Finding Interpolating Curves Minimizing $L^\infty$ Acceleration in the Euclidean Space via Optimal Control Theory

We study the problem of finding an interpolating curve passing through prescribed points in the Euclidean space. The interpolating curve minimizes the pointwise maximum length, i.e., $L^\infty$-norm, of its acceleration. We reformulate the problem as an optimal control problem and employ simple but effective tools of optimal control theory. We characterize solutions associated with singular and nonsingular controls. Some of the results we obtain are new even for the scalar interpolating function case. We reduce the infinite-dimensional interpolation problem to an ensuing finite-dimensional one and derive closed form expressions for interpolating curves. Consequently we devise efficient numerical techniques and illustrate them with examples.