This paper is concerned with the issues of stability analysis and control synthesis for a class of discrete-time semi-Markov jump linear systems (S-MJLSs). Motivated by the fact that the statistic characteristics of sojourn time and mode transitions are often difficult to acquire in practice, the sojourn-time probability density functions (ST-PDFs) and the transition probabilities (TPs) for jump instants are considered to be partially accessed. The systems under consideration is more general, which not only relaxes the conventional hypothesis on S-MJLSs that all the ST-PDFs and TPs are completely known but also covers systems with completely known and completely unknown ST-PDFs or TPs as special cases. By introducing the upper bound of sojourn time for each system mode, numerically testable stability criteria are established for S-MJLSs with incompletely available ST-PDFs and/or TPs in the sense of mean-square stability (MSS), and the existence conditions of desired stabilizing controller are developed to guarantee the MSS of closed-loop S-MJLSs. The theoretical results are testified by several numerical examples and a practical example of space robot manipulator, to demonstrate the effectiveness, superiority, and applicability of the developed control methodology.