UDC 531.36 Application of Method of Conjugate Equations to Research of Loss Stability of Shell Under the Action of Moving Loads A. N. Kudinov, E. V. Chusova Department of Mathematical Modelling Tver State University 33, Zhelyabova str., Tver, Russia, 170100 Actual problem of the theory of stability is creation of strict and effective methods of research of the loss of stability of movement of systems with distributed parameters, in particular for continuous environments. <...> This problem has a huge impact on both theoretical and applied uses. <...> In the article is presented the research of the loss of stability of the equations which describe mathematical models of soil massifs and the bases. <...> These models reflect the character of the soil under load, are based on the laws of structural mechanics and the theory of elasticity. <...> The adjoin method offered by Kudinov A.N. was applied to research of the loss of stability. <...> The main advantage of the adjoin method is that for use to the problems of dynamic stability studies in various fields of science and technology, if their equations can be reduced to the equations of second order, don’t demand the introduction of Lyapunov functions. <...> The developed algorithm allows to find positions of balance of the decision and to check up stability of system by adjoin method. <...> The research of stability of the solution of linear systems by using Lyapunov’s method of first approximation is presented in the article. <...> Key words and phrases: stability, dynamic criterion, Lyapunov’s first method, rocks, soil. 1. <...> Introduction differential equations, systems of partial differential equations, integral and integral– differential equations. <...> The system must have a dynamic and structural stability. <...> The results of the fundamental works of Nikolaja Pavlovicha Epygina, Viktora Dynamic processes are described by nonlinear autonomous systems of ordinary Alekcandpovicha Plica, Nikolaja Nikolaevicha Kpacovckogo, Aleksandra Aleksandrovicha Andronova, Anatolija Arkad’evicha Pepvozvanckogo, Marka Aponovicha Yzerman, Vladimira Vasil’evicha Bolotina and Arnol’da Sergeevicha Vol’mira research of complex dynamic systems described by nonlinear systems of differential equations, derived from the introduction and analysis of Lyapunov functions – is qualitative methods. <...> The proof of theorems, which represent a solution to the problem of Yzerman in the sense of establishing the necessary conditions for stability systems: – the dissemination of necessary and sufficient conditions <...>