On the long time behavior of second order differential equations with asymptotically small dissipation

By Alexandre Cabot, Hans Engler, Sébastien Gadat. 2007

138 citations recorded by the source corpus.

We investigate the asymptotic properties as $ t\to \infty$ of the following differential equation in the Hilbert space $ H$: $$(\mathcal{S})\qquad\qquad\qquad\ddot{x}(t)+a(t)\dot{x}(t)+ \nabla G(x(t))=0, \quad t\geq 0$$ where the map $ a:\mathbb{R}_+\to \mathbb{R}_+$ is nonincreasing and the potential $ G:H\to \mathbb{R}$ is of class $ \mathcal{C}^1$. If the coefficient $ a(t)$ is constant and positive, we recover the so-called ``Heavy Ball with Friction'' system. On the other hand, when $ a(t)=1/(t+1)$ we obtain the trajectories associated to some averaged gradient system. Our analysis is mainly based on the existence of some suitable energy function. When the function $ G$ is convex, the condition $ \int_0^\infty a(t) dt =\infty$ guarantees that the energy function converges toward its minimum. The more stringent condition $ \int_0^{\infty} e^{-\int_0^t a(s) ds}dt<\infty$ is necessary to obtain the convergence of the trajectories of $ (\mathcal{S})$ toward some minimum point of $ G$. In the one-dimensional setting, a precise description of the convergence of solutions is given for a general nonconvex function $ G$. We show that in this case the set of initial conditions for which solutions converge to a local minimum is open and dense.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/340846f4020b1cc1eb45f8bb26525668bf5cd7b6

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity (2018, background): “Fast minimization of function Φ: Equation (1) is a particular case of the inertial gradient-like system (3) ẍ(t) + a(t)ẋ(t) +∇Φ(x(t)) = 0, with asymptotic vanishing damping, studied by Cabot, Engler and Gadat in [17, 18].”
On damped second-order gradient systems (2014, background): “In the one-dimensional case, convergence can be obtained with much more general forms of damping, see [15].”
Fast optimization via inertial dynamics with closed-loop damping (2020, unknown): “To give a first answer to this question, we rely on the results of Cabot-Engler-Gaddat [51], AttouchCabot [18] and Attouch-Chbani-Riahi [24] which concern the asymptotic stabilization of inertial gradient dynamics with general time-dependent viscosity coefficient γ(t).”
Stochastic Heavy ball (2016, methodology): “In particular, as highlighted in [CEG09a], NAGD may be seen as a specific case of HBF after a time rescaling t “ ? s, thus making the acceleration explicit through this change of variable, as well as being closely linked to the modified Bessel functions when f is quadratic.”
Optimal non-asymptotic bound of the Ruppert-Polyak averaging without strong convexity (2017, background): “This phenomenon also occurs for the kinetic diﬀusion dynamics [Vil09, GM13]) and for the Nesterov accelerated gradient descent [Nes83, CEG09] even though we do not claim that such a clear common point exists be-tween these methods.”