We consider a method, which permits to harness the classic linear-quadratic optimal control theory for solution of certain specific nonconvex problems of global multicriteria constrained optimization. This method enables one to find the set of all the Slater optimal solutions and consists in a certain combination of the weighted average method and the method of Lagrange duality. New criteria for the above method to be applicable are established. The theory developed is illustrated by examples concerning diverse optimal control problems.