Discrete-time control systems approach for optimal smoothing splines

We consider the problem of designing optimal smoothing spline curves by employing an approach based on linear control systems. First, the problem is formulated using continuous-time, time-invariant systems with piecewise constant inputs. Then by introducing discrete time-varying systems, the solutions for optimal splines including periodic splines are derived. The existence conditions for unique optimal solutions are established, where the concepts of controllability and observability play central roles. The computational procedures for the optimal splines are straightforward. The design method for periodic splines is applied to a shape synthesizing problem using jellyfish as the example.