Controllability problems of differential equations appear in many situations or phenomena for which one is interested in finding a mechanism for bringing a given state into a desired one. Their resolutions often involve constrained minimization problems governed by differential equations systems. This paper is particularly interested in a null controllability problem for backward differential equations systems. One develop a numerical scheme by first approximating the control space by a space of piecewise continuous functions and by transforming the controllability problem into a classical minimization problem with constraints in finite dimension space. Next, one proceed to an adapted implementation of the numerical scheme in Matlab using some of its built-in functions. One then construct a sequence of codes written in Matlab allowing to robustly compute an approximation of the null control at a lower cost. To validate the numerical approach adopted in this paper, two numerical examples are presented. The first ones concerns the controllability of a backward ordinary differential quations system and the second, the controllability of a partial differential heat equation. In both cases, the numerical results obtained are very satisfactory and show that the numerical approach with Matlab developed in this paper leads to new insights for a large class of PDE control problems.
| Published in | Mathematics and Computer Science (Volume 7, Issue 3) |
| DOI | 10.11648/j.mcs.20220703.12 |
| Page(s) | 40-47 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Control of Ordinary Differential Systems, Null Control, Approximation Scheme, Implementation with Matlab
| [1] | P. den Boef, P. B. Cox, R. Toth, “LPVcore: MATLAB Toolbox for LPV Modelling, Identification and Control,” IFAC Papers OnLine, 54 (7), 385-390, 2021. |
| [2] | F. Camilli, L. Grune, and F. Wirth, “Control Lyapunov Functions and Zubov’s Method,” SIAM Journal on Control and Optimization, Vol. 47I, 1, 541-552, 2008. |
| [3] | J. M. Coron, L. Grune, and K. Worthmann, “Model Predictive Control, Cost Controllability, and Homogeneity,” SIAM Journal on Control and Optimization, Vol. 58 5, pp. 2711-3018, 2020. |
| [4] | J. M. Coron, “Control and nonlinearity,” Mathematical Surveys and Monographs, Vol 136, AMS, Providence Rhode Island, 2007. |
| [5] | E. Fernández-Cara, ”Remarks on the approximation and null controllability of semilinear parabolic equations,” ESAIM: Proceedings, Vol. 4, 73-81, 1998. |
| [6] | G. C. García, A. Osses and J. P. Puel, ”A null controllability data assimilation methodology applied to a large scale ocean circulation model,” Mathematical Modelling and Numerical Analysis, ESAIM: M2AN vol 45, Number 2, 361-386, 2011. |
| [7] | M. Gunzburger, L. S. Hou and L. Ju, “A Numerical Method for Exact Bondary Controllability Problems for the Wave Equation,” Elsevier, Computers and Mathematics with Applications 51, 721-750, 2006. |
| [8] | S. Leyendecker, S. Ober-Blobaum, J. E. Marsden and M. Ortiz, ”Discrete mechanics and optimal control for constrained systems,” Optim. Control Appl. Meth., 31, 6, 505-528, 2010. |
| [9] | N. I. Mahmudov, “Finite-Apprximate controllability of evolution equation,” Appl. Comput. Math., V. 16, N. 2, pp. 159-167, 2017. |
| [10] | J. C. Polking, “MATLAB Manual for Ordinary Differential Equations,” Printice-hall, Engle-Wood Cliffs, NJ, 1995. |
| [11] | H. Shuo, L. Hanbing, L. Ping, “Null controllability and global blowup controllability of ordinary differential equations with feedback controls,” Journal of Mathematical Analysis and Applications, Vol 493 (1), 2021. |
| [12] | L. N. Trefethen, “Spectral Methods in Matlab,” SIAM, 2000. |
| [13] | X. Wang, and M. Feckan, “Contollability of conformable differential systems,” Nonlinear Analysis: Modeling and control, Vol 25 (4), pp 658-674, 2020. |
| [14] | F. Zonghong, L. Fengying, L. Ying and Z. Shiqing, “A note on Cauchy-Lipschitz-Picard theorem,” Journal of Inequalities and Applications, 271, 2016. |
| [15] | E. Zuazua, “Control and numerical approximation of the wave and heat equations,” International Congress of Mathematician, Madrid, Spain, Vol III, pp. 1389-1417, 2006. |
APA Style
