American Institute of Mathematical Sciences

• Previous Article
Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect
• EECT Home
• This Issue
• Next Article
Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction
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 and 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. [2] H. Attouch, Z. Chbani, J. 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. [3] H. Attouch, X. 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. [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. [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. [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. [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. [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. [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. [10] W. Su, S. 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.

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. [2] H. Attouch, Z. Chbani, J. 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. [3] H. Attouch, X. 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. [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. [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. [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. [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. [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. [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. [10] W. Su, S. 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.
 [1] Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial and Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381 [2] Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259 [3] Julián Fernández Bonder, Analía Silva, Juan F. Spedaletti. Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2125-2140. doi: 10.3934/dcds.2020355 [4] Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202 [5] Marek Fila, Michael Winkler. Sharp rate of convergence to Barenblatt profiles for a critical fast diffusion equation. Communications on Pure and Applied Analysis, 2015, 14 (1) : 107-119. doi: 10.3934/cpaa.2015.14.107 [6] Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835 [7] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [8] Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial and Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333 [9] Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485 [10] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [11] Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557 [12] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [13] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [14] Wen-Chiao Cheng, Yun Zhao, Yongluo Cao. Pressures for asymptotically sub-additive potentials under a mistake function. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 487-497. doi: 10.3934/dcds.2012.32.487 [15] Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107 [16] Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial and Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199 [17] Shahad Al-azzawi, Jicheng Liu, Xianming Liu. Convergence rate of synchronization of systems with additive noise. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 227-245. doi: 10.3934/dcdsb.2017012 [18] Armand Bernou. A semigroup approach to the convergence rate of a collisionless gas. Kinetic and Related Models, 2020, 13 (6) : 1071-1106. doi: 10.3934/krm.2020038 [19] Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1 [20] Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503

2020 Impact Factor: 1.081