Existence of planar curves minimizing length and curvature

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 the 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 nonexistence of minimizers in cases in which initial and final directions are considered with orientation. In this case, minimizing sequences of trajectories may converge to curves with angles. We instead prove the existence of minimizers for the “time-reparametrized” functional $\int\|\dot\gamma(t)\|\sqrt{1+K_\gamma^2}\,dt$ for all boundary conditions if the initial and final directions are considered regardless of orientation. In this case, minimizers may present cusps (at most two) but not angles.