On the Convergence of Bounded Solutions of Non Homogeneous Gradient-like Systems

Phuong Minh Tran,Nhan Thanh Nguyen

doi:10.25073/jaec.201711.50

Abstract

We study the long time behavior of the bounded solutions of non homogeneous gradient-like system which admits a strict Lyapunov function. More precisely, we show that any bounded solution of the gradient-like system converges to an accumulation point as time goes to infinity under some mild hypotheses. As in homogeneous case, the key assumptions for this system are also the angle condition and the Kurdyka-Lojasiewicz inequality. The convergence result will be proved under a L1 -condition of the perturbation term. Moreover, if the Lyapunov function satisfies a Lojasiewicz inequality then the rate of convergence will be even obtained.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Highlights

We are interested in the long time behavior of bounded solutions of the rst order non homogeneous gradient-like system u (t) + G (t) = f (t), t 0 (1)
The most simple situation of (2) is the case of gradient system where G = ∇F. This system has been studied by many authors such as Absil & Kurdyka [1], Chill [5], Haraux & Jendoubi [11], [12] or Simon [17]. They have proved that if F satises a Lojasiewicz inequality the bounded solution converges to an equilibrium as t goes to innity
Chill et al [6], the authors gave an abstract result which guarantees that the convergence result holds for the gradient-like system (2)

Summary

Introduction

This system has been studied by many authors such as Absil & Kurdyka [1], Chill [5], Haraux & Jendoubi [11], [12] or Simon [17] They have proved that if F satises a Lojasiewicz inequality the bounded solution converges to an equilibrium as t goes to innity. This condition shows that the forcing term f (t) quickly decays to zero as t goes to innity Their results have been generalized to some second order systems in [2], [3], [4], [9] or [10]. Ghisi et al have estimated the decay rates for solutions of semi linear dissipative equations in [8] Motivated by these works, we establish the convergence results for the rst order nonhomogeneous gradient-like system (1) under a weaker assumption. We present some notations and definitions that we use through the whole of the

Some Denitions

Conclusion