The improvement of industrial systems' reliability and production quality is a real problem of the modern industry. The automatic systems of technological processes management enable solving this problem. The local technical control constitutes an important part of the system.In spite of the diversity and high level of checkout and measurable instruments， the problem oflatent failures detection and elimination is still significant.
　　The latent failure is a failure which cannot be detected by standard methods or visually， but by maintenance or special methods of diagnostics only.In this monograph， by the latent failure we refer to one which can be detected during control execution only.
　　In complex industrial systems， the periodical control is applied. The reason is the difficulty of checHng individual operation of all units and details(components). It means the control is carried out at fixed (in general case random) time periods， which should be optimal for the whole system by ensuring its maximum reliability and efficiency. The problem can be solved by constructing mathematical models of control of restorable systems with latent failures.
　　The present monograph is dedicated to building such models on the basis of the theory of semi-Markov processes with arbitrary phase state space， as well as to the definition of optimal periodicity of latent failures control. The problems of application of the results obtained are considered.
　　The authors express deep gratitude to professor V.J. Kopp and professor A.I.Peschansky for valuable comments， contributing to the monograph quality improvement， and to A.I. Kovalenko for the monograph translation into English.