Convergence and rate of convergence of a non-autonomous gradient system on Hadamard manifolds

Abstract

In this paper we consider the following nonhomogeneous gradient system on a Hadamard manifold M

$\left\{ \begin{gathered} - x'(t) = grad\phi (x(t)) + e(t), \hfill \\ x(0) = x_0 \hfill \\ \end{gathered} \right. $

where φ: M → ℝ is a geodesically convex function of class C ² with argmin φ ≠ Ø. We prove global convergence of solutions of the gradient systemto a minimum point of φ. We also discuss on the rate of convergence of φ(x(t)) to the minimum value of φ as well as the rate of convergence of ‖x′(t)‖ and d(x(t), p) to zero, where p is the minimum point of φ. Finally, we present some problems and future directions to study.