Continuous-time H-2-control problem for the class of linear systems with Markovian jumps (MJLS) using convex analysis is considered in this paper. The definition of the H-2-norm for continuous-time MJLS is presented and related to the appropriate observability and controllability Gramians. A convex programming formulation for the H-2-control problem of MJLS is developed. That enables us to tackle the optimization problem of MJLS under the assumption that the transition rate matrix Pi = [pi(ij)] for the Markov chain may not be exactly known, but belongs to an appropriate convex set. An equivalence between the convex formulation when Pi is exactly known and the usual dynamic programming approach of quadratic optimal control of MJLS is established. It is shown that there exists a solution for the convex programming problem if and only if there exists the mean-square stabilizing solution for a set of coupled algebraic Riccati equations. These results are compared with other related works in the current literature.