Input-to-state practical stability criteria of non-autonomous infinite-dimensional systems

Abstract

We develop tools for the investigation of input-to-state practical stability (ISpS) and integral input-to-state practical stability (iISpS) of non-autonomous infinite-dimensional systems in Banach spaces. Sufficient conditions of ISpS and iISpS are given based on indefinite Lyapunov functions. The practical stability analysis is accomplished with the help of scalar practical stable functions. Then, a construction of ISpS Lyapunov function for a class of non-autonomous evolutions equations is provided in Hilbert spaces. We propose the ISpS Lyapunov methodology to make it suitable for the analysis of ISpS w.r.t. inputs from $L_p$-spaces. Furthermore, we illustrate the theory with an example of a semi-linear reaction-diffusion equation.

Keywords:

• Non-autonomous infinite-dimensional systems
• input-to-state practical stability (ISpS)
• integral input-to-state practical stability (iISpS)
• Lyapunov methods

Acknowledgments

The authors are grateful to the reviewers for their valuable and insightful comments that contribute to improving the quality of the manuscript.

Data availability statement

Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Hanen Damak

Hanen Damak received her PhD in Mathematics from the University of Sfax in 2013 and her Habilitation (H.D.R) in 2021 from the University of Sfax. She is currently Assistant-Professor at the I.P.E.I.Sfax. Her research interests are around nonlinear control theory, evolution equations, partial differential equations (PDEs), ordinary differential equations (ODEs) and control of infinite-dimensional systems.

Mohamed Ali Hammami

Mohamed Ali Hammami received his PhD in Mathematics from the University of Metz (France), the Habilitation from University of Sfax (Tunisia). He is currently Professor at the Faculty of Sciences of Sfax in the Department of Mathematics (member of Stability and Control Systems and Nonlinear PDE laboratory). His research interests include nonlinear control systems and differential equations (stability of time varying systems, stabilisation of impulsive systems, time delay and fuzzy systems, stochastic evolution equations, observability and observer).

Rahma Heni

Rahma Heni received the Master Degree in Mathematics at the University of Kairouan and is currently pursuing the Doctorate Degree at the Faculty of Sciences of Sfax. Her current research interests include stability and stabilisation of infinite-dimensional nonlinear systems.