<p>A nonconvex linear-quadratic optimal control problem with indefinite matrices of quadratic forms is considered. The transformation and parametrization of the target functional are performed; this leads to a family of identical problems in the sense of a global solution. Conditions on the parameters that distinguish convex problems are obtained. A parametric optimization procedure is implemented according to the criterion of proximity to the original problem. As a result, convex linear-quadratic problems are constructed, which provide the possibility of improving extremal controls in a nonconvex problem.</p>