Constrained data smoothing via optimal control

The article considers a problem of best smoothing in strip, where the objective is to find function f : [ 0 , 1 ] → ℝ $$ f:\kern0.3em \left[0,1\right]\to \mathbb{R} that satisfies bilateral constraints on its values, d ( t ) ≤ e d(t)\le f(t)\le e(t) for all 0\le t\le and minimizes weighted sum L 2 {L}_2 -norm second derivative squared deviations from specified y i {y}_i at discrete points = < ⋯ N + 0={t}_1<{t}_2<\cdots <{t}_{N+2} . We assume d(t) are continuous functions linear each interval \left[{t}_i,{t}_{i+1}\right] … i=1,\dots, N+1 connect this state-constrained optimal control double integrator, give conditions existence uniqueness solution under which we also show cubic spline with knots {t}_i no more than two additional \left({t}_i,{t}_{i+1}\right) propose numerical algorithm solving based stage minimization, outer loop optimization finite-dimensional convex, while inner admits easy compute. Numerical results efficacy proposed approach reported.