Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence

We study the asymptotic behavior of the solutions of evolution equations of the form $\dot u(t)\in -\partial f(u(t),r(t))$, where $f(\cdot,r)$ is a one-parameter family of approximations of a convex function $f(\cdot)$ we wish to minimize. We investigate sufficient conditions on the parametrization $r(t)$ ensuring that the integral curves $u(t)$ converge when $t\rightarrow\infty$ towards a particular minimizer $u_\infty$ of $f$. The speed of convergence is also investigated, and a result concerning the continuity of the limit point $u_\infty$ with respect to the parametrization $r(\cdot)$ is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.