This paper is devoted to a specific LQ-type optimal control problem (OCP) in the presence of additional control constraints. We consider control processes governed by linear differential equations with a priori known control switchings. The piecewise constant structure of the admissible controls under consideration is motivated by variety of concrete engineering applications and moreover, can be interpreted as a result of a quantization procedure applied to the original dynamics. We propose an new implementable numeric algorithm that makes it possible to calculate a consistent approximating solution to the initial constrained LQ-type OCPs. The contribution discusses some theoretic aspects of the obtained computational scheme and also contains a numerical example.