This paper deals with a further development of analytic techniques for Optimal Control Problems (OCPs) involving differential systems with the state suprema. Differential equations evolving with state suprema (maxima) provide a useful modelling framework for various real-world applications, namely, in electrical engineering and in biology. The corresponding dynamic models lead to Functional Differential Equations (FDEs) in the presence of state-dependent delays. We study some particular (but important) cases of optimal control processes governed by systems with sup-operator in the right hand sides of the differential equations and obtain constructive characterizations of optimal solutions. The constrained OCPs we examine are formulated assuming the (linear) feedback-type control law. The case study presented in this article constitutes a formal extension of the concept of positive dynamic systems to differential systems with the state suprema.