  For a given $p\in[2,+\infty)$, we define the $p$-elastic energy $\mathscr{E}$
of a closed curve $\gamma:\mathbb{S}^1\to M$ immersed in a complete Riemannian
manifold $(M,g)$ as the sum of the length of the curve and the $L^p$--norm of
its curvature (with respect to the length measure). We are interested in the
convergence of the $(L^p,L^{p'})$--gradient flow of these energies to critical
points. By means of parabolic estimates, it is usually possible to prove
sub-convergence of the flow, that is, convergence to critical points up to
reparametrizations and, more importantly, up to isometry of the ambient.
Assuming that the flow sub-converges, we are interested in proving the smooth
convergence of the flow, that is, the existence of the full limit of the
evolving flow. We first give an overview of the general strategy one can apply
for proving such a statement. The crucial step is the application of a
Lojasiewicz-Simon gradient inequality, of which we present a versatile version.
Then we apply such strategy to the flow of $\mathscr{E}$ of curves into
manifolds, proving the desired improvement of sub-convergence to full smooth
convergence of the flow to critical points. As corollaries, we obtain the
smooth convergence of the flow for $p=2$ in the Euclidean space $\mathbb{R}^n$,
in the hyperbolic plane $\mathbb{H}^2$, and in the two-dimensional sphere
$\mathbb{S}^2$. In particular, the result implies that such flow in
$\mathbb{R}^n$ or $\mathbb{H}^2$ remains in a bounded region of the space for
any time.
