Smoothing spline via optimal control under uncertainty

Abstract

• Theoretic spline model is formulated as a linear quadratic optimal control problem. • The measurement data are corrupted by noise with knowledge of its distribution. • The uncertain objective function is transformed into an equivalent deterministic one. • Control parameterization technique is used to solve the deterministic problem. • The solution of the approximate optimal control converges in the weak* topology. In this paper, we consider a class of control theoretic spline model, which can be formulated as a linear quadratic optimal control problem. The unknown initial condition and the control are to be chosen optimally such that the output best fits a set of measurement data which are corrupted by noise with crucial knowledge of its distribution. We first transform the uncertain objective function into a deterministic objective function. The solution method is based on the control parameterization technique. We show that the approximate optimal controls obtained from the approximate finite dimensional problems converge to the optimal control of the original control problem in the weak ⋆ topology of L ∞ ( [ 0 , T ] , R r ) . Numerical results show that the proposed method is effective.