In this paper, we consider parametric optimal control problems with linear state equations and control constraints.
Applying the KKM fixed point theorem and the Gronwall Lemma, conditions of the existence of solutions to such problems are gained. Under suitable tools and techniques, we formulate sufficient conditions of a Hölder property of solution maps to reference problems without assuming conditions related to differentiability properties. Also, we suggest and investigate a concept of Hölder well-posedness for these problems. Finally, as applications, employing the main results, we discuss the Hölder calmness property of the solution maps to two practical situations, namely linear quadratic regulator and fuel-optimal frictionless horizontal motion of a mass point problems.