This paper studies a class of singular Optimal Control Problems (OCPs) governed by general switched - mode control processes. We develop a new constructive formalism for this class of switched dynamic systems and study OCPs with quadratic cost functionals. The initially given optimization problem is replaced by an auxiliary ”weakly relaxed” OCP. The main aim of our contribution is with the formal proof of the local convexity property of the obtained auxiliary OCP. This fundamental convexity property makes it possible to apply powerful and relatively simple numerical optimization schemes to the initially given sophisticated singular OCP. The conceptual optimal control design framework we propose includes an optimal switching times selection (”optimal timing”) as well as optimal modes scheduling (”optimal sequencing”).