic operator $$\tau \partial_t -\sigma \Delta, \tau, \sigma > 0$$. More precisely, this controllability result is obtained uniformly with respect to the parameters $$(\tau, \sigma) \in (0, 1) \times (1, +\infty)$$. Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit $$(\tau, \sigma) \rightarrow (0, +\infty)$$. Finally, we illustrate these results by numerical simulations.

### MSC:

 35K58 Semilinear parabolic equations 93B05 Controllability 93B07 Observability 93C20 Control/observation systems governed by partial differential equations 35K20 Initial-boundary value problems for second-order parabolic equations
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### References:

 [1] Ammar-Khodja, F.; Benabdallah, A.; González-Burgos, M.; de Teresa, L., Recent results on the controllability of linear coupled parabolic problems: a survey, Math. Control Relat. Fields, 1, 3, 267-306 (2011) · Zbl 1235.93041 [2] Barbu, V., Exact controllability of the superlinear heat equation, Appl. Math. Optim., 42, 1, 73-89 (2000) · Zbl 0964.93046 [3] Biccari, U.; Hernández-Santamaría, V., Null controllability of linear and semilinear nonlocal heat equations with an additive integral kernel, SIAM J. Control Optim., 57, 4, 2924-2938 (2019) · Zbl 1420.35137 [4] Boyer, F.: On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems. In: CANUM 2012, 41e Congrès National d’Analyse Numérique volume 41 of ESAIM Proc., pp. 15-58. EDP Sci., Les Ulis (2013) · Zbl 1329.49044 [5] Chaves-Silva, FW; Guerrero, S.; Puel, JP, Controllability of fast diffusion coupled parabolic systems., Math. Control Relat. Fields, 4, 4, 465-479 (2014) · Zbl 1319.35100 [6] Chaves-Silva, FW; Bendahmane, M., Uniform null controllability for a degenerating reaction-diffusion system approximating a simplified cardiac model, SIAM J. Control Optim., 53, 6, 3483-3502 (2015) · Zbl 1328.35090 [7] Chaves-Silva, FW; Guerrero, S., A uniform controllability result for the Keller-Segel system, Asymp. Anal., 92, 3-4, 313-338 (2015) · Zbl 1328.93057 [8] Clark, HR; Fernández-Cara, E.; Limaco, J.; Medeiros, LA, Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities, Appl. Math. Comput., 223, 483-505 (2013) · Zbl 1329.93083 [9] de Teresa, L., Insensitizing controls for a semilinear heat equation, Commun. Part. Differ. Equ., 25, 1-2, 39-72 (2000) · Zbl 0942.35028 [10] Ekeland, I., Témam, R.: Convex analysis and variational problems, volume 28 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, english edition (1999). Translated from the French · Zbl 0939.49002 [11] Evans, LC, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics (2010), Providence: American Mathematical Society, Providence [12] Fernández-Cara, E.; Guerrero, S., Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45, 4, 1399-1446 (2006) · Zbl 1121.35017 [13] Fernández-Cara, E.; Zuazua, E., Null and approximate controllability for weakly blowing up semilinear heat equations., Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, 5, 583-616 (2000) · Zbl 0970.93023 [14] Fernández-Cara, E.; González-Burgos, M.; Guerrero, S.; Puel, J-P, Null controllability of the heat equation with boundary Fourier conditions: the linear case, ESAIM Control Optim. Calc. Var., 12, 3, 442-465 (2006) · Zbl 1106.93009 [15] Fernández-Cara, E.; Limaco, J.; de Menezes, SB, Null controllability for a parabolic equation with nonlocal nonlinearities, Syst. Control Lett., 61, 1, 107-111 (2012) · Zbl 1250.93031 [16] Fernández-Cara, E.; Lü, Q.; Zuazua, E., Null controllability of linear heat and wave equations with nonlocal spatial terms, SIAM J. Control Optim., 54, 4, 2009-2019 (2016) · Zbl 1346.93071 [17] Fernández-Cara, E.; Límaco, J.; Nina-Huaman, D.; Núñez Chávez, MR, Exact controllability to the trajectories for parabolic PDEs with nonlocal nonlinearities, Math. Control Signals Syst., 31, 3, 415-431 (2019) · Zbl 1422.93015 [18] Fursikov, A.V., Imanuvilov, O Y.: Controllability of Evolution Equations, Volume 34 of Lecture Notes Series. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996) · Zbl 0862.49004 [19] González-Burgos, M.; de Teresa, L., Controllability results for cascade systems of $$m$$ coupled parabolic PDEs by one control force, Port. Math., 67, 1, 91-113 (2010) · Zbl 1183.93042 [20] Hernández-Santamaría, V.; Zuazua, E., Controllability of shadow reaction-diffusion systems, J. Differ. Equ., 268, 7, 3781-3818 (2020) · Zbl 1477.35292 [21] Hilhorst, D.; Rodrigues, J-F, On a nonlocal diffusion equation with discontinuous reaction, Adv. Differ. Equ., 5, 4-6, 657-680 (2000) · Zbl 0990.35058 [22] Kavallaris, NI; Suzuki, T., Non-local Partial Differential Equations for Engineering and Biology, Volume 31 of Mathematics for Industry (Tokyo). Mathematical Modeling and Analysis (2018), Cham: Springer, Cham · Zbl 1387.00004 [23] Kokotović, P., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control, Volume 25 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1999). Analysis and design, Corrected reprint of the 1986 original [24] Le Balc’h, K.: Controllability of nonlinear reaction-diffusion sytems. Theses, École normale supérieure de Rennes (2019) [25] Le Balc’h, K., Global null-controllability and nonnegative-controllability of slightly superlinear heat equations, J. de Math. Pures et Appl., 135, 103-139 (2020) · Zbl 1436.93065 [26] Lebeau, G.; Robbiano, L., Contrôle exact de l’équation de la chaleur, Commun. Part. l Differ. Equ., 20, 1-2, 335-356 (1995) · Zbl 0819.35071 [27] Lissy, P.; Zuazua, E., Internal controllability for parabolic systems involving analytic non-local terms, Chin. Ann. Math. Ser. B, 39, 2, 281-296 (2018) · Zbl 1391.35187 [28] Liu, Y.; Takahashi, T.; Tucsnak, M., Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19, 1, 20-42 (2013) · Zbl 1270.35259 [29] Lohéac, J.; Trélat, E.; Zuazua, E., Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27, 9, 1587-1644 (2017) · Zbl 1370.35156 [30] Micu, S.; Takahashi, T., Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differ. Equ., 264, 5, 3664-3703 (2018) · Zbl 1377.93047 [31] Montoya, C.; de Teresa, L., Robust Stackelberg controllability for the Navier-Stokes equations, NoDEA Nonlinear Differ. Equ. Appl., 25, 5, 46 (2018) · Zbl 1407.35154 [32] Perthame, B.: Parabolic equations in biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Growth, reaction, movement and diffusion. Springer, Cham (2015) [33] Rodrigues, J.-F.: Reaction-diffusion: from systems to nonlocal equations in a class of free boundary problems. Number 1249. 2002. In: International Conference on Reaction-Diffusion Systems: Theory and Applications, pp. 72-89. Kyoto (2001) [34] Simon, J., Compact sets in the space $$L^p(0, T;B)$$, Ann. Mater. Pura Appl., 4, 146, 65-96 (1987) · Zbl 0629.46031
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