The paper is concerned with a control system described by linear ordinary differential equations with time-variable coefficients which are defined by a homogeneous Markov process. The control parameters in the form of linear functions of state variables are found by minimizing the mean value of a quadratic functional. To solve the stated problem the well- known Bellman principle is used. As a result, determination of the controls in question is reduced to the solution of a special system of ordinary differential equations. These are some generalization of the Riccati equations arising in deterministic linear problems of optimal control with quadratic performance index.