In the article, based on the estimate of the Euclidean norm of the deviation of the coordinates of the transition and stationary states of the dynamic system, the compression condition of the generalized projection operator of the dynamic system with restrictions is derived. From the principle of contracting mappings, taking into account the derived compression condition of the projection operator, estimates are obtained for the sufficient condition for the stability of the dynamic system of stabilization of the equilibrium position and program motions. The obtained estimates generalize the previously obtained results. Ensuring the stability of the operator of a limited dynamic system is demonstrated experimentally.

Keywords: sufficient condition for stability, projection operator, stabilization of equilibrium position. stabilization of program motions, SimInTech

The paper considers a mathematical model of the ion-beam etching process. A nonlinear differential equation of first-order ion-beam etching is considered. It has been established that the model equation for ion-beam etching can be reduced to a homogeneous Monge-Ampere equation. Some classes of exact solutions are presented for this equation. A power-law solution is obtained by the method of functional separation of variables, which depends only on a set of constants and does not contain arbitrary functions. Solutions are also found that linearly depend on arbitrary functions of the coordinate variable and the time variable. Assumptions and explicit conditions are formulated on how to select solutions from the families of solutions of the Monge-Ampere equation that correspond to the model process under consideration. A class of nonlinear equations in partial derivatives of the first order is indicated, which can also be reduced to the Monge-Ampere equation. Limitations on the etching rate are established, which allow the ion-beam etching equation to be reduced to a second-order linear hyperbolic equation, for which, by separation of variables, it is possible to obtain a solution in the form of a Fourier series.

Keywords: ion-beam etching equation, Monge-Ampere equation, model solutions, exact solutions