Optimal control problem governed by an infinite dimensional two-nilpotent bilinear systems

Aziza Aib; Naceurdine Bensalem

doi:10.3934/naco.2022018

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2023, Volume 13, Issue 2: 314-327. Doi: 10.3934/naco.2022018

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Optimal control problem governed by an infinite dimensional two-nilpotent bilinear systems

Aziza Aib^, and
Naceurdine Bensalem

Numerical and Fundamental Mathematics Laboratory, Ferhat Abbas University Setif 1. Setif 19000, Algeria

^*Corresponding author: Aziza Aib
^*Corresponding author: Aziza Aib

Received: June 2021

Revised: June 2022

Early access: July 2022

Published: January 2023

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Abstract

The object of this work is to construct an explicit linear operators $ A $ and $ B $ which generate a nilpotent Lie algebra for the bracket $ \left[A,B\right] = AB-BA $ of degree 2 in infinite dimensional spaces. This construction can be applied to give an exact optimal solution for a class of infinite dimensional bilinear systems.

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Mathematics Subject Classification: 49J15, 47A07, 22E27.

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References

[1]	A. Aib and N. Bensalem, Optimal control problem governed by an infinite dimensional one-nilpotent bilinear systems, Bull. Math. Soc. Sci. Math. Roumanie, 55 (2012), 107-128.
[2]	M. S. Aronna, J. F. Bonnans and A. Kröner, Optimal control of infinite dimensional bilinear systems: Application to the heat and wave equations, Mathematical Programming, 168 (2018), 717-757. doi: 10.1007/s10107-016-1093-4.
[3]	A. Berrabah, N. Bensalem and F. Pelletier, Optimality problem for infinite dimensional bilinear systems, Bulletin des Sciences Mathématiques, 130 (2006), 442-466. doi: 10.1016/j.bulsci.2006.03.006.
[4]	S. P. Banks and K. Dinesh, Approximate optimal control and stability of nonlinear finite- and infinite-dimensional systems, Annals of Operations Research, 98 (2000), 19-44. doi: 10.1023/A:1019279617898.
[5]	M. Bergounioux, Optimisation et Contrôle des Systèmes Linéaires, Dunod, 2001.
[6]	C. Brian Hall, Lie Groups, Lie Algebras, and Representations, Springer, 2015. doi: 10.1007/978-3-319-13467-3.
[7]	N. Bourbaki, Lie Groups and Lie Algebras, Springer, 1989.
[8]	S. P. Banks and M. K. Yew, On a class of suboptimal controls for infinite-dimensionnal bilinear systems, Systems and Control Letters, 5 (1985), 327-333. doi: 10.1016/0167-6911(85)90030-1.
[9]	S. P. Banks and M. K. Yew, On the optimal control of bilinear systems and its relation to Lie algebras, Int. J. Control, 43 (1986), 891-900. doi: 10.1080/00207178608933510.
[10]	R. F. Curtain and A. J. pritchard, Infinite Dimensional Linear Systems Theory, Springer-Verlag, New York, 1978.
[11]	D. Dochain, Contribution to the Analysis and Control of Distributed Parameter Systems with Application to (Bio)Chimical Processes and Robotics, Thèse, Univ. Cath. Louvain, Belgium, 1994.
[12]	N. EL Alami, Analyse et commande optimale des systèmes bilinéaires à paramètres distribués- Application aux procédés énergétiques, Thèse, Univ. de Perpignan, France, 1986.
[13]	F. R. Gantmacher, Théorie des Matrices, Dunod, Paris, 1966.
[14]	J. P. Gauthier, C. Z. Xu and A. Bounat, An observer for infinite dimensional skew-adjoint bilinear systems, J. Math. Systems Estim. Control, 5 (1995), 119-122.
[15]	C. Z. Xu, P. Ligarius and J. P. Gauthier, An observer for infinite dimensional dissipative bilinear systems, Comput. Math. Appl., 29 (1995), 13-21. doi: 10.1016/0898-1221(95)00014-P.
[16]	P. Ligarius, Observateurs de systèmes bilinéaires à paramètres répartis - Application à un échangeur thermique, Thesis, Univ. of Rouen, France, 1995.
[17]	P. Ligarius, J. P. Gauthier and C. Z. Xu, A simple observer for distributed systems. Application on a heat exchanger, J. Math. Systems Estim. Control, 8 (1998), 1-23.
[18]	R. R. Mohler, Bilinear Control Processes, Academic Press, New York, 1973.
[19]	P. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[20]	L. S. Pontryagin, V. G. Bbltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, ED. Mir, tard. Francaise, 1974.
[21]	M. Popescu and F. Pelletier, Contrôle optimal pour une classe de systèmes bilinéaires, Revue Roumaine des Sciences Techniques, Série de Mécanique Appliquée, 50 (2005), 77-86.
[22]	S. G. Tzafestas, K. E. Anagnostou and T. G. Pimenides, Stabilizing optimal control of bilinear systems with a generalized cost, Optimal Control Applications and Methods, 5 (1984), 111-117. doi: 10.1002/oca.4660050204.
[23]	S. Yahyaoui, L. Lafhim and M. Ouzahra, Problem of optimal control for bilinear systems with endpoint constraint, Optimization and Control, (2021), arXiv: 2107.12663.