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June  2017, 6(2): 177-186. doi: 10.3934/eect.2017010

Asymptotic for the perturbed heavy ball system with vanishing damping term

1. 

Institut Préparatoire aux Etude Scientifiques et Techniques, Université de Carthage, Bp 51 La Marsa, Tunisia

2. 

Faculté des Sciences de Tunis, Laboratoire EDP-LR03ES04, Université de Tunis El Manar Tunis, Tunisia

3. 

College of Sciences, Department of Mathematics and Statistics, King Faisal University, P.O. 400 Al Ahsaa 31982, Kingdom of Saudi Arabia

*Corresponding author: Ramzi May

Received  September 2016 Revised  February 2017 Published  April 2017

Fund Project: The authors are grateful to the Deanship of Scientific Research at King Faisal University for financially and morally supporting this work under Project 160052

We investigate the long time behavior of solutions to the differential equation:
$\ddot{x}(t)+\frac{c}{{{\left( 1+t \right)}^{\alpha }}}\dot{x}(t)+\nabla \Phi \left( x(t) \right)=g(t),~t\ge 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)$
where
$c$
is nonnegative constant,
$α∈\lbrack0,1[,Φ \ \ {\rm{is \ \ a}}\ \ C^{1}$
convex function defined on a Hilbert space
$\mathcal{H}$
and
$g∈ L^{1}(0,+∞;\mathcal{H}).$
We obtain sufficient conditions on the source term
$g(t)$
ensuring the weak or the strong convergence of any trajectory
$x(t)$
of (1) as
$t\to ∞$
to a minimizer of the function
$Φ$
if one exists.
Citation: Mounir Balti, Ramzi May. Asymptotic for the perturbed heavy ball system with vanishing damping term. Evolution Equations & Control Theory, 2017, 6 (2) : 177-186. doi: 10.3934/eect.2017010
References:
[1]

F. Alvarez, On the minimizing properties of a second order dissipative system in Hilbert spaces, SIAM J. Cont. Optim., 38 (2000), 1102-1119.  doi: 10.1137/S0363012998335802.  Google Scholar

[2]

H. AttouchZ. ChbaniJ. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math Program Ser B, (2016), 1-53.  doi: 10.1007/s10107-016-0992-8.  Google Scholar

[3]

H. AttouchX. Goudou and P. Redont, The heavy ball with friction method, Ⅰ: The continuous dynamical system: Global exploration of the the local minima of a real valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.  doi: 10.1142/S0219199700000025.  Google Scholar

[4]

M. Balti and R. May, Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source, Submitted, arXiv: 1608. 08760v1. Google Scholar

[5]

A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differential Equations, 252 (2012), 294-322.  doi: 10.1016/j.jde.2011.09.012.  Google Scholar

[6]

A. Haraux and M. A. Jendoubi, On a second order dissipative ODE in Hilbert space with an integrable source term, Acta Mathematica Scientia, 32 (2012), 155-163.  doi: 10.1016/S0252-9602(12)60009-5.  Google Scholar

[7]

M. A. Jendoubi and R. May, Asymptotics for a second-order differential equation with non-autonomous damping and an integrable source term, Applicable Analysis, 94 (2015), 436-444.  doi: 10.1080/00036811.2014.903569.  Google Scholar

[8]

R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, J. Math. Anal. Appl., 430 (2015), 410-416.  doi: 10.1016/j.jmaa.2015.04.067.  Google Scholar

[9]

Z. Opial, Weak convergence of the sequence of successive aproximation for nonexpansive mapping, Bull. Amer. Math. Soc., 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[10]

W. SuS. Boyd and E. Candes, A differential equations for modeling Nestrov's accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 17 (2016), 1-43.   Google Scholar

show all references

References:
[1]

F. Alvarez, On the minimizing properties of a second order dissipative system in Hilbert spaces, SIAM J. Cont. Optim., 38 (2000), 1102-1119.  doi: 10.1137/S0363012998335802.  Google Scholar

[2]

H. AttouchZ. ChbaniJ. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math Program Ser B, (2016), 1-53.  doi: 10.1007/s10107-016-0992-8.  Google Scholar

[3]

H. AttouchX. Goudou and P. Redont, The heavy ball with friction method, Ⅰ: The continuous dynamical system: Global exploration of the the local minima of a real valued function by asymptotic analysis of a dissipative dynamical system, Commun. Contemp. Math., 2 (2000), 1-34.  doi: 10.1142/S0219199700000025.  Google Scholar

[4]

M. Balti and R. May, Asymptotic for a semilinear hyperbolic equation with asymptotically vanishing damping term, convex potential, and integrable source, Submitted, arXiv: 1608. 08760v1. Google Scholar

[5]

A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-autonomous damping, J. Differential Equations, 252 (2012), 294-322.  doi: 10.1016/j.jde.2011.09.012.  Google Scholar

[6]

A. Haraux and M. A. Jendoubi, On a second order dissipative ODE in Hilbert space with an integrable source term, Acta Mathematica Scientia, 32 (2012), 155-163.  doi: 10.1016/S0252-9602(12)60009-5.  Google Scholar

[7]

M. A. Jendoubi and R. May, Asymptotics for a second-order differential equation with non-autonomous damping and an integrable source term, Applicable Analysis, 94 (2015), 436-444.  doi: 10.1080/00036811.2014.903569.  Google Scholar

[8]

R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, J. Math. Anal. Appl., 430 (2015), 410-416.  doi: 10.1016/j.jmaa.2015.04.067.  Google Scholar

[9]

Z. Opial, Weak convergence of the sequence of successive aproximation for nonexpansive mapping, Bull. Amer. Math. Soc., 73 (1967), 591-597.  doi: 10.1090/S0002-9904-1967-11761-0.  Google Scholar

[10]

W. SuS. Boyd and E. Candes, A differential equations for modeling Nestrov's accelerated gradient method: Theory and insights, Journal of Machine Learning Research, 17 (2016), 1-43.   Google Scholar

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