A Note on the Synthesis of Nonquadratic Optimal Control in a One-Dimensional Linear System

The synthesis problem of the optimal control is investigated in a one-dimensional linear system with a cost function which is H¨olderian with respect to the control and convex with respect to the final state. Under some conditions it is proved that the closed-loop synthesis is given by a nonlinear state feedback defined in terms of the solution of a Bernoulli type operator equation. It is also shown that the feedback operator can be explicitly expressed by solutions of associated algebraic equations