We consider the problem of the perturbation of a class of linear-quadratic systems where the change from one structure (for the dynamics and costs) to another are governed by a finite-state Markov process where the transition are perturbed. The problem above lead to the analysis of some perturbed linearly coupled set of quadratic equations (Riccati Equation). We show that the matrix obtained as the solution of the equations, which determine the optimal value and control, has Taylor expansion in the perturbation parameter. We compute explicitly the term of this expansion.