MATHEMATICAL MODELING OF THE SPECTRAL REPRESENTATION FOR EVOLUTION OPERATORS AND INTERNAL TIME IN THE THEORY OF DURABILITY OF SYSTEMS WITH LONG PERIODS OF ACTIVE EXISTENCE

Abstract

Background. Modern and designed complex, highly dangerous and critical systems today should fulfill their functionality for 30–50 years or more (nuclear power, cosmonautics, transport systems, including oil and gas trunk pipelines. Therefore, for such complexes it is extremely important to know the durability indicators in advance (resource, service life and their residual values) and, therefore, be able to evaluate their values by analyzing the distribution function of the state ρ at any time of the intended use. The purpose of the article is to study the dependences of durability indicators using modern methods of functional analysis. Materials and methods. Spectral transformations of operators of evolution and internal time of systems, statistical methods of reliability theory. Results and conclusions. The spectral transformations of the Liouville and internal time operators are analyzed, as well as the physical essence of the transition of studies from the Hilbert space to the equipped spaces.

Keywords

• unstable systems
• evolution and internal time operators
• longevity
• resource
• service life