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Alain Haraux - One of the best experts on this subject based on the ideXlab platform.

  • The universal bound property for a class of second order ODEs
    Portugaliae Mathematica, 2019
    Co-Authors: Mama Abdelli, Alain Haraux
    Abstract:

    We consider the scalar second order ODE u + |u | α u + |u| β u = 0, where α, β are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 0, u (t) 2 + |u(t)| β+2 ≤ C max{t − 2 α , t − (α+1)(β+2) β−α }.

  • The universal bound property for a class of second order ODEs
    arXiv: Dynamical Systems, 2018
    Co-Authors: Mama Abdelli, Alain Haraux
    Abstract:

    We consider the scalar second order ODE u + |u | $\alpha$ u + |u| $\beta$ u = 0, where $\alpha$, $\beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 0, u (t) 2 + |u(t)| $\beta$+2 $\le$ C max{t -- 2 $\alpha$ , t -- ($\alpha$+1)($\beta$+2) $\beta$--$\alpha$ }.

  • The universal bound property for a class of second order ODEs
    2018
    Co-Authors: Mama Abdelli, Alain Haraux
    Abstract:

    We consider the scalar second order ODE u + |u | α u + |u| β u = 0, where α, β are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 < α < β and moreover there is a constant C independent of the initial data such that ∀t > 0, u (t) 2 + |u(t)| β+2 ≤ C max{t − 2 α , t − (α+1)(β+2) β−α }.

  • Global behavior of the solutions to a class of nonlinear, singular second order ODE
    Nonlinear Analysis: Theory Methods & Applications, 2014
    Co-Authors: Mama Abdelli, Alain Haraux
    Abstract:

    Abstract In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: ( | u ′ | l u ′ ) ′ + c | u ′ | α u ′ + d | u | β u = 0 , where α , β , l , c , d are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.

  • Global behavior of the Solutions to a Class of Nonlinear, Singular Second Order ODE
    arXiv: Classical Analysis and ODEs, 2013
    Co-Authors: Mama Abdelli, Alain Haraux
    Abstract:

    In this paper the initial value problem and global properties of solutions are studied for the scalar second order ODE: $ (|u'|^{l}u')' + c|u'|^{\alpha}u' + d|u|^\beta u=0$, where $\alpha,\beta,l,c, d$ are positive constants. In particular, existence, uniqueness and regularity as well as optimal decay rates of solutions to 0 are obtained depending on the various parameters, and the oscillatory or non-oscillatory behavior is elucidated.

Cristian Vladimirescu - One of the best experts on this subject based on the ideXlab platform.

Marek Kossowski - One of the best experts on this subject based on the ideXlab platform.

  • Fiber completions, contact singularities and single valued solutions for $Csp infty$-second order ODE
    Canadian Journal of Mathematics, 1996
    Co-Authors: Marek Kossowski
    Abstract:

    AbstractAn implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

Harri Lähdesmäki - One of the best experts on this subject based on the ideXlab platform.

  • ODE$^2$VAE: Deep generative second order ODEs with Bayesian neural networks
    arXiv: Machine Learning, 2019
    Co-Authors: Cagatay Yildiz, Markus Heinonen, Harri Lähdesmäki
    Abstract:

    We present Ordinary Differential Equation Variational Auto-EncODEr (ODE$^2$VAE), a latent second order ODE mODEl for high-dimensional sequential data. Leveraging the advances in deep generative mODEls, ODE$^2$VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex continuous-time latent dynamics. Our mODEl explicitly decomposes the latent space into momentum and position components and solves a second order ODE system, which is in contrast to recurrent neural network (RNN) based time series mODEls and recently proposed black-box ODE techniques. In order to account for uncertainty, we propose probabilistic latent ODE dynamics parameterized by deep Bayesian neural networks. We demonstrate our approach on motion capture, image rotation and bouncing balls datasets. We achieve state-of-the-art performance in long term motion prediction and imputation tasks.

  • NeurIPS - ODE2VAE: Deep generative second order ODEs with Bayesian neural networks
    2019
    Co-Authors: Cagatay Yildiz, Markus Heinonen, Harri Lähdesmäki
    Abstract:

    We present Ordinary Differential Equation Variational Auto-EncODEr (ODE2VAE), a latent second order ODE mODEl for high-dimensional sequential data. Leveraging the advances in deep generative mODEls, ODE2VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex continuous-time latent dynamics. Our mODEl explicitly decomposes the latent space into momentum and position components and solves a second order ODE system, which is in contrast to recurrent neural network (RNN) based time series mODEls and recently proposed black-box ODE techniques. In order to account for uncertainty, we propose probabilistic latent ODE dynamics parameterized by deep Bayesian neural networks. We demonstrate our approach on motion capture, image rotation, and bouncing balls datasets. We achieve state-of-the-art performance in long term motion prediction and imputation tasks.

Zhongcheng Zhou - One of the best experts on this subject based on the ideXlab platform.

  • Stabilization of a coupled second order ODE-wave system
    2016 35th Chinese Control Conference (CCC), 2016
    Co-Authors: Zhiyuan Zhen, Ke Wang, Zhongcheng Zhou, Ryan Loxton
    Abstract:

    This paper considers the stabilization of a coupled second order ODE-wave system, where the ODE dynamics contain the solution of the wave equation at an intermediate point. We design a stabilizing feedback controller by choosing a suitable target system and backstepping transformation. The backstepping transformation is defined in terms of several kernel functions, for which we establish existence, uniqueness and smoothness properties. We also prove exponential stability for the resulting closed-loop system. Finally, the effectiveness of the proposed feedback controller is verified via a numerical example.

  • Stabilization of a second order ODE-heat system coupling at intermediate point
    Automatica, 2015
    Co-Authors: Zhongcheng Zhou
    Abstract:

    This paper considers the stabilization of a second order ODE-heat system coupling at an intermediate point under natural and checkable assumptions, which is motivated by the thermoelastic coupling physics arising in microelectromechanical systems (MEMS). A novel backstepping transformation form is proposed in this work and the feedback gain for the lumped-parameter component is constructed using the eigenvector of coefficient matrix in the ODE system. Then, we prove the existence of smooth kernels with second order continuously derivative for the forward and inverse transformations in the backstepping feedback control law design. At the same time, we show that the forward and inverse transformations are mutually invertible transformation pair. Finally, the effectiveness of the stabilization feedback controller design is shown with some numerical examples.