Existence of planar curves minimizing length and curvature

In this paper we consider the problem of reconstructing a curve that is partially hidden or corrupted by minimizing the functional $\int \sqrt{1+K_\gamma^2} \,ds$, depending both on length and curvature $K$. We fix starting and ending points as well as initial and final directions. For this functional we discuss the problem of existence of minimizers on various functional spaces. We find non-existence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories can converge to curves with angles. We instead prove existence of minimizers for the ``time-reparameterized'' functional $$\int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} \,dt$$ for all boundary conditions if initial and final directions are considered regardless to orientation. In this case, minimizers can present cusps (at most two) but not angles.