Deryl Nathan Bonazebi Yindoula, Randhall M’pemba Massaka, Cyr S. Ngamouyih Moussata, Benjamin Mampassi. (2022). Solving Controllability Problems of Backward Ordinary Differential Systems with Matlab. Mathematics and Computer Science, 7(3), 40-47. https://doi.org/10.11648/j.mcs.20220703.12
ACS Style
Deryl Nathan Bonazebi Yindoula; Randhall M’pemba Massaka; Cyr S. Ngamouyih Moussata; Benjamin Mampassi. Solving Controllability Problems of Backward Ordinary Differential Systems with Matlab. Math. Comput. Sci. 2022, 7(3), 40-47. doi: 10.11648/j.mcs.20220703.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.mcs.20220703.12,
author = {Deryl Nathan Bonazebi Yindoula and Randhall M’pemba Massaka and Cyr S. Ngamouyih Moussata and Benjamin Mampassi},
title = {Solving Controllability Problems of Backward Ordinary Differential Systems with Matlab},
journal = {Mathematics and Computer Science},
volume = {7},
number = {3},
pages = {40-47},
doi = {10.11648/j.mcs.20220703.12},
url = {https://doi.org/10.11648/j.mcs.20220703.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20220703.12},
abstract = {Controllability problems of differential equations appear in many situations or phenomena for which one is interested in finding a mechanism for bringing a given state into a desired one. Their resolutions often involve constrained minimization problems governed by differential equations systems. This paper is particularly interested in a null controllability problem for backward differential equations systems. One develop a numerical scheme by first approximating the control space by a space of piecewise continuous functions and by transforming the controllability problem into a classical minimization problem with constraints in finite dimension space. Next, one proceed to an adapted implementation of the numerical scheme in Matlab using some of its built-in functions. One then construct a sequence of codes written in Matlab allowing to robustly compute an approximation of the null control at a lower cost. To validate the numerical approach adopted in this paper, two numerical examples are presented. The first ones concerns the controllability of a backward ordinary differential quations system and the second, the controllability of a partial differential heat equation. In both cases, the numerical results obtained are very satisfactory and show that the numerical approach with Matlab developed in this paper leads to new insights for a large class of PDE control problems.},
year = {2022}
}
TY - JOUR T1 - Solving Controllability Problems of Backward Ordinary Differential Systems with Matlab AU - Deryl Nathan Bonazebi Yindoula AU - Randhall M’pemba Massaka AU - Cyr S. Ngamouyih Moussata AU - Benjamin Mampassi Y1 - 2022/05/13 PY - 2022 N1 - https://doi.org/10.11648/j.mcs.20220703.12 DO - 10.11648/j.mcs.20220703.12 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 40 EP - 47 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20220703.12 AB - Controllability problems of differential equations appear in many situations or phenomena for which one is interested in finding a mechanism for bringing a given state into a desired one. Their resolutions often involve constrained minimization problems governed by differential equations systems. This paper is particularly interested in a null controllability problem for backward differential equations systems. One develop a numerical scheme by first approximating the control space by a space of piecewise continuous functions and by transforming the controllability problem into a classical minimization problem with constraints in finite dimension space. Next, one proceed to an adapted implementation of the numerical scheme in Matlab using some of its built-in functions. One then construct a sequence of codes written in Matlab allowing to robustly compute an approximation of the null control at a lower cost. To validate the numerical approach adopted in this paper, two numerical examples are presented. The first ones concerns the controllability of a backward ordinary differential quations system and the second, the controllability of a partial differential heat equation. In both cases, the numerical results obtained are very satisfactory and show that the numerical approach with Matlab developed in this paper leads to new insights for a large class of PDE control problems. VL - 7 IS - 3 ER -
Information
Email: