The tests of the stability of one class of autonomous differential“pseudo-linear” equations of the first order with autoregulated delay

In the article effective tests of exponential stability are obtained for some classes of autonomous differential equations of first order with autoregulated delay. An overview of works on this topic from the cities of Perm and Ivanovo is made. The criteria of S.A. Gusarenko (on the continuity of the operator with autoregulated delay) and of V.P. Maksimov (on the complete continuity of the operator with autoregulated delay) are given. Sufficient conditions for the existence and continuability of solutions are formulated. Theorems on stability of the system due to its first approximation are given, too. These propositions are based on theorems from the book and from the articles N.V. Azbelev and P.M. Simonov. Theorems on stability in the first approximation, although resembling the well-known Lyapunov's theorems, in reality differ significantly from the latter. Lyapunov's theorems for ordinary differential or functional differential equations give a technique for investigating stability. By means of linearization, the question of the nonlinear equation’s stability reduces to the question of linear equation’s stability. For this problem effective stability criteria are already proved. In our case it is not possible to linearize the nonlinear parts of the equations, and therefore the above technique is not applicable here. In the article, replacing the process of linearization with “pseudo-linearization”, and also using the results of V.V. Malygina, we obtained some analogues of theorems on the first approximation for scalar, autonomous equations with autoregulated delay. The main conclusions obtained on the basis of this idea can be formalized as follows: autonomous differential equations with autoregulated delay have stability properties similar to the properties of corresponding equations with concentrated delay.