I'm trying to find the rest shape of a paper strip with fixed endpoints (without gravity). Of course, the arc length of the resulting curve is given by the length of the strip, also the position and derivative at the endpoint is fixed. With these constraints, the elasticity of the paper implies the curve should be maximally smooth. Is there an analytic solution to this problem? More formally, the optimization problem looks something like this:
\begin{align} \arg\min_c \int_a^b \left|c''(t)\right|_2^2\ dt \text{ where } & c: [a,b] \rightarrow \mathbb{R}^2,\\ &\forall t \in [a,b]: \left|c'(t)\right| = 1\\ & c(a) = \mathbf{x}_a \\ & c'(a) = \mathbf{v}_a \\ & c(b) = \mathbf{x}_b \\ & c'(b) = \mathbf{v}_b \end{align}
If the endpoint constraints on $c'$ are dropped, the solution should be a circular arc (or so I think). If the arc length requirement is dropped, a candidate set of potential solutions is given by cubic Bézier curves. But since there is no analytic formula for the arc length of a Bézier curve, it might be difficult to continue from there.
Even if there is no analytical solution, is there a good approximation that can be discretized and refined using numerical optimization?