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Small-Time Extinction with Decay Estimate of Bilinear Systems on Hilbert Space

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Abstract

This paper considers the stabilization problem of bilinear systems in small time by various feedback laws. Then, under some reasonable assumptions on the system and control operator, we prove the global polynomial stabilization of the bilinear system, at hand, in a small time by unbounded feedback. A decay rate of the stabilized state is explicitly estimated. Moreover, we use an observability condition to prove a partial stabilization in a prescribed time by time-varying feedback. Examples of heat, transport and wave equations are revisited.

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Notes

  1. We precise here the following sets \({\mathbb {S}}_d (1):={\{u\in X: ||u||_d=1}\}\), \( B_d (1):={\{u\in X: ||u||_d\leqslant 1}\}.\)

  2. Note that, in \({\mathbb {R}}\) the equation \(\dot{z}=-ksgn(z)|z|^{a}\), with \(z(0)=z_0\not =0\) and \(\theta \in (0,\,1)\) admits the solution \(z(t)=sgn(z_0)\big (|z_0|^{1-\theta }-k(1-\theta )t\big )^{1/(1-\theta )})\) if \(t\leqslant \frac{|z_0|^{1-\theta }}{k(1-\theta )}\), and \(z(t)=0\) if else.

  3. 1. A function \(\alpha :{\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) is said in class \({\mathcal {K}}\) if \(\alpha (0)=0\) and \(\alpha \) is continuous and strictly increasing. 2. A function \(\alpha :{\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) is said in class \({\mathcal {K}}_{\infty }\), if \(\alpha \in {\mathcal {K}}\) and \(\alpha (t)\rightarrow +\infty \) when \(t\rightarrow +\infty .\) 3. A function \(\beta :{\mathbb {R}}_{+}\times {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}\) is belongs to class \({{\mathcal {K}}}{{\mathcal {L}}}\) if for every fixed \(t\geqslant 0,\,\beta (.,\,t)\in {\mathcal {K}}_{\infty }\) and for each fixed \(s\in {\mathbb {R}}_{+}, \beta (s,\,t)\rightarrow 0\) as \(t\rightarrow +\infty .\)

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Acknowledgements

The authors would like to thank the referees who read the work with great care, and made interesting remarks and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. Ecole Nationale d’Ingénieurs de Tunis- Université Tunis El-Manar, Tunis, Tunisia

    Chaker Jammazi

  2. Laboratoire d’Ingénierie Mathématique (LIM), Ecole Polytechnique de Tunisie, Université de Carthage, Carthage, Tunisia

    Chaker Jammazi

  3. Laboratoire de Mathématiques, Modélisation et Physique Appliquée (MMPA), Fez, Morocco

    Mohamed Ouzahra

  4. Faculté des Sciences de Bizerte, Laboratoire d’Ingénierie Mathématique (LIM), Ecole Polytechnique de Tunisie, Université de Carthage, Carthage, Tunisia

    Mohamed Sogoré

  5. Ecole Normale d’Enseignement Technique et Professionnel de Bamako (ENETP), Bamako, Mali

    Mohamed Sogoré

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  1. Chaker Jammazi
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  2. Mohamed Ouzahra
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  3. Mohamed Sogoré
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Correspondence to Chaker Jammazi.

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Communicated by Anthony Bloch.

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Jammazi, C., Ouzahra, M. & Sogoré, M. Small-Time Extinction with Decay Estimate of Bilinear Systems on Hilbert Space. J Nonlinear Sci 33, 54 (2023). https://doi.org/10.1007/s00332-023-09914-0

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