Optimal problems on Hölder function spaces (Q1776288)

Existence, uniqueness and properties of optimal trajectories are considered for two optimal problems \( \{\min \int_0^ {\infty} x^2 (t) \,dt\) subject to \(\dot x = y\), \(y \in \sum_{[0, \infty]} (1; \alpha)\), \(x(0)=x_0\), \(y(0) = y_0 \}\) and \( \{\min T\) subject to \(\dot x=y\), \(y\in \sum_{[0, \infty]} (1; \alpha)\), \(x(0)=x_0\), \(y(0) = y_0\), \(x(T) = 0\), \(y(T) = 0 \}\) where \( \sum_{[0, \infty]} (1; \alpha) = \{ f:| t_1 -t_2 |^ {\alpha} \geq | f(t_1) - f(t_2) |\); \(t_1, t_2 \in [0, \infty] \} \). 4 problems of nonparametric statistics reducing the first problem are listed and its importance for optimal control theory is presented. Its solutions are spline-functions that alternate maximally increasing and maximally decreasing modes. The results obtained are also applied to inequalities for derivatives.