Convergence to equilibrium for solutions of some forced discretized second-order gradient-like systems

Haythem Cheikh

doi:10.3934/mcrf.2023025

Article Contents

2024, Volume 14, Issue 3: 800-820. Doi: 10.3934/mcrf.2023025

This issue Previous Article Next Article Approximate controllability for some retarded integrodifferential inclusions using a version of the Leray-Schauder fixed point theorem for the multivalued maps

Convergence to equilibrium for solutions of some forced discretized second-order gradient-like systems

Haythem Cheikh^{1, 2,}

1.
Laboratoire Équations aux Dérivées Partielles, LR03ES04, Faculté des sciences de Tunis, Université Tunis El Manar, 2092 El Manar, Tunisia
2.
Laboratoire M2N, CNAM, 292 rue Saint-Martin, 75003 Paris, France

Received: May 2023

Revised: July 2023

Early access: August 2023

Published: September 2024

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Abstract

In this paper we prove that the solutions of an implicit scheme converge to equilibrium points under certain conditions on the potential function $ F $ and the forcing term $ g $. For this purpose, we use a Lojasiewicz inequality and introduce a discretized energy and a Lyapunov function. In addition, we provide some numerical simulations that support our findings.

Keywords:

Mathematics Subject Classification: Primary: 34E10, 37N30, 40A05; Secondary: 35L05, 35L70, 65L07.

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Figure 1. Graph of $ E^{n} $ versus time in logarithmic scale on the x-axis

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Figure 2. Visualization of the convergence of $ (u_{n}) $

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Figure 3. Graph of $ E^n $ as a function of $ u_n $

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Figure 4. Graph of $ E^{n} $ versus time in logarithmic scale on the x-axis

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Figure 5. Visualization of the convergence of $ (u_{n}) $

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Figure 6. Graph of $ E^{n} $ versus time

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Figure 7. Graph of $ E^{n} $ versus time in logarithmic scale on the x-axis

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Figure 8. Visualization of the convergence of $ (u_{n}) $

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Figure 9. Graph of $ E^n $ as a function of $ u_n $

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Figure 10. Graph of $ E^{n} $ versus time in logarithmic scale on the x-axis

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Figure 11. Visualization of the convergence of $ (u_{n}) $

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Figure 12. Graph of $ E^n $ as a function of $ u_n $

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References

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