This paper is a new step of a project on the null controllability of the 1D heat equation by strategic profile. We have obtained boundary and internal results of controllability by a new approach using a linear, continuous and surjective operator built from the solution of the heat system. The minimum time for null controllability of the 1D heat equation is improved by using the notion of strategic zone actuators. Through an innovative approach by adding strategic zone profiles, we obtain the best time and, in turn, a better cost of null controllability for the 1D heat equation.

control, null controllability, moment methods, estimations, strategic actuator, profile function.

Received: August 19, 2022;  Revised: October 27, 2022;  Accepted: November 3, 2022; Published: November 7, 2022

How to cite this article: Cheikh Seck, Mame Libasse Laye Ane and Babacar Diouf, A new approach to find the best cost of null controllability for the 1D heat equation by a strategic zone profile, Universal Journal of Mathematics and Mathematical Sciences 17 (2022), 47-65. http://dx.doi.org/10.17654/2277141722011

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Ration. Mech. Anal. 43(4) (1971), 272-292.
[2] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 2, Recherches en Mathématiques Appliquées, Research in Applied Mathematics, Volume 9, Perturbations, Masson, Paris, 1988.
[3] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Recherches en Mathématiques Appliquées, Research in Applied Mathematics, Volume 8, Paris, 1988.
[4] M. T. Niane, Contrôlabilité spectrale élargie des systèmes distribués par une action sur une partie analytique arbitraire de la frontière, C. R. Acad. Sci. Paris, t. 309(1) (1989), 335-340.
[5] C. Seck, Exact Controllability of the Heat Equation by Temporarily Strategic Actuators Borders, Journal of Mathematical Research, Published by Canadian Centre of Science and Education, 2019. https://doi.org/10.5539/jmr.v11n6p53
[6] A. Guesmia, M. Kafini and N. E. Tatar, General stability results for the translational problem of memory-type in porous thermoelasticity of type III, J. Nonlinear Funct. Anal. 2020 (2020), Article ID 49.
https://doi.org/10.23952/jnfa.2020.49
[7] V. Y. Glizer, Euclidean space controllability conditions and minimum energy problem for time delay system with a high gain control, J. Nonlinear Var. Anal. 2(1) (2018), 63-90. Available online at http://jnva.biemdas.com
https://doi.org/10.23952/jnva.2.2018.1.06
[8] A. Anguraj and K. A. Ramkumar, Global existence and controllability to a stochastic integro-differential equation with Poisson jumps, Commun. Optim. Theory (2018), Article ID 22. https://doi.org/10.23952/cot.2018.22
[9] C. Seck and M. L. L. Ane, Boundary exact controllability of the heat equation in 1D by strategic actuators and a linear surjective compact operator, Applied Mathematics 11 (2020), 991-999. https://doi.org/10.4236/am.2020.1110065
[10] C. Seck, A. Sène and M. T. Niane, Estimates related to the extended spectral control of the wave equation, Journal of Mathematics Research 10(4) (2018). Published by Canadian Center of Science and Education.
https://doi.org/10.5539/jmr.v10n4p156
[11] C. Seck, Minimal time of null controllability for the 1D heat equation by a strategic zone profile, Journal of Applied Mathematics and Physics 9 (2020), 1707-1717. https://doi.org/10.4236/jamp.2021.97113
[12] D. L. Russel and H. O. Fattorini, Exact controllability for linear parabolic equation in one space dimension, Arch. Mat. Mech. Anal. 4 (1971), 272-292.
[13] G. Lebeau and L. Robbiano, Contrôle exact de l’équation de la Chaleur, Prépublications Univ. Paris-Sud., 1994.
[14] A. V. Fursikov and O. Yu Imanuvilov, Controllability of Evolution Equations, Séoul National University, 1996.
[15] F. Ammar Khodja, Teresa Luz De, Assia Benabdallah and Manuel Gonzalez-Burgos, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, Journal of Functional Analysis, Elsevier, 267(7) (2014), 2077-2151. https://doi.org/10.1016/j.jfa.2014.07.024
[16] F. A. Khodja, A. Benabdallah, M. Gonzàlez-Burgo and L. D. Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, Journal of Functional Analysis, Elsevier, 267(7) (2014), pp. 2077-2151.
[17] P. Lissy and M. Guéye, Singular optimal control of a 1-D parabolic-hyperbolic, degenerate equation, Archives-ouvertes de France (2015), HAL Id: hal-01240264, https://hal.archives-ouvertes.fr/hal-01240264
[18] P. Lissy, A link between the cost of fast controls for the 1-D heat equation and the uniform controllability of a 1-D transport-diffusion equation, C. R. Acad. Sci. Paris, n° 591595, Ser. I 350 (2012).
[19] Marius Tucsnak and Gerald Tenenbaum, New blow-up rates for fast controls of Schrödinger and heat equations, Journal of Differential Equations, Elsevier, 243(1) (2007), https://doi.org/10.1016/j.jde.2007.06.019.
[20] A. El Jai, Quelques problèmes de contrôle propres aux systèms distribués, Annals of University of Craiova, Math. Comp. Sci. Ser. 30 (2003), 137-153.
[21] A. El Jai, June, Analyse régionale des systèmes distributes, ESAIM: Control, Optimisation and Calculus of Variation 8 (2002), 663-692.
[22] L. The Analysis of Linear Partial Differential Operators I, Springer Verlag, 1990.