This paper investigates the problem of robust and non-fragile $${\mathcal{H}}_{\infty }$$ static output feedback (SOF) controller design for a class of continuous-time semi-Markovian jump linear systems (S-MJLSs) with norm-bounded uncertainties. The systems under consideration are subject to a semi-Markov stochastic process with time-varying transition rates (TRs), which relax the traditional assumption on Markovian jump linear systems (MJLSs) that all of the TRs are constant. Based on a semi-Markovian Lyapunov function together with the property of cumulative distribution function, a sufficient condition for robust $${\mathcal{H}}_{\infty }$$ performance analysis is first derived. Then via some matrix inequality linearization procedures combined with sojourn-time decomposition idea, two approaches, namely, the convex linearization approach and iterative approach, are developed to solve the robust and non-fragile SOF controller synthesis problem. It is shown that the controller gains can be obtained by solving a set of strict linear matrix inequalities (LMIs) or a sequential minimization problem subject to LMI constraints. Finally, illustrative examples are provided to illustrate the effectiveness of the proposed approaches.