This paper is devoted to a general optimal control problem (OCP) associated with a family of nonlinear continuous-time switched systems in the presence of additional control constraints. The step wise control restrictions under consideration can usually be interpreted as a result of specific quantization procedures applied to the conventional systems inputs. We consider dynamic systems with a priori known switching strategies and a quadratic costs functional. Our aim is to elaborate a constructive implementable algorithm that makes it possible to calculate approximating solutions for the above-mentioned class of switched OCPs. The paper discuss such a computational scheme and also contains a numerical example. © 2013 IEEE.